**Part 1**

**Dentistry, Dental Implantology and Teeth Restoration** 

**1**

*1,2USA 3Brazil*

**Past, Present and Future of Finite**

*North Carolina State University Engineering School, Raleigh,* 

Ching-Chang Ko1,2,\*, Eduardo Passos Rocha1,3 and Matt Larson1

*Department of Dental Materials and Prosthodontics, Araçatuba, Saõ Pauló,* 

Biomechanics is fundamental to any dental practice, including dental restorations, movement of misaligned teeth, implant design, dental trauma, surgical removal of impacted teeth, and craniofacial growth modification. Following functional load, stresses and strains are created inside the biological structures. Stress at any point in the construction is critical and governs failure of the prostheses, remodeling of bone, and type of tooth movement. However, *in vivo* methods that directly measure internal stresses without altering the tissues do not currently exist. The advances in computer modeling techniques provide another option to realistically estimate stress distribution. Finite element analysis (FEA), a computer simulation technique, was introduced in the 1950s using the mathematical matrix analysis of structures to continuum bodies (Zienkiewicz and Kelly 1982). Over the past 30 years, FEA has become widely used to predict the biomechanical performance of various medical devices and biological tissues due to the ease of assessing irregular-shaped objects composed of several different materials with mixed boundary conditions. Unlike other methods (e.g., strain gauge) which are limited to points on the surface, the finite element method (FEM) can quantify stresses and displacement throughout the anatomy of a three

The FEM is a numerical approximation to solve partial differential equations (PDE) and integral equations (Hughes 1987, Segerlind 1984) that are formulated to describe physics of complex structures (like teeth and jaw joints). Weak formulations (virtual work principle) (Lanczos 1962) have been implemented in FEM to solve the PDE to provide stress-strain solutions at any location in the geometry. Visual display of solutions in graphic format adds attractive features to the method. In the first 30 years (1960-1990), the development of FEM

**1. Introduction** 

dimensional structure.

Corresponding Author

\*

**Element Analysis in Dentistry**

*University of North Carolina School of Dentistry, 2Department of Material Sciences and Engineering,* 

*3Faculty of Dentistry of Araçatuba, UNESP,* 

*1Department of Orthodontics,* 

## **Past, Present and Future of Finite Element Analysis in Dentistry**

Ching-Chang Ko1,2,\*, Eduardo Passos Rocha1,3 and Matt Larson1 *1Department of Orthodontics, University of North Carolina School of Dentistry, 2Department of Material Sciences and Engineering, North Carolina State University Engineering School, Raleigh, 3Faculty of Dentistry of Araçatuba, UNESP, Department of Dental Materials and Prosthodontics, Araçatuba, Saõ Pauló, 1,2USA 3Brazil*

#### **1. Introduction**

Biomechanics is fundamental to any dental practice, including dental restorations, movement of misaligned teeth, implant design, dental trauma, surgical removal of impacted teeth, and craniofacial growth modification. Following functional load, stresses and strains are created inside the biological structures. Stress at any point in the construction is critical and governs failure of the prostheses, remodeling of bone, and type of tooth movement. However, *in vivo* methods that directly measure internal stresses without altering the tissues do not currently exist. The advances in computer modeling techniques provide another option to realistically estimate stress distribution. Finite element analysis (FEA), a computer simulation technique, was introduced in the 1950s using the mathematical matrix analysis of structures to continuum bodies (Zienkiewicz and Kelly 1982). Over the past 30 years, FEA has become widely used to predict the biomechanical performance of various medical devices and biological tissues due to the ease of assessing irregular-shaped objects composed of several different materials with mixed boundary conditions. Unlike other methods (e.g., strain gauge) which are limited to points on the surface, the finite element method (FEM) can quantify stresses and displacement throughout the anatomy of a three dimensional structure.

The FEM is a numerical approximation to solve partial differential equations (PDE) and integral equations (Hughes 1987, Segerlind 1984) that are formulated to describe physics of complex structures (like teeth and jaw joints). Weak formulations (virtual work principle) (Lanczos 1962) have been implemented in FEM to solve the PDE to provide stress-strain solutions at any location in the geometry. Visual display of solutions in graphic format adds attractive features to the method. In the first 30 years (1960-1990), the development of FEM

<sup>\*</sup> Corresponding Author

Past, Present and Future of Finite Element Analysis in Dentistry 5

Ko et al (1999) to stimulate optical scattering of the incipient caries (e.g., white spot lesion). The simulated image of the lesion surface was consistent with the true image captured *in clinic* (Figure 1). Linear fit of the image brightness between the FE and clinical images was 85% matched, indicating the feasibility of using numerical model to interpret clinical white spot lesions. The similar probability method was recently used to predict healing bone adaptation in tibia (Byrne et al., 2011). Recognition of the importance of 3D models and

(A) (B) (C) Fig. 1. **A.** Finite element mesh of *in vivo* carious tooth used for Monte Carlo simulation; **B.** Image rendered from Monte Carlo 3D simulation, **C.** True image of carious tooth

As advancements have been made in computer and software capability, more complex 3D structures (e.g., occlusal surfaces, pulp, dentin, enamel) have been simulated in greater detail. Many recent FE studies have demonstrated accurate 3D anatomic structures of a sectioned jaw-teeth complex using μCT images. Increased mathematical functions in 3D computer-aid-design (CAD) have allowed accurate rendition of dental anatomy and prosthetic components such as implant configuration and veneer crowns (Figure 2). Fine meshing and high CPU computing power appeared to allow calculation of mechanical fields (e.g., stress, strain, energy) accounting for anatomic details and hierarchy interfaces between different tissues (e.g., dentin, PDL, enamel) that were offered by the CAD program. It was also recognized that inclusion of complete dentition is necessary to accurately predict stressstrain fields for functional treatment and jaw function (Field et al. 2009). Simplified models containing only a single tooth overlooked the effect of tooth-tooth contacts that is important in specified biomechanical problems such as orthodontic tooth movement and traumatic tooth injury. CAD software such as SolidWorks© (Waltham, MA, USA), Pro Engineering© (Needham, MA), and Geomagic (Triangle Park, NC, USA) have been adapted to construct dentofacial compartments and prostheses. These CAD programs output solid models that are then converted to FE programs (e.g., Abaqus, Ansys, Marc, Mimics) for meshing and solving. The automeshing capability of FE programs significantly improved during this era.

specific solutions were the major contributions in this era.

obtained from a patient's premolar using an intra-oral camera.

**2.3 "2000-2010" age of proliferation, 3D with CAD** 

programs focused on stability of the solution including minimization of numerical errors and improvement of computational speed. During the past 20 years, 3D technologies and non-linear solutions have evolved. These developments have directly affected automobile and aerospace evolutions, and gradually impacted bio-medicine. Built upon engineering achievement, dentistry shall take advantage of FEA approaches with emphasis on mechanotherapy. The following text will review history of dental FEA and validation of models, and show two examples.

#### **2. History of dental FEA**

#### **2.1 1970-1990: Enlightenment stage -2D modeling**

Since Farah's early work in restorative dentistry in 1973, the popularity of FEA has grown. Early dental models were two dimensional (2D) and often limited by the high number of calculations necessary to provide useful analysis (Farah and Craig 1975, Peters et al., 1983, Reinhardt et al., 1983, Thresher and Saito 1973). During 1980-1990, the plane-stress and plane-strain assumptions were typically used to construct 2D tooth models that did not contain the hoop structures of dentin because typically either pulp or restorative material occupied the central axis of the tooth (Anusavice et al., 1980). Additional constraints (e.g., side plate and axisymmetric) were occasionally used to patch these physical deficiencies (hoop structures) to prevent the separation of dentin associated with the 2D models (Ko, 1989). As such a reasonable biomechanical prediction was derived to aid designs of the endodontic post (Ko et al., 1992). Axisymmetric models were also used to estimate stress distribution of the dental implants with various thread designs (Rieger et al., 1990). Validation of the FE models was important in this era because assumptions and constraints were added to overcome geometric discontinuity in the models, leading to potential mathematical errors.

#### **2.2 "1990-2000" beginnings stage of 3D modeling**

As advancements have been made in imaging technologies, 3D FEA was introduced to dentistry. Computer tomography (CT) data provide stacks of sectional geometries of human jaws that could be digitized and reconstructed into the 3D models. Manual and semiautomatic meshing was gradually evolved during this time. The 3D jaw models and tooth models with coarse meshes were analyzed to study chewing forces (Korioth 1992, Korioth and Versluis 1997, Jones et al., 2001) and designs of restorations (Lin et al., 2001). In general, the element size was relatively large due to the immature meshing techniques at that time, which made models time consuming to build. Validation was required to check accuracy of the stress-strain estimates associated with the coarse-meshed models. In addition to the detail of 3D reconstruction, specific solvers (e.g., poroelasticity, homogenization theory, dynamic response) were adapted from the engineering field to study dental problems that involved heterogeneous microstructures and time-dependent properties of tissues. Interfacial micromechanics and bone adaptation around implants were found to be highly non-uniform, which may dictate osseointegration patterns of dental implants (Hollister et al., 1993; Ko 1994). The Monte Carlo model (probability prediction), with incorporation of the finite element method for handling irregular tooth surface, was developed by Wang and

programs focused on stability of the solution including minimization of numerical errors and improvement of computational speed. During the past 20 years, 3D technologies and non-linear solutions have evolved. These developments have directly affected automobile and aerospace evolutions, and gradually impacted bio-medicine. Built upon engineering achievement, dentistry shall take advantage of FEA approaches with emphasis on mechanotherapy. The following text will review history of dental FEA and validation of

Since Farah's early work in restorative dentistry in 1973, the popularity of FEA has grown. Early dental models were two dimensional (2D) and often limited by the high number of calculations necessary to provide useful analysis (Farah and Craig 1975, Peters et al., 1983, Reinhardt et al., 1983, Thresher and Saito 1973). During 1980-1990, the plane-stress and plane-strain assumptions were typically used to construct 2D tooth models that did not contain the hoop structures of dentin because typically either pulp or restorative material occupied the central axis of the tooth (Anusavice et al., 1980). Additional constraints (e.g., side plate and axisymmetric) were occasionally used to patch these physical deficiencies (hoop structures) to prevent the separation of dentin associated with the 2D models (Ko, 1989). As such a reasonable biomechanical prediction was derived to aid designs of the endodontic post (Ko et al., 1992). Axisymmetric models were also used to estimate stress distribution of the dental implants with various thread designs (Rieger et al., 1990). Validation of the FE models was important in this era because assumptions and constraints were added to overcome geometric discontinuity in the models, leading to potential

As advancements have been made in imaging technologies, 3D FEA was introduced to dentistry. Computer tomography (CT) data provide stacks of sectional geometries of human jaws that could be digitized and reconstructed into the 3D models. Manual and semiautomatic meshing was gradually evolved during this time. The 3D jaw models and tooth models with coarse meshes were analyzed to study chewing forces (Korioth 1992, Korioth and Versluis 1997, Jones et al., 2001) and designs of restorations (Lin et al., 2001). In general, the element size was relatively large due to the immature meshing techniques at that time, which made models time consuming to build. Validation was required to check accuracy of the stress-strain estimates associated with the coarse-meshed models. In addition to the detail of 3D reconstruction, specific solvers (e.g., poroelasticity, homogenization theory, dynamic response) were adapted from the engineering field to study dental problems that involved heterogeneous microstructures and time-dependent properties of tissues. Interfacial micromechanics and bone adaptation around implants were found to be highly non-uniform, which may dictate osseointegration patterns of dental implants (Hollister et al., 1993; Ko 1994). The Monte Carlo model (probability prediction), with incorporation of the finite element method for handling irregular tooth surface, was developed by Wang and

models, and show two examples.

**2.1 1970-1990: Enlightenment stage -2D modeling** 

**2.2 "1990-2000" beginnings stage of 3D modeling** 

**2. History of dental FEA** 

mathematical errors.

Ko et al (1999) to stimulate optical scattering of the incipient caries (e.g., white spot lesion). The simulated image of the lesion surface was consistent with the true image captured *in clinic* (Figure 1). Linear fit of the image brightness between the FE and clinical images was 85% matched, indicating the feasibility of using numerical model to interpret clinical white spot lesions. The similar probability method was recently used to predict healing bone adaptation in tibia (Byrne et al., 2011). Recognition of the importance of 3D models and specific solutions were the major contributions in this era.

Fig. 1. **A.** Finite element mesh of *in vivo* carious tooth used for Monte Carlo simulation; **B.** Image rendered from Monte Carlo 3D simulation, **C.** True image of carious tooth obtained from a patient's premolar using an intra-oral camera.

#### **2.3 "2000-2010" age of proliferation, 3D with CAD**

As advancements have been made in computer and software capability, more complex 3D structures (e.g., occlusal surfaces, pulp, dentin, enamel) have been simulated in greater detail. Many recent FE studies have demonstrated accurate 3D anatomic structures of a sectioned jaw-teeth complex using μCT images. Increased mathematical functions in 3D computer-aid-design (CAD) have allowed accurate rendition of dental anatomy and prosthetic components such as implant configuration and veneer crowns (Figure 2). Fine meshing and high CPU computing power appeared to allow calculation of mechanical fields (e.g., stress, strain, energy) accounting for anatomic details and hierarchy interfaces between different tissues (e.g., dentin, PDL, enamel) that were offered by the CAD program. It was also recognized that inclusion of complete dentition is necessary to accurately predict stressstrain fields for functional treatment and jaw function (Field et al. 2009). Simplified models containing only a single tooth overlooked the effect of tooth-tooth contacts that is important in specified biomechanical problems such as orthodontic tooth movement and traumatic tooth injury. CAD software such as SolidWorks© (Waltham, MA, USA), Pro Engineering© (Needham, MA), and Geomagic (Triangle Park, NC, USA) have been adapted to construct dentofacial compartments and prostheses. These CAD programs output solid models that are then converted to FE programs (e.g., Abaqus, Ansys, Marc, Mimics) for meshing and solving. The automeshing capability of FE programs significantly improved during this era.

Past, Present and Future of Finite Element Analysis in Dentistry 7

Although segmentations may initially appear very accurate (Figure 3A), there are often many small irregularities that must be addressed (Figure 3B). Obviously, organic objects will have natural irregularities that may be important to model, but defects from the scanning and segmentation process must be removed. Automated processes in Geomagic such as mesh doctor can identify problematic areas (Figure 3B) and fix many minor problems. For larger defects, defeaturing may be required. Once the gaps in the surface have been filled, some amount of smoothing is typically beneficial. Excess surface detail that will not affect results only increases the file size, meshing times, mesh density, and solution times. To improve surfacing, a surface mesh on the order of 200,000 polygons is recommended. Geomagic has a tool ("optimize for surfacing") that redistributes the polygons nodes on the surface to create a more ideal distribution for surfacing (Figure 3C). Following these optimization steps, it is important to compare the final surface to the initial

Fig. 3. Although initial geometry following segmentation can appear smooth (A), many small defects are present that Geomagic will highlight in red using "mesh doctor" as potentially problematic (B). Following closing gaps, smoothing, minor defeaturing, and

With the optimized surfaces prepared using the previous steps, closed solid bodies can be created. Although the actual final bodies with the interior and exterior surfaces can be created at this stage, we have observed that closing each surface independently and using Booleans in the CAD program typically improves results. For example, this forces the interior surface of the enamel to be the identical surface as the exterior of the dentin. If the Boolean operations are done prior to surfacing, minor differences in creating NURB surfaces

optimization for surfacing, the polygon mesh is greatly improved (C).

surface to verify that no significant changes were made.

Fig. 2. Fine finite element mesh generated for ceramics veneer simulation.

#### **3. Current development of 3D dental solid models using CAD programs**

Currently, solid models have been created from datasets of computer tomography (CT) images, microCT images, or magnetic resonance images (MRI). To create a solid model from an imaging database, objects first need to be segregated by identifying interfaces. This is performed through the creation of non-manifold assemblies either through sequential 2D sliced or through segmentation of 3D objects. For this type of model reconstruction, the interfaces between different bodies are precisely specified, ensuring the existence of common nodes between different objects of the contact area. This provides a realistic simulation of load distribution within the object. For complex interactions, such as boneimplant interfaces or modeling the periodontal ligament (PDL), creation of these coincident nodes is essential.

When direct engineering (forward engineering) cannot be applied, reverse engineering is useful for converting stereolithographic (STL) objects into CAD objects (.iges). Despite minor loss of detail, this was the only option for creation of 3D organic CAD objects until the development of 3D segmentation tools and remains a common method even today. The creation of STL layer-by-layer objects requires segmentation tools, such as ITK-SNAP (Yushkevich et al., 2006) to segment structures in 3D medical images. SNAP provides semiautomatic segmentation using active contour methods, as well as manual delineation and image navigation.

Following segmentation, additional steps are required to prepare a model to be imported into CAD programs. FEA requires closed solid bodies – in other words, each part of the model should be able to hold water. Typical CT segmentations yield polygon surfaces with irregularities and possible holes. A program capable of manipulating these polygons and creating solid CAD bodies is required, such as Geomagic (Triangle Park, NC, USA).

Fig. 2. Fine finite element mesh generated for ceramics veneer simulation.

nodes is essential.

image navigation.

**3. Current development of 3D dental solid models using CAD programs** 

Currently, solid models have been created from datasets of computer tomography (CT) images, microCT images, or magnetic resonance images (MRI). To create a solid model from an imaging database, objects first need to be segregated by identifying interfaces. This is performed through the creation of non-manifold assemblies either through sequential 2D sliced or through segmentation of 3D objects. For this type of model reconstruction, the interfaces between different bodies are precisely specified, ensuring the existence of common nodes between different objects of the contact area. This provides a realistic simulation of load distribution within the object. For complex interactions, such as boneimplant interfaces or modeling the periodontal ligament (PDL), creation of these coincident

When direct engineering (forward engineering) cannot be applied, reverse engineering is useful for converting stereolithographic (STL) objects into CAD objects (.iges). Despite minor loss of detail, this was the only option for creation of 3D organic CAD objects until the development of 3D segmentation tools and remains a common method even today. The creation of STL layer-by-layer objects requires segmentation tools, such as ITK-SNAP (Yushkevich et al., 2006) to segment structures in 3D medical images. SNAP provides semiautomatic segmentation using active contour methods, as well as manual delineation and

Following segmentation, additional steps are required to prepare a model to be imported into CAD programs. FEA requires closed solid bodies – in other words, each part of the model should be able to hold water. Typical CT segmentations yield polygon surfaces with irregularities and possible holes. A program capable of manipulating these polygons and

creating solid CAD bodies is required, such as Geomagic (Triangle Park, NC, USA).

Although segmentations may initially appear very accurate (Figure 3A), there are often many small irregularities that must be addressed (Figure 3B). Obviously, organic objects will have natural irregularities that may be important to model, but defects from the scanning and segmentation process must be removed. Automated processes in Geomagic such as mesh doctor can identify problematic areas (Figure 3B) and fix many minor problems. For larger defects, defeaturing may be required. Once the gaps in the surface have been filled, some amount of smoothing is typically beneficial. Excess surface detail that will not affect results only increases the file size, meshing times, mesh density, and solution times. To improve surfacing, a surface mesh on the order of 200,000 polygons is recommended. Geomagic has a tool ("optimize for surfacing") that redistributes the polygons nodes on the surface to create a more ideal distribution for surfacing (Figure 3C). Following these optimization steps, it is important to compare the final surface to the initial surface to verify that no significant changes were made.

Fig. 3. Although initial geometry following segmentation can appear smooth (A), many small defects are present that Geomagic will highlight in red using "mesh doctor" as potentially problematic (B). Following closing gaps, smoothing, minor defeaturing, and optimization for surfacing, the polygon mesh is greatly improved (C).

With the optimized surfaces prepared using the previous steps, closed solid bodies can be created. Although the actual final bodies with the interior and exterior surfaces can be created at this stage, we have observed that closing each surface independently and using Booleans in the CAD program typically improves results. For example, this forces the interior surface of the enamel to be the identical surface as the exterior of the dentin. If the Boolean operations are done prior to surfacing, minor differences in creating NURB surfaces

Past, Present and Future of Finite Element Analysis in Dentistry 9

Fig. 5. The solid model of a maxillary central incisor was created through the following steps. (A) Multiple sketches were created in various slices of the microCT data. The sketch defined the contour of the root. (B) Sequential contours were used to reconstruct outer surface of dentin and other parts (e.g., enamel and pulp - not shown). (C) All parts (enamel,

dentin and pulp) were combined to form the solid model of the central incisor. All

procedures were performed using SolidWorks software.

A.

B.

C.

may affect the connectivity of the objects. Some research labs (Bright and Rayfield 2011) will simply transfer the polygon surfaces over to a FEA program for analysis without using a CAD program. This can be very effective for relatively simple models, but when multiple solid bodies are included and various mesh densities are required this process becomes cumbersome.

To use a CAD program with organic structures, the surface cannot be a polygon mesh, but rather needs to have a mathematical approximation of the surface. This is typically done with NURB surfaces, so the solid can be saved as an .iges or .step file. This process involves multiple steps – laying out patches, creating grids within these patches, optimizing the surface detail, and finally creating the NURB surface (Figure 4). Surfacing must be done carefully as incorrectly laying out the patches on the surface or not allowing sufficient detail may severely distort the surface. In the end, the surfaced body should not have problematic geometry, such as sliver faces, small faces, or small edges.

Fig. 4. Process of NURB surface generation using Geomagic. (A) Contour lines are defined that follow the natural geometry - in this case, line angles were used. (B) Patches are constructed and shuffled to create a clean grid pattern. (C) Grids are created within each patch. (D) NURB surfaces are created by placing control points along the created grids.

CAD programs allow the incorporation of high definition materials or parts from geometry files (e.g. .iges, .step), such as dentures, prosthesis, orthodontics appliances, dental restorative materials, surgical plates and dental implants. They even allow partial modification of the solid model obtained by CT or μCT to more closely reproduce accurate organic geometry. Organic modeling (biomodeling) extensively uses splines and curves to model the complex geometry. FE software or other platforms with limited CAD tools typically do not provide the full range of features required to manipulate these complicated organic models. Therefore, the use of a genuine CAD program is typically preferred for detailed characterization of the material and its contact correlation with surrounding structures. This is especially true for models that demand strong modification of parts or incorporation of multiple different bodies.

When strong modification is required, the basic parts of the model such as bone, skin or basic structures can be obtained in .stl format. They are then converted to a CAD file allowing modification and/or incorporation of new parts before the FE analysis. It is also possible to use the CT or microCT dataset to directly create a solid in the CAD program.

may affect the connectivity of the objects. Some research labs (Bright and Rayfield 2011) will simply transfer the polygon surfaces over to a FEA program for analysis without using a CAD program. This can be very effective for relatively simple models, but when multiple solid bodies are included and various mesh densities are required this process becomes

To use a CAD program with organic structures, the surface cannot be a polygon mesh, but rather needs to have a mathematical approximation of the surface. This is typically done with NURB surfaces, so the solid can be saved as an .iges or .step file. This process involves multiple steps – laying out patches, creating grids within these patches, optimizing the surface detail, and finally creating the NURB surface (Figure 4). Surfacing must be done carefully as incorrectly laying out the patches on the surface or not allowing sufficient detail may severely distort the surface. In the end, the surfaced body should not have problematic

Fig. 4. Process of NURB surface generation using Geomagic. (A) Contour lines are defined that follow the natural geometry - in this case, line angles were used. (B) Patches are constructed and shuffled to create a clean grid pattern. (C) Grids are created within each patch. (D) NURB surfaces are created by placing control points along the created grids.

CAD programs allow the incorporation of high definition materials or parts from geometry files (e.g. .iges, .step), such as dentures, prosthesis, orthodontics appliances, dental restorative materials, surgical plates and dental implants. They even allow partial modification of the solid model obtained by CT or μCT to more closely reproduce accurate organic geometry. Organic modeling (biomodeling) extensively uses splines and curves to model the complex geometry. FE software or other platforms with limited CAD tools typically do not provide the full range of features required to manipulate these complicated organic models. Therefore, the use of a genuine CAD program is typically preferred for detailed characterization of the material and its contact correlation with surrounding structures. This is especially true for models that demand strong modification of parts or

When strong modification is required, the basic parts of the model such as bone, skin or basic structures can be obtained in .stl format. They are then converted to a CAD file allowing modification and/or incorporation of new parts before the FE analysis. It is also possible to use the CT or microCT dataset to directly create a solid in the CAD program.

geometry, such as sliver faces, small faces, or small edges.

incorporation of multiple different bodies.

cumbersome.

Fig. 5. The solid model of a maxillary central incisor was created through the following steps. (A) Multiple sketches were created in various slices of the microCT data. The sketch defined the contour of the root. (B) Sequential contours were used to reconstruct outer surface of dentin and other parts (e.g., enamel and pulp - not shown). (C) All parts (enamel, dentin and pulp) were combined to form the solid model of the central incisor. All procedures were performed using SolidWorks software.

Past, Present and Future of Finite Element Analysis in Dentistry 11

The validity of the dental FEA has been a concern for decades. Two review articles (Korioth and Versluis, 1997; Geng et al., 2001) in dentistry provided thorough discussions about effects of geometry, element type and size, material properties, and boundary conditions on the accuracy of solutions. In general these discussions echoed an earlier review by Huiskes and Cao (1983). The severity of these effects has decreased as the technologies and knowledge evolved in the field. In the present CAD-FEA era, the consideration of FEA accuracy in relation to loading, boundary (constraint) conditions, and validity of material

The static loading such as bite forces is usually applied as point forces to study prosthetic designs and dental restorations. The bite force, however, presents huge variations (both magnitude and direction) based on previous experimental measures (Proffit et al., 1983; Proffit and Field 1983). Fortunately, FEA allows for easy changes in force magnitudes and directions to approximate experimental data, which can serve as a reasonable parametric study to assess different loading effects. On the other hand, loading exerted by devices such as orthodontic wires is unknown or never measured experimentally, and should be

The boundary condition is a constraint applied to the model, from which potential energy and solutions are derived. False solutions can be associated at the areas next to the constraints. As a result, most dental models set constraints far away from the areas of interest. Based on the Saint-Venant's principle, the effects of constraints at sufficiently large distances become negligible. However, some modeling applies specific constraints to study particular physical phenomenon. For example, the homogenization theory was derived to resolve microstructural effects in composite by applying periodic constraints (Ko et al., 1996). It was reported that using homogenization theory to estimate boneimplant interfacial stresses by accounting for microstructural effects might introduce up

Mechanical properties of biological tissues remain a major concern for the FE approach because of the viscoelastic nature of biological tissues that prevents full characterization of its time-dependent behaviors. Little technology is available to measure oral tissue properties. Most FE studies in dentistry use the linear elastic assumption. Data based on density from CT images can be used to assign heterogeneous properties. Few researches attempting to predict non-linear behaviors using bilinear elastic constants aroused risks for a biased result (Cattaneo et al., 2009). Laboratory tests excluding tissues (e.g., PDL) were also found to result in less accurate data than computer predictions (Chi et al., 2011). Caution must be used when laboratory data is applied to validate the model. To our knowledge, the most valuable data for validation resides on clinical assessments such as

measuring tooth movement (Yoshida et al., 1998; Brosh et al., 2002).

**4.2 Validity of the models** 

properties are described as follows:

simulated with caution (see the section 5.2)

**4.2.2 Boundary Condition (BC)** 

to 20% error (Ko 1994).

**4.2.3 Material properties** 

**4.2.1 Loading** 

Initially, this procedure might be time-consuming. However, it is useful for quickly and efficiently making changes in parts, resizing multiple parts that are already combined, and incorporating new parts. This also allows for serial reproduction of unaltered parts of the model, such as loading areas and unaltered support structures, keeping their dimensions and Cartesian coordinates.

This procedure involves the partial or full use of the dataset, serially organized, to create different parts. (Figures 5 A & B) In models with multiple parts, additional tools such as lofts, sweeps, surfaces, splines, reference planes, and lines can be used to modify existing solids or create new solids (Figures 5C). Different parts may be combined through Boolean operations to generate a larger part, to create spaces or voids, or to modify parts. The parts can be also copied, moved, or mirrored in order to reproduce different scenarios without creating an entirely new model.

#### **4. Finite element analysis of the current dental models**

#### **4.1 Meshing**

For descritization of the solid model, most FE software has automated mesh generating features that produce rather dense meshes. However, it is important to enhance the controller that configures the elements including types, dimensions, and relations to better fit the analysis to a particular case and its applications. Most of current FE software is capable of assessing the quality of the mesh according to element aspect ratio and the adaptive method. The ability of the adaptive method to automatically evaluate and modify the contact area between two objects overlapping the same region and to refine the mesh locally in areas of greater importance and complexity has profoundly improved the accuracy of the solution. Although automated mesh generation has greatly improved, note that it still requires careful oversight based on the specific analysis being performed. For example, when examining stresses produced in the periodontal ligament with orthodontic appliances, the mesh will be greatly refined in the small geometry of the orthodontic bracket, but may be too coarse in the periodontal ligament – the area of interest.

The validation that was concerned with meshing errors and morphological inaccuracy during 1970 ─ 2000 is no longer a major concern as the CAD and meshing technology evolves. However, numerical convergence (Huang et al., 2007) is still required, which is frequently neglected in dental simulations (Tanne et al., 1987; Jones et al., 2001; Liang et al., 2009; Kim et al., 2010). Some biologists ignore all results from FEA, requesting an unreasonable level of validation for each model, but overlooking the valuable contributions of engineering principles. A rational request should recognize evolution of the advanced technologies but focus on numerical convergence. The numerical convergence is governed by two factors, continuity and approximation methods, and can be classified to strong convergence ||Xn|| ||X|| as n ∞ and weak convergence ∫(Xn) ∫(X) as n ∞ where Xn represents physical valuables such as displacement, temperature, and velocity. X represents the exact solution and ∫ indicates the potential energy. It is recommended that all dental FE models should test meshing convergence prior to analyses.

#### **4.2 Validity of the models**

The validity of the dental FEA has been a concern for decades. Two review articles (Korioth and Versluis, 1997; Geng et al., 2001) in dentistry provided thorough discussions about effects of geometry, element type and size, material properties, and boundary conditions on the accuracy of solutions. In general these discussions echoed an earlier review by Huiskes and Cao (1983). The severity of these effects has decreased as the technologies and knowledge evolved in the field. In the present CAD-FEA era, the consideration of FEA accuracy in relation to loading, boundary (constraint) conditions, and validity of material properties are described as follows:

#### **4.2.1 Loading**

10 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Initially, this procedure might be time-consuming. However, it is useful for quickly and efficiently making changes in parts, resizing multiple parts that are already combined, and incorporating new parts. This also allows for serial reproduction of unaltered parts of the model, such as loading areas and unaltered support structures, keeping their dimensions

This procedure involves the partial or full use of the dataset, serially organized, to create different parts. (Figures 5 A & B) In models with multiple parts, additional tools such as lofts, sweeps, surfaces, splines, reference planes, and lines can be used to modify existing solids or create new solids (Figures 5C). Different parts may be combined through Boolean operations to generate a larger part, to create spaces or voids, or to modify parts. The parts can be also copied, moved, or mirrored in order to reproduce different scenarios without

For descritization of the solid model, most FE software has automated mesh generating features that produce rather dense meshes. However, it is important to enhance the controller that configures the elements including types, dimensions, and relations to better fit the analysis to a particular case and its applications. Most of current FE software is capable of assessing the quality of the mesh according to element aspect ratio and the adaptive method. The ability of the adaptive method to automatically evaluate and modify the contact area between two objects overlapping the same region and to refine the mesh locally in areas of greater importance and complexity has profoundly improved the accuracy of the solution. Although automated mesh generation has greatly improved, note that it still requires careful oversight based on the specific analysis being performed. For example, when examining stresses produced in the periodontal ligament with orthodontic appliances, the mesh will be greatly refined in the small geometry of the orthodontic bracket, but may be too coarse in the periodontal ligament – the area of

The validation that was concerned with meshing errors and morphological inaccuracy during 1970 ─ 2000 is no longer a major concern as the CAD and meshing technology evolves. However, numerical convergence (Huang et al., 2007) is still required, which is frequently neglected in dental simulations (Tanne et al., 1987; Jones et al., 2001; Liang et al., 2009; Kim et al., 2010). Some biologists ignore all results from FEA, requesting an unreasonable level of validation for each model, but overlooking the valuable contributions of engineering principles. A rational request should recognize evolution of the advanced technologies but focus on numerical convergence. The numerical convergence is governed by two factors, continuity and approximation methods, and can be classified to strong convergence ||Xn|| ||X|| as n ∞ and weak convergence ∫(Xn) ∫(X) as n ∞ where Xn represents physical valuables such as displacement, temperature, and velocity. X represents the exact solution and ∫ indicates the potential energy. It is recommended that all dental FE models should test meshing convergence

and Cartesian coordinates.

creating an entirely new model.

**4.1 Meshing** 

interest.

prior to analyses.

**4. Finite element analysis of the current dental models** 

The static loading such as bite forces is usually applied as point forces to study prosthetic designs and dental restorations. The bite force, however, presents huge variations (both magnitude and direction) based on previous experimental measures (Proffit et al., 1983; Proffit and Field 1983). Fortunately, FEA allows for easy changes in force magnitudes and directions to approximate experimental data, which can serve as a reasonable parametric study to assess different loading effects. On the other hand, loading exerted by devices such as orthodontic wires is unknown or never measured experimentally, and should be simulated with caution (see the section 5.2)

#### **4.2.2 Boundary Condition (BC)**

The boundary condition is a constraint applied to the model, from which potential energy and solutions are derived. False solutions can be associated at the areas next to the constraints. As a result, most dental models set constraints far away from the areas of interest. Based on the Saint-Venant's principle, the effects of constraints at sufficiently large distances become negligible. However, some modeling applies specific constraints to study particular physical phenomenon. For example, the homogenization theory was derived to resolve microstructural effects in composite by applying periodic constraints (Ko et al., 1996). It was reported that using homogenization theory to estimate boneimplant interfacial stresses by accounting for microstructural effects might introduce up to 20% error (Ko 1994).

#### **4.2.3 Material properties**

Mechanical properties of biological tissues remain a major concern for the FE approach because of the viscoelastic nature of biological tissues that prevents full characterization of its time-dependent behaviors. Little technology is available to measure oral tissue properties. Most FE studies in dentistry use the linear elastic assumption. Data based on density from CT images can be used to assign heterogeneous properties. Few researches attempting to predict non-linear behaviors using bilinear elastic constants aroused risks for a biased result (Cattaneo et al., 2009). Laboratory tests excluding tissues (e.g., PDL) were also found to result in less accurate data than computer predictions (Chi et al., 2011). Caution must be used when laboratory data is applied to validate the model. To our knowledge, the most valuable data for validation resides on clinical assessments such as measuring tooth movement (Yoshida et al., 1998; Brosh et al., 2002).

Past, Present and Future of Finite Element Analysis in Dentistry 13

incomplete reflections of normal human anatomy. The purpose of this study was to construct a more anatomically accurate FE model to evaluate miniscrew biomechanics.

A posterior segment was sectioned from the full maxillary model. Borders of the model were established as follows: the mesial boundary was at the interproximal region between the maxillary right canine and first premolar; the distal boundary used the distal aspect of the maxillary tuberosity; the inferior boundary was the coronal anatomy of all teeth ; and the superior boundary was all maxillary structures (including sinus and zygoma) up to

An orthodontic miniscrew (TOMAS®, 8mm long, 1.6mm diameter) was created using Solidworks CAD software. The miniscrew outline was created using the Solidworks sketch function and revolved into three dimensions. The helical sweep function was used to create a continuous, spiral thread. Subtraction cuts were used to create the appropriate head configuration after hexagon ring placement. The miniscrew was inserted into the maxillary model from the buccal surface between the second premolar and first molar using Solidworks. The miniscrew was inserted sequentially at angles of 90°, 60° and 45° vertically relative to the surface of the cortical bone (Figure 6B), and was placed so that the miniscrew neck/thread interface was coincident with the external contour of the cortical bone. For each angulation, the point of intersection between the cortical bone surface and the central axis of the miniscrew was maintained constant to ensure consistency between models. Boolean operations were

performed and a completed model assembly was created at each angulation.

 (A) (B) (C) Fig. 6. The FE model of the orthodontic miniscrew used in the present study. (A) The solid model of four maxillary teeth plus the miniscrew was created using SolidWorks. (B) Close look of the miniscrew inserted to the bone. (C) FE mesh was generated by Ansys Workbench

The IGES format file of each finished 3D model was exported to ANSYS 10.0 Workbench (Swanson Analysis Inc., Huston, PA, USA), and FE models with 10-node tetrahedral helements were generated for each assembly. The final FE mesh generated for each model contained approximately 91,500 elements, which was sufficient to obtain solution convergence. Following FE mesh generation, the model was fixed at the palatal, mesial, and

10.0. F indicates the force (1.47 N = 150gm) applied to the miniscrew.

Variations of miniscrew insertion angulations and implant materials were analyzed.

**5.1.2 Methods** 

15mm superior to tooth apices (Figure 6A).

#### **4.3 Solution/principle**

The weak form of the equilibrium equation for classic mechanics is given below:

() () *cijkl ij kl v ud d t vi i* , where represents the total domain of the object, and *ti* represents tractions. *ε* is obtained by applying the small strain-displacement relationship <sup>1</sup> () ( ) <sup>2</sup> *j i ij j i u u u x x* . Stresses will be obtained by the constitutive law *ij Eijkl kl* . Using the variational formulation and mesh descritization, this equilibrium equation can be assembled by the individual element *<sup>t</sup> <sup>e</sup> B D Bd e* . plus the boundary integral where B is the shape function and D is element stiffness matrix. The element stiffness matrix represents material property of either a linear or non-linear function. As mentioned above, mechanical properties of oral tissues are poorly characterized. The most controversial oral tissue is the PDL due to its importance in supporting teeth and regulating alveolar bone remodeling. To date, studies conducted to characterize non-linear behaviors of the PDL are not yet conclusive. One approximation of PDL properties assumes zero stiffness under low compression resulting in very low stress under compression (Cattaneo et al., 2009). Interpretation of such non-linear models must be approached with cautious. Consequently, linear elastic constants are frequently used for dental simulations to investigate initial responses under static loading.

In addition to the commonly used point forces, the tractions (*t*i) in dental simulations should consider preconditions (e.g., residual stress, polymerization shrinkage and unloading of orthodontic archwire). Previously, investigation of composite shrinkage yielded valuable contributions to restorative dentistry (Magne et al., 1999). In the following section, we will demonstrate two applications using submodels from a full dentition CAD model: one with static point loading and the other with deactivated orthodontic archwire bending.

#### **5. Examples of dental FEA**

As described in Section 3, a master CAD model with full dentition was developed. The model separates detailed anatomic structures such as PDL, pulp, dentin, enamel, lamina dura, cortical bone, and trabecular bone. This state-of-the-art model contains high order NURB surfaces that allow for fine meshing, with excellent connectivity so the model can be conformally meshed with concurrent nodes at all interfaces. Many submodels can be isolated from this master model to study specified biomechanical questions. Two examples presented here are the first series of applications: orthodontic miniscrews and orthodontic archwires for tooth movement.

#### **5.1 Orthodontic miniscrews**

#### **5.1.1 Introduction**

The placement of miniscrews has become common in orthodontic treatment to enhance tooth movement and to prevent unwanted anchorage loss. Unfortunately, the FE biomechanical miniscrew models reported to date have been oversimplified or show incomplete reflections of normal human anatomy. The purpose of this study was to construct a more anatomically accurate FE model to evaluate miniscrew biomechanics. Variations of miniscrew insertion angulations and implant materials were analyzed.

#### **5.1.2 Methods**

12 Finite Element Analysis – From Biomedical Applications to Industrial Developments

 

*ti* represents tractions. *ε* is obtained by applying the small strain-displacement relationship

the variational formulation and mesh descritization, this equilibrium equation can be

is the shape function and D is element stiffness matrix. The element stiffness matrix represents material property of either a linear or non-linear function. As mentioned above, mechanical properties of oral tissues are poorly characterized. The most controversial oral tissue is the PDL due to its importance in supporting teeth and regulating alveolar bone remodeling. To date, studies conducted to characterize non-linear behaviors of the PDL are not yet conclusive. One approximation of PDL properties assumes zero stiffness under low compression resulting in very low stress under compression (Cattaneo et al., 2009). Interpretation of such non-linear models must be approached with cautious. Consequently, linear elastic constants are frequently used for dental simulations to investigate initial

In addition to the commonly used point forces, the tractions (*t*i) in dental simulations should consider preconditions (e.g., residual stress, polymerization shrinkage and unloading of orthodontic archwire). Previously, investigation of composite shrinkage yielded valuable contributions to restorative dentistry (Magne et al., 1999). In the following section, we will demonstrate two applications using submodels from a full dentition CAD model: one with

As described in Section 3, a master CAD model with full dentition was developed. The model separates detailed anatomic structures such as PDL, pulp, dentin, enamel, lamina dura, cortical bone, and trabecular bone. This state-of-the-art model contains high order NURB surfaces that allow for fine meshing, with excellent connectivity so the model can be conformally meshed with concurrent nodes at all interfaces. Many submodels can be isolated from this master model to study specified biomechanical questions. Two examples presented here are the first series of applications: orthodontic miniscrews and orthodontic

The placement of miniscrews has become common in orthodontic treatment to enhance tooth movement and to prevent unwanted anchorage loss. Unfortunately, the FE biomechanical miniscrew models reported to date have been oversimplified or show

static point loading and the other with deactivated orthodontic archwire bending.

*<sup>e</sup> B D Bd e*

represents the total domain of the object, and

. plus the boundary integral where B

*ij Eijkl*

*kl* . Using

The weak form of the equilibrium equation for classic mechanics is given below:

. Stresses will be obtained by the constitutive law

**4.3 Solution/principle** 

 

<sup>1</sup> () ( ) <sup>2</sup>

*ij*

*u*

() () *cijkl ij kl v ud d t vi i*

*j i*

*j i u u*

 

*x x*

responses under static loading.

**5. Examples of dental FEA** 

archwires for tooth movement.

**5.1 Orthodontic miniscrews** 

**5.1.1 Introduction** 

, where

assembled by the individual element *<sup>t</sup>*

A posterior segment was sectioned from the full maxillary model. Borders of the model were established as follows: the mesial boundary was at the interproximal region between the maxillary right canine and first premolar; the distal boundary used the distal aspect of the maxillary tuberosity; the inferior boundary was the coronal anatomy of all teeth ; and the superior boundary was all maxillary structures (including sinus and zygoma) up to 15mm superior to tooth apices (Figure 6A).

An orthodontic miniscrew (TOMAS®, 8mm long, 1.6mm diameter) was created using Solidworks CAD software. The miniscrew outline was created using the Solidworks sketch function and revolved into three dimensions. The helical sweep function was used to create a continuous, spiral thread. Subtraction cuts were used to create the appropriate head configuration after hexagon ring placement. The miniscrew was inserted into the maxillary model from the buccal surface between the second premolar and first molar using Solidworks. The miniscrew was inserted sequentially at angles of 90°, 60° and 45° vertically relative to the surface of the cortical bone (Figure 6B), and was placed so that the miniscrew neck/thread interface was coincident with the external contour of the cortical bone. For each angulation, the point of intersection between the cortical bone surface and the central axis of the miniscrew was maintained constant to ensure consistency between models. Boolean operations were performed and a completed model assembly was created at each angulation.

Fig. 6. The FE model of the orthodontic miniscrew used in the present study. (A) The solid model of four maxillary teeth plus the miniscrew was created using SolidWorks. (B) Close look of the miniscrew inserted to the bone. (C) FE mesh was generated by Ansys Workbench 10.0. F indicates the force (1.47 N = 150gm) applied to the miniscrew.

The IGES format file of each finished 3D model was exported to ANSYS 10.0 Workbench (Swanson Analysis Inc., Huston, PA, USA), and FE models with 10-node tetrahedral helements were generated for each assembly. The final FE mesh generated for each model contained approximately 91,500 elements, which was sufficient to obtain solution convergence. Following FE mesh generation, the model was fixed at the palatal, mesial, and

Past, Present and Future of Finite Element Analysis in Dentistry 15

Composite SS Titanium Material

Composite SS Titanium Material

Composite SS Titanium Material

Composite SS Titanium Material

Composite SS Titanium Material

Composite SS Titanium Material

Fig. 8. Plots of mean stress (MPa) averaged over angulations (left column), materials (middle), and cross-action between angulation and materials (right). Symbols - MaxPS: maximum principal stress; MinPS: minimum principal stress; S: screw; B: Bone.



45

60 90

456090

45

60 90

45

60 90

> 45 60 90

90

60

Composite SS Titanium Material

Composite SS Titanium Material

Composite SS Titanium Material

Composite SS Titanium Material

Composite SS Titanium Material

Composite SS Titanium Material




MaxPS S



Von Mises S

Von Mises B

MinPS B

MinPS S

MaxPS B

45 60 90 Angulation

45 60 90 Angulation

45 60 90 Angulation

45 60 90 Angulation

45 60 90 Angulation

45 60 90 Angulation

superior boundaries. A 150 grams loading force to the mesial was then applied to the miniscrew to simulate distalization of anterior teeth (Figure 6C). All materials were linear and isotropic (Table 1), and the miniscrew/bone interface was assumed to be rigidly bonded. Three material properties (stainless steel, titanium, and composite) were used for the miniscrew. Each model was solved under the small displacement assumption. Two-way ANOVA was used to compare effects of angulation and material.


Table 1. Computer model component material properties (O'Brien, 1997)

#### **5.1.3 Results**

#### **Angle effect**

Stress patterns in both cortical bone and the miniscrew from each simulation were concentrated in the second premolar/first molar area immediately around the implant/bone interface (Figure 7). Peak stress values for each model simulation are listed in Table 2. Peak maximum principal stress (MaxPS) within the miniscrew was greatest when angle placement was 45°. Peak MaxPS was lowest at the 60° placement angle. Peak MaxPS in cortical bone was greatest at 45° angulation, except for the stainless steel implant. In each angulation, the location of greatest maximum principle stress was located at the distal aspect of the miniscrew/cortical bone interface. Similarly, peak minimum principal stress (MinPS) was lowest on the miniscrew at 60° and greatest at 45°. Figure 8 shows mean stress plots for all angulations and materials. Angulation difference was statistically significant for miniscrews at 45° compared to 60° and 90° for all stress types analyzed (MaxPS p=0.01, MinPS p=0.01, von Mises stress (vonMS) p=0.01). There is no significant difference in cortical bone stress at any angulation.

Fig. 7. Stress distributions of the orthodontic miniscrew showed that stresses concentrated in the neck region of the miniscrew at the interface between bone and the screw.

superior boundaries. A 150 grams loading force to the mesial was then applied to the miniscrew to simulate distalization of anterior teeth (Figure 6C). All materials were linear and isotropic (Table 1), and the miniscrew/bone interface was assumed to be rigidly bonded. Three material properties (stainless steel, titanium, and composite) were used for the miniscrew. Each model was solved under the small displacement assumption. Two-way

13,700 1370 175 18,000 77,900 175 190,000 113,000 20,000

Ratio 0.3 0.3 0.4 0.3 0.3 0.4 0.3 0.3 0.3

Stress patterns in both cortical bone and the miniscrew from each simulation were concentrated in the second premolar/first molar area immediately around the implant/bone interface (Figure 7). Peak stress values for each model simulation are listed in Table 2. Peak maximum principal stress (MaxPS) within the miniscrew was greatest when angle placement was 45°. Peak MaxPS was lowest at the 60° placement angle. Peak MaxPS in cortical bone was greatest at 45° angulation, except for the stainless steel implant. In each angulation, the location of greatest maximum principle stress was located at the distal aspect of the miniscrew/cortical bone interface. Similarly, peak minimum principal stress (MinPS) was lowest on the miniscrew at 60° and greatest at 45°. Figure 8 shows mean stress plots for all angulations and materials. Angulation difference was statistically significant for miniscrews at 45° compared to 60° and 90° for all stress types analyzed (MaxPS p=0.01, MinPS p=0.01, von Mises stress (vonMS) p=0.01).

Fig. 7. Stress distributions of the orthodontic miniscrew showed that stresses concentrated in

the neck region of the miniscrew at the interface between bone and the screw.

Stainless Steel (SS)

Titanium

(Ti) Composite


ANOVA was used to compare effects of angulation and material.

Cortical Trabecular PDL Dentin Enamel Pulp

Table 1. Computer model component material properties (O'Brien, 1997)

There is no significant difference in cortical bone stress at any angulation.

Young's Modulus (MPa)

Poisson's

**5.1.3 Results Angle effect** 

Fig. 8. Plots of mean stress (MPa) averaged over angulations (left column), materials (middle), and cross-action between angulation and materials (right). Symbols - MaxPS: maximum principal stress; MinPS: minimum principal stress; S: screw; B: Bone.

Past, Present and Future of Finite Element Analysis in Dentistry 17

these results to other orthodontic studies using FEA is challenging due to several differences between models. Many of the studies available in the literature do not model human anatomy ( Gracco et al., 2009; Motoyoshi et al., 2005) and are not 3-dimensional (Brettin et al., 2008), or require additional resolution (Jones et al., 2001). Cattaneo et al (2009) produced a similar high-quality model of two teeth and surrounding bone for evaluation of orthodontic tooth movement and resulting periodontal stresses. Both linear and non-linear PDL mechanical properties were simulated. In a different study, Motoyoshi found peak bone vonMS in their model between 4-33MPa, similar to the levels in the current study (9.43-15.13MPa) However, Motoyoshi used a 2N (~203gm) force applied obliquely at 45°,

From the results of the present study, an angulation of 60° is more favorable than either 45° or 90° for all three stress types generated relative to the stress on the miniscrew. Conversely, all three stress types have levels at 60° which are comparable to 45° and 90° in bone, so varied angulation within the range evaluated in the present study may not have a marked effect on the bone. However, miniscrews at 60° do have slightly higher MinPS (compressive) values than either 45° or 90°, which could have an effect on the rate or extent of biological

A third area of focus in the present study was the effect of using different miniscrew materials by comparing stress levels generated by popular titanium miniscrews with rarelyor never used stainless steel and composite miniscrews. Although no studies were found that compare Ti and/or SS miniscrews to composite miniscrews, one published study compared some of the mechanical properties of Ti and SS miniscrews (Carano et al., 2005). Carano et al reported that Ti and SS miniscrews could both safely be used as skeletal anchorage, and that Ti and SS miniscrew bending is >0.02mm at 1.471N (150gm) equivalent to the load applied in the present study. There was deformation of >0.01mm noted in the present study. However, the geometry in the study by Carano et al was otherwise not

There are no studies available which compare Ti and SS stresses or stress pattern generation. Therefore, to compare stresses generated from the use of composite to Ti and SS in the present study, the modulus of the miniscrew in each Ti model was varied to reflect that of SS and composite, with a subsequent test at each angulation. The fact that the general stress pattern for composite is dissimilar to Ti and SS when the miniscrew angulation is changed may be of clinical importance (Pollei 2009). Because composite has a much lower modulus than Ti or SS, it may be that stress shielding does not happen as much in composite miniscrews, and therefore more stress is transferred to the adjacent cortical bone in both compression and tension scenarios. As a result of increased compressive and tensile forces, especially in the 45° model, biological activity related to remodeling may be increased relative to other models with lower stress levels, and therefore have a more significant impact on long-term miniscrew stability and success. Another potential undesirable effect of using composite miniscrews in place of either Ti or SS is the increased deformation and distortion that is inherent due to the decreased modulus of composite relative to titanium or steel. Mechanical or design improvements need to be made to allow for composite

different in both magnitude and orientation from that in this study.

activity and remodeling.

comparable to that in the present study.

miniscrews to be a viable alternative in clinical practice.


Table 2. Peak mean stress for each model.

#### **Material effect**

There is a noticeable (p=0.05) difference between material types with composite miniscrews having a higher average MaxPS and MinPS in cortical bone than Ti or SS. Peak MinPS is lowest on the miniscrew at 60° for Ti and SS miniscrews, and similar at 60° and 90° for composite. MinPS is greatest at 45° for all three materials. Peak MinPS is approximately the same in cortical bone for all three miniscrew materials at 90° (range -10.29 to -11.93MPa) but at 45° and 60° MinPS in cortical bone is higher for composite than Ti or SS (-33.26 & - 29.83MPa respectively for composite vs. -11.68/-10.05MPa & -16.23/-12.62MPa for Ti/SS). When comparing the MinPS pattern generated for 45°, 60°, and 90° angulations, the composite miniscrew does not mimic the Ti or SS pattern. Rather, MinMS is substantially lower in both the miniscrew and bone at 90° than Ti or SS. Peak vonMS was lowest on the miniscrew at 60° for all three miniscrew materials relative to the other angulations. As with MinPS, the vonMS for the composite miniscrew differs from the Ti and SS pattern generated for 45°, 60°, and 90° angulations and is substantially lower in both miniscrew and bone at 90° than Ti or SS.

#### **5.1.4 Discussion**

One of the primary areas of interest in the present study related to the construction of a human model that is both realistic and of sufficient detail to clinically valuable. Comparing

Bone Mini-

45 Titanium 89.3 17.93 -117.63 -11.68 107.54 12.89

60 Titanium 40.31 16.55 -32.18 -16.23 31.56 15.13

90 Titanium 49.73 16.01 -75.82 -10.29 67.24 9.43

There is a noticeable (p=0.05) difference between material types with composite miniscrews having a higher average MaxPS and MinPS in cortical bone than Ti or SS. Peak MinPS is lowest on the miniscrew at 60° for Ti and SS miniscrews, and similar at 60° and 90° for composite. MinPS is greatest at 45° for all three materials. Peak MinPS is approximately the same in cortical bone for all three miniscrew materials at 90° (range -10.29 to -11.93MPa) but at 45° and 60° MinPS in cortical bone is higher for composite than Ti or SS (-33.26 & - 29.83MPa respectively for composite vs. -11.68/-10.05MPa & -16.23/-12.62MPa for Ti/SS). When comparing the MinPS pattern generated for 45°, 60°, and 90° angulations, the composite miniscrew does not mimic the Ti or SS pattern. Rather, MinMS is substantially lower in both the miniscrew and bone at 90° than Ti or SS. Peak vonMS was lowest on the miniscrew at 60° for all three miniscrew materials relative to the other angulations. As with MinPS, the vonMS for the composite miniscrew differs from the Ti and SS pattern generated for 45°, 60°, and 90° angulations and is substantially lower in both miniscrew and bone at

One of the primary areas of interest in the present study related to the construction of a human model that is both realistic and of sufficient detail to clinically valuable. Comparing

screw

Composite 82.99 39.94 -102.34 -33.26 110.09 30.33

Stainless Steel 82.75 13.26 -109.24 -10.05 94.49 9.47

Composite 46.79 31.26 -40.29 -29.83 32.02 30.43

Stainless Steel 47.53 12.74 -32.24 -12.62 35.35 11.42

Composite 27.91 19.81 -39.67 -11.25 33.55 10.99

Stainless Steel 58.38 14.96 -88.9 -11.93 78.45 10

Bone Mini-

screw

Bone

**Angulation Material MaxPS MinPS vonMS** 

Miniscrew

**Material effect** 

90° than Ti or SS.

**5.1.4 Discussion** 

Table 2. Peak mean stress for each model.

these results to other orthodontic studies using FEA is challenging due to several differences between models. Many of the studies available in the literature do not model human anatomy ( Gracco et al., 2009; Motoyoshi et al., 2005) and are not 3-dimensional (Brettin et al., 2008), or require additional resolution (Jones et al., 2001). Cattaneo et al (2009) produced a similar high-quality model of two teeth and surrounding bone for evaluation of orthodontic tooth movement and resulting periodontal stresses. Both linear and non-linear PDL mechanical properties were simulated. In a different study, Motoyoshi found peak bone vonMS in their model between 4-33MPa, similar to the levels in the current study (9.43-15.13MPa) However, Motoyoshi used a 2N (~203gm) force applied obliquely at 45°, different in both magnitude and orientation from that in this study.

From the results of the present study, an angulation of 60° is more favorable than either 45° or 90° for all three stress types generated relative to the stress on the miniscrew. Conversely, all three stress types have levels at 60° which are comparable to 45° and 90° in bone, so varied angulation within the range evaluated in the present study may not have a marked effect on the bone. However, miniscrews at 60° do have slightly higher MinPS (compressive) values than either 45° or 90°, which could have an effect on the rate or extent of biological activity and remodeling.

A third area of focus in the present study was the effect of using different miniscrew materials by comparing stress levels generated by popular titanium miniscrews with rarelyor never used stainless steel and composite miniscrews. Although no studies were found that compare Ti and/or SS miniscrews to composite miniscrews, one published study compared some of the mechanical properties of Ti and SS miniscrews (Carano et al., 2005). Carano et al reported that Ti and SS miniscrews could both safely be used as skeletal anchorage, and that Ti and SS miniscrew bending is >0.02mm at 1.471N (150gm) equivalent to the load applied in the present study. There was deformation of >0.01mm noted in the present study. However, the geometry in the study by Carano et al was otherwise not comparable to that in the present study.

There are no studies available which compare Ti and SS stresses or stress pattern generation. Therefore, to compare stresses generated from the use of composite to Ti and SS in the present study, the modulus of the miniscrew in each Ti model was varied to reflect that of SS and composite, with a subsequent test at each angulation. The fact that the general stress pattern for composite is dissimilar to Ti and SS when the miniscrew angulation is changed may be of clinical importance (Pollei 2009). Because composite has a much lower modulus than Ti or SS, it may be that stress shielding does not happen as much in composite miniscrews, and therefore more stress is transferred to the adjacent cortical bone in both compression and tension scenarios. As a result of increased compressive and tensile forces, especially in the 45° model, biological activity related to remodeling may be increased relative to other models with lower stress levels, and therefore have a more significant impact on long-term miniscrew stability and success. Another potential undesirable effect of using composite miniscrews in place of either Ti or SS is the increased deformation and distortion that is inherent due to the decreased modulus of composite relative to titanium or steel. Mechanical or design improvements need to be made to allow for composite miniscrews to be a viable alternative in clinical practice.

Past, Present and Future of Finite Element Analysis in Dentistry 19

To assess the effect of proper PDL modeling, two separate models were generated from our master model. In both models, the upper right central incisor, lateral incisor, canine, and first premolar were segmented from the full model. Brackets (0.022" slot) were ideally placed on the facial surface of each tooth and an archwire was created that had a 0.5 mm intrusive step on the lateral incisor. Passive stainless steel ligatures were placed on each bracket keep the archwire seated. For one model, all bodies were assumed to be stainless steel (mimicking the laboratory setup by Badawi and Major 2009), while suitable material properties were assigned to each body in the second model (Figure 9). Each model was meshed using tetrahedral elements, except for swept hexahedral elements in the archwire, and consisted of 238758 nodes and 147747 elements. The ends of the archwire and the sectioned faces of bone were rigidly

The static equilibrium equations were solved under large displacement assumptions. The final displacement in each model is shown in Figure 10, showing dramatically increased displacement in the PDL model. Notice that in both models, the lateral experienced unpredicted distal displacement due to the interaction of the archwire. This highlights the importance of accurate loading conditions in FEA. In addition to increased overall displacement, the center of rotation of the lateral incisor also moves apically and facially in the stainless steel model (Figure 11). Therefore, any results generated without properly modeling the PDL should be taken with caution – this includes laboratory testing of

Fig. 10. Displacement viewed from the occlusal in the A) PDL model and B) stainless steel model after placement of a wire with 0.5 mm intrusive step bend. Note the different color scales and that both models have 7.1 times the actually displacement visually displayed.

The stress and strain distributions in the PDL also show variations in both magnitude and distribution (Figures 12 and 13). Unsurprisingly, the PDL shows increased strain when accurately modeled as a less stiff material than stainless steel. In this PDL model, the strain is also concentrated to the PDL, as opposed to more broadly distributed in the stainless steel model. Due to the increased rigidity in the stainless steel model, higher stresses were

fixed. The contacts between the wire and the brackets were assumed frictionless.

continuous archwire mechanics (Badawi and Major 2009).

A B

generated by the same displacement in the archwire.

**5.2.2 Methods** 

**5.2.3 Results and discussion** 

#### **5.2 Orthodontic archwire**

#### **5.2.1 Introduction**

Currently, biomechanical analysis of orthodontic force systems is typically limited to simple 2D force diagrams with only 2 or 3 teeth. Beyond this point, the system often becomes indeterminate. Recent laboratory developments (Badawi et al., 2009) allow investigation of the forces and moments generated with continuous archwires. However, this laboratory technique has 3 significant limitations: interbracket distance is roughly doubled, the PDL is ignored, and only a single resultant force and moment is calculated for each tooth.

The complete dentition CAD model assembled in our lab includes the PDL for each tooth and calculates the resultant stress-strain at any point in the model, improving on the limitations of the laboratory technique. However, the accuracy of this technique depends on the 3 factors mentioned previously: material properties, boundary conditions, and loading conditions. The considerations for material properties and boundary conditions are similar to the other models discussed above, but loading conditions with orthodontic archwires deserves closer attention.

Previously studies in orthodontics have typically used point-forces to load teeth, but fixed appliances rarely generate pure point-forces. In order to properly model a wide range of orthodontic movement, a new technique was developed which stores residual stresses during the insertion (loading) stage of the archwire, followed by a deactivation stage where the dentition is loaded equivalently to intraoral archwire loading. This method provides a new way to investigate orthodontic biomechanics (Canales 2011).

Fig. 9. Four tooth model used for FEA of continuous orthodontic archwires. A. Model with accurate material properties assigned to each body. B. Model with all bodies assumed to be stainless steel.

#### **5.2.2 Methods**

18 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Currently, biomechanical analysis of orthodontic force systems is typically limited to simple 2D force diagrams with only 2 or 3 teeth. Beyond this point, the system often becomes indeterminate. Recent laboratory developments (Badawi et al., 2009) allow investigation of the forces and moments generated with continuous archwires. However, this laboratory technique has 3 significant limitations: interbracket distance is roughly doubled, the PDL is

The complete dentition CAD model assembled in our lab includes the PDL for each tooth and calculates the resultant stress-strain at any point in the model, improving on the limitations of the laboratory technique. However, the accuracy of this technique depends on the 3 factors mentioned previously: material properties, boundary conditions, and loading conditions. The considerations for material properties and boundary conditions are similar to the other models discussed above, but loading conditions with orthodontic archwires

Previously studies in orthodontics have typically used point-forces to load teeth, but fixed appliances rarely generate pure point-forces. In order to properly model a wide range of orthodontic movement, a new technique was developed which stores residual stresses during the insertion (loading) stage of the archwire, followed by a deactivation stage where the dentition is loaded equivalently to intraoral archwire loading. This method provides a

Fig. 9. Four tooth model used for FEA of continuous orthodontic archwires. A. Model with accurate material properties assigned to each body. B. Model with all bodies assumed to be

new way to investigate orthodontic biomechanics (Canales 2011).

A B

ignored, and only a single resultant force and moment is calculated for each tooth.

**5.2 Orthodontic archwire** 

deserves closer attention.

stainless steel.

**5.2.1 Introduction** 

To assess the effect of proper PDL modeling, two separate models were generated from our master model. In both models, the upper right central incisor, lateral incisor, canine, and first premolar were segmented from the full model. Brackets (0.022" slot) were ideally placed on the facial surface of each tooth and an archwire was created that had a 0.5 mm intrusive step on the lateral incisor. Passive stainless steel ligatures were placed on each bracket keep the archwire seated. For one model, all bodies were assumed to be stainless steel (mimicking the laboratory setup by Badawi and Major 2009), while suitable material properties were assigned to each body in the second model (Figure 9). Each model was meshed using tetrahedral elements, except for swept hexahedral elements in the archwire, and consisted of 238758 nodes and 147747 elements. The ends of the archwire and the sectioned faces of bone were rigidly fixed. The contacts between the wire and the brackets were assumed frictionless.

#### **5.2.3 Results and discussion**

The static equilibrium equations were solved under large displacement assumptions. The final displacement in each model is shown in Figure 10, showing dramatically increased displacement in the PDL model. Notice that in both models, the lateral experienced unpredicted distal displacement due to the interaction of the archwire. This highlights the importance of accurate loading conditions in FEA. In addition to increased overall displacement, the center of rotation of the lateral incisor also moves apically and facially in the stainless steel model (Figure 11). Therefore, any results generated without properly modeling the PDL should be taken with caution – this includes laboratory testing of continuous archwire mechanics (Badawi and Major 2009).

Fig. 10. Displacement viewed from the occlusal in the A) PDL model and B) stainless steel model after placement of a wire with 0.5 mm intrusive step bend. Note the different color scales and that both models have 7.1 times the actually displacement visually displayed.

The stress and strain distributions in the PDL also show variations in both magnitude and distribution (Figures 12 and 13). Unsurprisingly, the PDL shows increased strain when accurately modeled as a less stiff material than stainless steel. In this PDL model, the strain is also concentrated to the PDL, as opposed to more broadly distributed in the stainless steel model. Due to the increased rigidity in the stainless steel model, higher stresses were generated by the same displacement in the archwire.

Past, Present and Future of Finite Element Analysis in Dentistry 21

Fig. 13. Equivalent (von-Mises) stress in MPa for the A) PDL model and B) stainless steel

Although FEA techniques have greatly improved over the past few decades, further developments remain. More robust solid models, like the one demonstrated in Figure 14, with increased capability to manipulate CAD objects would allow increased research in this area. The ability to fix minor problematic geometry and easily create models with minor variations would greatly reduce the time required to model different biomechanical

Fig. 14. Full Jaw Orthodontic Dentition (FJORD, UNC Copyright) Model: Isometric view of the solid models of mandible and maxillary arches and dentitions that was reconstructed and combined in SolidWorks software in our lab. The left image renders transparency of the

gingiva and bone to reveal internal structures.

model after placement of a wire with 0.5 mm intrusive step bend.

**A B**

**6. Future of dental FEA** 

The results clearly show the importance of the PDL in modeling orthodontic loading. We aim to further improve this model, adding additional teeth, active ligatures, and friction.

Fig. 11. Displacement viewed from the distal in the A) PDL model and B) stainless steel model after placement of a wire with 0.5 mm intrusive step bend. Note the center of rotation (red dot) in the stainless steel model moves apically and facially.

Fig. 12. Equivalent (von-Mises) elastic strain for the A) PDL model and B) stainless steel model after placement of a wire with 0.5 mm intrusive step bend.

Fig. 13. Equivalent (von-Mises) stress in MPa for the A) PDL model and B) stainless steel model after placement of a wire with 0.5 mm intrusive step bend.

#### **6. Future of dental FEA**

20 Finite Element Analysis – From Biomedical Applications to Industrial Developments

The results clearly show the importance of the PDL in modeling orthodontic loading. We aim to further improve this model, adding additional teeth, active ligatures, and friction.

Fig. 11. Displacement viewed from the distal in the A) PDL model and B) stainless steel model after placement of a wire with 0.5 mm intrusive step bend. Note the center of rotation

Fig. 12. Equivalent (von-Mises) elastic strain for the A) PDL model and B) stainless steel

(red dot) in the stainless steel model moves apically and facially.

**A B**

model after placement of a wire with 0.5 mm intrusive step bend.

Although FEA techniques have greatly improved over the past few decades, further developments remain. More robust solid models, like the one demonstrated in Figure 14, with increased capability to manipulate CAD objects would allow increased research in this area. The ability to fix minor problematic geometry and easily create models with minor variations would greatly reduce the time required to model different biomechanical

Fig. 14. Full Jaw Orthodontic Dentition (FJORD, UNC Copyright) Model: Isometric view of the solid models of mandible and maxillary arches and dentitions that was reconstructed and combined in SolidWorks software in our lab. The left image renders transparency of the gingiva and bone to reveal internal structures.

Past, Present and Future of Finite Element Analysis in Dentistry 23

Farah JW, Craig RG, Sikarskie DL. Photoelastic and finite element stress analysis of a

Field C, Ichim I, Swain MV, Chan E, Darendeliler MA, Li W, et al. Mechanical responses to

Geng J-P, Tan KBC, Liu G-R. Application of finite element analysis in implant dentistry:A

Gracco A, Cirignaco A, Cozzani M, Boccaccio A, Pappalettere C, Vitale G.

Hollister SJ, Ko CC, Kohn DH. Bone density around screw thread dental implants predicted

Huang H-L, Chang C-H, Hsu J-T, Fallgatter AM, Ko CC. Comparisons of Implant Body

Element Analysis. The Int. J Oral & Maxillofac Implants. 2007; 22: 551–562. Huang HM, Tsai CY, Lee HF et al. Damping effects on the response of maxillary incisor

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Huiskes R, Chao EYS. A survey of finite element analysis in orthopedic biomechanics: The

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Kim T, Suh J, Kim N, Lee M. Optimum conditions for parallel translation of maxillary

Ko CC, Chu CS, Chung KH, Lee MC. Effects of posts on dentin stress distribution in

Ko CC, Kohn DH, and Hollister SJ: Effective anisotropic elastic constants of bimaterial

Ko CC. Mechanical characteristics of implant/tissue interphases. PhD Thesis. University of

Ko CC. Stress analysis of pulpless tooth: effects of casting post on dentin stress distribution.

Korioth TWP, Versluis A. Modeling the mechanical behavior of the jaws and their related structures by finite element analysis. Crit Rev Oral Biol Med. 1997; 8(l):90-104. Korioth TWP. Finite element modelling of human mandibular biomechanics (PhD thesis).

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situations. Additionally, adding frictional boundaries conditions between teeth and active ligations for orthodontic appliances will continue to increase the accuracy of these models. Three dimensional dynamic simulations for assessing tooth injury, similar to those demonstrated in 2D studies (Huang et al., 2006; Miura and Maeda, 2008), should be reevaluated. While techniques will continually be optimized to improve numerical approximations, this does not negate the value of finite element techniques in dentistry. These techniques use proven engineering principles to model aspects of dentistry that are unable to be efficiently investigated using clinical techniques, and will continue to provide valuable clinical insights regarding dental biomechanics.

#### **7. Acknowledgement**

This study was supported, in part, by AAOF, NIH/NIDCR K08DE018695, NC Biotech Center, and UNC Research Council. We also like to thank Geomagic for providing software and technique supports to our studies.

#### **8. References**


situations. Additionally, adding frictional boundaries conditions between teeth and active ligations for orthodontic appliances will continue to increase the accuracy of these models. Three dimensional dynamic simulations for assessing tooth injury, similar to those demonstrated in 2D studies (Huang et al., 2006; Miura and Maeda, 2008), should be reevaluated. While techniques will continually be optimized to improve numerical approximations, this does not negate the value of finite element techniques in dentistry. These techniques use proven engineering principles to model aspects of dentistry that are unable to be efficiently investigated using clinical techniques, and will continue to provide

This study was supported, in part, by AAOF, NIH/NIDCR K08DE018695, NC Biotech Center, and UNC Research Council. We also like to thank Geomagic for providing software

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**7. Acknowledgement** 

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**2** 

**Finite Element Analysis in Dentistry –** 

Carlos José Soares1, Antheunis Versluis2, Andréa Dolores Correia Miranda Valdivia1, Aline Arêdes Bicalho1, Crisnicaw Veríssimo1,

*1Federal University of Uberlândia,* 

*2University of Tennessee,* 

*1Brazil 2USA* 

**Improving the Quality of Oral Health Care** 

Bruno de Castro Ferreira Barreto1 and Marina Guimarães Roscoe1

The primary function of the human dentition is preparation and processing of food through a biomechanical process of biting and chewing. This process is based on the transfer of masticatory forces, mediated through the teeth (Versluis & Tantbirojn, 2011). The intraoral environment is a complex biomechanical system. Because of this complexity and limited access, most biomechanical research of the oral environment such as restorative, prosthetic, root canal, orthodontic and implant procedures has been performed in vitro (Assunção et al., 2009). In the in vitro biomechanical analysis of tooth structures and restorative materials, destructive mechanical tests for determination of fracture resistance and mechanical properties are important means of analyzing tooth behavior. These tests, however, are limited with regard to obtaining information about the internal behavior of the structures studied. Furthermore, biomechanics are not only of interest at the limits of fracture or failure, but biomechanics are also important during normal function, for understanding property-structure relationships, and for tissue response to stress and strain. For a more precise interrogation of oral biomechanical systems, analysis by means of computational

When loads are applied to a structure, structural strains (deformation) and stresses are generated. This is normal, and is how a structure performs its structural function. But if such stresses become excessive and exceed the elastic limit, structural failure may result. In such situations, a combination of methodologies will provide the means for sequentially analyzing continuous and cyclic failure processes (Soares et al., 2008). Stresses represent how masticatory forces are transferred through a tooth or implant structure (Versluis & Tantbirojn, 2011). These stresses cannot be measured directly, and for failure in complex structures it is not easy to understand why and when a failure process is initiated, and how we can optimize the strength and longevity of the components of the stomatognathic system. The relationship between stress and strain is expressed in constitutive equations

**1. Introduction** 

techniques is desirable.


### **Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care**

Carlos José Soares1, Antheunis Versluis2, Andréa Dolores Correia Miranda Valdivia1, Aline Arêdes Bicalho1, Crisnicaw Veríssimo1, Bruno de Castro Ferreira Barreto1 and Marina Guimarães Roscoe1 *1Federal University of Uberlândia, 2University of Tennessee, 1Brazil 2USA* 

#### **1. Introduction**

24 Finite Element Analysis – From Biomedical Applications to Industrial Developments

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angulation and material type. Master Thesis. University of North Carolina- Chapel

The primary function of the human dentition is preparation and processing of food through a biomechanical process of biting and chewing. This process is based on the transfer of masticatory forces, mediated through the teeth (Versluis & Tantbirojn, 2011). The intraoral environment is a complex biomechanical system. Because of this complexity and limited access, most biomechanical research of the oral environment such as restorative, prosthetic, root canal, orthodontic and implant procedures has been performed in vitro (Assunção et al., 2009). In the in vitro biomechanical analysis of tooth structures and restorative materials, destructive mechanical tests for determination of fracture resistance and mechanical properties are important means of analyzing tooth behavior. These tests, however, are limited with regard to obtaining information about the internal behavior of the structures studied. Furthermore, biomechanics are not only of interest at the limits of fracture or failure, but biomechanics are also important during normal function, for understanding property-structure relationships, and for tissue response to stress and strain. For a more precise interrogation of oral biomechanical systems, analysis by means of computational techniques is desirable.

When loads are applied to a structure, structural strains (deformation) and stresses are generated. This is normal, and is how a structure performs its structural function. But if such stresses become excessive and exceed the elastic limit, structural failure may result. In such situations, a combination of methodologies will provide the means for sequentially analyzing continuous and cyclic failure processes (Soares et al., 2008). Stresses represent how masticatory forces are transferred through a tooth or implant structure (Versluis & Tantbirojn, 2011). These stresses cannot be measured directly, and for failure in complex structures it is not easy to understand why and when a failure process is initiated, and how we can optimize the strength and longevity of the components of the stomatognathic system. The relationship between stress and strain is expressed in constitutive equations

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 27

Fig. 1. Upper central incisor restored with cast post-and-cores. A) 2D FEA model, B) 3D FEA model with different cutting planes, showing internal stress distributions (ANSYS 12

In dental research, FEA has been used effectively in many research studies. For example, FEA has been used to analyze stress generation during the polymerization process of composite materials and stress analyses associated with different restorative protocols like tooth implant, root post canal, orthodontic approaches (Versluis et al., 1996; Versluis et al., 1998; Ausiello et al., 2001; Lin et al., 2001; Ausiello et al., 2002; Versluis et al., 2004; Misra et al., 2005; Meira et al., 2007; Witzel et al., 2007; Meira et al., 2010). This chapter will discuss the application and potential of finite element analysis in biomechanical studies, and how

this method has been instrumental in improving the quality of oral health care.

**2. Application of finite element analysis in dentistry - Modeling steps: Geometry,** 

The FEA procedure consists of three steps: pre-processing, processing and post-processing.

Pre-processing involves constructing the "model". A model consists of: (1) the geometrical representation, (2) the definition of the material properties, and (3) the determination of what loads and restraints are applied and where. Model construction is often difficult, because biological structures have irregular shapes, consist of different materials and/or compositions, and the exact loading conditions can have a large effect on the outcome. Therefore, the correct construction of a model to obtain accurate results from a FEA is very important. The development of FEA models can follow different protocols, depending on the aim of the study. Models used to analyze laboratory test parameters, like microtensile bond tests, flexural tests, or push-out tests usually have the simplest geometries and can be generated directly into the FEA software (Fig. 2.). Modeling of 2D and 3D biological structures are often more intricate, and may have to be performed with Computer Aided Design (CAD) or Bio-CAD software. This chapter mainly discusses 3D Bio-CAD modeling.

Workbench - Ansys Inc., Houston, USA). (Santos-Filho, 2008).

**properties, and boundary conditions** 

**2.1 Pre-processing: Building a model** 

according to universal physical laws. When dealing with physically and geometrically complex systems, an engineering concept that uses a numerical analysis to solve such equations becomes inevitable. Finite Element Analysis (FEA) is a widely used numerical analysis that has been applied successfully in many engineering and bioengineering areas since the 1950s. This computational numerical analysis can be considered the most comprehensive method currently available to calculate the complex conditions of stress distributions as are encountered in dental systems (Versluis & Tantbirojn, 2009).

The concept of FEA is obtaining a solution to a complex physical problem by dividing the problem domain into a collection of much smaller and simpler domains in which the field variables can be interpolated with the use of shape functions. The structure is discretized into so called "elements" connected through nodes. In FEA choosing the appropriate mathematical model, element type and degree of discretization are important to obtain accurate as well as time and cost effective solutions. Given the right model definition, FEA is capable of computationally simulating the stress distribution and predicting the sites of stress concentrations, which are the most likely points of failure initiation within a structure or material. Other advantages of this method compared with other research methodologies are the low operating costs, reduced time to carry out the investigation and it provides information that cannot be obtained by experimental studies (Soares et al. 2008).

However, FEA studies cannot replace the traditional laboratory studies. FEA needs laboratory validation to prove its results. The properties and boundary conditions dentistry is dealing with are complex and often little understood, therefore requiring assumptions and simplifications in the modeling of the stress-strain responses. Furthermore, large anatomical variability precludes conclusions based on unique solutions. The most powerful application of FEA is thus when it is conducted together with laboratory studies. For example, the finite element method can be performed before a laboratory study as a way to design and conduct the experimental research, to predict possible errors, and serve as a pilot study for the standardization protocols. The use of this methodology can also occur after laboratory experimental tests in order to explain ultra-structural phenomena that cannot be detected or isolated. The identification of stress fields and their internal and external distribution in the specimens may therefore help answer a research hypothesis (Ausiello et al., 2001).

The complexity of a FEA can differ depending on the modeled structure, research question, and available knowledge or operator experience. For example, FEA can be performed using two-dimensional (2D) or three-dimensional (3D) models. The choice between these two models depends on many inter-related factors, such as the complexity of the geometry, material properties, mode of analysis, required accuracy and the applicability of general findings, and finally the time and costs involved (Romeed et al., 2004; Poiate et al., 2011). 2D FEA is often performed in dental research (Soares et al., 2008; Silva et al., 2009; Soares et al., 2010). The advantage of a 2D-analysis is that it provides significant results and immediate insight with relatively low operating cost and reduced analysis time. However, the results of 2D models also have limitations regarding the complexity of some structural problems. In contrast, 3D FEA has the advantage of more realistic 3D stress distributions in complex 3D geometries (Fig. 1). However, creating a 3D model can be considered more costly, because it is more labor-intensive and time-consuming and may require additional technology for acquiring 3D geometrical data and generation of models (Santos-Filho et al., 2008).

according to universal physical laws. When dealing with physically and geometrically complex systems, an engineering concept that uses a numerical analysis to solve such equations becomes inevitable. Finite Element Analysis (FEA) is a widely used numerical analysis that has been applied successfully in many engineering and bioengineering areas since the 1950s. This computational numerical analysis can be considered the most comprehensive method currently available to calculate the complex conditions of stress

The concept of FEA is obtaining a solution to a complex physical problem by dividing the problem domain into a collection of much smaller and simpler domains in which the field variables can be interpolated with the use of shape functions. The structure is discretized into so called "elements" connected through nodes. In FEA choosing the appropriate mathematical model, element type and degree of discretization are important to obtain accurate as well as time and cost effective solutions. Given the right model definition, FEA is capable of computationally simulating the stress distribution and predicting the sites of stress concentrations, which are the most likely points of failure initiation within a structure or material. Other advantages of this method compared with other research methodologies are the low operating costs, reduced time to carry out the investigation and it provides

However, FEA studies cannot replace the traditional laboratory studies. FEA needs laboratory validation to prove its results. The properties and boundary conditions dentistry is dealing with are complex and often little understood, therefore requiring assumptions and simplifications in the modeling of the stress-strain responses. Furthermore, large anatomical variability precludes conclusions based on unique solutions. The most powerful application of FEA is thus when it is conducted together with laboratory studies. For example, the finite element method can be performed before a laboratory study as a way to design and conduct the experimental research, to predict possible errors, and serve as a pilot study for the standardization protocols. The use of this methodology can also occur after laboratory experimental tests in order to explain ultra-structural phenomena that cannot be detected or isolated. The identification of stress fields and their internal and external distribution in the specimens may therefore help answer a research hypothesis (Ausiello et

The complexity of a FEA can differ depending on the modeled structure, research question, and available knowledge or operator experience. For example, FEA can be performed using two-dimensional (2D) or three-dimensional (3D) models. The choice between these two models depends on many inter-related factors, such as the complexity of the geometry, material properties, mode of analysis, required accuracy and the applicability of general findings, and finally the time and costs involved (Romeed et al., 2004; Poiate et al., 2011). 2D FEA is often performed in dental research (Soares et al., 2008; Silva et al., 2009; Soares et al., 2010). The advantage of a 2D-analysis is that it provides significant results and immediate insight with relatively low operating cost and reduced analysis time. However, the results of 2D models also have limitations regarding the complexity of some structural problems. In contrast, 3D FEA has the advantage of more realistic 3D stress distributions in complex 3D geometries (Fig. 1). However, creating a 3D model can be considered more costly, because it is more labor-intensive and time-consuming and may require additional technology for

acquiring 3D geometrical data and generation of models (Santos-Filho et al., 2008).

distributions as are encountered in dental systems (Versluis & Tantbirojn, 2009).

information that cannot be obtained by experimental studies (Soares et al. 2008).

al., 2001).

Fig. 1. Upper central incisor restored with cast post-and-cores. A) 2D FEA model, B) 3D FEA model with different cutting planes, showing internal stress distributions (ANSYS 12 Workbench - Ansys Inc., Houston, USA). (Santos-Filho, 2008).

In dental research, FEA has been used effectively in many research studies. For example, FEA has been used to analyze stress generation during the polymerization process of composite materials and stress analyses associated with different restorative protocols like tooth implant, root post canal, orthodontic approaches (Versluis et al., 1996; Versluis et al., 1998; Ausiello et al., 2001; Lin et al., 2001; Ausiello et al., 2002; Versluis et al., 2004; Misra et al., 2005; Meira et al., 2007; Witzel et al., 2007; Meira et al., 2010). This chapter will discuss the application and potential of finite element analysis in biomechanical studies, and how this method has been instrumental in improving the quality of oral health care.

#### **2. Application of finite element analysis in dentistry - Modeling steps: Geometry, properties, and boundary conditions**

The FEA procedure consists of three steps: pre-processing, processing and post-processing.

#### **2.1 Pre-processing: Building a model**

Pre-processing involves constructing the "model". A model consists of: (1) the geometrical representation, (2) the definition of the material properties, and (3) the determination of what loads and restraints are applied and where. Model construction is often difficult, because biological structures have irregular shapes, consist of different materials and/or compositions, and the exact loading conditions can have a large effect on the outcome. Therefore, the correct construction of a model to obtain accurate results from a FEA is very important. The development of FEA models can follow different protocols, depending on the aim of the study. Models used to analyze laboratory test parameters, like microtensile bond tests, flexural tests, or push-out tests usually have the simplest geometries and can be generated directly into the FEA software (Fig. 2.). Modeling of 2D and 3D biological structures are often more intricate, and may have to be performed with Computer Aided Design (CAD) or Bio-CAD software. This chapter mainly discusses 3D Bio-CAD modeling.

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 29

Fig. 3. Images used to build a two-dimensional model. A) Photographs, B) Section plane of

These images can be imported into different software programs that can digitize reference points of the structure, such as Image J (available at http://imagej.nih.gov). These points can be exported as a list of coordinates, which can subsequently be imported into a finite element program, for example MENTAT-MARC package (MSC. Software Corporation, Santa Ana, CA, USA), or CAD software, such as Mechanical Desktop (Autodesk, San Rafael, CA, USA) that can generate IGES-files that can be read by most FEA software. The imported reference points can be used to outline the shape of the modeled structure or materials, and

NURBS Modeling (Non Uniform Rational Bezier Spline) is one of several methods applied for building 3D models. This methodology involves a model creation from a base geometry in STL (stereolithography) format. Obtaining an STL-file, consisting of a mesh of triangular surfaces created from a distribution of surface points, is a critical step for 3D modeling. Several methods have been described in the literature (Magne, 2007; Soares et al., 2008). The

Fig. 4. NURBS modeling. A) STL model (stereolithography), B) NURBS-based geometry

computed tomography, C) Radiograph.

hence the finite element mesh.

created from the STL (Santos-Filho, 2008).

Fig. 2. Finite element models of test specimens made directly in FEA software.

#### **2.2 Bio-CAD protocol for 3D modeling of organic structures**

The modeling technique often used in bioengineering studies is called Bio-CAD, and consists of obtaining a virtual geometric model of a structure from anatomical references (Protocol developed in the Center for Information Technology Renato Archer, Travassos, 2010). The obtained geometrical model consists of closed volumes or solid shapes, in which a mesh distribution of discrete elements can be generated. The shape of the object of study can be reconstructed as close to reality as possible, for example, by reducing the size of the elements in regions that require more details. However, higher detail and thus reducing the element sizes will increase the total number of elements and consequently, the computational requirements. Modeling Bio-CAD involves the stages of obtaining the basegeometry, creation of reference curves, construction of surface areas, union of surfaces for generation of solids and exportation of the model to FEA software.

#### **2.2.1 Obtaining the base-geometry**

References for model creation, whether 2D or 3D, are images of the structure that is modeled. Modeling of biological structures for the finite element method usually requires CAD techniques. For 2D models, the modeling is made from the images or planar sections of a structure (photograph, tomography or radiograph) (Fig. 3).

Fig. 2. Finite element models of test specimens made directly in FEA software.

The modeling technique often used in bioengineering studies is called Bio-CAD, and consists of obtaining a virtual geometric model of a structure from anatomical references (Protocol developed in the Center for Information Technology Renato Archer, Travassos, 2010). The obtained geometrical model consists of closed volumes or solid shapes, in which a mesh distribution of discrete elements can be generated. The shape of the object of study can be reconstructed as close to reality as possible, for example, by reducing the size of the elements in regions that require more details. However, higher detail and thus reducing the element sizes will increase the total number of elements and consequently, the computational requirements. Modeling Bio-CAD involves the stages of obtaining the basegeometry, creation of reference curves, construction of surface areas, union of surfaces for

References for model creation, whether 2D or 3D, are images of the structure that is modeled. Modeling of biological structures for the finite element method usually requires CAD techniques. For 2D models, the modeling is made from the images or planar sections

**2.2 Bio-CAD protocol for 3D modeling of organic structures** 

generation of solids and exportation of the model to FEA software.

of a structure (photograph, tomography or radiograph) (Fig. 3).

**2.2.1 Obtaining the base-geometry** 

Fig. 3. Images used to build a two-dimensional model. A) Photographs, B) Section plane of computed tomography, C) Radiograph.

These images can be imported into different software programs that can digitize reference points of the structure, such as Image J (available at http://imagej.nih.gov). These points can be exported as a list of coordinates, which can subsequently be imported into a finite element program, for example MENTAT-MARC package (MSC. Software Corporation, Santa Ana, CA, USA), or CAD software, such as Mechanical Desktop (Autodesk, San Rafael, CA, USA) that can generate IGES-files that can be read by most FEA software. The imported reference points can be used to outline the shape of the modeled structure or materials, and hence the finite element mesh.

NURBS Modeling (Non Uniform Rational Bezier Spline) is one of several methods applied for building 3D models. This methodology involves a model creation from a base geometry in STL (stereolithography) format. Obtaining an STL-file, consisting of a mesh of triangular surfaces created from a distribution of surface points, is a critical step for 3D modeling. Several methods have been described in the literature (Magne, 2007; Soares et al., 2008). The

Fig. 4. NURBS modeling. A) STL model (stereolithography), B) NURBS-based geometry created from the STL (Santos-Filho, 2008).

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 31

both bodies are identical and coincide using commands such as *Boolean Operations*, or

Fig. 5. Creation of surfaces and solids from reference curves in Rhinoceros 3D (Robert

The export model is usually saved in STEP (STP) format or IGES (IGS) format. The choice of format depends on the compatibility with the pre-processing software of the FEA program. Also be aware of the units (chosen at the beginning of modeling, usually in millimeters) before importing the model into the pre-processing software. Before exporting, it is recommended to carefully re-check the model to avoid rework: ensure that solids are closed, check for acute angles in surfaces or discontinuities, check for very short edges, check for surfaces that are too narrow or small, and inspect intersections between solids (Travassos,

Material properties can be determined by means of mechanical tests and applied for any material with the same characteristics. Specimens and procedures can be carried out following agreed testing standards (ASTM - American Society for Testing and Materials). The minimum properties required for most linear elastic isotropic finite element analyses

The elastic modulus (E) represents the inherent stiffness of a material within the elastic range, and describes the relationship between stress and strain. The elastic modulus can thus be determined from the slope of a stress/strain curve. Such relationship can be acquired by means of a uniaxial tensile test in the elastic regime (Chabrier et al., 1999). The

copying common surfaces.

McNeel & Associates, USA).

**2.2.3 Exporting the solids** 

**2.3 Material properties** 

are the elastic modulus and Poisson's ratio.

modulus of elasticity is defined as:

**2.3.1 Methods for obtaining material properties used in FEA** 

2010).

STL file can be obtained by computed tomography, Micro-CT, magnetic resonance imaging (MRI) or optical, contact or laser scanning. Using CAD software, NURBS curves can be defined that follow the anatomical details of the structure. This transformation from surface elements to a NURBS-based representation allows for greater control of the shape and quality of the resulting finite element mesh (Fig. 4).

Our research group has used this strategy to create models of the tooth. First the outer shape of an intact tooth is scanned using a laser scanner (LPX 600, Roland DG, Osaka, Japan). Next, the enamel is removed by covering the root surface with a thin layer of nail polish, and immersing the tooth in 10% citric acid for 10 minutes in an ultra-sound machine. Using a stereomicroscope (40X) the complete removal of enamel can be confirmed. Then the tooth is scanned again and the two shapes (sound tooth and dentin) are fit using PixForm Pro II software (Roland DG, Osaka, Japan). The pulp geometry is generated by two X-ray images obtained from the tooth positioned bucco-lingually and mesio-distally. These images are exported to Image J software where the pulp shapes are traced and digitized, and eventually merged with the scanned tooth and dentin surfaces.

#### **2.2.2 NURBS Modeling: Creation of the curves, surfaces and solids**

NURBS Modeling or irregular surface modeling begins with planning the number and position of curves that will represent the main anatomic landmarks of the models, justifying the level of detail in each case. From these curves surfaces and volumes (solids) will be created. The NURBS curves will determine the quality of the model, and consequently, the quality of the finite element mesh. The modeling strategy begins with knowledge of the anatomy of the structure to be studied. The curves should be as regular as possible, and should not form a very small or narrow area with sharp angles, as this would hinder the formation of meshes. The boundary conditions, defined by external restrictions, contact structures and loading definition, must already be defined at the time of construction of lines and surfaces. The curves should provide continuity to ensure that the model will result in closed volumes. If models are made up of multiple solids, NURBS curves from adjacent solids should have the same point of origin to facilitate the formation of a regular mesh across the solid boundaries.

After curves have been defined, surfaces can be created using three or four curves each. The formation of surfaces should follow a chess pattern to prevent wrinkling of the end surfaces caused by the assigned tangency between the surfaces (Fig. 5). This makes it possible to choose the form of tangency between the surfaces and avoid creases in the models that would become areas of mesh complications and consequently locations of erroneous stress concentrations in the final finite element model. It is recommended that there is continuity of curvature between the surfaces. Finally the surfaces should be joined to form a closed NURBS volume.

Most cases involve more than one solid, with different materials and contact areas defined, among other features. In these cases, a classification is assigned to *multi-bodies*. Another important requirement is that there can be no intersection between bodies. There should also be no empty spaces between solids in contact, which in contact analysis would cause single contacts with associated stress peaks, or would cause gaps for intended bonded interfaces. In order to avoid these problems, it is recommended that the contact surfaces of both bodies are identical and coincide using commands such as *Boolean Operations*, or copying common surfaces.

Fig. 5. Creation of surfaces and solids from reference curves in Rhinoceros 3D (Robert McNeel & Associates, USA).

#### **2.2.3 Exporting the solids**

30 Finite Element Analysis – From Biomedical Applications to Industrial Developments

STL file can be obtained by computed tomography, Micro-CT, magnetic resonance imaging (MRI) or optical, contact or laser scanning. Using CAD software, NURBS curves can be defined that follow the anatomical details of the structure. This transformation from surface elements to a NURBS-based representation allows for greater control of the shape and

Our research group has used this strategy to create models of the tooth. First the outer shape of an intact tooth is scanned using a laser scanner (LPX 600, Roland DG, Osaka, Japan). Next, the enamel is removed by covering the root surface with a thin layer of nail polish, and immersing the tooth in 10% citric acid for 10 minutes in an ultra-sound machine. Using a stereomicroscope (40X) the complete removal of enamel can be confirmed. Then the tooth is scanned again and the two shapes (sound tooth and dentin) are fit using PixForm Pro II software (Roland DG, Osaka, Japan). The pulp geometry is generated by two X-ray images obtained from the tooth positioned bucco-lingually and mesio-distally. These images are exported to Image J software where the pulp shapes are traced and digitized, and eventually

NURBS Modeling or irregular surface modeling begins with planning the number and position of curves that will represent the main anatomic landmarks of the models, justifying the level of detail in each case. From these curves surfaces and volumes (solids) will be created. The NURBS curves will determine the quality of the model, and consequently, the quality of the finite element mesh. The modeling strategy begins with knowledge of the anatomy of the structure to be studied. The curves should be as regular as possible, and should not form a very small or narrow area with sharp angles, as this would hinder the formation of meshes. The boundary conditions, defined by external restrictions, contact structures and loading definition, must already be defined at the time of construction of lines and surfaces. The curves should provide continuity to ensure that the model will result in closed volumes. If models are made up of multiple solids, NURBS curves from adjacent solids should have the same point of origin to facilitate the formation of a regular mesh

After curves have been defined, surfaces can be created using three or four curves each. The formation of surfaces should follow a chess pattern to prevent wrinkling of the end surfaces caused by the assigned tangency between the surfaces (Fig. 5). This makes it possible to choose the form of tangency between the surfaces and avoid creases in the models that would become areas of mesh complications and consequently locations of erroneous stress concentrations in the final finite element model. It is recommended that there is continuity of curvature between the surfaces. Finally the surfaces should be joined to form a closed

Most cases involve more than one solid, with different materials and contact areas defined, among other features. In these cases, a classification is assigned to *multi-bodies*. Another important requirement is that there can be no intersection between bodies. There should also be no empty spaces between solids in contact, which in contact analysis would cause single contacts with associated stress peaks, or would cause gaps for intended bonded interfaces. In order to avoid these problems, it is recommended that the contact surfaces of

quality of the resulting finite element mesh (Fig. 4).

merged with the scanned tooth and dentin surfaces.

across the solid boundaries.

NURBS volume.

**2.2.2 NURBS Modeling: Creation of the curves, surfaces and solids** 

The export model is usually saved in STEP (STP) format or IGES (IGS) format. The choice of format depends on the compatibility with the pre-processing software of the FEA program. Also be aware of the units (chosen at the beginning of modeling, usually in millimeters) before importing the model into the pre-processing software. Before exporting, it is recommended to carefully re-check the model to avoid rework: ensure that solids are closed, check for acute angles in surfaces or discontinuities, check for very short edges, check for surfaces that are too narrow or small, and inspect intersections between solids (Travassos, 2010).

#### **2.3 Material properties**

Material properties can be determined by means of mechanical tests and applied for any material with the same characteristics. Specimens and procedures can be carried out following agreed testing standards (ASTM - American Society for Testing and Materials). The minimum properties required for most linear elastic isotropic finite element analyses are the elastic modulus and Poisson's ratio.

#### **2.3.1 Methods for obtaining material properties used in FEA**

The elastic modulus (E) represents the inherent stiffness of a material within the elastic range, and describes the relationship between stress and strain. The elastic modulus can thus be determined from the slope of a stress/strain curve. Such relationship can be acquired by means of a uniaxial tensile test in the elastic regime (Chabrier et al., 1999). The modulus of elasticity is defined as:

$$E = \sigma / \varepsilon \tag{1}$$

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 33

example, that the assumption of isotropic properties for enamel did not change the

The other mandatory property for a FEA, the Poisson's ratio, is the ratio of lateral contraction and longitudinal elongation of a material subjected to a uniaxial load (Chabrier et al., 1999). Among the static methods are tensile and compression tests, in which a uniaxial stress is applied to the material and the Poisson's ratio is calculated from the resulting axial and transverse strains. Another method uses ultrasound (resonance), where the Poisson's ratio is obtained from the speed or natural frequency of the generated longitudinal and

The type of structural analysis depends on the subject that is being modeled. Depending on the model, the FEA can be linear or nonlinear. Linear or nonlinear analysis refers to the proportionality of the solutions. A solution is linear if the outcome is independent of its loading history. For example, an analysis is linear if the outcome will be the same irrespective of if the load is applied in one or multiple increments. Some conditions are inherently nonlinear, such as nonlinear material responses (e.g., rate-dependent properties or viscoelasticity, plastic deformation), time-dependent boundary conditions (e.g., contact analysis where independent bodies interact), or geometric instabilities (e.g., buckling). Sometimes linear conditions become nonlinear when general assumptions become invalid. For example, the stress-strain responses are generally based on the assumption of small displacements. When large deformations occur, the numerical solution procedures must be adjusted. Most high-end FEA software programs have the capability to resolve nonlinear equation systems. For the end-user, the difference between submitting a linear or nonlinear analysis is minimal, and usually only involves the prescription of multiple increments or invoking an alternative solver for the ensuing nonlinear solution. Since nonlinear systems potentially have multiple solutions, nonlinear analyses should also be checked more thoroughly for the convergence to the correct solution. Nonlinear solutions require more computational iterations to converge to a final solution, therefore nonlinear analyses are

Nonlinear FEA is a powerful tool to predict stress and strain within structures in situations that cannot be simulated in conventional linear static models. However the determination of elastic, plastic, and viscoelastic material behavior of the materials involved requires accurate mechanical testing prior to FEA. The experimental determination of mechanical properties continues to be a major challenge and impediment for more accurate FEA. For example, periodontal ligament (PDL) is a dental tissue structure with significant viscoelastic behavior, and simulation using nonlinear analysis would be more realistic. However, due to its complex structure; the exact mechanical properties of PDL must still be considered poorly understood. It such case it can be argued that using incorrect or questionable nonlinear mechanical properties in a FEA may be more obscuring than a well defined and understood

An example of the need for a nonlinear analysis is the simulation of the mechanical behavior across an interface. Interfacial areas are among the most important areas for the performance of materials or structures. Interfaces between different materials can often be

conclusions of a shrinkage stress analysis (Versluis & Tantbirojn, 2011).

**2.3.2 Type of structural analysis: Linear and nonlinear analysis** 

more costly in terms of computation and time.

transverse waves.

simplification.

where () is the stress and () is the strain (ratio between amount of deformation and original dimension).

Various methods have been used to measure the elastic modulus (Chung et al., 2005; Vieira et al., 2006; Boaro et al., 2010; Suwannaroop et al., 2011). For dental materials and tissues, the classical uniaxial tensile test is often problematic due to small specimen dimensions dictated by size, cost, and/or manufacturing limitations. Therefore, other methods such as 3-point bending, indentation, nanoindentation and ultrasonic waves have been used to determine the elastic modulus. Using a Knoop hardness setup, the elastic modulus of composites can be estimated with an empirical relationship, yielding a simple and low cost method (Marshall et al., 1982). Using the dimensions of the short and long diagonals of the indentation, the elastic modulus (GPa) can be determined using the following equation:

$$E = 0.45 \text{ KHN/((0.140647 \text{-} d/D) 100)}\tag{2}$$

where KHN is the Knoop Hardness (kg/mm2), d is the short diagonal of the indentation, D is the long diagonal of the indentation, and 0.140647 is the ratio of the short and long diagonals of the Knoop indenter (1/7.11). Nanoindentation systems have also been used for this purpose. The elastic modulus from nanoindentation is obtained from the data generated in the load-displacement curve by means of the equation (Suwannaroop et al., 2011):

$$1/E^\* = (1\text{-v}\mathcal{D})/E + (1\text{-v}\mathcal{D})/E^\prime \tag{3}$$

where E\* is the reduced modulus from the nanoindenter, E is the modulus of the Berkovich diamond indenter (1,050 GPa ), E' is the modulus of the specimen, υ is the Poisson's ratio for the indenter (0.07)28, and υ' is the Poisson's ratio for the specimens.

The ISO 4049 (Dentistry - Resin based dental fillings) provides a standard for the use of three-point bending tests for determining the flexural modulus (elastic modulus) for composites. Generally, the preparation of specimens for microindentation tests is easier, specimen size is smaller and it has been suggested that their results are more consistent than with an ISO 4049 three-point bending test (Chung et al., 2005).

The analysis of anisotropic materials (i.e., materials with different stress-strain responses in different directions) requires the application of elastic moduli and Poisson's ratios in 3 directions (2 in case of orthotropy), as well as shear moduli in those directions. It is well accepted that enamel is not isotropic, but the anisotropy of dentin is less well established. Analyzing the effect of anisotropy in dentin, the presence and direction of dentinal tubules were not found to affect the mechanical response, indicating that dentin behaved homogeneous and isotropic (Peyton et al., 1952). More recently, some heterogeneity and anisotropy was demonstrated for dentin. However, the stiffness response seems to be only mildly anisotropic (Wang & Weiner, 1998; Kinney et al., 2004; Huo, 2005). Therefore, dentin properties in FEA are usually assumed to be isotropic. Potential simplifications such as the assumption of linear-elastic isotropic material behavior may be necessary in FEA simulations due to the difficulty of obtaining the correct directional properties, or the need to reduce the complexity of an analysis. As in other research approaches, some simplifications and assumptions are also common in FEA, and are permissible provided that their impact on the conclusions is carefully taken into account. It has been shown, for

where () is the stress and () is the strain (ratio between amount of deformation and

Various methods have been used to measure the elastic modulus (Chung et al., 2005; Vieira et al., 2006; Boaro et al., 2010; Suwannaroop et al., 2011). For dental materials and tissues, the classical uniaxial tensile test is often problematic due to small specimen dimensions dictated by size, cost, and/or manufacturing limitations. Therefore, other methods such as 3-point bending, indentation, nanoindentation and ultrasonic waves have been used to determine the elastic modulus. Using a Knoop hardness setup, the elastic modulus of composites can be estimated with an empirical relationship, yielding a simple and low cost method (Marshall et al., 1982). Using the dimensions of the short and long diagonals of the indentation, the elastic modulus (GPa) can be determined using the following equation:

where KHN is the Knoop Hardness (kg/mm2), d is the short diagonal of the indentation, D is the long diagonal of the indentation, and 0.140647 is the ratio of the short and long diagonals of the Knoop indenter (1/7.11). Nanoindentation systems have also been used for this purpose. The elastic modulus from nanoindentation is obtained from the data generated

where E\* is the reduced modulus from the nanoindenter, E is the modulus of the Berkovich diamond indenter (1,050 GPa ), E' is the modulus of the specimen, υ is the Poisson's ratio for

The ISO 4049 (Dentistry - Resin based dental fillings) provides a standard for the use of three-point bending tests for determining the flexural modulus (elastic modulus) for composites. Generally, the preparation of specimens for microindentation tests is easier, specimen size is smaller and it has been suggested that their results are more consistent than

The analysis of anisotropic materials (i.e., materials with different stress-strain responses in different directions) requires the application of elastic moduli and Poisson's ratios in 3 directions (2 in case of orthotropy), as well as shear moduli in those directions. It is well accepted that enamel is not isotropic, but the anisotropy of dentin is less well established. Analyzing the effect of anisotropy in dentin, the presence and direction of dentinal tubules were not found to affect the mechanical response, indicating that dentin behaved homogeneous and isotropic (Peyton et al., 1952). More recently, some heterogeneity and anisotropy was demonstrated for dentin. However, the stiffness response seems to be only mildly anisotropic (Wang & Weiner, 1998; Kinney et al., 2004; Huo, 2005). Therefore, dentin properties in FEA are usually assumed to be isotropic. Potential simplifications such as the assumption of linear-elastic isotropic material behavior may be necessary in FEA simulations due to the difficulty of obtaining the correct directional properties, or the need to reduce the complexity of an analysis. As in other research approaches, some simplifications and assumptions are also common in FEA, and are permissible provided that their impact on the conclusions is carefully taken into account. It has been shown, for

in the load-displacement curve by means of the equation (Suwannaroop et al., 2011):

the indenter (0.07)28, and υ' is the Poisson's ratio for the specimens.

with an ISO 4049 three-point bending test (Chung et al., 2005).

original dimension).

E = / (1)

E = 0.45 KHN/((0.140647-d/D) 100) (2)

1/E\*=(1-v2)/E+(1-v2)/E' (3)

example, that the assumption of isotropic properties for enamel did not change the conclusions of a shrinkage stress analysis (Versluis & Tantbirojn, 2011).

The other mandatory property for a FEA, the Poisson's ratio, is the ratio of lateral contraction and longitudinal elongation of a material subjected to a uniaxial load (Chabrier et al., 1999). Among the static methods are tensile and compression tests, in which a uniaxial stress is applied to the material and the Poisson's ratio is calculated from the resulting axial and transverse strains. Another method uses ultrasound (resonance), where the Poisson's ratio is obtained from the speed or natural frequency of the generated longitudinal and transverse waves.

#### **2.3.2 Type of structural analysis: Linear and nonlinear analysis**

The type of structural analysis depends on the subject that is being modeled. Depending on the model, the FEA can be linear or nonlinear. Linear or nonlinear analysis refers to the proportionality of the solutions. A solution is linear if the outcome is independent of its loading history. For example, an analysis is linear if the outcome will be the same irrespective of if the load is applied in one or multiple increments. Some conditions are inherently nonlinear, such as nonlinear material responses (e.g., rate-dependent properties or viscoelasticity, plastic deformation), time-dependent boundary conditions (e.g., contact analysis where independent bodies interact), or geometric instabilities (e.g., buckling). Sometimes linear conditions become nonlinear when general assumptions become invalid. For example, the stress-strain responses are generally based on the assumption of small displacements. When large deformations occur, the numerical solution procedures must be adjusted. Most high-end FEA software programs have the capability to resolve nonlinear equation systems. For the end-user, the difference between submitting a linear or nonlinear analysis is minimal, and usually only involves the prescription of multiple increments or invoking an alternative solver for the ensuing nonlinear solution. Since nonlinear systems potentially have multiple solutions, nonlinear analyses should also be checked more thoroughly for the convergence to the correct solution. Nonlinear solutions require more computational iterations to converge to a final solution, therefore nonlinear analyses are more costly in terms of computation and time.

Nonlinear FEA is a powerful tool to predict stress and strain within structures in situations that cannot be simulated in conventional linear static models. However the determination of elastic, plastic, and viscoelastic material behavior of the materials involved requires accurate mechanical testing prior to FEA. The experimental determination of mechanical properties continues to be a major challenge and impediment for more accurate FEA. For example, periodontal ligament (PDL) is a dental tissue structure with significant viscoelastic behavior, and simulation using nonlinear analysis would be more realistic. However, due to its complex structure; the exact mechanical properties of PDL must still be considered poorly understood. It such case it can be argued that using incorrect or questionable nonlinear mechanical properties in a FEA may be more obscuring than a well defined and understood simplification.

An example of the need for a nonlinear analysis is the simulation of the mechanical behavior across an interface. Interfacial areas are among the most important areas for the performance of materials or structures. Interfaces between different materials can often be

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 35

useful in bioengineering, which often deals with irregular geometries. Since few modeled geometries have perfectly square dimensions or even straight edges, element shapes must be adapted to fit. Note that the accuracy of elements deteriorates the further they are distorted from their ideal basic shape. Besides their basic geometrical shapes, elements can differ in the way they are solved, such as linear or quadratic interpolation. This refers to

Fig. 7. Radiography of maxillary premolar (A); Lines plotted in the MARC/MENTAT software (B); Manual creation of mesh (C); Final subdivision for improving the mesh quality

discontinuities) while creating coarser mesh distributions in regions of less interest.

In finite element modeling, a finer (denser) mesh should allow a more accurate solution. However, as a mesh is made finer and the element count increases, the computation time also increases. How can you get a mesh that balances accuracy and computing resources? One way is to perform a convergence study. This process involves the creation and analysis of multiple mesh distributions with increasing number of elements or refinements. When results with the various models are plotted, a convergence to a particular solution can be found. Based on this convergence data, an estimation of the error can be made for the

Most FEA software provides *automesh* or automatic mesh generation options. The program may suggests the size and number of elements or allow manual control for generating the element meshes. Manual mesh generation can give good results for 2D models (Fig. 7). However, most 3D models rely on automated mesh generators because manual creation of 3D models is very time-consuming. Still, various aspects of the 3D automeshing need to be controlled manually, such as the number of elements required in a given pre-selected area, the distribution of elements, the range of element sizes within a model, uniting or dividing elements, etc. The manual controls also allows selective distribution of elements, for example, more refined meshes in special regions of interest (contact interfaces, geometric

how stress and strain is interpolated within an element.

(D).

modeled as a perfect bond, where nodes are shared across the interface (Wakabayashi et al., 2008). The simulations of such interface can normally be conducted in a linear analysis. However, depending the actual conditions at a simulated interface, such perfect fusion can occasionally lead to unrealistic results. Fig. 6 shows a 2D FEA model of a root filled tooth restored with a cast post and core and a fiberglass post and composite core. When perfect bonding was assumed in a linear analysis, the stress distribution indicated higher stress concentrations in the cast post and core compared to the fiberglass post, while the stress distribution in the root dentin was nearly identical between these two models. Experimental failure data showed, however, that the failure modes of the cast post and core group were more catastrophic and involved longitudinal root fractures while all fractures of the root with fiberglass post were coronal fractures. Simulating the interface more realistically with friction between resin cement and cast post and core (requiring a nonlinear analysis) rather than a perfect fusion, the stress distribution changed substantially between the two post types (Fig. 6), and yielded more realistic results when compared with the experimental observations.

Fig. 6. Nonlinear FEA of endodontic treated tooth restored with A. Cast post and core and B. Fiberglass post.

#### **2.4 Mesh generation**

In FEA the whole domain is divided into smaller elements. The collection and distribution of these elements is called a mesh. Elements are interconnected by nodes, which are thus the only points though which elements interact with each other. The process of creating an element mesh is referred to as "discretization" of the problem domain (Geng et al., 2001).

There are many different types of elements. One of the differences can be their basic shape, such as triangular, tetrahedral, hexahedral, etc. Triangular or tetrahedral elements are popular because automatic meshing software routines are easier to develop and thus more advanced for those shapes. Automatic generation of element distributions is especially

modeled as a perfect bond, where nodes are shared across the interface (Wakabayashi et al., 2008). The simulations of such interface can normally be conducted in a linear analysis. However, depending the actual conditions at a simulated interface, such perfect fusion can occasionally lead to unrealistic results. Fig. 6 shows a 2D FEA model of a root filled tooth restored with a cast post and core and a fiberglass post and composite core. When perfect bonding was assumed in a linear analysis, the stress distribution indicated higher stress concentrations in the cast post and core compared to the fiberglass post, while the stress distribution in the root dentin was nearly identical between these two models. Experimental failure data showed, however, that the failure modes of the cast post and core group were more catastrophic and involved longitudinal root fractures while all fractures of the root with fiberglass post were coronal fractures. Simulating the interface more realistically with friction between resin cement and cast post and core (requiring a nonlinear analysis) rather than a perfect fusion, the stress distribution changed substantially between the two post types (Fig. 6), and yielded more realistic results when compared with the experimental

Fig. 6. Nonlinear FEA of endodontic treated tooth restored with A. Cast post and core and B.

In FEA the whole domain is divided into smaller elements. The collection and distribution of these elements is called a mesh. Elements are interconnected by nodes, which are thus the only points though which elements interact with each other. The process of creating an element mesh is referred to as "discretization" of the problem domain (Geng et al., 2001).

There are many different types of elements. One of the differences can be their basic shape, such as triangular, tetrahedral, hexahedral, etc. Triangular or tetrahedral elements are popular because automatic meshing software routines are easier to develop and thus more advanced for those shapes. Automatic generation of element distributions is especially

observations.

Fiberglass post.

**2.4 Mesh generation** 

useful in bioengineering, which often deals with irregular geometries. Since few modeled geometries have perfectly square dimensions or even straight edges, element shapes must be adapted to fit. Note that the accuracy of elements deteriorates the further they are distorted from their ideal basic shape. Besides their basic geometrical shapes, elements can differ in the way they are solved, such as linear or quadratic interpolation. This refers to how stress and strain is interpolated within an element.

Fig. 7. Radiography of maxillary premolar (A); Lines plotted in the MARC/MENTAT software (B); Manual creation of mesh (C); Final subdivision for improving the mesh quality (D).

Most FEA software provides *automesh* or automatic mesh generation options. The program may suggests the size and number of elements or allow manual control for generating the element meshes. Manual mesh generation can give good results for 2D models (Fig. 7). However, most 3D models rely on automated mesh generators because manual creation of 3D models is very time-consuming. Still, various aspects of the 3D automeshing need to be controlled manually, such as the number of elements required in a given pre-selected area, the distribution of elements, the range of element sizes within a model, uniting or dividing elements, etc. The manual controls also allows selective distribution of elements, for example, more refined meshes in special regions of interest (contact interfaces, geometric discontinuities) while creating coarser mesh distributions in regions of less interest.

In finite element modeling, a finer (denser) mesh should allow a more accurate solution. However, as a mesh is made finer and the element count increases, the computation time also increases. How can you get a mesh that balances accuracy and computing resources? One way is to perform a convergence study. This process involves the creation and analysis of multiple mesh distributions with increasing number of elements or refinements. When results with the various models are plotted, a convergence to a particular solution can be found. Based on this convergence data, an estimation of the error can be made for the

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 37

Fig. 8. Load application A. Contact analysis using antagonist tooth (Marc/Mentat software);

FEA results can be intimidating, the amount of generated data (displacements, strains, stresses, temperatures, etc) is almost unlimited. However, one of the most powerful features of FEA is that results can be easily visualized and made accessible. Visualization of the results can be done by showing data distributions using a colour scale, where each colour corresponds to a range of values. Furthermore, deformations and displacements can be shown by comparing the original unstressed model outlines with the outline of the model under stress. Based on this immediate visualized output, the operator can investigate the displacement of the structure, the type of movement that was performed, which region has a higher dislodgement, or how to redistribute the stresses in the analyzed structure in either three dimensions or in two-dimensions. Such a structural analysis allows the determination of stress and strain resulting from external force, pressure, thermal change, and other factors. This section discusses how results from a finite element analysis should be evaluated, starting with a check of the model, followed by checking the outcome. Then the relationship between finite element and experimental will be discussed with respect to the

All finite element analyses should first be checked for coherence (or sanity). The first step of a coherence analysis is to visualize the displacements and deformations to verify that the simulated model moves in the expected direction. The second step of a coherence analysis is

After the model definition is confirmed to be sound, the validity of the outcome still has to be validated. A finite element analysis is modelled based on geometric, property, and

B. Point load application (Ansys Inc., Houston, USA).

**3. Evaluation of finite element analysis** 

limitations of either method.

**3.1 Analysis of coherency** 

**3.2 Validation of the outcome** 

to analyze if the distribution stresses is as expected.

various mesh distributions. A mesh convergence study can thus be used to find a balance between an efficient mesh distribution and an acceptably accurate solution within the limitations of the computing resources. Moreover, a convergence test can verify if an obtained solution is true or if it was an artifact of a particular element distribution.

#### **2.5 Boundary conditions**

The boundary conditions define the external influences on a modeled structure, usually loading and constraints. Boundary conditions are associated with six degrees of freedom (DOF). The combination of all boundary conditions of a FEA model must represent the procedural conditions to which the actual structure that is simulated is subjected. The choice and application of boundary conditions is extremely important, because they determine the outcome of the FEA.

#### **2.5.1 Prescribed displacement – Fixation and symmetry**

In a simple way, restrictions can be summarized as the imposition of displacements and rotations on a finite element model, which can be either null or have fixed values or rates. These restrictions concern three rotations (around X, Y, Z-axes) and three translations (in X, Y, Z-directions). Static analysis requires sufficient fixation of a model to remain in place. Insufficient fixation will lead to instability and failure to reach a numerical solution in the FEA. Since nodes are the points through which elements communicate, boundary conditions are usually applied to nodes, where in a 3D model each free node has 6 degrees of freedom (3 translations and 3 rotations). Although some FEA software may allow application of boundary conditions to element edges or surfaces, they are extrapolated to the associated nodes. To achieve the fixation mimicking a support system in real life, for example complete immobilization of a modeled specimen in a test fixture, displacement constraints can be applied to nodes located in a region equivalent to those of the real support system. Symmetry can be viewed as a form of fixation. Since all displacements are mirrored, the displacement across the symmetry-axis is zero.

#### **2.5.2 Load application**

The application of loads in a FEA model must also represent the external loading situations to which the modeled structure is subjected. These loads can be tensile, compressive, shear, torque, etc. To simulate the masticatory forces, loads have been applied using different methods, for example point loads, distributed loads across a specific area, and by means of a simulated opposing cusp of the antagonist tooth (Fig. 8). A point load application may result in high stress concentrations around the loaded nodes, creating unrealistic stress concentrations. In reality, a masticatory contact force is likely to be distributed across certain contact areas on both the buccal and lingual cuspal inclines. However, the most realistic load application is not always the best choice for all research questions. Contact areas move depending on stiffness and thus deformation of both opposing teeth. If contact areas change, contact loads change also, which can have significant effect on the stress distribution. When a research question requires well-defined load conditions, point loads or prescribed distributed loading may be better choices than the seemingly more realistic simulated tooth contact.

Fig. 8. Load application A. Contact analysis using antagonist tooth (Marc/Mentat software); B. Point load application (Ansys Inc., Houston, USA).

#### **3. Evaluation of finite element analysis**

36 Finite Element Analysis – From Biomedical Applications to Industrial Developments

various mesh distributions. A mesh convergence study can thus be used to find a balance between an efficient mesh distribution and an acceptably accurate solution within the limitations of the computing resources. Moreover, a convergence test can verify if an

The boundary conditions define the external influences on a modeled structure, usually loading and constraints. Boundary conditions are associated with six degrees of freedom (DOF). The combination of all boundary conditions of a FEA model must represent the procedural conditions to which the actual structure that is simulated is subjected. The choice and application of boundary conditions is extremely important, because they determine the

In a simple way, restrictions can be summarized as the imposition of displacements and rotations on a finite element model, which can be either null or have fixed values or rates. These restrictions concern three rotations (around X, Y, Z-axes) and three translations (in X, Y, Z-directions). Static analysis requires sufficient fixation of a model to remain in place. Insufficient fixation will lead to instability and failure to reach a numerical solution in the FEA. Since nodes are the points through which elements communicate, boundary conditions are usually applied to nodes, where in a 3D model each free node has 6 degrees of freedom (3 translations and 3 rotations). Although some FEA software may allow application of boundary conditions to element edges or surfaces, they are extrapolated to the associated nodes. To achieve the fixation mimicking a support system in real life, for example complete immobilization of a modeled specimen in a test fixture, displacement constraints can be applied to nodes located in a region equivalent to those of the real support system. Symmetry can be viewed as a form of fixation. Since all displacements are mirrored, the

The application of loads in a FEA model must also represent the external loading situations to which the modeled structure is subjected. These loads can be tensile, compressive, shear, torque, etc. To simulate the masticatory forces, loads have been applied using different methods, for example point loads, distributed loads across a specific area, and by means of a simulated opposing cusp of the antagonist tooth (Fig. 8). A point load application may result in high stress concentrations around the loaded nodes, creating unrealistic stress concentrations. In reality, a masticatory contact force is likely to be distributed across certain contact areas on both the buccal and lingual cuspal inclines. However, the most realistic load application is not always the best choice for all research questions. Contact areas move depending on stiffness and thus deformation of both opposing teeth. If contact areas change, contact loads change also, which can have significant effect on the stress distribution. When a research question requires well-defined load conditions, point loads or prescribed distributed loading may be better choices than the seemingly more realistic simulated tooth

obtained solution is true or if it was an artifact of a particular element distribution.

**2.5.1 Prescribed displacement – Fixation and symmetry** 

displacement across the symmetry-axis is zero.

**2.5 Boundary conditions** 

outcome of the FEA.

**2.5.2 Load application** 

contact.

FEA results can be intimidating, the amount of generated data (displacements, strains, stresses, temperatures, etc) is almost unlimited. However, one of the most powerful features of FEA is that results can be easily visualized and made accessible. Visualization of the results can be done by showing data distributions using a colour scale, where each colour corresponds to a range of values. Furthermore, deformations and displacements can be shown by comparing the original unstressed model outlines with the outline of the model under stress. Based on this immediate visualized output, the operator can investigate the displacement of the structure, the type of movement that was performed, which region has a higher dislodgement, or how to redistribute the stresses in the analyzed structure in either three dimensions or in two-dimensions. Such a structural analysis allows the determination of stress and strain resulting from external force, pressure, thermal change, and other factors. This section discusses how results from a finite element analysis should be evaluated, starting with a check of the model, followed by checking the outcome. Then the relationship between finite element and experimental will be discussed with respect to the limitations of either method.

#### **3.1 Analysis of coherency**

All finite element analyses should first be checked for coherence (or sanity). The first step of a coherence analysis is to visualize the displacements and deformations to verify that the simulated model moves in the expected direction. The second step of a coherence analysis is to analyze if the distribution stresses is as expected.

#### **3.2 Validation of the outcome**

After the model definition is confirmed to be sound, the validity of the outcome still has to be validated. A finite element analysis is modelled based on geometric, property, and

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 39

2010). Sometimes interfacial interactions are more complex, such as areas where two materials may contact but do not bond. In such cases contact analysis needs to be simulated

Most research protocols, including finite element analyses, will have limitations. Some limitations in finite element modelling are a deliberately choice. For example, although teeth are 3D structures, often they are modelled in 2D. Two-dimensional models offer excellent visual access for pre- and post-processing, improving their didactic potential. Furthermore, because of the reduced dimensions, more computational capacity can be preserved for improvements in element and simulation quality or functional processes such as masticatory movements. On the other hand, 3D models, although geometrically more realistic, may give a false impression of accuracy, because they are generally more coarse, contain elements with compromised shapes, and examination or improvement of the model

The finite element method is sometimes viewed as a less time-consuming process than experimental research, and therefore could minimize laboratory testing requirements. For some applications, finite element analysis may provide faster solutions, for example for the testing of parameters, which can be changed more easily in FEA than in laboratory experiments. However, due to the complexity of shape, properties, and boundary conditions of dental structures, comprehensive modelling can also quickly becomes very complex and time-consuming. Finite element analysis should be viewed in combination with experimental methods, not as a substitute. Finite element analysis can provide information that would be difficult or impossible to obtain with experimental observations, but at the same time, finite element analysis cannot be performed without experimental input and

Compared with experiments, FEA has clear limitations. These limitations are mainly due to the many factors that contribute to the mechanical response but are still poorly understood. Such lack of understanding usually does not affect experiments, if their outcomes are simply considered as phenomena. However, FEA is the compilation of our understanding of physical laws and material properties, expressed in a theoretical model that describes the interactions between the various factors. Therefore, phenomena are no acceptable input for a theoretical model. Limitations in FEA therefore most often refer back to our own lack of understanding the reality. In other words, our own limitations in understanding are the cause of limitations in FEA. Our inability to accurately describe and simulate biomechanical dynamics and properties of a tooth and its supporting structures limits the accuracy of our FEA models. Fortunately, even imperfect experimental or FEA testing methods can improve our insight and continuously expand our understanding of reality. Therefore, although certain differences may remain between reality and the analyses we conduct using the finite element method, the numerical approach can approximate, for example, otherwise inaccessible stress distributions within a tooth-restoration complex. Furthermore, the ability to visualize many of the results from finite element analyses has also undoubtedly helped researchers to more clearly convey their data, and helped expand the discussion and

dissemination of research findings that have contributed to improve oral health.

which will require FEA software that can perform nonlinear analyses.

**3.4 Relationship between finite element and experimental analysis** 

is far more difficult (Korioth & Versluis, 1997).

validation.

boundary conditions, each of which may have required significant assumptions or simplifications. The purpose of validation is thus to confirm that the general response of the model is realistic. Unlike stress, which cannot be measured directly, deformation or displacement can be directly measured. Therefore, displacement is often a good choice for comparing the simulated behaviour with the behaviour observed in reality, even if the simulation was a "stress" analysis. Validation can be achieved by comparing the outcome with published results from validated analyses or with laboratory measurements. Examples are strain gauge measurement, cuspal flexure, bending displacements, etc. Effects of stresses can also be indirectly validated through the observation of their expected effects, such as crack initiation and fractures. It is not realistic to expect an exact fit between experimental and numerical results, because even between experimental results there will not be an exact fit due to natural and experimental variations. Therefore it is important to remember that it is not an exact fit that validates a finite element analysis but rather the similarity in general tendencies (Versluis et al., 1997).

#### **3.3 Interpretation of the results**

After a finite element analysis has been checked and validated, it can be used to interpret the research question. Most finite element analysis results should be interpreted qualitatively. Quantitative interpretation can sometimes be justified, provided all input is verifiable and quantitatively validated. The current state of the art of the use of finite element modelling in dentistry indicates that predictive results are still best viewed in a qualitative manner. It is the search for optimal balance between the objectives of a study, computational efforts (accuracy and efficiency), and practical limitations that ultimately determines the value of a finite element model. Since most finite element models are linear, errors in magnitudes of the loads will not have a direct effect on qualitative predictions. However, small changes in types of boundary conditions such as the location of the loading can substantially alter even the qualitative performance predictions.

Biomechanical performance involves efficient function as well as failure. One of the failure mechanisms is loss of structural integrity, which can eventually result in loss of function. FEA can be used in the determination of fracture mechanics parameters, and examination of experimental failure test methods. In the dental FEA literature, failure is usually extrapolated from maximum stress values, where stress concentrations are identified as possible locations for failure initiation and relative concentration values are interpreted as related to the failure risk. When using this process for interpreting failure behaviour, it is important to carefully assess the stress concentration locations because they may depend heavily on the chosen modelling and boundary condition options (Korioth & Versluis, 1997). Furthermore, stress distributions change when a crack propagates. Therefore, researchers should be extremely cautious about extrapolating crack behaviour based on the distribution of stress concentrations from a static analysis.

Interfacial stress was previously noted as an important area that needs careful interpretation in FEA. For example, stress analyses of the tooth–restoration complex have been performed to predict failure risks at the interfaces as well as stresses transferred across such interfaces. Usually such interfaces are modelled as perfectly bonded, where tooth and restoration elements share the same node. Depending on the accuracy of this assumption, this may lead to erroneous interpretation of the results of a finite element analysis (Srirekha & Bashetty,

boundary conditions, each of which may have required significant assumptions or simplifications. The purpose of validation is thus to confirm that the general response of the model is realistic. Unlike stress, which cannot be measured directly, deformation or displacement can be directly measured. Therefore, displacement is often a good choice for comparing the simulated behaviour with the behaviour observed in reality, even if the simulation was a "stress" analysis. Validation can be achieved by comparing the outcome with published results from validated analyses or with laboratory measurements. Examples are strain gauge measurement, cuspal flexure, bending displacements, etc. Effects of stresses can also be indirectly validated through the observation of their expected effects, such as crack initiation and fractures. It is not realistic to expect an exact fit between experimental and numerical results, because even between experimental results there will not be an exact fit due to natural and experimental variations. Therefore it is important to remember that it is not an exact fit that validates a finite element analysis but rather the similarity in general

After a finite element analysis has been checked and validated, it can be used to interpret the research question. Most finite element analysis results should be interpreted qualitatively. Quantitative interpretation can sometimes be justified, provided all input is verifiable and quantitatively validated. The current state of the art of the use of finite element modelling in dentistry indicates that predictive results are still best viewed in a qualitative manner. It is the search for optimal balance between the objectives of a study, computational efforts (accuracy and efficiency), and practical limitations that ultimately determines the value of a finite element model. Since most finite element models are linear, errors in magnitudes of the loads will not have a direct effect on qualitative predictions. However, small changes in types of boundary conditions such as the location of the loading can substantially alter even

Biomechanical performance involves efficient function as well as failure. One of the failure mechanisms is loss of structural integrity, which can eventually result in loss of function. FEA can be used in the determination of fracture mechanics parameters, and examination of experimental failure test methods. In the dental FEA literature, failure is usually extrapolated from maximum stress values, where stress concentrations are identified as possible locations for failure initiation and relative concentration values are interpreted as related to the failure risk. When using this process for interpreting failure behaviour, it is important to carefully assess the stress concentration locations because they may depend heavily on the chosen modelling and boundary condition options (Korioth & Versluis, 1997). Furthermore, stress distributions change when a crack propagates. Therefore, researchers should be extremely cautious about extrapolating crack behaviour based on the distribution

Interfacial stress was previously noted as an important area that needs careful interpretation in FEA. For example, stress analyses of the tooth–restoration complex have been performed to predict failure risks at the interfaces as well as stresses transferred across such interfaces. Usually such interfaces are modelled as perfectly bonded, where tooth and restoration elements share the same node. Depending on the accuracy of this assumption, this may lead to erroneous interpretation of the results of a finite element analysis (Srirekha & Bashetty,

tendencies (Versluis et al., 1997).

**3.3 Interpretation of the results** 

the qualitative performance predictions.

of stress concentrations from a static analysis.

2010). Sometimes interfacial interactions are more complex, such as areas where two materials may contact but do not bond. In such cases contact analysis needs to be simulated which will require FEA software that can perform nonlinear analyses.

Most research protocols, including finite element analyses, will have limitations. Some limitations in finite element modelling are a deliberately choice. For example, although teeth are 3D structures, often they are modelled in 2D. Two-dimensional models offer excellent visual access for pre- and post-processing, improving their didactic potential. Furthermore, because of the reduced dimensions, more computational capacity can be preserved for improvements in element and simulation quality or functional processes such as masticatory movements. On the other hand, 3D models, although geometrically more realistic, may give a false impression of accuracy, because they are generally more coarse, contain elements with compromised shapes, and examination or improvement of the model is far more difficult (Korioth & Versluis, 1997).

#### **3.4 Relationship between finite element and experimental analysis**

The finite element method is sometimes viewed as a less time-consuming process than experimental research, and therefore could minimize laboratory testing requirements. For some applications, finite element analysis may provide faster solutions, for example for the testing of parameters, which can be changed more easily in FEA than in laboratory experiments. However, due to the complexity of shape, properties, and boundary conditions of dental structures, comprehensive modelling can also quickly becomes very complex and time-consuming. Finite element analysis should be viewed in combination with experimental methods, not as a substitute. Finite element analysis can provide information that would be difficult or impossible to obtain with experimental observations, but at the same time, finite element analysis cannot be performed without experimental input and validation.

Compared with experiments, FEA has clear limitations. These limitations are mainly due to the many factors that contribute to the mechanical response but are still poorly understood. Such lack of understanding usually does not affect experiments, if their outcomes are simply considered as phenomena. However, FEA is the compilation of our understanding of physical laws and material properties, expressed in a theoretical model that describes the interactions between the various factors. Therefore, phenomena are no acceptable input for a theoretical model. Limitations in FEA therefore most often refer back to our own lack of understanding the reality. In other words, our own limitations in understanding are the cause of limitations in FEA. Our inability to accurately describe and simulate biomechanical dynamics and properties of a tooth and its supporting structures limits the accuracy of our FEA models. Fortunately, even imperfect experimental or FEA testing methods can improve our insight and continuously expand our understanding of reality. Therefore, although certain differences may remain between reality and the analyses we conduct using the finite element method, the numerical approach can approximate, for example, otherwise inaccessible stress distributions within a tooth-restoration complex. Furthermore, the ability to visualize many of the results from finite element analyses has also undoubtedly helped researchers to more clearly convey their data, and helped expand the discussion and dissemination of research findings that have contributed to improve oral health.

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 41

except for fracture of enamel that was undermined by the NCCL (Hur et al., 2011). Since the location of the lesion will affect the stress conditions, combining clinical observations and finite element modeling will be essential to determine the stress factor in the initiation and

In the case of dental caries, the decay process can continue until the destruction of the tooth and the compromise of adjacent tissues. As the caries process progresses without some type of intervention, the pulp ultimately becomes involved and the root canal therapy is required (Vargas & Arevalo, 2009). One of the steps in root canal treatment is to completely fill the root canal system. During root canal preparation, many variables are outside the control of the clinician (natural root morphology, canal shape and size, dentine thickness) other factors can be addressed during treatment to reduce fracture susceptibility. Using finite element analysis, Versluis et al. (2006) demonstrated that the potential for fracture susceptibility may be reduced by ensuring round canal profiles and smooth canal taper (Fig. 10). Even when fins were not contacted by the instrument, stresses within the root were lower and more evenly distributed than before preparation. Rundquist & Versluis (2006) also used FEA to demonstrate that with increasing taper, root stresses decreased during root filling but tended to increase slightly during a masticatory load. Based on the simulation of vertical warm gutta-percha compaction and a subsequent occlusal load, they suggested that root fracture originating at the apical third was likely initiated during filling, whilst fracture originating in the cervical portion was likely caused by occlusal loads. Gutta-percha is the

Fig. 10. Stress distribution during obturation pressure in a root with oval canal, cleaned with

development of NCCL.

**4.2 Endodontic treatment** 

ProTaper F1 (Versluis et al., 2006).

#### **4. Impact of finite element analysis on dentistry - How FE analyses have contributed to improved oral health**

Oral health is important to an individual's well-being and overall health. In dentistry, most oral diseases are neither self-limiting nor self-repairing (Vargas & Arevalo, 2009). Therefore, prompt professional care is fundamental, given that oral diseases follow a downward spiral: incipient diseases requiring minimum dental care, if untreated, progress into diseases that require increasingly more complex and expensive treatments; increases in complexity and cost usually make the treatment even more out of reach for a large proportion of the population (Vargas & Ronzio, 2002). In this context, finite element analysis has been applied in various areas in dentistry (1) to improve the understanding of these complex processes and (2) to help to design better procedures.

#### **4.1 Non-Carious Cervical Lesions (NCCL)**

FEA has been used in the investigation of NCCL (Michael et al., 2009). Although the etiology of NCCL remains a controversial subject, there is a general consensus that the process is multi-factorial, and that stress can be one of the factors. Goel et al. (1991) investigated stresses arising at the dentino-enamel junction during function and noted that the shape of the dentino-enamel junction was different under working cusps than nonworking cusps. Tensile stresses were elevated toward cervical enamel where the mechanical inter-locking between enamel and dentin is weaker than in other areas of the tooth, making it susceptible to cracking, which could contribute to cervical caries (Goel et al., 1991). Finite element analyses have usually assumed the NCCL across the CEJ (Fig. 9). A recent study, however, did not find clinical evidence of enamel loss above the occlusal margin of NCCL,

Fig. 9. A. 3D Model of FEA analysis of a non-carious cervical lesions not restored; B. 3D Model of FEA analysis of a non-carious cervical lesions restored with composite resin; C. Maximum principal stress distribution at the unrestored non-carious cervical lesion (Pereira FA, 2011).

except for fracture of enamel that was undermined by the NCCL (Hur et al., 2011). Since the location of the lesion will affect the stress conditions, combining clinical observations and finite element modeling will be essential to determine the stress factor in the initiation and development of NCCL.

#### **4.2 Endodontic treatment**

40 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Oral health is important to an individual's well-being and overall health. In dentistry, most oral diseases are neither self-limiting nor self-repairing (Vargas & Arevalo, 2009). Therefore, prompt professional care is fundamental, given that oral diseases follow a downward spiral: incipient diseases requiring minimum dental care, if untreated, progress into diseases that require increasingly more complex and expensive treatments; increases in complexity and cost usually make the treatment even more out of reach for a large proportion of the population (Vargas & Ronzio, 2002). In this context, finite element analysis has been applied in various areas in dentistry (1) to improve the understanding of these complex processes

FEA has been used in the investigation of NCCL (Michael et al., 2009). Although the etiology of NCCL remains a controversial subject, there is a general consensus that the process is multi-factorial, and that stress can be one of the factors. Goel et al. (1991) investigated stresses arising at the dentino-enamel junction during function and noted that the shape of the dentino-enamel junction was different under working cusps than nonworking cusps. Tensile stresses were elevated toward cervical enamel where the mechanical inter-locking between enamel and dentin is weaker than in other areas of the tooth, making it susceptible to cracking, which could contribute to cervical caries (Goel et al., 1991). Finite element analyses have usually assumed the NCCL across the CEJ (Fig. 9). A recent study, however, did not find clinical evidence of enamel loss above the occlusal margin of NCCL,

Fig. 9. A. 3D Model of FEA analysis of a non-carious cervical lesions not restored; B. 3D Model of FEA analysis of a non-carious cervical lesions restored with composite resin; C. Maximum principal stress distribution at the unrestored non-carious cervical lesion (Pereira

**4. Impact of finite element analysis on dentistry - How FE analyses have** 

**contributed to improved oral health** 

and (2) to help to design better procedures.

**4.1 Non-Carious Cervical Lesions (NCCL)** 

FA, 2011).

In the case of dental caries, the decay process can continue until the destruction of the tooth and the compromise of adjacent tissues. As the caries process progresses without some type of intervention, the pulp ultimately becomes involved and the root canal therapy is required (Vargas & Arevalo, 2009). One of the steps in root canal treatment is to completely fill the root canal system. During root canal preparation, many variables are outside the control of the clinician (natural root morphology, canal shape and size, dentine thickness) other factors can be addressed during treatment to reduce fracture susceptibility. Using finite element analysis, Versluis et al. (2006) demonstrated that the potential for fracture susceptibility may be reduced by ensuring round canal profiles and smooth canal taper (Fig. 10). Even when fins were not contacted by the instrument, stresses within the root were lower and more evenly distributed than before preparation. Rundquist & Versluis (2006) also used FEA to demonstrate that with increasing taper, root stresses decreased during root filling but tended to increase slightly during a masticatory load. Based on the simulation of vertical warm gutta-percha compaction and a subsequent occlusal load, they suggested that root fracture originating at the apical third was likely initiated during filling, whilst fracture originating in the cervical portion was likely caused by occlusal loads. Gutta-percha is the

Fig. 10. Stress distribution during obturation pressure in a root with oval canal, cleaned with ProTaper F1 (Versluis et al., 2006).

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 43

thus be indicated for endodontically treated teeth that have suffered excessive coronary structure loss (Yu et al., 2006). Research studies using FEA concluded that the use of post systems that have an elastic modulus similar to that of dentine result in a mechanically homogenous units with better biomechanical performance (Barjau-Escribano et al., 2006; Silva et al., 2009). Some studies have concluded that the attributes of carbon and glass fiber dowels make them suitable for dowel restoration (Glazer, 2000; Lanza et al., 2005). Dowel length, size, and design have also been shown to influence the biomechanics and stress distribution of restored teeth (Barjau-Escribano et al., 2006). Using finite element analysis it is possible to evaluate the influence of the type of material (carbon and glass fiber) and the external configuration of the dowel (smooth and serrated) on the stress distribution of teeth restored with varying dowel systems (Soares et al., 2009). Moreover, the difference in elastic modulus between dentin, intraradicular retainers, and cements could result in stress concentrations at the restoration interface when the tooth is in function (Soares et al., 2010,

In the field of operative dentistry, FEA seems to be an appropriate method for obtaining answers about the interferences caused by the restorative process in a complete structure, for optimizing the design of dental restorations and for evaluating stress distributions in relation to different designs. Many materials are available for dental restorations. The selection and indications for direct and indirect restorative materials involve esthetic, financial, and anatomic considerations, as well as important factors such as analysis of the biomechanical characteristics of the restorative materials, and the amount and state of remaining tooth structure (Soares et al., 2008). In recent years, the demand for nonmetal dental restorations has grown considerably. Metal-free reinforced restorative systems have become popular because of the less favorable esthetic appearance of metal ceramic crowns (Gardner et al., 1997). The primary advantages of nonmetal alternatives (composite resins and ceramics) are improved esthetics, the avoidance of mercury, and cost effectiveness (Stein et al., 2005). Composite resin and ceramic restorations retained with an adhesive resin are the most popular restorations currently used. Composite resins have mechanical properties similar to dentin (Willams et al., 1992) while ceramic has an elastic modulus

The conservation of dental structure is crucial to offering fracture resistance, since the removal of dentin reduces the structural integrity of a tooth and causes alteration in stress distributions (Soares et al., 2008b). In this context, the use of adhesive restorations is recommended for reinforcing remaining dental structure (Soares et al., 2008b, Versluis & Tantbirojn, 2011). By using the finite element analysis, stress distributions could be accessed within endodontically treated maxillary premolars that lost tooth structure and the effect of the type of restorative material used for restorations could be studied (Soares et al., 2008). The use of directly placed adhesive restorative materials, such as composite resin, and indirectly placed restorations, such as ceramic inlays, cemented with adhesive materials, generally reduced stress concentrations in comparison with amalgam restorations (Soares et al., 2008). Although indirect restorations may be recommended, the dentist still faces to the

choice of geometric configuration of the cavity preparation (Soares et al., 2003).

Silva et al., 2011).

**4.4 Restorative procedures** 

similar to that of enamel (Albakry et al., 2003).

most common core material used (Er et al., 2007). Although the softening of gutta-percha by heat is a widely used technique, the use of high levels of heat can lead to complications. When heat compaction techniques are used, the procedure should not harm the periodontal ligament (Budd et al., 1991). The use of the technique may result in an unintentional transmission of excessive heat to the surrounding periodontal tissues (Er et al., 2007). Excessive heat during obturation techniques may cause irreversible injury to tissues (Atrizadeh et al., 1971; Albrektsson et al., 1986). By using a three-dimensional thermal finite element analysis the distribution and temperatures were evaluated in a virtual model of a maxillary canine and surrounding tissues during a simulated continuous heat obturation procedure (Er et al., 2007).

#### **4.3 Restoration of root filled teeth**

Endodontically treated teeth are compromised by coronal destruction from dental caries (Ross, 1980), fractures (Soares et al., 2007), previous restorations (Schatz et al., 2001), and endodontic access (Soares et al., 2007). How these compromised teeth should be reconstructed to regain their original fracture resistance has been the subject of many studies investigating restoration types and benefits of posts (Fokkinga et al., 2005; Salameh et al., 2006; Salameh et al., 2007). It is not sufficient to only measure an endpoint such as fracture resistance to fully understand the effect of restoration type and post application. A more comprehensive analysis is thus needed to determine the optimal procedures for reconstructing endodontically treated teeth (Soares et al., 2008). The biomechanical conditions that lead to fracture are characterized by the stress state in a tooth, which can be assessed by finite element analysis (Fig. 11). Soares et al. (2008) therefore used FEA to investigate the stress distribution in an endodontically treated premolar restored with composite resin with or without a glass fiber post system and concluded that the use of glass fiber posts did not reinforce the tooth-restoration complex. Intraradicular retention should

Fig. 11. A. 3D Model of FEA analysis of a 3 elements fixed prosthesis regarding the effect of post type (B. fiber glass post; C. Cast post and core) (Silva GR, 2011).

thus be indicated for endodontically treated teeth that have suffered excessive coronary structure loss (Yu et al., 2006). Research studies using FEA concluded that the use of post systems that have an elastic modulus similar to that of dentine result in a mechanically homogenous units with better biomechanical performance (Barjau-Escribano et al., 2006; Silva et al., 2009). Some studies have concluded that the attributes of carbon and glass fiber dowels make them suitable for dowel restoration (Glazer, 2000; Lanza et al., 2005). Dowel length, size, and design have also been shown to influence the biomechanics and stress distribution of restored teeth (Barjau-Escribano et al., 2006). Using finite element analysis it is possible to evaluate the influence of the type of material (carbon and glass fiber) and the external configuration of the dowel (smooth and serrated) on the stress distribution of teeth restored with varying dowel systems (Soares et al., 2009). Moreover, the difference in elastic modulus between dentin, intraradicular retainers, and cements could result in stress concentrations at the restoration interface when the tooth is in function (Soares et al., 2010, Silva et al., 2011).

#### **4.4 Restorative procedures**

42 Finite Element Analysis – From Biomedical Applications to Industrial Developments

most common core material used (Er et al., 2007). Although the softening of gutta-percha by heat is a widely used technique, the use of high levels of heat can lead to complications. When heat compaction techniques are used, the procedure should not harm the periodontal ligament (Budd et al., 1991). The use of the technique may result in an unintentional transmission of excessive heat to the surrounding periodontal tissues (Er et al., 2007). Excessive heat during obturation techniques may cause irreversible injury to tissues (Atrizadeh et al., 1971; Albrektsson et al., 1986). By using a three-dimensional thermal finite element analysis the distribution and temperatures were evaluated in a virtual model of a maxillary canine and surrounding tissues during a simulated continuous heat obturation

Endodontically treated teeth are compromised by coronal destruction from dental caries (Ross, 1980), fractures (Soares et al., 2007), previous restorations (Schatz et al., 2001), and endodontic access (Soares et al., 2007). How these compromised teeth should be reconstructed to regain their original fracture resistance has been the subject of many studies investigating restoration types and benefits of posts (Fokkinga et al., 2005; Salameh et al., 2006; Salameh et al., 2007). It is not sufficient to only measure an endpoint such as fracture resistance to fully understand the effect of restoration type and post application. A more comprehensive analysis is thus needed to determine the optimal procedures for reconstructing endodontically treated teeth (Soares et al., 2008). The biomechanical conditions that lead to fracture are characterized by the stress state in a tooth, which can be assessed by finite element analysis (Fig. 11). Soares et al. (2008) therefore used FEA to investigate the stress distribution in an endodontically treated premolar restored with composite resin with or without a glass fiber post system and concluded that the use of glass fiber posts did not reinforce the tooth-restoration complex. Intraradicular retention should

Fig. 11. A. 3D Model of FEA analysis of a 3 elements fixed prosthesis regarding the effect of

post type (B. fiber glass post; C. Cast post and core) (Silva GR, 2011).

procedure (Er et al., 2007).

**4.3 Restoration of root filled teeth** 

In the field of operative dentistry, FEA seems to be an appropriate method for obtaining answers about the interferences caused by the restorative process in a complete structure, for optimizing the design of dental restorations and for evaluating stress distributions in relation to different designs. Many materials are available for dental restorations. The selection and indications for direct and indirect restorative materials involve esthetic, financial, and anatomic considerations, as well as important factors such as analysis of the biomechanical characteristics of the restorative materials, and the amount and state of remaining tooth structure (Soares et al., 2008). In recent years, the demand for nonmetal dental restorations has grown considerably. Metal-free reinforced restorative systems have become popular because of the less favorable esthetic appearance of metal ceramic crowns (Gardner et al., 1997). The primary advantages of nonmetal alternatives (composite resins and ceramics) are improved esthetics, the avoidance of mercury, and cost effectiveness (Stein et al., 2005). Composite resin and ceramic restorations retained with an adhesive resin are the most popular restorations currently used. Composite resins have mechanical properties similar to dentin (Willams et al., 1992) while ceramic has an elastic modulus similar to that of enamel (Albakry et al., 2003).

The conservation of dental structure is crucial to offering fracture resistance, since the removal of dentin reduces the structural integrity of a tooth and causes alteration in stress distributions (Soares et al., 2008b). In this context, the use of adhesive restorations is recommended for reinforcing remaining dental structure (Soares et al., 2008b, Versluis & Tantbirojn, 2011). By using the finite element analysis, stress distributions could be accessed within endodontically treated maxillary premolars that lost tooth structure and the effect of the type of restorative material used for restorations could be studied (Soares et al., 2008). The use of directly placed adhesive restorative materials, such as composite resin, and indirectly placed restorations, such as ceramic inlays, cemented with adhesive materials, generally reduced stress concentrations in comparison with amalgam restorations (Soares et al., 2008). Although indirect restorations may be recommended, the dentist still faces to the choice of geometric configuration of the cavity preparation (Soares et al., 2003).

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 45

based on type of preparation (inlay or onlay), and restorative material (composite resin, resin laboratory, reinforced ceramic with lithium disilicate or reinforced ceramic with leucite). Materials with higher modulus of elasticity transfer less stress into the tooth structure. However, materials with modulus of elasticity much larger than the dental structure caused more severe stress concentrations. The models that used reinforced ceramics with leucite showed a behavior that was biomechanically closest to healthy teeth

Resin-composite materials have been widely and increasingly used today in adhesive dental restorative procedures (Fagundes et al., 2009). An important advantage over metallic filling materials is the well-known possibility of bonding the restoration to dental tissues (Marques de Melo et al., 2008) and a significant disadvantage of many of these materials are still the polymerization shrinkage (Pereira et al., 2008). The clinical concern about polymerization shrinkage is evident from the large number of publications and large number of controversial opinions about this topic (Versluis et al., 2004). Shrinkage stress has been associated with various clinical symptoms, including fracture propagation, microleakage and post-operative sensitivity, none of which are direct measures of shrinkage stress. Since stress cannot be measured directly, the presence of shrinkage stresses can only be quantified through indirect manifestations, in particular tooth deformation (Tantbirojn et al., 2004).

Various methods have been used to estimate residual shrinkage stresses, ranging from extrapolated shrinkage or load measurements in vitro to stress analyses in tooth shaped anatomies using photoelastic or finite element methods (Kinomoto et al., 1999; Ausiello et al., 2001). Determination of shrinkage stress is difficult, because it is a transient and nonlinear process. The amount of stress after polymerization therefore depends on the correct description of all changes in mechanical properties and their sequence. Moreover, stress is not a material property or even a structural value, because stress is a threedimensional local tensor (system of related vectors) that is determined by the combination of multiple material properties and local conditions. Since finite element analysis performs its calculation based on such input (mechanical properties, geometry, boundary conditions), it is eminently suitable for studying residual shrinkage stress in dental systems. On the other hand, as the input for especially the mechanical properties remains to be determined more comprehensively, any polymerization shrinkage predicted by finite element analysis should be validated experimentally using indirect factors that can be measured, such as

Using such validated finite element analyses, shrinkage stresses in restored teeth (enamel and dentin) were found to increase with increasing restoration size, while stresses in the restoration and along the tooth-restoration interface decreased (Versluis et al., 2004). This outcome was explained by the change in tooth stiffness: removal of dental hard tissue decreases the stiffness of the tooth, causing the tooth to be deformed more by the shrinkage stresses (higher stress in the tooth) and causing less resistance to the composite shrinkage (lower stress in the composite). As this example shows, shrinkage stresses are generated in the adhesive interface as well as in the composite and in the residual tooth structure

(Reis et al., 2010).

displacement.

(Versluis et al., 2004; Ausiello et al., 2011).

**4.5 Composite and resin cement shrinkage** 

Inlays and onlays are the 2 technical choices for indirect restorations (Fig. 12). Some studies have shown that after endodontic treatment, teeth restored with intracoronal restorations show more severe fracture patterns (Hannig et al., 2005; Soares et al., 2008c). However, it is unclear whether bonded intracoronal restorations should be used for large defects and which material is the most indicated. In this context, Soares et al. (2003) evaluated the cavity preparation influence on the stress distribution of molar teeth restored with esthetic indirect restorations. The stress distribution pattern of the sound tooth was compared to several different extensions of preparation for inlay, onlay and overlay restored with ceramic or ceromer materials. The cavity preparation extension was significant only for onlays covering one cusp and for overlays. Ceramic restorations had higher stress concentrations, while ceromer restorations caused higher stresses in the tooth structure (Soares et al., 2003).

Fig. 12. FEA of different cavity restoration designs for ceramic indirect restorations. A. Intact tooth, B. Inlay restoration; C. Onlay convering buccal cusps; D. Overlay ceramic. (Von mises Stress distribution)

The routine use of metal-free crowns has resulted in an increasing number of fractured restorations (Bello & Jarvis, 1997). Increased fracture resistance of ceramic systems when metal reinforcement was eliminated, has been obtained by the addition of chemical components such as aluminum oxide, leucite, and lithium disilicate (Mak et al., 1997; Drummond et al., 2000). Considering that any restoration has a risk of fracture, the finite element analysis provides a method to evaluate stress distributions in different ceramic systems under occlusal forces. Various studies investigating the performance of ceramic restorations have been performed. Using the finite element analysis method, some investigators (Hubsch et al., 2000; Magne et al., 2002; Magne, 2007; Dejak & Mlotkowski, 2008) demonstrated that ceramic inlays reduced tension at the dentin-adhesive interface and may offer better protection against debonding at the dentin restoration interface, compared with the composite resin inlay. In this context, Reis et al. (2010) investigated, through a 3D finite element analysis, the biomechanical behavior of indirect restored maxillary premolars based on type of preparation (inlay or onlay), and restorative material (composite resin, resin laboratory, reinforced ceramic with lithium disilicate or reinforced ceramic with leucite). Materials with higher modulus of elasticity transfer less stress into the tooth structure. However, materials with modulus of elasticity much larger than the dental structure caused more severe stress concentrations. The models that used reinforced ceramics with leucite showed a behavior that was biomechanically closest to healthy teeth (Reis et al., 2010).

#### **4.5 Composite and resin cement shrinkage**

44 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Inlays and onlays are the 2 technical choices for indirect restorations (Fig. 12). Some studies have shown that after endodontic treatment, teeth restored with intracoronal restorations show more severe fracture patterns (Hannig et al., 2005; Soares et al., 2008c). However, it is unclear whether bonded intracoronal restorations should be used for large defects and which material is the most indicated. In this context, Soares et al. (2003) evaluated the cavity preparation influence on the stress distribution of molar teeth restored with esthetic indirect restorations. The stress distribution pattern of the sound tooth was compared to several different extensions of preparation for inlay, onlay and overlay restored with ceramic or ceromer materials. The cavity preparation extension was significant only for onlays covering one cusp and for overlays. Ceramic restorations had higher stress concentrations, while ceromer restorations caused higher stresses in the tooth structure (Soares et al., 2003).

Fig. 12. FEA of different cavity restoration designs for ceramic indirect restorations. A. Intact tooth, B. Inlay restoration; C. Onlay convering buccal cusps; D. Overlay ceramic. (Von mises

The routine use of metal-free crowns has resulted in an increasing number of fractured restorations (Bello & Jarvis, 1997). Increased fracture resistance of ceramic systems when metal reinforcement was eliminated, has been obtained by the addition of chemical components such as aluminum oxide, leucite, and lithium disilicate (Mak et al., 1997; Drummond et al., 2000). Considering that any restoration has a risk of fracture, the finite element analysis provides a method to evaluate stress distributions in different ceramic systems under occlusal forces. Various studies investigating the performance of ceramic restorations have been performed. Using the finite element analysis method, some investigators (Hubsch et al., 2000; Magne et al., 2002; Magne, 2007; Dejak & Mlotkowski, 2008) demonstrated that ceramic inlays reduced tension at the dentin-adhesive interface and may offer better protection against debonding at the dentin restoration interface, compared with the composite resin inlay. In this context, Reis et al. (2010) investigated, through a 3D finite element analysis, the biomechanical behavior of indirect restored maxillary premolars

Stress distribution)

Resin-composite materials have been widely and increasingly used today in adhesive dental restorative procedures (Fagundes et al., 2009). An important advantage over metallic filling materials is the well-known possibility of bonding the restoration to dental tissues (Marques de Melo et al., 2008) and a significant disadvantage of many of these materials are still the polymerization shrinkage (Pereira et al., 2008). The clinical concern about polymerization shrinkage is evident from the large number of publications and large number of controversial opinions about this topic (Versluis et al., 2004). Shrinkage stress has been associated with various clinical symptoms, including fracture propagation, microleakage and post-operative sensitivity, none of which are direct measures of shrinkage stress. Since stress cannot be measured directly, the presence of shrinkage stresses can only be quantified through indirect manifestations, in particular tooth deformation (Tantbirojn et al., 2004).

Various methods have been used to estimate residual shrinkage stresses, ranging from extrapolated shrinkage or load measurements in vitro to stress analyses in tooth shaped anatomies using photoelastic or finite element methods (Kinomoto et al., 1999; Ausiello et al., 2001). Determination of shrinkage stress is difficult, because it is a transient and nonlinear process. The amount of stress after polymerization therefore depends on the correct description of all changes in mechanical properties and their sequence. Moreover, stress is not a material property or even a structural value, because stress is a threedimensional local tensor (system of related vectors) that is determined by the combination of multiple material properties and local conditions. Since finite element analysis performs its calculation based on such input (mechanical properties, geometry, boundary conditions), it is eminently suitable for studying residual shrinkage stress in dental systems. On the other hand, as the input for especially the mechanical properties remains to be determined more comprehensively, any polymerization shrinkage predicted by finite element analysis should be validated experimentally using indirect factors that can be measured, such as displacement.

Using such validated finite element analyses, shrinkage stresses in restored teeth (enamel and dentin) were found to increase with increasing restoration size, while stresses in the restoration and along the tooth-restoration interface decreased (Versluis et al., 2004). This outcome was explained by the change in tooth stiffness: removal of dental hard tissue decreases the stiffness of the tooth, causing the tooth to be deformed more by the shrinkage stresses (higher stress in the tooth) and causing less resistance to the composite shrinkage (lower stress in the composite). As this example shows, shrinkage stresses are generated in the adhesive interface as well as in the composite and in the residual tooth structure (Versluis et al., 2004; Ausiello et al., 2011).

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 47

rankings and may draw contradictory conclusions about polymerization stress, shrinkage or modulus (Meira et al., 2011). Finite element analysis can help to better understand the test mechanics that cause such divergences among studies. Using an FEA approach, a commonly used test apparatus was simulated with different compliance levels defined by the bonding substrate (steel, glass, composite, or acrylic). The authors showed that when shrinkage and modulus increased simultaneously, stress increased regardless of the substrate. However, if shrinkage and modulus were inversely related, their magnitudes and

Another oral problem with high prevalence, mainly in adults, is periodontal disease. ''Periodontal disease'' is a generic term describing diseases affecting the gums and tissues that support the teeth (Thomson et al., 2004). A periodontal compromised tooth can be diagnosed from probing depth, mobility, supporting bone volume, crown-to-root ratio, and root form (Grossmann & Sadan, 2005). It is generally accepted that a reduction of periodontal support worsens the prognosis of a tooth. However, the morphology of the periodontum with reduced structural support has not been well understood in relation to clinical functions, such as load-bearing capability (Ona & Wakabayashi, 2006). To determine the interaction of reduced periodontal support with mechanical function, one must determine the stress and strain created in the periodontum in accordance with the morphologic alteration of the structures (Ona & Wakabayashi, 2006). Finite element analysis can be used for such assessment, and of the influence of progressive reduction of alveolar support on stress distributions in periodontal structures (Ona & Wakabayashi, 2006). The stress in the periodontum could also predict the potential pain and damage that may occur

Fig. 14. FEA analysis of implant prosthesis demonstrating the stress concentration on the mesial region of the interface between implant and prosthesis. B. FEA analysis of canine

restored with fiber glass post and its effect on bone loss (Roscoe MG, 2010).

interaction with rod material determined the stress response (Meira et al., 2011).

**4.6 Periodontology and implantology** 

under functional bite force (Kawarizadeh et al., 2004).

Restoration placement, techniques are widely recognized as a major factor in the modification of shrinkage stresses. Various techniques, ranging from incremental composite placement to light-exposure regimes, have been advocated to reduce shrinkage stress effects on a restored tooth. Using finite element analysis, it was shown that even during restoration, cavities deform, and thus that incremental application of composite may end up with a higher tooth deformation than a bulk filling (Versluis et al., 1996). Recently the interaction between incremental filling technique, elastic modulus, and post-gel shrinkage of different dental composites was investigated in a restored premolar. Sixteen composites, indicated for restoring posterior teeth, were analyzed. Two incremental techniques, horizontal or oblique, were applied in a finite element model using experimentally determined properties. The calculated shrinkage stress showed a strong correlation with post-gel shrinkage and a weaker correlation was found with elastic modulus. The oblique incremental filling technique resulted in slightly lower residual shrinkage stress along the enamel/composite interface compared to the horizontal technique. However horizontal incremental filling resulted in slightly lower stresses along the dentin/composite interface compared to the oblique technique (Soares et al., 2011). FEA has been used also to analyze the residual shrinkage stress of resin cement used to cement a ceramic inlay, recently we proved that resin cement polymerized immediately after cementation produced significantly more residual stress than when was delayed for 5 minutes after setting ceramic inlay and polymerization (Fig. 13).

Fig. 13. FEA of residual shrinkage stress of resin cement used to cement a ceramic inlay. A. resin cement polymerized immediately after cementation; B. Reduction of shrinkage stress with delay for 5 minutes after setting ceramic inlay and polymerization.

An often used experimental test for measuring shrinkage forces uses a cylindrical composite specimen bonded between two flat surfaces of steel, glass, composite, or acrylic rods. Even for such seemingly simple experimental tests, understanding the outcome can be difficult. Although one may expect that for a specific experimental set-up, differences in the measured force could be attributed to the composite properties, particularly shrinkage and elastic modulus, it was found that the relative ranking of a series of materials was affected by differences in system compliance. As a result, different studies may show different rankings and may draw contradictory conclusions about polymerization stress, shrinkage or modulus (Meira et al., 2011). Finite element analysis can help to better understand the test mechanics that cause such divergences among studies. Using an FEA approach, a commonly used test apparatus was simulated with different compliance levels defined by the bonding substrate (steel, glass, composite, or acrylic). The authors showed that when shrinkage and modulus increased simultaneously, stress increased regardless of the substrate. However, if shrinkage and modulus were inversely related, their magnitudes and interaction with rod material determined the stress response (Meira et al., 2011).

#### **4.6 Periodontology and implantology**

46 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Restoration placement, techniques are widely recognized as a major factor in the modification of shrinkage stresses. Various techniques, ranging from incremental composite placement to light-exposure regimes, have been advocated to reduce shrinkage stress effects on a restored tooth. Using finite element analysis, it was shown that even during restoration, cavities deform, and thus that incremental application of composite may end up with a higher tooth deformation than a bulk filling (Versluis et al., 1996). Recently the interaction between incremental filling technique, elastic modulus, and post-gel shrinkage of different dental composites was investigated in a restored premolar. Sixteen composites, indicated for restoring posterior teeth, were analyzed. Two incremental techniques, horizontal or oblique, were applied in a finite element model using experimentally determined properties. The calculated shrinkage stress showed a strong correlation with post-gel shrinkage and a weaker correlation was found with elastic modulus. The oblique incremental filling technique resulted in slightly lower residual shrinkage stress along the enamel/composite interface compared to the horizontal technique. However horizontal incremental filling resulted in slightly lower stresses along the dentin/composite interface compared to the oblique technique (Soares et al., 2011). FEA has been used also to analyze the residual shrinkage stress of resin cement used to cement a ceramic inlay, recently we proved that resin cement polymerized immediately after cementation produced significantly more residual stress than when was delayed for 5 minutes after setting ceramic inlay and

Fig. 13. FEA of residual shrinkage stress of resin cement used to cement a ceramic inlay. A. resin cement polymerized immediately after cementation; B. Reduction of shrinkage stress

An often used experimental test for measuring shrinkage forces uses a cylindrical composite specimen bonded between two flat surfaces of steel, glass, composite, or acrylic rods. Even for such seemingly simple experimental tests, understanding the outcome can be difficult. Although one may expect that for a specific experimental set-up, differences in the measured force could be attributed to the composite properties, particularly shrinkage and elastic modulus, it was found that the relative ranking of a series of materials was affected by differences in system compliance. As a result, different studies may show different

with delay for 5 minutes after setting ceramic inlay and polymerization.

polymerization (Fig. 13).

Another oral problem with high prevalence, mainly in adults, is periodontal disease. ''Periodontal disease'' is a generic term describing diseases affecting the gums and tissues that support the teeth (Thomson et al., 2004). A periodontal compromised tooth can be diagnosed from probing depth, mobility, supporting bone volume, crown-to-root ratio, and root form (Grossmann & Sadan, 2005). It is generally accepted that a reduction of periodontal support worsens the prognosis of a tooth. However, the morphology of the periodontum with reduced structural support has not been well understood in relation to clinical functions, such as load-bearing capability (Ona & Wakabayashi, 2006). To determine the interaction of reduced periodontal support with mechanical function, one must determine the stress and strain created in the periodontum in accordance with the morphologic alteration of the structures (Ona & Wakabayashi, 2006). Finite element analysis can be used for such assessment, and of the influence of progressive reduction of alveolar support on stress distributions in periodontal structures (Ona & Wakabayashi, 2006). The stress in the periodontum could also predict the potential pain and damage that may occur under functional bite force (Kawarizadeh et al., 2004).

Fig. 14. FEA analysis of implant prosthesis demonstrating the stress concentration on the mesial region of the interface between implant and prosthesis. B. FEA analysis of canine restored with fiber glass post and its effect on bone loss (Roscoe MG, 2010).

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 49

Finite element analysis has also been used to study the biomechanics of tooth movement, which allows accurate assessment of appliance systems and materials without the need to go to animal or other less representative models (Srirekha & Bashetty, 2010). Orthodontic tooth movement is a biomechanical process, because the remodeling processes of the alveolar support structures that result in the tooth movement are triggered by orthodontic forces and moments and their consequences for the stress strain distribution in the periodontium. The redistribution of stresses and strains causes site-specific resorption and formation of the alveolar bone and with it the translation and rotation of the associated tooth (Cattaneo et al., 2009). Finite element analysis can provide insight into the stress and strain distributions around teeth with orthodontic loading to help orthodontists define a loading regime that results in a maximal rate of tooth movement with a minimum of adverse side-effects. The main challenges for the application of finite element analysis in orthodontics has been the definition of the mechanical properties of the periodontal ligament (Toms et al., 2002) and to move beyond the currently most common practice of

It is often commented that finite element analysis is a powerful tool for the interpretation of complex biomechanical systems. Yet, all clinicians and dental researchers are acutely aware of the complexity of oral tissues and their interactions, and hence of the limitations of any theoretical model that depends on input from our incomplete knowledge. The reason why FEA is nonetheless considered such a powerful tool is that it does not need perfect input to be already extremely useful. FEA helps researchers and clinicians formulate the right research questions, design appropriate experiments, and through the underlying universal physics that form the basis of FEA it provides an almost instant insight into complex biomechanical relationships (cause and effect) that cannot be easily obtained or communicated with any other method. The expanded insight and understanding of mechanical responses have undeniably been of direct significance for justifying

As the preceding examples show, finite element analysis not only offers solutions for the engineering problems, but it has been instrumental in the progress in many areas of dentistry. Finite element analysis has improved the understanding of complex processes and has assisted researchers and clinicians in designing better procedures to maintain oral health. Finite element simulation provides unique advantages for dental research, such as its precision and its ability to solve complex biomechanical problems for which other research methods are too cumbersome or even impossible (Ersoz, 2000). Finite element simulation allows more comprehensive prediction and analysis of medical processes or treatments because in a process where many variables need to be considered, it allows for manipulation of single parameters, making it possible to isolate and study the influence of each parameter with more precision (Sun et al., 2008). Thanks to the highly graphic pre- and post-processing features, finite element analysis has also brought researchers and clinicians closer together. It can be argued that without such visualization, stress and strain development would remain mostly academic. The visual interface has improved the communication and collaboration between clinical and research expertise, and is likely to have had a significant

**4.8 Summary: How FE analysis contributes to improve oral health** 

experimental questions and improving clinical treatments.

impact on the current state of the art in dentistry.

static finite element models.

Historically, periodontal disease is one of the main causes of tooth loss (Deng et al., 2010). Traditionally, patients with severe periodontitis have ultimately had all teeth removed due to severe alveolar bone resorption and high risks for systemic infections (Deng et al., 2010). In this context implant therapy has been applied successfully for three decades, and proven to be a successful means for oral rehabilitation (Albrektsson et al., 1986). The knowledge of physiologic values of alveolar stresses provides a guideline reference for the design of dental implants and it is also important for the understanding of stress-related bone remodeling and osseointegration (Srirekha & Bashetty, 2010). Stiffness of the tissue-implant interface and implant-supporting tissues is considered the main determinant factor in osseointegration (Ramp & Jeffcoat, 2001; Turkyilmaz et al., 2009). Finite element analysis has been used extensively in the field of implant research over the past 2 decades (Geng et al., 2001). It has been used to investigate the impact of implant geometry (Himmlova et al., 2004), material properties of implants (Yang & Xiang, 2007), quality of implant-supporting tissues (Petrie & Williams, 2007), fixture-prosthesis connections (Akca et al., 2003), and of implant loading conditions (Natali et al., 2006).

#### **4.7 Trauma and orthodontics**

Beyond caries and periodontal disease, orofacial trauma is also considered a public health problem (Ferrari & Ferreria de Mederios, 2002). Finite element analysis has also been widely used for dental trauma analysis (Huang et al., 2005). In the real world traumatic injuries to teeth typically result from a dynamic force (Huang et al., 2005). Therefore, for traumatic analysis of a tooth, it has been recommended to simulate time-dependent behavior and analyze different rates of loading (Natali et al., 2004). Finite element analysis can provide insight into the process of impact stresses and fracture propagation in teeth subjected to dynamic impact loads in various directions.

Fig. 15. FEA analysis of orthodontic intrusion movement of a maxillary canine.

Historically, periodontal disease is one of the main causes of tooth loss (Deng et al., 2010). Traditionally, patients with severe periodontitis have ultimately had all teeth removed due to severe alveolar bone resorption and high risks for systemic infections (Deng et al., 2010). In this context implant therapy has been applied successfully for three decades, and proven to be a successful means for oral rehabilitation (Albrektsson et al., 1986). The knowledge of physiologic values of alveolar stresses provides a guideline reference for the design of dental implants and it is also important for the understanding of stress-related bone remodeling and osseointegration (Srirekha & Bashetty, 2010). Stiffness of the tissue-implant interface and implant-supporting tissues is considered the main determinant factor in osseointegration (Ramp & Jeffcoat, 2001; Turkyilmaz et al., 2009). Finite element analysis has been used extensively in the field of implant research over the past 2 decades (Geng et al., 2001). It has been used to investigate the impact of implant geometry (Himmlova et al., 2004), material properties of implants (Yang & Xiang, 2007), quality of implant-supporting tissues (Petrie & Williams, 2007), fixture-prosthesis connections (Akca et al., 2003), and of

Beyond caries and periodontal disease, orofacial trauma is also considered a public health problem (Ferrari & Ferreria de Mederios, 2002). Finite element analysis has also been widely used for dental trauma analysis (Huang et al., 2005). In the real world traumatic injuries to teeth typically result from a dynamic force (Huang et al., 2005). Therefore, for traumatic analysis of a tooth, it has been recommended to simulate time-dependent behavior and analyze different rates of loading (Natali et al., 2004). Finite element analysis can provide insight into the process of impact stresses and fracture propagation in teeth subjected to

Fig. 15. FEA analysis of orthodontic intrusion movement of a maxillary canine.

implant loading conditions (Natali et al., 2006).

dynamic impact loads in various directions.

**4.7 Trauma and orthodontics** 

Finite element analysis has also been used to study the biomechanics of tooth movement, which allows accurate assessment of appliance systems and materials without the need to go to animal or other less representative models (Srirekha & Bashetty, 2010). Orthodontic tooth movement is a biomechanical process, because the remodeling processes of the alveolar support structures that result in the tooth movement are triggered by orthodontic forces and moments and their consequences for the stress strain distribution in the periodontium. The redistribution of stresses and strains causes site-specific resorption and formation of the alveolar bone and with it the translation and rotation of the associated tooth (Cattaneo et al., 2009). Finite element analysis can provide insight into the stress and strain distributions around teeth with orthodontic loading to help orthodontists define a loading regime that results in a maximal rate of tooth movement with a minimum of adverse side-effects. The main challenges for the application of finite element analysis in orthodontics has been the definition of the mechanical properties of the periodontal ligament (Toms et al., 2002) and to move beyond the currently most common practice of static finite element models.

#### **4.8 Summary: How FE analysis contributes to improve oral health**

It is often commented that finite element analysis is a powerful tool for the interpretation of complex biomechanical systems. Yet, all clinicians and dental researchers are acutely aware of the complexity of oral tissues and their interactions, and hence of the limitations of any theoretical model that depends on input from our incomplete knowledge. The reason why FEA is nonetheless considered such a powerful tool is that it does not need perfect input to be already extremely useful. FEA helps researchers and clinicians formulate the right research questions, design appropriate experiments, and through the underlying universal physics that form the basis of FEA it provides an almost instant insight into complex biomechanical relationships (cause and effect) that cannot be easily obtained or communicated with any other method. The expanded insight and understanding of mechanical responses have undeniably been of direct significance for justifying experimental questions and improving clinical treatments.

As the preceding examples show, finite element analysis not only offers solutions for the engineering problems, but it has been instrumental in the progress in many areas of dentistry. Finite element analysis has improved the understanding of complex processes and has assisted researchers and clinicians in designing better procedures to maintain oral health. Finite element simulation provides unique advantages for dental research, such as its precision and its ability to solve complex biomechanical problems for which other research methods are too cumbersome or even impossible (Ersoz, 2000). Finite element simulation allows more comprehensive prediction and analysis of medical processes or treatments because in a process where many variables need to be considered, it allows for manipulation of single parameters, making it possible to isolate and study the influence of each parameter with more precision (Sun et al., 2008). Thanks to the highly graphic pre- and post-processing features, finite element analysis has also brought researchers and clinicians closer together. It can be argued that without such visualization, stress and strain development would remain mostly academic. The visual interface has improved the communication and collaboration between clinical and research expertise, and is likely to have had a significant impact on the current state of the art in dentistry.

Finite Element Analysis in Dentistry – Improving the Quality of Oral Health Care 51

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Finite element analysis is not perfect. But we should not expect our theoretical models to be perfect because our understanding of dental properties and processes is still developing. Finite element analysis, however, will continue to improve along with our own understanding about reality. Such continuous improvement will happen as long as we keep comparing reality with theory, and use the insight we gain from these comparisons for improving the theory. The past decades have shown how finite element simulation, which is an expression of our theoretical understanding of biomechanics, has moved from mainly static and linear conditions to more dynamic or transient and nonlinear conditions (Wakabayashi et al., 2008; Srirekha & Bashetty, 2010), thus reflecting the gains that were made in dental science with support from finite element analysis.

#### **5. Acknowledgment**

The authors are indebted to financial support granted by FAPEMIG and CNPq.

#### **6. References**


Finite element analysis is not perfect. But we should not expect our theoretical models to be perfect because our understanding of dental properties and processes is still developing. Finite element analysis, however, will continue to improve along with our own understanding about reality. Such continuous improvement will happen as long as we keep comparing reality with theory, and use the insight we gain from these comparisons for improving the theory. The past decades have shown how finite element simulation, which is an expression of our theoretical understanding of biomechanics, has moved from mainly static and linear conditions to more dynamic or transient and nonlinear conditions (Wakabayashi et al., 2008; Srirekha & Bashetty, 2010), thus reflecting the gains that were

made in dental science with support from finite element analysis.

The authors are indebted to financial support granted by FAPEMIG and CNPq.

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**3**

*Brazil* 

**FEA in Dentistry: A Useful Tool to**

**of Implant Supported Prosthesis** 

*Univ Estadual Paulista (UNESP), Aracatuba Dental School, Univ of Sao Paulo (USP), Dental School of Ribeirao Preto,* 

**Investigate the Biomechanical Behavior** 

Wirley Gonçalves Assunção, Valentim Adelino Ricardo Barão,

Érica Alves Gomes, Juliana Aparecida Delben and Ricardo Faria Ribeiro

The use of dental implants is widespread and has been successfully applied to replace missing teeth (Amoroso et al., 2006). Although high success rate has been reported by several clinical studies, early or late dental implants failures are still inevitable. During mastication, overstress around dental implants may cause bone resorption, which leads to infection on the peri-implant region and failure of oral rehabilitation (Kopp, 1990). The way in which bone is loaded may influence its response (Koca et al., 2005). The results of cyclic loading into the bone differ from those of static loading (Papavasiliou et al., 1996). In case of repetitive cyclic load application, stress microfractures in bone may occur (Koca et al., 2005) and may induce osteoclastic activity to remove the damaged bone (Papavasiliou et al., 1996). So far, it is imperative to understand where the maximum stresses occur during mastication

Considering that stress/strain distribution at bone level is hard to be clinically assessed, the finite element analysis (FEA) has been extensively used in Dentistry to understand the biomechanical behavior of implant-supported prosthesis. To date, FEA was first used in the Implant Dentistry field by Weinstein et al. (1976) to evaluate the stress distribution of porous rooted dental implants. Nowadays, owing to the geometric complexity of implantbone-prosthesis system, FEA has been viewed as a suitable tool for analyzing stress distribution into this system and to predict its performance clinically. Such analysis has the advantage of allowing several conditions to be changed easily and allows measurement of stress distribution around implants at optional points that are difficult to be clinically

Therefore, this chapter provides the current status of using FEA to investigate the biomechanical behavior of implant-supported prosthesis. The modeling of complex structures that represents the oral cavity is described, and comparisons between twodimensional (2D) and three-dimensional (3D) modeling techniques are discussed. Additionally, the application of microcomputer tomography to develop complex and more

realistic FE models are assessed. Some sensitive cases are also illustrated.

around the implants in order to avoid these complications (Nagasao et al., 2003).

**1. Introduction** 

examined.


### **FEA in Dentistry: A Useful Tool to Investigate the Biomechanical Behavior of Implant Supported Prosthesis**

Wirley Gonçalves Assunção, Valentim Adelino Ricardo Barão, Érica Alves Gomes, Juliana Aparecida Delben and Ricardo Faria Ribeiro *Univ Estadual Paulista (UNESP), Aracatuba Dental School, Univ of Sao Paulo (USP), Dental School of Ribeirao Preto, Brazil* 

#### **1. Introduction**

56 Finite Element Analysis – From Biomedical Applications to Industrial Developments

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Wang R, Weiner S. Human root dentin: Structural anisotropy and Vickers microhardness

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Witzel MF, Ballester RY, Meira JB, Lima RG, Braga RR. Composite shrinkage stress as a

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post and root canal wall by finite element simulation. Int Endod J. 2006;39(12):959-

The use of dental implants is widespread and has been successfully applied to replace missing teeth (Amoroso et al., 2006). Although high success rate has been reported by several clinical studies, early or late dental implants failures are still inevitable. During mastication, overstress around dental implants may cause bone resorption, which leads to infection on the peri-implant region and failure of oral rehabilitation (Kopp, 1990). The way in which bone is loaded may influence its response (Koca et al., 2005). The results of cyclic loading into the bone differ from those of static loading (Papavasiliou et al., 1996). In case of repetitive cyclic load application, stress microfractures in bone may occur (Koca et al., 2005) and may induce osteoclastic activity to remove the damaged bone (Papavasiliou et al., 1996). So far, it is imperative to understand where the maximum stresses occur during mastication around the implants in order to avoid these complications (Nagasao et al., 2003).

Considering that stress/strain distribution at bone level is hard to be clinically assessed, the finite element analysis (FEA) has been extensively used in Dentistry to understand the biomechanical behavior of implant-supported prosthesis. To date, FEA was first used in the Implant Dentistry field by Weinstein et al. (1976) to evaluate the stress distribution of porous rooted dental implants. Nowadays, owing to the geometric complexity of implantbone-prosthesis system, FEA has been viewed as a suitable tool for analyzing stress distribution into this system and to predict its performance clinically. Such analysis has the advantage of allowing several conditions to be changed easily and allows measurement of stress distribution around implants at optional points that are difficult to be clinically examined.

Therefore, this chapter provides the current status of using FEA to investigate the biomechanical behavior of implant-supported prosthesis. The modeling of complex structures that represents the oral cavity is described, and comparisons between twodimensional (2D) and three-dimensional (3D) modeling techniques are discussed. Additionally, the application of microcomputer tomography to develop complex and more realistic FE models are assessed. Some sensitive cases are also illustrated.

FEA in Dentistry: A Useful Tool to Investigate

blue (lowest) to red (highest).

(Fig. 2).

the Biomechanical Behavior of Implant Supported Prosthesis 59

the stress decreased at the supporting tissues as mucosa thickness and resiliency increased

Fig. 1. First principal stress distribution (in MPa). (a) conventional complete denture. (b) overdenture – bar-clip system. (c) overdenture – o'ring system). (d) overdenture – barclip associated with distally placed o'ring system. Colors indicate level of stress from dark

Fig. 2. Distribution of first principal stress (MPa) in supporting tissues for groups BC (barclip) and BC-C (bar-clip associated with two-distally placed o'rings) considering different

mucosa thickness (1, 3 and 5mm) and resilience (hard, resilient and soft).

#### **2. Biomechanical behavior of implant-supported prosthesis**

In order to enhance treatment longevity, it is important to understand the biomechanics of implant-supported prosthesis during masticatory loading. And the way that the stress/strain is transmitted and distributed to the bone tissue dictates whether the implant treatments will failure or succeed (Geng et al., 2001). Several variables affect the stress/strain distribution on the implant/bone complex such as prosthesis type, implant type, veneering and framework materials, bone quality, and presence of misfit.

#### **2.1 Prosthesis and implant types**

The implant-supported prosthesis can be classified as single- or multi- unit prosthesis. From a biomechanical point of view, the multi-unit prosthesis is subdivided into implantsupported overdentures and implant-supported fixed prosthesis (cantilevered design or not). The nature of FEA studies for these prosthesis designs is much more complex than for single-unit design (Geng et al., 2001).

Implant-retained overdentures are considered a simple, cost-effective, viable, less invasive and successful treatment option for edentulous patients (Assuncao et al., 2008; Barao et al., 2009). However, controversies toward the design of attachment systems for overdentures still exist (Bilhan et al., 2011). Our previous study (Barao et al., 2009) used a 2D FEA to investigate the effect of different designs of attachment systems on the stress distribution of implant-retained mandibular overdentures. The bar-clip attachment system showed the greatest stress values followed by bar-clip associated with two distally placed o'ring attachment systems, and o'ring attachment system (Fig. 1). Other 2D (Meijer et al., 1992) and 3D FEA studies (Menicucci et al., 1998) also showed stress optimization in overdenture with unsplinted implants (e.g. o'ring attachment system). The flexibility and resiliency provided by the o'ring rubber and the spacer in the o'ring system assembly may be the driven force toward the lower stress values with o'ring attachment system. Additionally, the stress breaking effect of the o'ring rubber can also decrease the stress in implants, prosthetics components and supporting tissues (Tokuhisa et al., 2003).

Tanino et al. (2007) evaluated the effect of stress-breaking attachments at the connections between maxillary palateless overdentures and implants using 3D models with two and four implants. Stress-breaking materials (with elastic modulus ranging from 1 to 3,000 MPa) connecting the implants and denture were included around each abutment. As the elastic modulus of the stress-breaking materials increased, the stress increased at the implant-bone interface and decreased at the cortical bone surface. Additionally, the 3-mm-thick stressbreaking material decreased the stress values at the implant-bone interface when compared to the 1-mm-thick material. Knowing that overdentures are retained by implants but are still supported by the mucosa, and facing the difference in displacement between implants (20- 30 µm) and soft tissue (about 500 µm), our previous study (Barao et al., 2008) investigated the influence of different mucosa thickness and resiliency on stress distribution of implantretained overdentures using a 2D FEA. Two models were designed: two-splinted-implants connected with bar-clip system and two-splinted-implants connected with bar-clip system associated with two-distally placed o'ring system. For each design, mucosa assumed three characteristics of thickness (1, 3 and 5 mm) varying its resiliencies (based on its Young's modulus) in hard (680 MPa), resilient (340 MPa) and soft (1 MPa), respectively. In general,

In order to enhance treatment longevity, it is important to understand the biomechanics of implant-supported prosthesis during masticatory loading. And the way that the stress/strain is transmitted and distributed to the bone tissue dictates whether the implant treatments will failure or succeed (Geng et al., 2001). Several variables affect the stress/strain distribution on the implant/bone complex such as prosthesis type, implant

The implant-supported prosthesis can be classified as single- or multi- unit prosthesis. From a biomechanical point of view, the multi-unit prosthesis is subdivided into implantsupported overdentures and implant-supported fixed prosthesis (cantilevered design or not). The nature of FEA studies for these prosthesis designs is much more complex than for

Implant-retained overdentures are considered a simple, cost-effective, viable, less invasive and successful treatment option for edentulous patients (Assuncao et al., 2008; Barao et al., 2009). However, controversies toward the design of attachment systems for overdentures still exist (Bilhan et al., 2011). Our previous study (Barao et al., 2009) used a 2D FEA to investigate the effect of different designs of attachment systems on the stress distribution of implant-retained mandibular overdentures. The bar-clip attachment system showed the greatest stress values followed by bar-clip associated with two distally placed o'ring attachment systems, and o'ring attachment system (Fig. 1). Other 2D (Meijer et al., 1992) and 3D FEA studies (Menicucci et al., 1998) also showed stress optimization in overdenture with unsplinted implants (e.g. o'ring attachment system). The flexibility and resiliency provided by the o'ring rubber and the spacer in the o'ring system assembly may be the driven force toward the lower stress values with o'ring attachment system. Additionally, the stress breaking effect of the o'ring rubber can also decrease the stress in implants, prosthetics

Tanino et al. (2007) evaluated the effect of stress-breaking attachments at the connections between maxillary palateless overdentures and implants using 3D models with two and four implants. Stress-breaking materials (with elastic modulus ranging from 1 to 3,000 MPa) connecting the implants and denture were included around each abutment. As the elastic modulus of the stress-breaking materials increased, the stress increased at the implant-bone interface and decreased at the cortical bone surface. Additionally, the 3-mm-thick stressbreaking material decreased the stress values at the implant-bone interface when compared to the 1-mm-thick material. Knowing that overdentures are retained by implants but are still supported by the mucosa, and facing the difference in displacement between implants (20- 30 µm) and soft tissue (about 500 µm), our previous study (Barao et al., 2008) investigated the influence of different mucosa thickness and resiliency on stress distribution of implantretained overdentures using a 2D FEA. Two models were designed: two-splinted-implants connected with bar-clip system and two-splinted-implants connected with bar-clip system associated with two-distally placed o'ring system. For each design, mucosa assumed three characteristics of thickness (1, 3 and 5 mm) varying its resiliencies (based on its Young's modulus) in hard (680 MPa), resilient (340 MPa) and soft (1 MPa), respectively. In general,

**2. Biomechanical behavior of implant-supported prosthesis** 

**2.1 Prosthesis and implant types** 

single-unit design (Geng et al., 2001).

components and supporting tissues (Tokuhisa et al., 2003).

type, veneering and framework materials, bone quality, and presence of misfit.

the stress decreased at the supporting tissues as mucosa thickness and resiliency increased (Fig. 2).

Fig. 1. First principal stress distribution (in MPa). (a) conventional complete denture. (b) overdenture – bar-clip system. (c) overdenture – o'ring system). (d) overdenture – barclip associated with distally placed o'ring system. Colors indicate level of stress from dark blue (lowest) to red (highest).

Fig. 2. Distribution of first principal stress (MPa) in supporting tissues for groups BC (barclip) and BC-C (bar-clip associated with two-distally placed o'rings) considering different mucosa thickness (1, 3 and 5mm) and resilience (hard, resilient and soft).

FEA in Dentistry: A Useful Tool to Investigate

(Fig. 3), mucosa and implant/prosthetic components.

the Biomechanical Behavior of Implant Supported Prosthesis 61

minimum (σmin) (compressive) principal stresses (in MPa) were obtained. BC-C group exhibited the highest stress values (σvM = 398.8, σmax = 580.5 and σmin = -455.2) while FD group showed the lowest one (σvM = 128.9, σmax = 185.9 and σmin = -172.1) in the implant/prosthetic components. Within overdenture groups, the use of unsplinted implants (OR group) reduced the stress level in the implant/prosthetic components (59.4% for σvM, 66.2% for σmax and 57.7% for σmin versus BC-C group) and supporting tissues (maximum stress reduction of 72% and 79.5% for σmax, and 15.7% and 85.7% for σmin on the cortical bone and the trabecular bone, respectively). The cortical bone exhibited greater stress concentration than the trabecular bone for all groups. We concluded that the use of fixed implant dentures and removable dentures retained by unsplinted implants to rehabilitate completely edentulous mandible reduced the stresses in the peri-implant cortical bone tissue

Fig. 3. von Mises stress (σvM), maximum (σmax) and minimum (σmin) principal stress distributions (in MPa) within cortical bone for o'ring (OR), bar-clip (BC), bar-clip with

Concerning the implant design, Ding et al. (2009) analyzed the stress distribution around immediately loaded implants of different diameters (3.3; 4., and 4.8 mm) using an accurate complete mandible model. The authors observed that with the increase of implant diameter, stress/strain on the implant-bone interface decreased, mainly when the diameter increased from 3.3 to 4.1 mm for both axial and oblique loading conditions. Other studies also showed more favorable stress distribution with the use of wide-diameter implants (Himmlova et al., 2004; Matsushita et al., 1990). Huang et al. (2008) analyzed the peri-implant bone stress and the implant-bone sliding as affected by different implant designs and implant sizes of immediately loaded implant with maxillary sinus augmentation. Twenty-four 3D FE models with four implant designs (cylindrical, threaded, stepped and step-thread implants) and three dimensions (standard, long and wide threaded implants) with a bonded and three levels of frictional contact of implant-bone interfaces were analyzed. The use of threaded implants decreased the bone stress and sliding distance about 30% as compared with non-

distally placed cantilever (BC-C) and fixed denture (FD) groups.

In relation to the implant-supported fixed prosthesis, the variety of factors that affect the stress distribution into the bone-implant complex comprise implant inclination, implant number and position, framework/veneering material properties, and cross-sectional design of the framework (Geng et al., 2001). The use of tilted implants mostly affected the stress concentration in the peri-implant bone tissue when compared to vertical implants (Canay et al., 1996). However, tilted implants have been used in case of atrophic jaw, to avoid maxillary sinus, and to reduce the cantilever extension (Silva et al., 2010). Caglar et al. (2006) investigated the effects of mesiodistal inclination of implants on the stress distribution of posterior maxillary implant-supported fixed prosthesis using a 3D FEA. Inclination of the implant in the molar region resulted in increased stress. Similar results were found by a Iplikcioglu & Akca (2002) who investigated the effect of buccolingual inclination in implantsupported fixed prosthesis applied to the posterior mandibular region using a 3D FEA. Bevilacqua et al. (2011) investigated the influence of cantilever length (13, 9, 5 and 0 mm) and implant inclination (0, 15, 30 and 45 degrees) on stress distribution in maxillary fixed dentures. This 3D FEA study showed that tilted implants, with consequent reduction of the posterior cantilevers, reduced the stress values in the peri-implant cortical bone.

Zarone et al. (2003) evaluated the relative deformations and stress distributions in six different designs of full-arch implant-supported fixed mandibular denture (six or four implants, cantilevered designed or not, cross-arch or midline-divided bar into two freestanding bridges) by means of 3D FEA. When the implants were rigidly connected by onepiece framework, the free bending of the mandible was hindered. The flexibility of the mandible was increased as the more distal implant supports were more mesially located. The use of two free-standing bars also reduced the overall stress on the bone/implant interface, fixtures and superstructure. Contradicting these findings, Yokoyama et al. (2005) observed that the use of single-unit superstructure was more effective in relining stress concentration in the edentulous mandibular bone than 3-unit superstructure. Other study (Silva et al., 2010), using a 3D FEA, assessed the biomechanical behavior of the "All-on-four" system with that of six-implant-supported maxillary prosthesis with tilted implants. The stress values were greater to the "All-on-four" concept, and the presence of cantilever increased the stress values about 100% in both models.

It is believed that loading distribution pattern in implant-retained overdentures differs from those in implant-supported fixed restorations (Tokuhisa et al., 2003). Our ongoing project has compared the effect of different designs of implant-retained overdentures and fixed fullarch implant-supported prosthesis on stress distribution in edentulous mandible by using a 3D-FEA based on a computerized tomography (CT). Four 3D FE models of an edentulous human mandible with mucosa and four implants placed in the interforamina area were constructed and restored with different designs of dentures. In the OR group, the mandible was restored with an overdenture retained by four unsplinted implants with O'ring attachment; in the BC-C and BC groups, the mandibles were restored with overdentures retained by four splinted implants with bar-clip anchor associated or not with two distally placed cantilevers, respectively; in the FD group, the mandible was restored with a fixed full-arch four-implant-supported prosthesis. The masticatory muscles and temporomandibular joints supported the models. A 100-N oblique load (30 degrees) was applied on the left first molar of each denture in a buccolingual direction. Qualitative and quantitative analysis based on the von Mises stress (σvM), the maximum (σmax) (tensile) and

In relation to the implant-supported fixed prosthesis, the variety of factors that affect the stress distribution into the bone-implant complex comprise implant inclination, implant number and position, framework/veneering material properties, and cross-sectional design of the framework (Geng et al., 2001). The use of tilted implants mostly affected the stress concentration in the peri-implant bone tissue when compared to vertical implants (Canay et al., 1996). However, tilted implants have been used in case of atrophic jaw, to avoid maxillary sinus, and to reduce the cantilever extension (Silva et al., 2010). Caglar et al. (2006) investigated the effects of mesiodistal inclination of implants on the stress distribution of posterior maxillary implant-supported fixed prosthesis using a 3D FEA. Inclination of the implant in the molar region resulted in increased stress. Similar results were found by a Iplikcioglu & Akca (2002) who investigated the effect of buccolingual inclination in implantsupported fixed prosthesis applied to the posterior mandibular region using a 3D FEA. Bevilacqua et al. (2011) investigated the influence of cantilever length (13, 9, 5 and 0 mm) and implant inclination (0, 15, 30 and 45 degrees) on stress distribution in maxillary fixed dentures. This 3D FEA study showed that tilted implants, with consequent reduction of the

posterior cantilevers, reduced the stress values in the peri-implant cortical bone.

increased the stress values about 100% in both models.

Zarone et al. (2003) evaluated the relative deformations and stress distributions in six different designs of full-arch implant-supported fixed mandibular denture (six or four implants, cantilevered designed or not, cross-arch or midline-divided bar into two freestanding bridges) by means of 3D FEA. When the implants were rigidly connected by onepiece framework, the free bending of the mandible was hindered. The flexibility of the mandible was increased as the more distal implant supports were more mesially located. The use of two free-standing bars also reduced the overall stress on the bone/implant interface, fixtures and superstructure. Contradicting these findings, Yokoyama et al. (2005) observed that the use of single-unit superstructure was more effective in relining stress concentration in the edentulous mandibular bone than 3-unit superstructure. Other study (Silva et al., 2010), using a 3D FEA, assessed the biomechanical behavior of the "All-on-four" system with that of six-implant-supported maxillary prosthesis with tilted implants. The stress values were greater to the "All-on-four" concept, and the presence of cantilever

It is believed that loading distribution pattern in implant-retained overdentures differs from those in implant-supported fixed restorations (Tokuhisa et al., 2003). Our ongoing project has compared the effect of different designs of implant-retained overdentures and fixed fullarch implant-supported prosthesis on stress distribution in edentulous mandible by using a 3D-FEA based on a computerized tomography (CT). Four 3D FE models of an edentulous human mandible with mucosa and four implants placed in the interforamina area were constructed and restored with different designs of dentures. In the OR group, the mandible was restored with an overdenture retained by four unsplinted implants with O'ring attachment; in the BC-C and BC groups, the mandibles were restored with overdentures retained by four splinted implants with bar-clip anchor associated or not with two distally placed cantilevers, respectively; in the FD group, the mandible was restored with a fixed full-arch four-implant-supported prosthesis. The masticatory muscles and temporomandibular joints supported the models. A 100-N oblique load (30 degrees) was applied on the left first molar of each denture in a buccolingual direction. Qualitative and quantitative analysis based on the von Mises stress (σvM), the maximum (σmax) (tensile) and minimum (σmin) (compressive) principal stresses (in MPa) were obtained. BC-C group exhibited the highest stress values (σvM = 398.8, σmax = 580.5 and σmin = -455.2) while FD group showed the lowest one (σvM = 128.9, σmax = 185.9 and σmin = -172.1) in the implant/prosthetic components. Within overdenture groups, the use of unsplinted implants (OR group) reduced the stress level in the implant/prosthetic components (59.4% for σvM, 66.2% for σmax and 57.7% for σmin versus BC-C group) and supporting tissues (maximum stress reduction of 72% and 79.5% for σmax, and 15.7% and 85.7% for σmin on the cortical bone and the trabecular bone, respectively). The cortical bone exhibited greater stress concentration than the trabecular bone for all groups. We concluded that the use of fixed implant dentures and removable dentures retained by unsplinted implants to rehabilitate completely edentulous mandible reduced the stresses in the peri-implant cortical bone tissue (Fig. 3), mucosa and implant/prosthetic components.

Fig. 3. von Mises stress (σvM), maximum (σmax) and minimum (σmin) principal stress distributions (in MPa) within cortical bone for o'ring (OR), bar-clip (BC), bar-clip with distally placed cantilever (BC-C) and fixed denture (FD) groups.

Concerning the implant design, Ding et al. (2009) analyzed the stress distribution around immediately loaded implants of different diameters (3.3; 4., and 4.8 mm) using an accurate complete mandible model. The authors observed that with the increase of implant diameter, stress/strain on the implant-bone interface decreased, mainly when the diameter increased from 3.3 to 4.1 mm for both axial and oblique loading conditions. Other studies also showed more favorable stress distribution with the use of wide-diameter implants (Himmlova et al., 2004; Matsushita et al., 1990). Huang et al. (2008) analyzed the peri-implant bone stress and the implant-bone sliding as affected by different implant designs and implant sizes of immediately loaded implant with maxillary sinus augmentation. Twenty-four 3D FE models with four implant designs (cylindrical, threaded, stepped and step-thread implants) and three dimensions (standard, long and wide threaded implants) with a bonded and three levels of frictional contact of implant-bone interfaces were analyzed. The use of threaded implants decreased the bone stress and sliding distance about 30% as compared with non-

FEA in Dentistry: A Useful Tool to Investigate

same group (P<.05, Student's t-test ).

the Biomechanical Behavior of Implant Supported Prosthesis 63

Fig. 4. Detorque mean value (N.cm) before and after mechanical cycling for all groups. Within each period, mean followed by different letters represent statistically significant difference (P<.05, Fisher's exact test ).\*denotes statistically significant difference within the

Fig. 5. von Mises stress distribution (MPa) within supporting bone for GP, GR, TP, TR and

ZP groups. Colors indicate level of stress from dark blue (lowest) to red (highest).

threaded (cylindrical and stepped) implants. With the increase of implant's length or diameter, the bone stress reduced around 13-26%. The immediately loaded implant with smooth machine surface increased the bone stress by 28-63% versus osseointegrated implants. The increase of implant's surface roughness did not reduce the bone stress but decrease the implant-bone interfacial sliding.

#### **2.2 Veneering and framework material**

The literature is scarce about the best material to fabricate superstructures of implantsupported prosthesis (Gomes et al., 2011). Originally, the protocol consisted of gold alloy framework and acrylic resin for denture base and acrylic resin or composite resin for artificial denture teeth (Zarb & Jansson, 1985 ). Rigid occlusal material such as porcelain on metal may increase the load transfer to the implant and surrounding bone tissue (Skalak, 1983). So far, the use of occlusal veneering based on resin material is indicated to absorb shock and consequently to reduce the stress on the implant-bone complex (Skalak, 1983). Gracis et al. (1991) stated that the use of harder and stiffer materials to fabricated implantsupported restorations increased the stress transmitted to the implant. On the other hand, some studies (Ciftci & Canay, 2001; Sertgoz, 1997) showed that the use of softer restorative materials lead to a higher stress on implants and supporting tissues.

Our previous studies (Delben et al., 2011; Gomes et al., 2011) evaluated the influence of different superstructures on preload maintenance of retention screw of single implantsupported crowns submitted to mechanical cycling and stress distribution through 3D FEA.

Twelve replicas for each group and 3D FEA models were created to simulate a single crown supported by external hexagon implant in premolar region. Five groups were obtained: gold abutment veneered with ceramic (GC) and resin (GR), titanium abutment veneered with ceramic (TC) and resin (TR), and zirconia abutment veneered with ceramic (ZC). During mechanical cycling, the replicas were submitted to dynamic vertical loading of 50 N at 2 Hz for detorque measurement after each period of 1x105 cycles up to 1x106 cycles. The FEA software generated the stress maps after vertical loading of 100 N on the contact points of the crowns. Significant difference (P<.05) between group TC (21.4 ± 1.78) and groups GC (23.9 ± 0,91), GR (24.1 ± 1.34) and TR (23.2 ± 1.33); and between group ZC (21,9 ± 2,68) and groups GC and GR for initial detorque mean (in N.cm) was noted. After mechanical cycling, there was significant difference (P<.05) between groups GR (23.8 ± 1.56) and TC (22.1 ± 1.86), and between group ZC (21.7 ± 2.02) and groups GR and TR (23.6 ± 1.30) (Fig. 4). The stress values and distribution in bone tissue were similar for groups GC, GR, TC and ZC (1574.3 MPa, 1574.3 MPa, 1574.3 MPa and 1574.2 MPa, respectively), except for group TR (1838.3 MPa) (Fig. 5). Group ZC transferred lower stress to the retention screw (785 MPa) than the other groups (939 MPa for GC, 961 MPa for GR, 1010 MPa for TC, and 1037 MPa for TR) We concluded that detorque reduction occurred for all superstructure materials but torque maintenance was enough to maintain joint stability in this study. The different materials did not affect stress distribution in bone. However, group ZC presented the best stress distribution for the retention screw. Previous study conducted by our research group also found similar stress distribution to single implant-supported prosthesis regardless of the type of veneering/framework material through a 2D FEA (Assuncao et al., 2010).

threaded (cylindrical and stepped) implants. With the increase of implant's length or diameter, the bone stress reduced around 13-26%. The immediately loaded implant with smooth machine surface increased the bone stress by 28-63% versus osseointegrated implants. The increase of implant's surface roughness did not reduce the bone stress but

The literature is scarce about the best material to fabricate superstructures of implantsupported prosthesis (Gomes et al., 2011). Originally, the protocol consisted of gold alloy framework and acrylic resin for denture base and acrylic resin or composite resin for artificial denture teeth (Zarb & Jansson, 1985 ). Rigid occlusal material such as porcelain on metal may increase the load transfer to the implant and surrounding bone tissue (Skalak, 1983). So far, the use of occlusal veneering based on resin material is indicated to absorb shock and consequently to reduce the stress on the implant-bone complex (Skalak, 1983). Gracis et al. (1991) stated that the use of harder and stiffer materials to fabricated implantsupported restorations increased the stress transmitted to the implant. On the other hand, some studies (Ciftci & Canay, 2001; Sertgoz, 1997) showed that the use of softer restorative

Our previous studies (Delben et al., 2011; Gomes et al., 2011) evaluated the influence of different superstructures on preload maintenance of retention screw of single implantsupported crowns submitted to mechanical cycling and stress distribution through 3D FEA. Twelve replicas for each group and 3D FEA models were created to simulate a single crown supported by external hexagon implant in premolar region. Five groups were obtained: gold abutment veneered with ceramic (GC) and resin (GR), titanium abutment veneered with ceramic (TC) and resin (TR), and zirconia abutment veneered with ceramic (ZC). During mechanical cycling, the replicas were submitted to dynamic vertical loading of 50 N at 2 Hz for detorque measurement after each period of 1x105 cycles up to 1x106 cycles. The FEA software generated the stress maps after vertical loading of 100 N on the contact points of the crowns. Significant difference (P<.05) between group TC (21.4 ± 1.78) and groups GC (23.9 ± 0,91), GR (24.1 ± 1.34) and TR (23.2 ± 1.33); and between group ZC (21,9 ± 2,68) and groups GC and GR for initial detorque mean (in N.cm) was noted. After mechanical cycling, there was significant difference (P<.05) between groups GR (23.8 ± 1.56) and TC (22.1 ± 1.86), and between group ZC (21.7 ± 2.02) and groups GR and TR (23.6 ± 1.30) (Fig. 4). The stress values and distribution in bone tissue were similar for groups GC, GR, TC and ZC (1574.3 MPa, 1574.3 MPa, 1574.3 MPa and 1574.2 MPa, respectively), except for group TR (1838.3 MPa) (Fig. 5). Group ZC transferred lower stress to the retention screw (785 MPa) than the other groups (939 MPa for GC, 961 MPa for GR, 1010 MPa for TC, and 1037 MPa for TR) We concluded that detorque reduction occurred for all superstructure materials but torque maintenance was enough to maintain joint stability in this study. The different materials did not affect stress distribution in bone. However, group ZC presented the best stress distribution for the retention screw. Previous study conducted by our research group also found similar stress distribution to single implant-supported prosthesis regardless of the

type of veneering/framework material through a 2D FEA (Assuncao et al., 2010).

materials lead to a higher stress on implants and supporting tissues.

decrease the implant-bone interfacial sliding.

**2.2 Veneering and framework material** 

Fig. 4. Detorque mean value (N.cm) before and after mechanical cycling for all groups. Within each period, mean followed by different letters represent statistically significant difference (P<.05, Fisher's exact test ).\*denotes statistically significant difference within the same group (P<.05, Student's t-test ).

Fig. 5. von Mises stress distribution (MPa) within supporting bone for GP, GR, TP, TR and ZP groups. Colors indicate level of stress from dark blue (lowest) to red (highest).

FEA in Dentistry: A Useful Tool to Investigate

misfit added stress to the implant hexagon.

the Biomechanical Behavior of Implant Supported Prosthesis 65

implant fracture) (Carlson & Carlsson, 1994; Dellinges & Tebrock, 1993) and biological (i.e. sensorial disturbances, soft tissue injuries, peri-implantitis, and bone loss) (Berglundh et al., 2002) complications of the implant treatment, a passive fit between the crown and implant should be achieved (Sahin & Cehreli, 2001). Previous studies (Assuncao et al., 2011; Kunavisarut et al., 2002; Natali et al., 2006) have showed an increase of stress on peri-

Our previous study (Assuncao et al., 2011) used a 3D FEA to investigate the effect of vertical and angular misfit in three-piece implant-supported screwed crown on the biomechanical behavior of peri-implant bone, implants, and prosthetic components. A total of four 3D models were fabricated to represent a posterior mandibular section with one implant in the region of the second premolar and another in the region of the second molar. The implants were splinted by a three-piece implant-supported metal-ceramic prosthesis and differed according to the type of misfit, as represented by four different models: control - prosthesis with complete fit to the implants; unilateral angular misfit - prosthesis presenting unilateral angular misfit of 100 μm in the mesial region of the second molar; unilateral vertical misfit prosthesis presenting unilateral vertical misfit of 100 μm in the mesial region of the second molar; and total vertical misfit - prosthesis presenting total vertical misfit of 100 μm in the platform of the framework in the second molar (Fig. 6). A vertical load of 400 N was distributed and applied on 12 centric points (a vertical load of 150 N was applied to each molar in the prosthesis and a vertical load of 100 N was applied at the second premolar). We observed that stress on the peri-implant cortical bone was slightly affected by the presence of misfit. Each type of misfit overloaded a specific region of the implant-supported system. The unilateral angular misfit was most harmful for the implant body and retention screw, the unilateral vertical misfit placed the most stress on the framework, and the total vertical

implant bone tissue under the presence of misfit in implant-supported prostheses.

Fig. 6. Mesh of the main model: cortical bone, trabecular bone, implant and crown

Another study conducted by our group (Gomes et al., 2009) assessed the influence misfit on the displacement and stress distribution in the bone-implant-prosthesis complex using a 2D FEA. A single-unit mandibular implant-supported prosthesis was fabricated. Different

(framework, veneering material and retention screw).

#### **2.3 Bone quality**

The bone quality is strongly correlated with the implant success as pointed out by several longitudinal clinical studies (Friberg et al., 1991; Jemt & Lekholm, 1995; van Steenberghe et al., 1990). The biomechanical behavior among the different types of bone (I, II, III or IV) differs substantially, which affect the ability of bone to support physiological loads (de Almeida et al., 2010). The poor quality bone type 4 has promoted greater failures of dental implants (Jaffin et al., 2004) owing to its reduced capability to bond the implant to the bone (Drage et al., 2007; Shapurian et al., 2006).

de Almeida et al. (2010) investigated the influence of different types of bone (types I to IV) on the stress distribution on the supporting tissue of a fixed full-arch implant-supported mandibular prosthesis based on a prefabricated bar by using a 3D FEA. Three unilateral posterior loads of 150 N were applied on the prosthesis: L1 – axial loading; L2 – oblique loading (buccolingual direction, 30 degrees); L3 – oblique loading (linguobuccal direction, 30 degrees). Type III and IV bones displayed the greatest stress values in the axial and buccolingual loading conditions, while stiffer bones (type I and II) exhibited the lowest. For the linguobuccal loading condition, the poorest quality cortical bone (type IV) had the highest stress concentration followed by types III, II and I.

Tada et al. (2003) evaluated whether bone quality affect the stress/strain distribution of single-unit implant-supported mandibular prosthesis with different implant type and length. Screw and cylinder implants with 9.2, 10.8, 12.4 and 14.0 mm length were used and virtually placed in 4 types of bone. Two different loads (axial and buccolingual forces) were applied to the occlusal surface at the center of the abutment. As the bone density decreased, the stress/strain into the bone increased. Under axial loading, the stress in the cancellous bone was lower with the screw-type implant when compared with cylinder-type implant. Additionally, longer implants displayed lower stress values. The bone quality also influences the stress distribution under buccolingual load. According to the authors, lowdensity bone presents reduced stiffness, which increases implant displacement. Under greater displacement, the bone is deformed and consequently higher stresses in the cortical and cancellous bone are expected.

Another 3D FEA study (Sevimay et al., 2005) examined the effect of the bone quality on stress distribution for an implant-supported mandibular crown. A 3D FE model of a mandibular section of bone with a missing second premolar and an implant to receive a crown restoration was used. A total vertical force of 300 N was applied from the buccal cusp (150 N) and distal fossa (150 N) in centric occlusion. Low-density bone (types III and IV) displayed the greatest stress values (163 and 180 MPa, respectively) mainly at the periimplant cortical bone. On the other hand, type I and II bones exhibited the lowest levels of stresses (150 and 152 MPa, respectively). Other similar study (Holmes & Loftus, 1997) found that the placement of implants in type I bone resulted in less micromotion and reduced stress concentration.

#### **2.4 Presence of misfit**

An increase of clinical failures has been correlated with misfit of implant-supported prostheses (Klineberg & Murray, 1985; Skalak, 1983). In order to prevent mechanical (i.e. retention screw and abutment screw loosening and fracture, superstructure mobility, and

The bone quality is strongly correlated with the implant success as pointed out by several longitudinal clinical studies (Friberg et al., 1991; Jemt & Lekholm, 1995; van Steenberghe et al., 1990). The biomechanical behavior among the different types of bone (I, II, III or IV) differs substantially, which affect the ability of bone to support physiological loads (de Almeida et al., 2010). The poor quality bone type 4 has promoted greater failures of dental implants (Jaffin et al., 2004) owing to its reduced capability to bond the implant to the bone

de Almeida et al. (2010) investigated the influence of different types of bone (types I to IV) on the stress distribution on the supporting tissue of a fixed full-arch implant-supported mandibular prosthesis based on a prefabricated bar by using a 3D FEA. Three unilateral posterior loads of 150 N were applied on the prosthesis: L1 – axial loading; L2 – oblique loading (buccolingual direction, 30 degrees); L3 – oblique loading (linguobuccal direction, 30 degrees). Type III and IV bones displayed the greatest stress values in the axial and buccolingual loading conditions, while stiffer bones (type I and II) exhibited the lowest. For the linguobuccal loading condition, the poorest quality cortical bone (type IV) had the

Tada et al. (2003) evaluated whether bone quality affect the stress/strain distribution of single-unit implant-supported mandibular prosthesis with different implant type and length. Screw and cylinder implants with 9.2, 10.8, 12.4 and 14.0 mm length were used and virtually placed in 4 types of bone. Two different loads (axial and buccolingual forces) were applied to the occlusal surface at the center of the abutment. As the bone density decreased, the stress/strain into the bone increased. Under axial loading, the stress in the cancellous bone was lower with the screw-type implant when compared with cylinder-type implant. Additionally, longer implants displayed lower stress values. The bone quality also influences the stress distribution under buccolingual load. According to the authors, lowdensity bone presents reduced stiffness, which increases implant displacement. Under greater displacement, the bone is deformed and consequently higher stresses in the cortical

Another 3D FEA study (Sevimay et al., 2005) examined the effect of the bone quality on stress distribution for an implant-supported mandibular crown. A 3D FE model of a mandibular section of bone with a missing second premolar and an implant to receive a crown restoration was used. A total vertical force of 300 N was applied from the buccal cusp (150 N) and distal fossa (150 N) in centric occlusion. Low-density bone (types III and IV) displayed the greatest stress values (163 and 180 MPa, respectively) mainly at the periimplant cortical bone. On the other hand, type I and II bones exhibited the lowest levels of stresses (150 and 152 MPa, respectively). Other similar study (Holmes & Loftus, 1997) found that the placement of implants in type I bone resulted in less micromotion and reduced

An increase of clinical failures has been correlated with misfit of implant-supported prostheses (Klineberg & Murray, 1985; Skalak, 1983). In order to prevent mechanical (i.e. retention screw and abutment screw loosening and fracture, superstructure mobility, and

**2.3 Bone quality** 

(Drage et al., 2007; Shapurian et al., 2006).

and cancellous bone are expected.

stress concentration.

**2.4 Presence of misfit** 

highest stress concentration followed by types III, II and I.

implant fracture) (Carlson & Carlsson, 1994; Dellinges & Tebrock, 1993) and biological (i.e. sensorial disturbances, soft tissue injuries, peri-implantitis, and bone loss) (Berglundh et al., 2002) complications of the implant treatment, a passive fit between the crown and implant should be achieved (Sahin & Cehreli, 2001). Previous studies (Assuncao et al., 2011; Kunavisarut et al., 2002; Natali et al., 2006) have showed an increase of stress on periimplant bone tissue under the presence of misfit in implant-supported prostheses.

Our previous study (Assuncao et al., 2011) used a 3D FEA to investigate the effect of vertical and angular misfit in three-piece implant-supported screwed crown on the biomechanical behavior of peri-implant bone, implants, and prosthetic components. A total of four 3D models were fabricated to represent a posterior mandibular section with one implant in the region of the second premolar and another in the region of the second molar. The implants were splinted by a three-piece implant-supported metal-ceramic prosthesis and differed according to the type of misfit, as represented by four different models: control - prosthesis with complete fit to the implants; unilateral angular misfit - prosthesis presenting unilateral angular misfit of 100 μm in the mesial region of the second molar; unilateral vertical misfit prosthesis presenting unilateral vertical misfit of 100 μm in the mesial region of the second molar; and total vertical misfit - prosthesis presenting total vertical misfit of 100 μm in the platform of the framework in the second molar (Fig. 6). A vertical load of 400 N was distributed and applied on 12 centric points (a vertical load of 150 N was applied to each molar in the prosthesis and a vertical load of 100 N was applied at the second premolar). We observed that stress on the peri-implant cortical bone was slightly affected by the presence of misfit. Each type of misfit overloaded a specific region of the implant-supported system. The unilateral angular misfit was most harmful for the implant body and retention screw, the unilateral vertical misfit placed the most stress on the framework, and the total vertical misfit added stress to the implant hexagon.

Fig. 6. Mesh of the main model: cortical bone, trabecular bone, implant and crown (framework, veneering material and retention screw).

Another study conducted by our group (Gomes et al., 2009) assessed the influence misfit on the displacement and stress distribution in the bone-implant-prosthesis complex using a 2D FEA. A single-unit mandibular implant-supported prosthesis was fabricated. Different

FEA in Dentistry: A Useful Tool to Investigate

presence of friction coefficient.

similar.

such as the finite element method.

the Biomechanical Behavior of Implant Supported Prosthesis 67

conditions of implant-bone contact (bonded interface and non-bonded interface – friction coefficient of 0.3) on peri-implant bone stress distribution and implant-bone interfacial sliding of different implant designs and implant sizes. The use of friction coefficient to represent the immediate loading condition of the implants increased the bone stress by 28- 63% when compared with the osseointegrated condition (bonded contact) for all implant designs and sizes. The interfacial sliding between bone and implant decreased with the

Thread configuration is an important objective in biomechanical optimization of dental implants (Valen & Locante, 2000). However, several 2D and 3D FEA studies have not considered the threads modeling of implants and retention screws or have modeled them using only concentric rings owing to the difficulty in modeling the thread helix (Sertgoz, 1997). Additionally, some justify the use of simplified model due to the difficulty in constructing a 3D complex model and the enormous increase in element numbers (Assuncao et al., 2009). As the oversimplication of implant complex geometry may affect the results of several FEA studies (Al-Sukhun et al., 2007), some authors (Lang et al., 2003; Sakaguchi & Borgersen, 1995) believe that the modeling of the perfect geometry of the implant, including the thread helix of the screw and the screw bore, is essential to finite element analysis simulation. Therefore, our previous study (Assuncao et al., 2009) investigated whether or not the representation of implant's threads would affect the outcome of a 2D FEA. Two models reproducing a frontal section of edentulous mandibular posterior bone were constructed. In the first models, the implants threads were accurately simulated (precise model) and, on the other implants with a smooth surface (press-fit implant) were used (simplified model). A load of 133 N was obliquely (30 degrees) applied with on the models. Precise model (1,45 MPa) showed higher maximum stress values than simplified model (1,2 MPa). Stress distribution and stress values in the cortical bone (292.95 MPa for precise model and 401.14 MPa for simplified model) and trabecular bone (19.35 MPa for precise model and 20.35 MPa for simplified) were similar, and the stresses were mostly located around implant neck and implant apex. We concluded that considering implant and screw analysis, remarkable differences in stress values were found between the models. Although models have showed differences on absolute stress values, the stress distribution was

**4. Two-dimensional (2D) versus three-dimensional (3D) analysis** 

models. Khera et al. (1988) were pioneers on 3D modeling for human mandible.

The biomechanical performance of mandibular and maxillary bones associated to other natural (teeth, periodontal ligament, gingiva, etc.) and artificial (prostheses, dental implants, etc.) structures has been considered clinically and biologically relevant for modern Dentistry. In this sense, several tools have been developed for biomechanical evaluations,

The first studies reported in the literature using the Finite Element Method in Dentistry presented simple 2D models (Takahashi et al., 1978; Weinstein et al., 1979; Yettram et al., 1976). However, the complexity of the oral environment required the development of 3D

Although the technological advancement allowed the use of accurate and fast software and hardware to obtain the images, the decision about 2D or 3D modeling for FEA remains

unilateral angular misfit (0 µm – control, 50 µm, 100 µm, and 200 µm) was represented on the contact region between the implant and the crown. An oblique (30 degrees) load of 133 N was applied at the opposite direction of misfit on the models. The greater the angular misfit, the higher the stress and displacement values in the bone-implant-prosthesis assembly. On the other hand, Spazzin et al. (2011) investigated the effect of different levels of vertical misfit (5, 25, 50, 100, 200 and 300 µm) between implant and bar framework in overdenture and showed that the presence of misfit did not influence the stress level at the peri-implant bone tissue, but stress on the prosthetic components (bar framework, retention screw) and implants increased with greater misfit levels.

Spazzin et al. (2011) also assessed the influence of horizontal misfit (10, 50, 100 and 200 µm) and bar framework material (gold alloy, silver-palladium alloy, commercially pure titanium and cobalt-chromium alloy) on the stress distribution in implant-retained mandibular overdenture associated with bar attachment system using a 3D FEA. The increase in horizontal misfit promoted an enhancement of stress levels in the inferior region of the bar, retention screw neck, cervical and medium third of the implant, and peri-implant cortical bone. The stiffer the bar material was, the greater the stress on the framework.

#### **3. Modeling complex structures**

As the oral cavity is very complex in nature, it is very hard to represent this structure with high accuracy by means FE models. For this reason, several simplifications are necessary to our reality. Most of the FEA studies in Dental field consider the materials as isotropic, homogeneous and linearly elastic. The modeling of biological tissues (e.g. bone) is a very difficult task because of their inherent heterogeneous and anisotropic character (Cowin & editor, 2001 ). The use of isotropic properties instead of anisotropic properties for bone tissue may affect the overall results of stress distribution (O'Mahony et al., 2001). In addition, the ultimate strain and Young modulus of bone under compression is different than those under tension (Geng et al., 2001). A previous study (Liao et al., 2008) investigated in what extend the anisotropic elastic properties affect the stress and strain distribution around implants under physiologic load in a complete mandible model based on CT. Models were loaded obliquely, and the principal stress and strain values in the peri-implant bone tissue were recorded. The authors observed an increase of up to 70% of stress/strain values for the anisotropic model versus isotropic model. O'Mahony et al. (2001) compared implant-bone interface stresses and peri-implant principal strains in anisotropic versus isotropic 3D models of the posterior mandible. Anisotropy increased by 20 to 30% the stress and strain in the cortical bone when compared with isotropic case. In the trabecular bone, the anisotropy enhanced by 3- to 4-fold the stress level versus isotropic condition. So far, anisotropy has significant effects on peri-implant stress and strain; therefore, careful consideration should be given to its use in biomechanical FE studies (Liao et al., 2008; O'Mahony et al., 2001).

The bone-implant contact in most of FEA studies is considered at a 100%; however, in the clinical scenario the implant-bone contact ranges from 30% to 70% (Geng et al., 2001). Additionally, the bone is not homogeneous and porosities are presented. Nowadays, it is possible to insert contact algorithms to simulate contacts (friction coefficients). In Dentistry, this factor is very important because it allows the representation of different degrees of osseointegration and the scenario of immediate loading. Huang et al. (2008) compared two

unilateral angular misfit (0 µm – control, 50 µm, 100 µm, and 200 µm) was represented on the contact region between the implant and the crown. An oblique (30 degrees) load of 133 N was applied at the opposite direction of misfit on the models. The greater the angular misfit, the higher the stress and displacement values in the bone-implant-prosthesis assembly. On the other hand, Spazzin et al. (2011) investigated the effect of different levels of vertical misfit (5, 25, 50, 100, 200 and 300 µm) between implant and bar framework in overdenture and showed that the presence of misfit did not influence the stress level at the peri-implant bone tissue, but stress on the prosthetic components (bar framework, retention

Spazzin et al. (2011) also assessed the influence of horizontal misfit (10, 50, 100 and 200 µm) and bar framework material (gold alloy, silver-palladium alloy, commercially pure titanium and cobalt-chromium alloy) on the stress distribution in implant-retained mandibular overdenture associated with bar attachment system using a 3D FEA. The increase in horizontal misfit promoted an enhancement of stress levels in the inferior region of the bar, retention screw neck, cervical and medium third of the implant, and peri-implant cortical

As the oral cavity is very complex in nature, it is very hard to represent this structure with high accuracy by means FE models. For this reason, several simplifications are necessary to our reality. Most of the FEA studies in Dental field consider the materials as isotropic, homogeneous and linearly elastic. The modeling of biological tissues (e.g. bone) is a very difficult task because of their inherent heterogeneous and anisotropic character (Cowin & editor, 2001 ). The use of isotropic properties instead of anisotropic properties for bone tissue may affect the overall results of stress distribution (O'Mahony et al., 2001). In addition, the ultimate strain and Young modulus of bone under compression is different than those under tension (Geng et al., 2001). A previous study (Liao et al., 2008) investigated in what extend the anisotropic elastic properties affect the stress and strain distribution around implants under physiologic load in a complete mandible model based on CT. Models were loaded obliquely, and the principal stress and strain values in the peri-implant bone tissue were recorded. The authors observed an increase of up to 70% of stress/strain values for the anisotropic model versus isotropic model. O'Mahony et al. (2001) compared implant-bone interface stresses and peri-implant principal strains in anisotropic versus isotropic 3D models of the posterior mandible. Anisotropy increased by 20 to 30% the stress and strain in the cortical bone when compared with isotropic case. In the trabecular bone, the anisotropy enhanced by 3- to 4-fold the stress level versus isotropic condition. So far, anisotropy has significant effects on peri-implant stress and strain; therefore, careful consideration should be given to its use in biomechanical FE studies (Liao et al., 2008;

The bone-implant contact in most of FEA studies is considered at a 100%; however, in the clinical scenario the implant-bone contact ranges from 30% to 70% (Geng et al., 2001). Additionally, the bone is not homogeneous and porosities are presented. Nowadays, it is possible to insert contact algorithms to simulate contacts (friction coefficients). In Dentistry, this factor is very important because it allows the representation of different degrees of osseointegration and the scenario of immediate loading. Huang et al. (2008) compared two

bone. The stiffer the bar material was, the greater the stress on the framework.

screw) and implants increased with greater misfit levels.

**3. Modeling complex structures** 

O'Mahony et al., 2001).

conditions of implant-bone contact (bonded interface and non-bonded interface – friction coefficient of 0.3) on peri-implant bone stress distribution and implant-bone interfacial sliding of different implant designs and implant sizes. The use of friction coefficient to represent the immediate loading condition of the implants increased the bone stress by 28- 63% when compared with the osseointegrated condition (bonded contact) for all implant designs and sizes. The interfacial sliding between bone and implant decreased with the presence of friction coefficient.

Thread configuration is an important objective in biomechanical optimization of dental implants (Valen & Locante, 2000). However, several 2D and 3D FEA studies have not considered the threads modeling of implants and retention screws or have modeled them using only concentric rings owing to the difficulty in modeling the thread helix (Sertgoz, 1997). Additionally, some justify the use of simplified model due to the difficulty in constructing a 3D complex model and the enormous increase in element numbers (Assuncao et al., 2009). As the oversimplication of implant complex geometry may affect the results of several FEA studies (Al-Sukhun et al., 2007), some authors (Lang et al., 2003; Sakaguchi & Borgersen, 1995) believe that the modeling of the perfect geometry of the implant, including the thread helix of the screw and the screw bore, is essential to finite element analysis simulation. Therefore, our previous study (Assuncao et al., 2009) investigated whether or not the representation of implant's threads would affect the outcome of a 2D FEA. Two models reproducing a frontal section of edentulous mandibular posterior bone were constructed. In the first models, the implants threads were accurately simulated (precise model) and, on the other implants with a smooth surface (press-fit implant) were used (simplified model). A load of 133 N was obliquely (30 degrees) applied with on the models. Precise model (1,45 MPa) showed higher maximum stress values than simplified model (1,2 MPa). Stress distribution and stress values in the cortical bone (292.95 MPa for precise model and 401.14 MPa for simplified model) and trabecular bone (19.35 MPa for precise model and 20.35 MPa for simplified) were similar, and the stresses were mostly located around implant neck and implant apex. We concluded that considering implant and screw analysis, remarkable differences in stress values were found between the models. Although models have showed differences on absolute stress values, the stress distribution was similar.

#### **4. Two-dimensional (2D) versus three-dimensional (3D) analysis**

The biomechanical performance of mandibular and maxillary bones associated to other natural (teeth, periodontal ligament, gingiva, etc.) and artificial (prostheses, dental implants, etc.) structures has been considered clinically and biologically relevant for modern Dentistry. In this sense, several tools have been developed for biomechanical evaluations, such as the finite element method.

The first studies reported in the literature using the Finite Element Method in Dentistry presented simple 2D models (Takahashi et al., 1978; Weinstein et al., 1979; Yettram et al., 1976). However, the complexity of the oral environment required the development of 3D models. Khera et al. (1988) were pioneers on 3D modeling for human mandible.

Although the technological advancement allowed the use of accurate and fast software and hardware to obtain the images, the decision about 2D or 3D modeling for FEA remains

FEA in Dentistry: A Useful Tool to Investigate

**4.1 Case sensitive** 

in a faster way.

the Biomechanical Behavior of Implant Supported Prosthesis 69

Two- and 3- D models were constructed to evaluate the behavior of different veneering materials of single implant-supported prostheses (Assuncao et al., 2010; Gomes et al., 2011). For both models, five types of finite element (FE) models were simulated according to the different framework (gold alloy, titanium, and zirconia) and veneering (porcelain and modified composite resin) materials. However, several differences between the models representation were introduced according to the limitations of each modeling process. First of all, it was observed differences about the geometry, mainly for the superstructure (veneering material and framework) and bone tissue (Fig. 8). In the 2D model, the superstructure was modeled with 8-mm in height and 8-mm in diameter and the surrounding bone model assumed the characteristic of a block. In the 3D model, the superstructure was modeled based on the characteristics of a left first premolar tooth and the bone tissue reproduced a segment of the maxilla with a missing left first premolar tooth. In relation to the loading, the load was applied at a 30-degree inclination and 2-mm off-axis in 2D simulation while a 100-N vertical force was applied to the contact points of the crowns in the 3D model. Additionally, the 3D model represented a contact element between the abutment and the implant to simulate the clinical situation. At the end, it was observed difference between the 2D and 3D models for both qualitative and quantitative analyses.

Fig. 8. Differences about the geometry representation of the superstructure and bone tissue for single implant-supported prostheses: 2D and 3D model from left to right, respectively.

Thus, the selection between 2D or 3D modeling should be guided by the researcher knowledge about both methods. The 3D models are similar to real structures but are timeconsuming for modeling and data processing even when powerful computers are used. The higher the model complexity, the higher the number of elements and the complexity of the analysis. Thus, a simple geometry may generate an accurate mesh with satisfactory results

Two-dimensional models offer excellent access for pre- and post-processing, and because of the reduced dimensions, computational capacity can be preserved for improvements in element and simulation quality. On the other hand, 3D models, although more realistic with

uncertain. It is important to understand that the biomechanical performance of complex oral structures depends on several factors. Therefore, the accuracy of the results may be influenced by: 1- complex geometry of the model; 2 – materials properties, such as isotropy or anisotropy; 3 – type, size and quantity of elements in the mesh; 4 – boundary and loading conditions similar to clinical scenario; and 5 – analysis mode (static or dynamic). In addition, the choice between 2D or 3D models should be guided by the expectances and applicability of the results. Thus, the researcher should understand the advantages and limitations of both modeling types (Romeed et al., 2006).

The 2D modeling has been continually applied in Dentistry since it is a simple, fast and lowcost approach. On the other hand, the 3D model is more accurate and may represent the details of a real condition (Fig. 7). However, a complex model is not worthy if it is misinterpreted considering that the higher the complexity of the model, the higher the density of the elements mesh. Thus, the following question should be answered before starting a FEA study "How can I represent the model accurately to obtain results within the ideal parameters?"

Fig. 7. Different types of FE implant models: Two-dimensional model and Threedimensional model from left to right, respectively.

Several studies (Poiate et al., 2011; Romeed et al., 2006) were conducted to compare the different models applied from Engineering to Dentistry to verify the effect of 2D or 3D modeling on the analysis of the biomechanical performance of complex structures. Romeed et al. (2006) evaluated the mechanical behavior of a second maxillary premolar restored with full-coverage crown under different occlusal schemes and observed similar stress distribution and minor differences for the stress values. On the other hand, Poiate et al. (2011) compared the biomechanical performance of a maxillary central incisor between 2D and 3D modeling and concluded that 2D models can be safely applied only for qualitative analysis since these models showed overestimated values for the quantitative analysis of the stress. The differences between 2D and 3D models have been attributed to the geometric representation of the model. Although 2D models are simplified and easier to be obtained than 3D models, the biaxial state may influence the reliability of the results since some important biomechanical aspects may be not reproduced (Gao et al., 2006).

#### **4.1 Case sensitive**

68 Finite Element Analysis – From Biomedical Applications to Industrial Developments

uncertain. It is important to understand that the biomechanical performance of complex oral structures depends on several factors. Therefore, the accuracy of the results may be influenced by: 1- complex geometry of the model; 2 – materials properties, such as isotropy or anisotropy; 3 – type, size and quantity of elements in the mesh; 4 – boundary and loading conditions similar to clinical scenario; and 5 – analysis mode (static or dynamic). In addition, the choice between 2D or 3D models should be guided by the expectances and applicability of the results. Thus, the researcher should understand the advantages and limitations of

The 2D modeling has been continually applied in Dentistry since it is a simple, fast and lowcost approach. On the other hand, the 3D model is more accurate and may represent the details of a real condition (Fig. 7). However, a complex model is not worthy if it is misinterpreted considering that the higher the complexity of the model, the higher the density of the elements mesh. Thus, the following question should be answered before starting a FEA study "How can I represent the model accurately to obtain results within the

Fig. 7. Different types of FE implant models: Two-dimensional model and Three-

important biomechanical aspects may be not reproduced (Gao et al., 2006).

Several studies (Poiate et al., 2011; Romeed et al., 2006) were conducted to compare the different models applied from Engineering to Dentistry to verify the effect of 2D or 3D modeling on the analysis of the biomechanical performance of complex structures. Romeed et al. (2006) evaluated the mechanical behavior of a second maxillary premolar restored with full-coverage crown under different occlusal schemes and observed similar stress distribution and minor differences for the stress values. On the other hand, Poiate et al. (2011) compared the biomechanical performance of a maxillary central incisor between 2D and 3D modeling and concluded that 2D models can be safely applied only for qualitative analysis since these models showed overestimated values for the quantitative analysis of the stress. The differences between 2D and 3D models have been attributed to the geometric representation of the model. Although 2D models are simplified and easier to be obtained than 3D models, the biaxial state may influence the reliability of the results since some

dimensional model from left to right, respectively.

both modeling types (Romeed et al., 2006).

ideal parameters?"

Two- and 3- D models were constructed to evaluate the behavior of different veneering materials of single implant-supported prostheses (Assuncao et al., 2010; Gomes et al., 2011). For both models, five types of finite element (FE) models were simulated according to the different framework (gold alloy, titanium, and zirconia) and veneering (porcelain and modified composite resin) materials. However, several differences between the models representation were introduced according to the limitations of each modeling process. First of all, it was observed differences about the geometry, mainly for the superstructure (veneering material and framework) and bone tissue (Fig. 8). In the 2D model, the superstructure was modeled with 8-mm in height and 8-mm in diameter and the surrounding bone model assumed the characteristic of a block. In the 3D model, the superstructure was modeled based on the characteristics of a left first premolar tooth and the bone tissue reproduced a segment of the maxilla with a missing left first premolar tooth. In relation to the loading, the load was applied at a 30-degree inclination and 2-mm off-axis in 2D simulation while a 100-N vertical force was applied to the contact points of the crowns in the 3D model. Additionally, the 3D model represented a contact element between the abutment and the implant to simulate the clinical situation. At the end, it was observed difference between the 2D and 3D models for both qualitative and quantitative analyses.

Fig. 8. Differences about the geometry representation of the superstructure and bone tissue for single implant-supported prostheses: 2D and 3D model from left to right, respectively.

Thus, the selection between 2D or 3D modeling should be guided by the researcher knowledge about both methods. The 3D models are similar to real structures but are timeconsuming for modeling and data processing even when powerful computers are used. The higher the model complexity, the higher the number of elements and the complexity of the analysis. Thus, a simple geometry may generate an accurate mesh with satisfactory results in a faster way.

Two-dimensional models offer excellent access for pre- and post-processing, and because of the reduced dimensions, computational capacity can be preserved for improvements in element and simulation quality. On the other hand, 3D models, although more realistic with

FEA in Dentistry: A Useful Tool to Investigate

reproduction of images and results.

**5.1 Converting CT images into FEA models** 

10) at the *ScanIP* segment.

Brazil), and ITK Snap (General Public License, USA).

models.

the Biomechanical Behavior of Implant Supported Prosthesis 71

However, some caution should be taken when these data are used. The X-ray images in grey scale must be recorded by a CT in DICOM (Digital Imaging Communications in Medicine) format. Thus, the procedure should be carefully indicated for the patients due to the exposition to radiation and the research project should be approved by the ethics committee ensuring that physical and geometrical parameters within safety limits will be adopted. Furthermore, the patient should sign an informed consent form to authorize the

The processing techniques used to extract this information from the CT data may be also frequently affected by no negligible errors that propagate in an unknown way through the various steps of the model generation, affecting the accuracy of the model (Taddei et al., 2006). The first source of geometric error and distortion is the resolution of the dataset that depends on the scan parameters setting (Taddei et al., 2006). The ones that yielded the best results for image quality were obtained in the regime of 120 kV, 150 mA, 512 × 512 matrix, 14 cm × 14 cm field of view, and slice thickness of 0.5 mm (Poiate et al., 2011). The second error may result from the segmentation process of the region of interest. Several segmentation algorithms have been proposed, with various level of automation, starting from complete manual contours extraction to complex fully automatic algorithms (Taddei et al., 2006). Considering that several softwares are currently available in the market, the professional should be trained to accurately use the tools to convert CT images into FE

Some steps to convert CT images into solid models will be presented in this section considering the availability of softwares, such as Mimics (Materialise, Leuven, Belgium), Simpleware (Simpleware Ltd, United Kingdom), InVesalius (Brazilian Public Software,

The use of Simpleware software to convert a maxillary CT images into a maxillary FEA model will be discussed. After submission by the Ethic Committee in Research of Dental School of Ribeirão Preto, University of São Paulo (Process CAAE – 0038.0.138.000-11 and SISNEP – FR – 430209), the CT images obtained from patient imaging with 219 cross sections were imported into the Simpleware 4.1 software (Simpleware Ltd, United Kingdom) (Fig.

In the *ScanIP*, the segmentation tool was used to identify the pixels value of the tomographic image. Then, it was possible to separate an object from other adjacent anatomical structures in different masks, such as cortical bone and trabecular bone (Fig. 11). According to its radio-density, expressed in Hounsfield unities, the program picked the values up and automatically created different masks. After that, fine adjustments were executed to further improve the quality of the model masks (Fig. 12). The program also provides tools to eliminate any interference of the tomographic image. By examining the image, a certain level of noise in the data could be corrected by filters. After obtaining the masks for the cortical and trabecular bone, the soft tissue was constructed with 2.0 mm in thickness for the whole model using a morphological filter tool in structuring element. Then, the step to

convert the CT image into a solid model was completed (Figs. 13 and 14).

respect to the dimensional properties, are generally more coarse, with elements that are far from their ideal shapes. Moreover, examination of the model is more difficult. Depending on the investigated structure and boundary conditions, 2D modeling may be justified as a reasonable or even sensible simplification (Korioth & Versluis, 1997). Additionally, combinations of 2D or 3D FEA may offer the best understanding of the biomechanical behavior of complex dental structures in certain situations (Romeed et al., 2006).

#### **5. Microcomputer tomography in FE models**

In the beginning, the FE models were obtained from sectional images of bone tissue, tooth, surrounding structures and other elements related with the study that would be executed. Khera et al. (1988) constructed a 3D human mandible based on a 2D model. Initially, a 2D model was obtained and, using the projection of several pictures in a magnifying monitor, a 3D model was generated and an axial *z*-axis was defined.

Another widely used technique to obtain 2D and 3D models is the embedment of structures in acrylic resin. Gomes et al. (2009) embedded a prosthesis/retention screw/implant system in resin and sectioned it longitudinally to investigate the effect of different levels of unilateral angular misfit prostheses in the assembly and surrounding bone using a 2D FEA. The embedded model was scanned to produce digitalized images that were imported into CAD image analysis software and placed within the supporting tissue based on literature data. The outline of the images was manually quoted and each point was converted into x and y coordinates. At the end, the coordinates were finally imported into the FE software as key points of the final images (Fig. 9).

Fig. 9. Scanned embedded model and finite element model.

Recently, microcomputer tomography (CT) images have been obtained as a useful tool to model the bone complex in FE models and have gained general consensus among researches. The 3D model simulated from CT images provides high fidelity to the anatomical dimensions and configuration of all oral structures because it is possible to define the geometry and the local tissue properties of the bone segment to be modeled.

respect to the dimensional properties, are generally more coarse, with elements that are far from their ideal shapes. Moreover, examination of the model is more difficult. Depending on the investigated structure and boundary conditions, 2D modeling may be justified as a reasonable or even sensible simplification (Korioth & Versluis, 1997). Additionally, combinations of 2D or 3D FEA may offer the best understanding of the biomechanical

In the beginning, the FE models were obtained from sectional images of bone tissue, tooth, surrounding structures and other elements related with the study that would be executed. Khera et al. (1988) constructed a 3D human mandible based on a 2D model. Initially, a 2D model was obtained and, using the projection of several pictures in a magnifying monitor, a

Another widely used technique to obtain 2D and 3D models is the embedment of structures in acrylic resin. Gomes et al. (2009) embedded a prosthesis/retention screw/implant system in resin and sectioned it longitudinally to investigate the effect of different levels of unilateral angular misfit prostheses in the assembly and surrounding bone using a 2D FEA. The embedded model was scanned to produce digitalized images that were imported into CAD image analysis software and placed within the supporting tissue based on literature data. The outline of the images was manually quoted and each point was converted into x and y coordinates. At the end, the coordinates were finally imported into the FE software as

Recently, microcomputer tomography (CT) images have been obtained as a useful tool to model the bone complex in FE models and have gained general consensus among researches. The 3D model simulated from CT images provides high fidelity to the anatomical dimensions and configuration of all oral structures because it is possible to define the geometry and the local tissue properties of the bone segment to be modeled.

behavior of complex dental structures in certain situations (Romeed et al., 2006).

**5. Microcomputer tomography in FE models** 

3D model was generated and an axial *z*-axis was defined.

Fig. 9. Scanned embedded model and finite element model.

key points of the final images (Fig. 9).

However, some caution should be taken when these data are used. The X-ray images in grey scale must be recorded by a CT in DICOM (Digital Imaging Communications in Medicine) format. Thus, the procedure should be carefully indicated for the patients due to the exposition to radiation and the research project should be approved by the ethics committee ensuring that physical and geometrical parameters within safety limits will be adopted. Furthermore, the patient should sign an informed consent form to authorize the reproduction of images and results.

The processing techniques used to extract this information from the CT data may be also frequently affected by no negligible errors that propagate in an unknown way through the various steps of the model generation, affecting the accuracy of the model (Taddei et al., 2006). The first source of geometric error and distortion is the resolution of the dataset that depends on the scan parameters setting (Taddei et al., 2006). The ones that yielded the best results for image quality were obtained in the regime of 120 kV, 150 mA, 512 × 512 matrix, 14 cm × 14 cm field of view, and slice thickness of 0.5 mm (Poiate et al., 2011). The second error may result from the segmentation process of the region of interest. Several segmentation algorithms have been proposed, with various level of automation, starting from complete manual contours extraction to complex fully automatic algorithms (Taddei et al., 2006). Considering that several softwares are currently available in the market, the professional should be trained to accurately use the tools to convert CT images into FE models.

#### **5.1 Converting CT images into FEA models**

Some steps to convert CT images into solid models will be presented in this section considering the availability of softwares, such as Mimics (Materialise, Leuven, Belgium), Simpleware (Simpleware Ltd, United Kingdom), InVesalius (Brazilian Public Software, Brazil), and ITK Snap (General Public License, USA).

The use of Simpleware software to convert a maxillary CT images into a maxillary FEA model will be discussed. After submission by the Ethic Committee in Research of Dental School of Ribeirão Preto, University of São Paulo (Process CAAE – 0038.0.138.000-11 and SISNEP – FR – 430209), the CT images obtained from patient imaging with 219 cross sections were imported into the Simpleware 4.1 software (Simpleware Ltd, United Kingdom) (Fig. 10) at the *ScanIP* segment.

In the *ScanIP*, the segmentation tool was used to identify the pixels value of the tomographic image. Then, it was possible to separate an object from other adjacent anatomical structures in different masks, such as cortical bone and trabecular bone (Fig. 11). According to its radio-density, expressed in Hounsfield unities, the program picked the values up and automatically created different masks. After that, fine adjustments were executed to further improve the quality of the model masks (Fig. 12). The program also provides tools to eliminate any interference of the tomographic image. By examining the image, a certain level of noise in the data could be corrected by filters. After obtaining the masks for the cortical and trabecular bone, the soft tissue was constructed with 2.0 mm in thickness for the whole model using a morphological filter tool in structuring element. Then, the step to convert the CT image into a solid model was completed (Figs. 13 and 14).

FEA in Dentistry: A Useful Tool to Investigate

the Biomechanical Behavior of Implant Supported Prosthesis 73

Fig. 13. Tomographic image of an edentulous maxilla processed in *ScanIP*. Determination of

Fig. 14. Edentulous maxilla generated in the *ScanIP* representing the cortical and medullary

After obtaining the solid model, the finite elements mesh was generated. The mesh can be created either in the software for image conversion or in the FE software. In this study, the Simpleware software generated the mesh. A mix of tetrahedral and hexahedral elements was obtained using gallery elements + FE Free (Fig. 15). Afterwards, the meshed model is ready to be exported to a FEA software in order to conduct stress and displacement

the soft tissue with 2.0mm in thickness in the whole extension.

bone and the soft tissue.

analysis.

Fig. 10. Tomographic image of an edentulous maxilla imported into *ScanIP*.

Fig. 11. Tomographic image of an edentulous maxilla processed in *ScanIP*. Determination of the cortical bone tissue.

Fig. 12. Tomographic image of an edentulous maxilla processed in *ScanIP*. Determination of the trabecular bone tissue.

Fig. 10. Tomographic image of an edentulous maxilla imported into *ScanIP*.

Fig. 11. Tomographic image of an edentulous maxilla processed in *ScanIP*. Determination of

Fig. 12. Tomographic image of an edentulous maxilla processed in *ScanIP*. Determination of

the cortical bone tissue.

the trabecular bone tissue.

Fig. 13. Tomographic image of an edentulous maxilla processed in *ScanIP*. Determination of the soft tissue with 2.0mm in thickness in the whole extension.

Fig. 14. Edentulous maxilla generated in the *ScanIP* representing the cortical and medullary bone and the soft tissue.

After obtaining the solid model, the finite elements mesh was generated. The mesh can be created either in the software for image conversion or in the FE software. In this study, the Simpleware software generated the mesh. A mix of tetrahedral and hexahedral elements was obtained using gallery elements + FE Free (Fig. 15). Afterwards, the meshed model is ready to be exported to a FEA software in order to conduct stress and displacement analysis.

FEA in Dentistry: A Useful Tool to Investigate

1040-1044, ISSN 0882-2786

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57, ISSN 1741-2358

Feb), pp. 36-44, ISSN 0882-2786

0905-7161

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Assuncao, W. G.; Tabata, L. F.; Barao, V. A. & Rocha, E. P. (2008). Comparison of Stress

Assuncao, W. G.; Gomes, E. A.; Barao, V. A.; Delben, J. A.; Tabata, L. F. & de Sousa, E. A.

Barao, V. A.; Assuncao, W. G.; Tabata, L. F.; de Sousa, E. A. & Rocha, E. P. (2008). Effect of

Barao, V. A.; Assuncao, W. G.; Tabata, L. F.; Delben, J. A.; Gomes, E. A.; de Sousa, E. A. &

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Distribution between Complete Denture and Implant-Retained Overdenture-2d Fea. *Journal of oral rehabilitation,* Vol.35, No.10, (Oct), pp. 766-774, ISSN 1365-2842 Assuncao, W. G.; Gomes, E. A.; Barao, V. A. & de Sousa, E. A. (2009). Stress Analysis in

Simulation Models with or without Implant Threads Representation. *The International journal of oral & maxillofacial implants,* Vol.24, No.6, (Nov-Dec), pp.

(2010). Effect of Superstructure Materials and Misfit on Stress Distribution in a Single Implant-Supported Prosthesis: A Finite Element Analysis. *The Journal of* 

Unilateral Misfit on Preload of Retention Screws of Implant-Supported Prostheses Submitted to Mechanical Cycling. *Journal of prosthodontic research,* Vol.55, No.1,

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Fig. 15. Generated mesh with parabolic tetrahedral interpolation solid elements by the Simpleware software. The meshed model is ready to be imported by the finite element analysis software to investigate de stress distribution into the bone tissue.

#### **6. Future perspectives**

Considering that computational power is exhibiting rapid progress and hardware costs are decreasing, the numerical techniques probably will increase its application over time. Thus, the finite element method will be increasingly applied in Dentistry to generate reliable results for the biomechanical investigation of dental and supporting structures at lower cost than other *in vitro* and *in vivo* approaches. In addition, this technique can be associated to clinical evaluations as a further tool for diagnosis and/or treatment planning. For instance, the numerical techniques of the finite element method are increasingly indicated to simulate dental movements induced by orthodontic force systems. Thus, this method may provide information to the orthodontist about the choice of individual therapy (Clement et al., 2004).

#### **7. Acknowledgment**

Authors express gratitude to Sao Paulo State Research Foundation (FAPESP – Process number 2010/09857-3) for the grant support provided.

#### **8. References**


Fig. 15. Generated mesh with parabolic tetrahedral interpolation solid elements by the Simpleware software. The meshed model is ready to be imported by the finite element

Considering that computational power is exhibiting rapid progress and hardware costs are decreasing, the numerical techniques probably will increase its application over time. Thus, the finite element method will be increasingly applied in Dentistry to generate reliable results for the biomechanical investigation of dental and supporting structures at lower cost than other *in vitro* and *in vivo* approaches. In addition, this technique can be associated to clinical evaluations as a further tool for diagnosis and/or treatment planning. For instance, the numerical techniques of the finite element method are increasingly indicated to simulate dental movements induced by orthodontic force systems. Thus, this method may provide information to the orthodontist about the choice of individual therapy (Clement et al., 2004).

Authors express gratitude to Sao Paulo State Research Foundation (FAPESP – Process

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**6. Future perspectives** 

**7. Acknowledgment** 

**8. References** 

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Implant Design and Bone Quality on Stress/Strain Distribution in Bone around Implants: A 3-Dimensional Finite Element Analysis. *The International journal of oral & maxillofacial implants,* Vol.18, No.3, (May-Jun), pp. 357-368, ISSN 0882-2786 Taddei, F.; Martelli, S.; Reggiani, B.; Cristofolini, L. & Viceconti, M. (2006). Finite-Element

Modeling of Bones from Ct Data: Sensitivity to Geometry and Material Uncertainties. *IEEE transactions on bio-medical engineering,* Vol.53, No.11, (Nov), pp.

Denture with a Blade-Vent Implant Abutment. *The Journal of prosthetic dentistry,*

Stress-Breaking Attachments on Maxillary Implant-Retained Overdentures. *The International journal of prosthodontics,* Vol.20, No.2, (Mar-Apr), pp. 193-198, ISSN

Takahashi, N.; Kitagami, T. & Komori, T. (1978). Analysis of Stress on a Fixed Partial

Tanino, F.; Hayakawa, I.; Hirano, S. & Minakuchi, S. (2007). Finite Element Analysis of

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**4**

*1USA 2Brazil* 

**Critical Aspects for Mechanical**

*New York University College of Dentistry, New York, NY, 2Department of Dental Materials and Prosthodontics,* 

*Sao Paulo State University College of Dentistry, Araçatuba, SP,* 

Dental implants have been widely used for the rehabilitation of completely and partially edentulous patients. (Branemark et al. 1969; Branemark et al. 1977; Adell et al. 1990; van Steenberghe et al. 1990) Despite the high success rates reported by a vast number of clinical studies, early or late implant failures are still unavoidable. (Esposito et al. 1998) Mechanical complications and failures have frequently been reported during prosthetic treatment.

From a biomechanical point of view, forces occurring either from functional or parafunctional occlusal contact may result in a physiologic adaptation of the supporting tissues since implants are rigidly anchored to the bone. However, if the stress generated is beyond the adaptive capacity of the host, the response of the supporting tissues and prosthetic components may result in failures. Therefore, the load magnitude and duration employed to implant-restoration systems play a significant role in biomechanical stress dissipation on the implant-prosthesis system and surrounding tissues. (Menicucci et al. 2002) Other factors that are known to affect the stress/strain distribution on bone surrounding implants are the implant position and angulations, implant-abutment connection and the magnitude and direction of the occlusal load. (Stanford and Brand 1999;

Amilcar C. Freitas Júnior3, Eduardo P. Rocha2, Roberto S. Pessoa4, Nikhil Gupta5, Nick Tovar1 and

*1Department of Biomaterials and Biomimetics, New York University College of Dentistry, New York, USA 2Department of Dental Materials and Prosthodontics, São Paulo State University College of Dentistry,* 

*3Postgraduate Program in Dentistry, Potiguar University, College of Dentistry – UnP, Natal, RN, Brazil 4Department of Mechanical Engineering - Federal University at Uberlândia,* 

*5Department of Mechanical and Aerospace Engineering, Polytechnic Institute of New York University, New* 

(Roberts 1970; Randow et al. 1986; Walton et al. 1986; Levine et al. 1999)

**1. Introduction** 

Kozlovsky et al. 2007; Lin et al. 2009)

*FEMEC - Uberlândia, Uberlândia, MG, Brazil* 

 \*

Paulo G. Coelho1

*Araçatuba, SP, Brazil* 

*York, USA* 

Erika O. Almeida1,2 et al.\*

*1Department of Biomaterials and Biomimetics,* 

**Simulation in Dental Implantology** 


### **Critical Aspects for Mechanical Simulation in Dental Implantology**

Erika O. Almeida1,2 et al.\*

*1Department of Biomaterials and Biomimetics, New York University College of Dentistry, New York, NY, 2Department of Dental Materials and Prosthodontics, Sao Paulo State University College of Dentistry, Araçatuba, SP, 1USA 2Brazil* 

#### **1. Introduction**

80 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Tokuhisa, M.; Matsushita, Y. & Koyano, K. (2003). In Vitro Study of a Mandibular Implant

Valen, M. & Locante, W. M. (2000). Laminoss Immediate-Load Implants: I. Introducing

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Zarb, G. A. & Jansson, T. (1985). Prosthodontic Procedures, In: T*issue-Integrated Prostheses:* 

*maxillofacial implants,* Vol.5, No.3, (Fall), pp. 272-281, ISSN 0882-2786 Weinstein, A. M.; Klawitter, J. J.; Anand, S. C. & Schuessler, R. (1976). Stress Analysis of

No.2, (Mar-Apr), pp. 128-134, ISSN 0893-2174

177-184, ISSN 0160-6972

pp. 772-777, ISSN 0022-0345

(Nov-Dec), pp. 1004-1011, ISSN 0022-0345

(Jul-Aug), pp. 578-583, ISSN 0882-2786

169-175, ISSN 0090-5488

114, ISSN 0905-7161

Overdenture Retained with Ball, Magnet, or Bar Attachments: Comparison of Load Transfer and Denture Stability. *The International journal of prosthodontics,* Vol.16,

Osteocompression in Dentistry. *The Journal of oral implantology,* Vol.26, No.3, pp.

Higuchi, K.; Laney, W.; Linden, U. & Astrand, P. (1990). Applicability of Osseointegrated Oral Implants in the Rehabilitation of Partial Edentulism: A Prospective Multicenter Study on 558 Fixtures. *The International journal of oral &* 

Porous Rooted Dental Implants. *Journal of Dental Research,* Vol.55, No.5, (Sep-Oct),

Implant Design. *Biomaterials, medical devices, and artificial organs,* Vol.7, No.1, pp.

Crowns of Normal and Restored Teeth. *Journal of Dental Research,* Vol.55, No.6,

Edentulous Mandibular Bone Supporting Implant-Retained 1-Piece or Multiple Superstructures. *The International journal of oral & maxillofacial implants,* Vol.20, No.4,

*Osseointegration in Clinical Dentistry*, Branemark, P.-I., Zarb, G. A. & Albrektsson, T. editors, 241-282, Quintessence, ISBN 0867151293 / 0-86715-129-3, Chicago, USA Zarone, F.; Apicella, A.; Nicolais, L.; Aversa, R. & Sorrentino, R. (2003). Mandibular Flexure

and Stress Build-up in Mandibular Full-Arch Fixed Prostheses Supported by Osseointegrated Implants. *Clinical oral implants research,* Vol.14, No.1, (Feb), pp. 103Dental implants have been widely used for the rehabilitation of completely and partially edentulous patients. (Branemark et al. 1969; Branemark et al. 1977; Adell et al. 1990; van Steenberghe et al. 1990) Despite the high success rates reported by a vast number of clinical studies, early or late implant failures are still unavoidable. (Esposito et al. 1998) Mechanical complications and failures have frequently been reported during prosthetic treatment. (Roberts 1970; Randow et al. 1986; Walton et al. 1986; Levine et al. 1999)

From a biomechanical point of view, forces occurring either from functional or parafunctional occlusal contact may result in a physiologic adaptation of the supporting tissues since implants are rigidly anchored to the bone. However, if the stress generated is beyond the adaptive capacity of the host, the response of the supporting tissues and prosthetic components may result in failures. Therefore, the load magnitude and duration employed to implant-restoration systems play a significant role in biomechanical stress dissipation on the implant-prosthesis system and surrounding tissues. (Menicucci et al. 2002) Other factors that are known to affect the stress/strain distribution on bone surrounding implants are the implant position and angulations, implant-abutment connection and the magnitude and direction of the occlusal load. (Stanford and Brand 1999; Kozlovsky et al. 2007; Lin et al. 2009)

<sup>\*</sup> Amilcar C. Freitas Júnior3, Eduardo P. Rocha2, Roberto S. Pessoa4, Nikhil Gupta5, Nick Tovar1 and Paulo G. Coelho1

*<sup>1</sup>Department of Biomaterials and Biomimetics, New York University College of Dentistry, New York, USA 2Department of Dental Materials and Prosthodontics, São Paulo State University College of Dentistry,* 

*Araçatuba, SP, Brazil* 

*<sup>3</sup>Postgraduate Program in Dentistry, Potiguar University, College of Dentistry – UnP, Natal, RN, Brazil 4Department of Mechanical Engineering - Federal University at Uberlândia,* 

*FEMEC - Uberlândia, Uberlândia, MG, Brazil* 

*<sup>5</sup>Department of Mechanical and Aerospace Engineering, Polytechnic Institute of New York University, New York, USA* 

Critical Aspects for Mechanical Simulation in Dental Implantology 83

been commonly used for 3D modeling as it allows the achievement of reliable analytic or

Fig. 1. Models of maxilla (A) and mandible (B) implant supported prosthesis based on (A) model reconstruction from imaging techniques and (B) manual construction on CAD

The manual input technique generate structures in appropriate aided design software such as AutoCAD (Autodesk Inc, San Rafael, CA, USA), SolidWorks (SolidWorks Corp., Concord, MA, USA), Pro/Engineer (Wildfire, PTC, Needham, MA, USA), Rhino 3D

Fig. 2. CT scan data as seen in Mimics 13.0 (Materialise, Leuven, Belgium). (A-C) The maxilla is presented in three different cross-sectional views. Masks have been applied according to voxel density thresholding to determine the regions of interest. (D) 3D

representation of maxilla as a result of segmentation in Mimics.

freeform parts based on an efficient management of curves and surfaces.

software (Solid Works Corp).

(McNeel North America, Seatle, WA, USA) (Figure 1B).

While randomized controlled clinical studies are strongly suggested (Bozini et al. 2011; Pieri et al. 2011; Turkyilmaz 2011) as the optimal approach to evaluate the performance of biomaterials and biomechanical aspects of dental implants and prosthetic components, often times these studies are not economically viable (especially in cases where multiple variables are to be included). Thus, well designed *in vitro* studies utilizing virtual models via finite element analysis (FEA) should be considered, as this method allows researchers to address a range of questions that are otherwise intractable due to the number of variations within clinical trials or the difficulty in solving analytically (Ross 2005).

Although there are several implant dentistry studies using the FEA method, (Rayfield 2004; Yokoyama et al. 2005; Galantucci et al. 2006; Huang et al. 2007; Assuncao et al. 2008; Lim et al. 2008; de Almeida et al. 2011) each step of the method deserves discussion in order to facilitate its understanding when applied to implant/restoration/bone system. Thus, the aim of this chapter is to conduct a critical review of simulated biomechanical scenarios describing essential basic aspects and approaches currently used in implantology.

#### **2. Model creation**

The first step in creating a finite element (FE) model is to determine the appropriate number of dimension to be utilized for evaluation (1, 2, or 3D). While three dimensions are more realistic for the complex anatomy/implant/restoration biomechanical interaction, it is proportionally more challenging when CAD modeling, solving, and software output interpretation is considered. (Richmond et al. 2005) For selected situations, 2D analyses are often adequate for the questions at hand. (Rayfield 2004; Assuncao et al. 2008; Freitas et al. 2010; Freitas Junior et al. 2010)

The choice between 2D and 3D FEA for investigating the biomechanical behavior of complex structures depends on several factors, including the complexity of the geometry, the type of analyses required, expectations in terms of accuracy as well as the general applicability of the results. (Romeed et al. 2006)

Two-dimensional FEA has previously been used in different areas of dental research. (Burak Ozcelik et al. ; Freitas, Rocha et al. 2010; Freitas Junior, Rocha et al. 2010) However, its main limitation is known to be the lower accuracy and reliability relative to 3D and its utilization has drastically fallen from favor when dental treatment biomechanical simulations are considered. (Yang et al. 1999; Yang et al. 2001) In contrast, 3D FEA has acceptable accuracy/reliability while properly capturing the geometry of complex structures. However, the higher the complexity of 3D FE models the higher the difficulty in generating appropriate mesh refinement for simulation, which is more easily achievable for 2D models. (Romeed, Fok et al. 2006)

It is general consensus that 3D reconstruction is essential to study the interaction between anatomy and mechanical behavior of restorative components and dental implants. (Galantucci, Percoco et al. 2006) For that purpose, models may be manually constructed or may be generated from imaging methods such as computed tomography (Figures 1 A and B). The choice for building a model using either a manual or automatic technique depends on the purpose of the study and the structure of interest. Considering the specific morphology of biological structures, the non-uniform rational basis spline (NURBS) has

While randomized controlled clinical studies are strongly suggested (Bozini et al. 2011; Pieri et al. 2011; Turkyilmaz 2011) as the optimal approach to evaluate the performance of biomaterials and biomechanical aspects of dental implants and prosthetic components, often times these studies are not economically viable (especially in cases where multiple variables are to be included). Thus, well designed *in vitro* studies utilizing virtual models via finite element analysis (FEA) should be considered, as this method allows researchers to address a range of questions that are otherwise intractable due to the number of variations within

Although there are several implant dentistry studies using the FEA method, (Rayfield 2004; Yokoyama et al. 2005; Galantucci et al. 2006; Huang et al. 2007; Assuncao et al. 2008; Lim et al. 2008; de Almeida et al. 2011) each step of the method deserves discussion in order to facilitate its understanding when applied to implant/restoration/bone system. Thus, the aim of this chapter is to conduct a critical review of simulated biomechanical scenarios

The first step in creating a finite element (FE) model is to determine the appropriate number of dimension to be utilized for evaluation (1, 2, or 3D). While three dimensions are more realistic for the complex anatomy/implant/restoration biomechanical interaction, it is proportionally more challenging when CAD modeling, solving, and software output interpretation is considered. (Richmond et al. 2005) For selected situations, 2D analyses are often adequate for the questions at hand. (Rayfield 2004; Assuncao et al. 2008; Freitas et al.

The choice between 2D and 3D FEA for investigating the biomechanical behavior of complex structures depends on several factors, including the complexity of the geometry, the type of analyses required, expectations in terms of accuracy as well as the general

Two-dimensional FEA has previously been used in different areas of dental research. (Burak Ozcelik et al. ; Freitas, Rocha et al. 2010; Freitas Junior, Rocha et al. 2010) However, its main limitation is known to be the lower accuracy and reliability relative to 3D and its utilization has drastically fallen from favor when dental treatment biomechanical simulations are considered. (Yang et al. 1999; Yang et al. 2001) In contrast, 3D FEA has acceptable accuracy/reliability while properly capturing the geometry of complex structures. However, the higher the complexity of 3D FE models the higher the difficulty in generating appropriate mesh refinement for simulation, which is more easily achievable for 2D models.

It is general consensus that 3D reconstruction is essential to study the interaction between anatomy and mechanical behavior of restorative components and dental implants. (Galantucci, Percoco et al. 2006) For that purpose, models may be manually constructed or may be generated from imaging methods such as computed tomography (Figures 1 A and B). The choice for building a model using either a manual or automatic technique depends on the purpose of the study and the structure of interest. Considering the specific morphology of biological structures, the non-uniform rational basis spline (NURBS) has

describing essential basic aspects and approaches currently used in implantology.

clinical trials or the difficulty in solving analytically (Ross 2005).

**2. Model creation** 

2010; Freitas Junior et al. 2010)

(Romeed, Fok et al. 2006)

applicability of the results. (Romeed et al. 2006)

been commonly used for 3D modeling as it allows the achievement of reliable analytic or freeform parts based on an efficient management of curves and surfaces.

Fig. 1. Models of maxilla (A) and mandible (B) implant supported prosthesis based on (A) model reconstruction from imaging techniques and (B) manual construction on CAD software (Solid Works Corp).

The manual input technique generate structures in appropriate aided design software such as AutoCAD (Autodesk Inc, San Rafael, CA, USA), SolidWorks (SolidWorks Corp., Concord, MA, USA), Pro/Engineer (Wildfire, PTC, Needham, MA, USA), Rhino 3D (McNeel North America, Seatle, WA, USA) (Figure 1B).

Fig. 2. CT scan data as seen in Mimics 13.0 (Materialise, Leuven, Belgium). (A-C) The maxilla is presented in three different cross-sectional views. Masks have been applied according to voxel density thresholding to determine the regions of interest. (D) 3D representation of maxilla as a result of segmentation in Mimics.

Critical Aspects for Mechanical Simulation in Dental Implantology 85

Material properties such as heat conductivity, linear and nonlinear elastic properties, and temperature-dependent elastic properties may be utilized in FEA. (Richmond, Wright et al. 2005) In dentistry, the majority of previous FE scientific communications in implant prosthodontics has considered material properties to be isotropic, homogeneous, and linear elastic. (Huang et al. 2008; Canay and Akca 2009; Eser et al. 2009; Li et al. 2009; Chang et al. 2010; Chou et al. 2010; Okumura et al. 2010; Sagat et al. 2010; Wu et al. 2010; Burak Ozcelik et al. 2011) An isotropic material indicates that the mechanical response is similar regardless of the stress field direction, requiring Young's modulus (E) and Poisson's ratio (ν) values for

The elastic, or Young's modulus (E), is defined as stress/strain (σ/Є) and is measured in simple extension or compression. It is a measure of material deformation under a given axial load. In other words, a numerical description of its stiffness. Poisson's ratio (ν) is the lateral strain divided by axial strain, thus representing how much the sides of a material as it is

Since bone is one of the structures to be simulated in FEA, (Peterson and Dechow 2003; Bozkaya et al. 2004; Danza et al. 2010; de Almeida et al. 2010) it is often treated as an anisotropic structure and three elastic modulus (E), three Poisson's ratio (ν) and three shear modulus (Ԍ) are required (Table 1). (Chen et al. 2010; Sotto-Maior et al. 2010) Anisotropic materials are characterized by different stress responses under forces applied in varied directions within the structure. (O'Mahony et al. 2001; Natali et al. 2009; Eraslan and Inan 2010) The elastic behavior in cortical bone approximates to orthotropic, which is a type of anisotropy in which the internal structure of the material results in unique elastic behavior along each of the three orthogonal axes of the material. In this case, three elastic (E) and shear modulus (Ԍ) and six Poisson's ratios (ν) are necessary for model input. (Richmond,

**MATERIAL PROPERTIES TRABECULLAR BONE CORTICAL BONE**  EX (MPa) 1,148 12,600 EY (MPa) 210 12,600 EZ (MPa) 1,148 19,400 GXY (MPa) 68 4,850 GYZ (MPa) 68 5,700 GXZ (MPa) 434 5,700 νYX 0,010 0,300 νZY 0,055 0,390 νZX 0,322 0,390 νXY 0,055 0,300 νYZ 0,010 0,253 νXZ 0,322 0,253

Table 1. Material properties used in an anisotropic model. The material axes correspond to the global coordinate system. E = Young's modulus. G = shear modulus. νyx = Poisson's ratio

for strain in the y-direction when loaded in the x-direction.

**3. Material properties and software input limitation** 

the FE calculation. (Richmond, Wright et al. 2005)

tensile tested. (Richmond, Wright et al. 2005)

Wright et al. 2005; Natali et al. 2010)

The imaging approach involves transforming available medical imaging files from computed tomography (CT) scans, magnetic resonance images (MRI), ultrasound, and laser digitizers into wireframe models that are then converted into FE models. (Romeed, Fok et al. 2006) While laser scans offer a high-resolution representation of the outer surface, these lack information about internal geometry. (Kappelman 1998) Creating models from CT or MRI is often time-consuming but can provide accurate models with fine structural details based on image density thresholding. (Cohen et al. 1999; Ryan and van Rietbergen 2005) The 3D object is automatically created in the form of masks by thresholding the region of interest on the entire stack of scans (Figure 2A-D). The degree of automation and high resolution make this model creation method attractive, but determining the appropriate thresholding algorithms to the bone-air boundary reliably throughout a structure with varying bone thicknesses and density can be challenging. (Fajardo et al. 2002)

When advanced imaging data is used to generate solid models, surface smoothing is advised in order to decrease the number of nodes and elements in the discretized FE model as such an approach generally decreases computation time. (Wang et al. 2005; Magne 2007) Fig. 3A shows an excessive number of elements in a dental implant that was subsequently reduced (Fig. 3B) by computer software (Materialise, Leuven, Belgium). However, it is also advisable that when surface smoothing is performed it does not over simplify the geometries, causing a decrease in solution accuracy.

Fig. 3. (A) 3D CAD of the Nobel Speed TM RP Implant (Nobel Biocare, CA, USA) based on the micro-CT (µCT; CT40, Scanco Medical, Bassersdorf, Switzerland). (B) The Mimics Remesh (Materialise) function "quality preserving reduce triangle" and "reduce triangles" were used to reduce the number of elements at the implant CAD.

The imaging approach involves transforming available medical imaging files from computed tomography (CT) scans, magnetic resonance images (MRI), ultrasound, and laser digitizers into wireframe models that are then converted into FE models. (Romeed, Fok et al. 2006) While laser scans offer a high-resolution representation of the outer surface, these lack information about internal geometry. (Kappelman 1998) Creating models from CT or MRI is often time-consuming but can provide accurate models with fine structural details based on image density thresholding. (Cohen et al. 1999; Ryan and van Rietbergen 2005) The 3D object is automatically created in the form of masks by thresholding the region of interest on the entire stack of scans (Figure 2A-D). The degree of automation and high resolution make this model creation method attractive, but determining the appropriate thresholding algorithms to the bone-air boundary reliably throughout a structure with varying bone

When advanced imaging data is used to generate solid models, surface smoothing is advised in order to decrease the number of nodes and elements in the discretized FE model as such an approach generally decreases computation time. (Wang et al. 2005; Magne 2007) Fig. 3A shows an excessive number of elements in a dental implant that was subsequently reduced (Fig. 3B) by computer software (Materialise, Leuven, Belgium). However, it is also advisable that when surface smoothing is performed it does not over simplify the

Fig. 3. (A) 3D CAD of the Nobel Speed TM RP Implant (Nobel Biocare, CA, USA) based on the micro-CT (µCT; CT40, Scanco Medical, Bassersdorf, Switzerland). (B) The Mimics Remesh (Materialise) function "quality preserving reduce triangle" and "reduce triangles"

were used to reduce the number of elements at the implant CAD.

thicknesses and density can be challenging. (Fajardo et al. 2002)

geometries, causing a decrease in solution accuracy.

#### **3. Material properties and software input limitation**

Material properties such as heat conductivity, linear and nonlinear elastic properties, and temperature-dependent elastic properties may be utilized in FEA. (Richmond, Wright et al. 2005) In dentistry, the majority of previous FE scientific communications in implant prosthodontics has considered material properties to be isotropic, homogeneous, and linear elastic. (Huang et al. 2008; Canay and Akca 2009; Eser et al. 2009; Li et al. 2009; Chang et al. 2010; Chou et al. 2010; Okumura et al. 2010; Sagat et al. 2010; Wu et al. 2010; Burak Ozcelik et al. 2011) An isotropic material indicates that the mechanical response is similar regardless of the stress field direction, requiring Young's modulus (E) and Poisson's ratio (ν) values for the FE calculation. (Richmond, Wright et al. 2005)

The elastic, or Young's modulus (E), is defined as stress/strain (σ/Є) and is measured in simple extension or compression. It is a measure of material deformation under a given axial load. In other words, a numerical description of its stiffness. Poisson's ratio (ν) is the lateral strain divided by axial strain, thus representing how much the sides of a material as it is tensile tested. (Richmond, Wright et al. 2005)

Since bone is one of the structures to be simulated in FEA, (Peterson and Dechow 2003; Bozkaya et al. 2004; Danza et al. 2010; de Almeida et al. 2010) it is often treated as an anisotropic structure and three elastic modulus (E), three Poisson's ratio (ν) and three shear modulus (Ԍ) are required (Table 1). (Chen et al. 2010; Sotto-Maior et al. 2010) Anisotropic materials are characterized by different stress responses under forces applied in varied directions within the structure. (O'Mahony et al. 2001; Natali et al. 2009; Eraslan and Inan 2010) The elastic behavior in cortical bone approximates to orthotropic, which is a type of anisotropy in which the internal structure of the material results in unique elastic behavior along each of the three orthogonal axes of the material. In this case, three elastic (E) and shear modulus (Ԍ) and six Poisson's ratios (ν) are necessary for model input. (Richmond, Wright et al. 2005; Natali et al. 2010)


Table 1. Material properties used in an anisotropic model. The material axes correspond to the global coordinate system. E = Young's modulus. G = shear modulus. νyx = Poisson's ratio for strain in the y-direction when loaded in the x-direction.

Critical Aspects for Mechanical Simulation in Dental Implantology 87

the interface; and shear stresses in parallel with the interface (Figure 4). (Hansson and

Fig. 4. The different types of stresses occurring at the implant–bone interface.

Linear static models have been employed extensively in previous FEA studies. (Kohal et al. 2002; Menicucci, Mossolov et al. 2002; Romanos 2004; Van Staden et al. 2006; Lim, Chew et al. 2008; Eser, Akca et al. 2009; Hsu et al. 2009; Li, Kong et al. 2009; Faegh and Muftu 2010; Freitas Junior et al. 2010; Hasan et al. 2011) These analyses usually assumed that all modeled volumes were bonded as one unit, which indicates that the trabecular and cortical bone are perfectly bonded to the implant interface. (Winter et al. 2004; Maeda et al. 2007; Fazel et al. 2009; Chang, Chen et al. 2010; Okumura, Stegaroiu et al. 2010). However, the validity of a linear static analysis may be questionable when the investigation aims to explore more realistic situations that are generally encountered in the dental implant field. Some actual implant clinical situations will give rise to nonlinearities, mainly related to the chang of interrelations between the simulated structures of a FE model. (Wakabayashi et al. 2008) Moreover, frictional contact mode potentially provides an improved accuracy with respect to the relative component's micromotion within the implant system, and, therefore, a more reasonable representation of the real implant clinical condition. (Merz et al. 2000) This configuration allows minor displacements between all components of the model without interpenetration. Under these conditions, the contact zones transfer pressure and tangential forces (i.e. friction), but not tension. Some FE analyses have shown remarkable differences in the values and even in the distribution of stresses between "fixed bond" and "non-linear contact" interface conditions. (Brunski 1992; Van Oosterwyck et al. 1998; Huang, Hsu et al. 2008) Not only the stress and strain levels but also the stress and strain highly affected by the interface state (Figure 5). Van Oosterwyck et al. (Van Oosterwyck, Duyck et al. 1998) argued that through the bonded interface the force was dissipated evenly in both the compressive and the tension site. However, on the contact interfaces, tensions are not transferred and force is only passed on through the compressive site, which results in excessive stresses. Additionally, the condition of the bone to implant interface also influences the strain distribution and level inside of the implant system. (Pessoa et al.

Halldin 2009)

2010)

Since in dental implantology bone quality has been related to the structural efficiency of the cortical and trabecular bone architecture and ratio (lower bone quality results in biomechanically challenged treatments), mechanical simulations of poor bone quality critical as clinical studies have shown that dental implants placed in regions of the jaw bones with lower density have a higher chance to fail than implants placed at regions with higher bone density. (Genna 2003)

To date, no consensus regarding the mechanical properties that are appropriate for simulating the different bone density scenarios clinically encountered in implant dentistry has been reached. For instance, the value of trabecular bone elastic modulus observed in the literature range from 0.3 to 9.5 GPa. (Zarone et al. 2003; Eskitascioglu et al. 2004; Sevimay et al. 2005; Yokoyama, Wakabayashi et al. 2005) A different approach has been employed by Tada and coworkers (Tada et al. 2003) who assigned different elastic moduli to bone depending on its density from most dense (Type 1) to least dense (Type IV). The moduli utilized were 9.5 GPa, 5.5 GPa, 1.6 GPa and 0.69 GPa for bone types I, II, III, and IV, respectively. Recent work has also been carried out using these bone property values for different simulations (Table 2).


Table 2. Mechanical properties of the most used structures in implant prosthodontics research through finite element analysis.

Considering the fact that the most common bone quality types in mandible are type I or II in the anterior region and type III in the posterior region, the elastic modulus that would best represent the mandible would be 9.5 GPa or 5.5 GPa for the anterior region and 1.6 GPa for the posterior region. For the maxillary bone, the most prevalent types of bone are type II or III in the anterior region and type IV in the posterior region. Therefore, the Young's modulus that would best represent these sites would be 5.5 GPa or 1.6 GPa for the anterior region and 0.69 GPa for the posterior region, respectively. For cortical bone, studies typically use the elastic modulus (E) of 13.7 GPa and the Poisson ratio (ν) similar to 0.3 for the trabecular and cortical bone (Table 2). (Sevimay, Turhan et al. 2005; Huang, Hsu et al. 2008)

#### **4. Bone-implant interface**

In implant therapy, the fact that bone quantity and related biomechanical behavior differs for each patient implies a challenge to model the percentage of osseointegration. A critical issue when evaluating a study is to analyze the conditions specified for the interface. What is to be specified is the ability of the interface to resist three different types of stresses: compressive stresses at a right angle to the interface; tensile stresses, also at a right angle to

Since in dental implantology bone quality has been related to the structural efficiency of the cortical and trabecular bone architecture and ratio (lower bone quality results in biomechanically challenged treatments), mechanical simulations of poor bone quality critical as clinical studies have shown that dental implants placed in regions of the jaw bones with lower density have a higher chance to fail than implants placed at regions with

To date, no consensus regarding the mechanical properties that are appropriate for simulating the different bone density scenarios clinically encountered in implant dentistry has been reached. For instance, the value of trabecular bone elastic modulus observed in the literature range from 0.3 to 9.5 GPa. (Zarone et al. 2003; Eskitascioglu et al. 2004; Sevimay et al. 2005; Yokoyama, Wakabayashi et al. 2005) A different approach has been employed by Tada and coworkers (Tada et al. 2003) who assigned different elastic moduli to bone depending on its density from most dense (Type 1) to least dense (Type IV). The moduli utilized were 9.5 GPa, 5.5 GPa, 1.6 GPa and 0.69 GPa for bone types I, II, III, and IV, respectively. Recent work has also been carried out using these bone property values for

**POISSON'S** 

Bone type I 9.5 0.3 Tada et al. (2003) Bone type II 5.5 0.3 Tada et al. (2003) Bone type III 1.6 0.3 Tada et al. (2003) Bone type III 0.69 0.3 Tada et al. (2003) Cortical bone 13.70 0.3 Shunmugasamy et al. (2011) Titanium 110 0.35 Huang et al. (2008)

Table 2. Mechanical properties of the most used structures in implant prosthodontics

Considering the fact that the most common bone quality types in mandible are type I or II in the anterior region and type III in the posterior region, the elastic modulus that would best represent the mandible would be 9.5 GPa or 5.5 GPa for the anterior region and 1.6 GPa for the posterior region. For the maxillary bone, the most prevalent types of bone are type II or III in the anterior region and type IV in the posterior region. Therefore, the Young's modulus that would best represent these sites would be 5.5 GPa or 1.6 GPa for the anterior region and 0.69 GPa for the posterior region, respectively. For cortical bone, studies typically use the elastic modulus (E) of 13.7 GPa and the Poisson ratio (ν) similar to 0.3 for the trabecular and cortical bone (Table 2). (Sevimay, Turhan et al. 2005; Huang, Hsu et al. 2008)

In implant therapy, the fact that bone quantity and related biomechanical behavior differs for each patient implies a challenge to model the percentage of osseointegration. A critical issue when evaluating a study is to analyze the conditions specified for the interface. What is to be specified is the ability of the interface to resist three different types of stresses: compressive stresses at a right angle to the interface; tensile stresses, also at a right angle to

**RATIO REFERENCES** 

higher bone density. (Genna 2003)

different simulations (Table 2).

**MATERIAL YOUNG'S** 

research through finite element analysis.

**4. Bone-implant interface** 

**MODULUS (GPa)** 

the interface; and shear stresses in parallel with the interface (Figure 4). (Hansson and Halldin 2009)

Fig. 4. The different types of stresses occurring at the implant–bone interface.

Linear static models have been employed extensively in previous FEA studies. (Kohal et al. 2002; Menicucci, Mossolov et al. 2002; Romanos 2004; Van Staden et al. 2006; Lim, Chew et al. 2008; Eser, Akca et al. 2009; Hsu et al. 2009; Li, Kong et al. 2009; Faegh and Muftu 2010; Freitas Junior et al. 2010; Hasan et al. 2011) These analyses usually assumed that all modeled volumes were bonded as one unit, which indicates that the trabecular and cortical bone are perfectly bonded to the implant interface. (Winter et al. 2004; Maeda et al. 2007; Fazel et al. 2009; Chang, Chen et al. 2010; Okumura, Stegaroiu et al. 2010). However, the validity of a linear static analysis may be questionable when the investigation aims to explore more realistic situations that are generally encountered in the dental implant field. Some actual implant clinical situations will give rise to nonlinearities, mainly related to the chang of interrelations between the simulated structures of a FE model. (Wakabayashi et al. 2008) Moreover, frictional contact mode potentially provides an improved accuracy with respect to the relative component's micromotion within the implant system, and, therefore, a more reasonable representation of the real implant clinical condition. (Merz et al. 2000) This configuration allows minor displacements between all components of the model without interpenetration. Under these conditions, the contact zones transfer pressure and tangential forces (i.e. friction), but not tension. Some FE analyses have shown remarkable differences in the values and even in the distribution of stresses between "fixed bond" and "non-linear contact" interface conditions. (Brunski 1992; Van Oosterwyck et al. 1998; Huang, Hsu et al. 2008) Not only the stress and strain levels but also the stress and strain highly affected by the interface state (Figure 5). Van Oosterwyck et al. (Van Oosterwyck, Duyck et al. 1998) argued that through the bonded interface the force was dissipated evenly in both the compressive and the tension site. However, on the contact interfaces, tensions are not transferred and force is only passed on through the compressive site, which results in excessive stresses. Additionally, the condition of the bone to implant interface also influences the strain distribution and level inside of the implant system. (Pessoa et al. 2010)

Critical Aspects for Mechanical Simulation in Dental Implantology 89

demonstrated that the value of µ shows no significant influence for increasing or decreasing the tensile and compressive stresses of bone. Nevertheless, increasing µ from 0.3 to 1, the interfacial sliding between implant and bone was mainly reduced from 20% to 30–60%,

Moreover, when an implant is surgically placed into the jawbone, the implant is mechanically screwed into a drilled hole of a smaller diameter. Large stresses will occur due to the torque applied in the process of implant insertion. As the implant stability and stress state around an immediately loaded implant may be influenced by such conditions, this should be also considered in FE simulation of immediately loaded implants. However, this phenomenon has not been thoroughly investigated to date. The implementation of such implant insertion stresses in FEA is still unclear and should therefore be a matter of further

Another commonly observed assumption in dental implantology FE models, perfect bonding between implant, abutment and abutment screw also is not the most realistic scenario. Non-linear contact analysis was proven to be the most effective interface condition for realistically simulating the relative micromotions occurring between different components within the implant system. (Williams 2000; Pessoa, Muraru et al. 2010; Pessoa et al. 2010) Therefore, for correct simulation of an implant-abutment connection, frictional contact should be defined between the implant components. Accordingly, between implant, abutment and abutment screw regions in contact, a frictional coefficient of 0.5 was generally assumed in non-linear simulations of implant-abutment connection. (Merz, Hunenbart et al. 2000; Lin et al. 2007; Pessoa, Muraru et al. 2010) When using a contact interface between implant components, for lateral or oblique loading conditions, specific parts can separate, or new parts that were initially not in contact can come into contact. Consequently, higher stress levels may be expected to occur in an implant-abutment connection simulated with contact interfaces, compared to a glued connection. In this regard, the pattern and magnitude of deformation in both periimplant bone and implant components will be

Since the components in a dental implant-bone system are complex from a geometric standpoint, FEA has been viewed as the most suitable tool for mechanically analyzing them. A mesh is needed in FEA to divide the whole domain into elements. The process of creating the mesh, elements, their respective nodes, and defining boundary conditions is called "discretization" of the problem domain. (Geng et al. 2001; Richmond, Wright et al. 2005)

The 2D structures are typically meshed with triangular or quadrilateral elements (Figure 6A). These elements may possess two or more nodes per side, with one node at each vertex. If nodes are only placed at the vertices, the element is called linear because a line function describes the element geometry and how the displacement field will vary along an element

Typical 3D elements include eight-node bricks, six-node wedges, five-node pyramids, and four-node tetrahedral (Figure 6B). With increasing numbers of elements and nodes, the model becomes more complex and computationally more difficult and lengthy. (Assuncao

depending on the implant design. (Natali, Carniel et al. 2009)

influenced by the implant connection design. (Pessoa, Vaz et al. 2010)

**5. Mesh and convergence analysis** 

edge. (Richmond, Wright et al. 2005)

et al. 2009)

investigations.

Fig. 5. Equivalent strain (µe) distribution for 100N loaded implant in a superior central incisor region, in a median buccopalatal plane. The bone to implant interfaces were assumed as fixed bond ("glued") (A) and frictional contact (B). The arrows indicate the loading direction for clarity.

Conventionally, the osseointegrated bone to implant interface is treated as fully bonded. (Kohal, Papavasiliou et al. 2002; Menicucci, Mossolov et al. 2002; Romanos 2004; Van Staden, Guan et al. 2006; Lim, Chew et al. 2008; Eser, Akca et al. 2009; Hsu, Fuh et al. 2009; Li, Kong et al. 2009; Faegh and Muftu 2010; Freitas Junior, Rocha et al. 2010; Hasan, Rahimi et al. 2011)

This assumption is supported by experimental investigations in which removal of rough implants frequently resulted in fractures in bone distant from the implant surface, (Gotfredsen et al. 2000) suggesting the existence of an implant-bone "bond". On the other hand, frictional contact elements are used to simulate a nonintegrated bone to implant interface (i.e. in immediately loaded protocols), which allows minor displacements between the implant and the bone. (Pessoa, Muraru et al. 2010) The occurrence of relative motion between implant and bone introduces a source of non-linearity in FEA, since the contact conditions will change during load application.

The friction coefficient (µ) to be used in such simulations depends on many factors including mechanical properties and the roughness of the contact interface, exposure to interfacial contaminants (Williams 2000) and in some cases the normal load. A µ = 0.3 was measured for interfaces between a smooth metal surface and bone, while a µ = 0.45, for interfaces between a rough metal surface and bone. (Rancourt et al. 1990) Frequently, the friction coefficient between bone and implant is assumed as being µ = 0.3. Huang et al. (Huang, Hsu et al. 2008) investigated the effects of different frictional coefficients (µ = 0.3, 0.45 and 1) on the stress and displacement of an immediately loaded implant simulation. The authors

Fig. 5. Equivalent strain (µe) distribution for 100N loaded implant in a superior central incisor region, in a median buccopalatal plane. The bone to implant interfaces were assumed

as fixed bond ("glued") (A) and frictional contact (B). The arrows indicate the loading

Conventionally, the osseointegrated bone to implant interface is treated as fully bonded. (Kohal, Papavasiliou et al. 2002; Menicucci, Mossolov et al. 2002; Romanos 2004; Van Staden, Guan et al. 2006; Lim, Chew et al. 2008; Eser, Akca et al. 2009; Hsu, Fuh et al. 2009; Li, Kong et al. 2009; Faegh and Muftu 2010; Freitas Junior, Rocha et al. 2010; Hasan, Rahimi et al. 2011)

This assumption is supported by experimental investigations in which removal of rough implants frequently resulted in fractures in bone distant from the implant surface, (Gotfredsen et al. 2000) suggesting the existence of an implant-bone "bond". On the other hand, frictional contact elements are used to simulate a nonintegrated bone to implant interface (i.e. in immediately loaded protocols), which allows minor displacements between the implant and the bone. (Pessoa, Muraru et al. 2010) The occurrence of relative motion between implant and bone introduces a source of non-linearity in FEA, since the contact

The friction coefficient (µ) to be used in such simulations depends on many factors including mechanical properties and the roughness of the contact interface, exposure to interfacial contaminants (Williams 2000) and in some cases the normal load. A µ = 0.3 was measured for interfaces between a smooth metal surface and bone, while a µ = 0.45, for interfaces between a rough metal surface and bone. (Rancourt et al. 1990) Frequently, the friction coefficient between bone and implant is assumed as being µ = 0.3. Huang et al. (Huang, Hsu et al. 2008) investigated the effects of different frictional coefficients (µ = 0.3, 0.45 and 1) on the stress and displacement of an immediately loaded implant simulation. The authors

direction for clarity.

conditions will change during load application.

demonstrated that the value of µ shows no significant influence for increasing or decreasing the tensile and compressive stresses of bone. Nevertheless, increasing µ from 0.3 to 1, the interfacial sliding between implant and bone was mainly reduced from 20% to 30–60%, depending on the implant design. (Natali, Carniel et al. 2009)

Moreover, when an implant is surgically placed into the jawbone, the implant is mechanically screwed into a drilled hole of a smaller diameter. Large stresses will occur due to the torque applied in the process of implant insertion. As the implant stability and stress state around an immediately loaded implant may be influenced by such conditions, this should be also considered in FE simulation of immediately loaded implants. However, this phenomenon has not been thoroughly investigated to date. The implementation of such implant insertion stresses in FEA is still unclear and should therefore be a matter of further investigations.

Another commonly observed assumption in dental implantology FE models, perfect bonding between implant, abutment and abutment screw also is not the most realistic scenario. Non-linear contact analysis was proven to be the most effective interface condition for realistically simulating the relative micromotions occurring between different components within the implant system. (Williams 2000; Pessoa, Muraru et al. 2010; Pessoa et al. 2010) Therefore, for correct simulation of an implant-abutment connection, frictional contact should be defined between the implant components. Accordingly, between implant, abutment and abutment screw regions in contact, a frictional coefficient of 0.5 was generally assumed in non-linear simulations of implant-abutment connection. (Merz, Hunenbart et al. 2000; Lin et al. 2007; Pessoa, Muraru et al. 2010) When using a contact interface between implant components, for lateral or oblique loading conditions, specific parts can separate, or new parts that were initially not in contact can come into contact. Consequently, higher stress levels may be expected to occur in an implant-abutment connection simulated with contact interfaces, compared to a glued connection. In this regard, the pattern and magnitude of deformation in both periimplant bone and implant components will be influenced by the implant connection design. (Pessoa, Vaz et al. 2010)

#### **5. Mesh and convergence analysis**

Since the components in a dental implant-bone system are complex from a geometric standpoint, FEA has been viewed as the most suitable tool for mechanically analyzing them. A mesh is needed in FEA to divide the whole domain into elements. The process of creating the mesh, elements, their respective nodes, and defining boundary conditions is called "discretization" of the problem domain. (Geng et al. 2001; Richmond, Wright et al. 2005)

The 2D structures are typically meshed with triangular or quadrilateral elements (Figure 6A). These elements may possess two or more nodes per side, with one node at each vertex. If nodes are only placed at the vertices, the element is called linear because a line function describes the element geometry and how the displacement field will vary along an element edge. (Richmond, Wright et al. 2005)

Typical 3D elements include eight-node bricks, six-node wedges, five-node pyramids, and four-node tetrahedral (Figure 6B). With increasing numbers of elements and nodes, the model becomes more complex and computationally more difficult and lengthy. (Assuncao et al. 2009)

Critical Aspects for Mechanical Simulation in Dental Implantology 91

(Figure 7). Some authors considered that the convergence criterion between meshes refinement was a change of less than 5 (Li, Kong et al. 2009; Lin et al. 2010) or 6% (Huang, Hsu et al. 2008; Huang et al. 2009) in the maximum simulated stress of the bone-implant edge. Alternatively, a mesh can be considered converged when the rate of density change

A attempting to successfully replicate the clinical situation that an implant might encounter in the oral environment, it is also important to understand and correctly reproduce the natural forces that are exerted throughout the system. These forces are mainly the result of the masticatory muscles action, and are related to the amount, frequency and duration of the

Forces acting on dental implants possess both magnitude and direction, and are referred to as vector quantities. For accurate predictions on the implant-bone behavior, it is essential to determine realistic in vivo loading magnitudes and directions. However, at each specific bite point, bite forces can be generated in a wide range of directions. Also, although bite forces are generally act downward, toward the apex of the implant and thereby tending to compress the implant into the alveolar bone, tensile forces and bending moments may also be present depending on where the bite force is applied relative to the implant-supported prosthesis. This fact is even more important when the investigation aims to simulate multiple-implant treatment modalities, because of the geometric factors involving the restorations that are linking the implants, such as the existence of distal cantilevers. (Mericske-Stern et al. 1996; Fontijn-Tekamp et al. 1998) Nevertheless, although for implants used in single-tooth replacement simulations the in vivo forces ought to replicate the forces exerted on natural teeth, factors such as the width of the crown occlusal table, the height of the abutment above the bone level, and the angulation of the implant with respect to the

was less than 10-5 between subsequent iterations. (Hasan, Rahimi et al. 2011)

occlusal plane will affect the value of the moment on the single-tooth implant.

A significant amount of investigations have assumed the direction of the load applied to the implant to be horizontal, vertical and oblique. However, the rationale for use of an oblique loading condition is based on finding the vertical (axial) forces directed to the implant system that are relatively low and well tolerated in comparison to oblique forces, which generate bending moments (Figure 8A). The combination of axial and transverse loading, termed as mixed loading, simulates practical conditions where the actual applied force may be inclined with respect to the implant axis and components can be solved in the longitudinal and transverse directions. (Shunmugasamy et al. 2011) These oblique forces have been considered more clinically realistic in FEA than vertical ones. (Chun et al. 2002; Pierrisnard et al. 2002)

Several investigators have tried to gain insight into implant loading magnitudes by performing tests using experimental, analytical, and computer-based simulations of various implant-supported prosthesis types. (Mericske-Stern, Piotti et al. 1996; Fontijn-Tekamp, Slagter et al. 1998; Duyck et al. 2000; Morneburg and Proschel 2002) Bite forces ranging from 50 to 400 N in the molar regions and 25 to 170 N in the incisor areas have been reported. These variations are influenced by patient's gender, muscle mass, exercise, diet, bite location, parafunction, number of teeth and implants, type of implant-supported prosthesis, physical status and age. (Duyck, Van Oosterwyck et al. 2000; Morneburg and Proschel 2002;

**6. Loading and boundary conditions** 

masticatory function.

Eser, Akca et al. 2009)

Fig. 6. (A) 2D Finite element model meshed with triangular elements. (B) 3D Finite element model meshed with tetrahedral elements.

To address this problem, researchers often devise strategies to minimize computational expense, such as taking advantage of symmetry when possible by modeling only half the structure, using a generally coarse mesh with finer elements only near regions of geometric complexity or high stress and strain, and/or using a 2D model when it suffices. (Richmond, Wright et al. 2005)

A convergence study of the model is always important to verify the mesh quality. A measurement of convergence is the degree of difference in the total strain energy between two successive mesh refinements. (Hasan, Rahimi et al. 2011) When the difference in energy is less than some tolerable limit specified by the user, the solution is considered converged

Fig. 7. Maximum principal stress (σmax) in hypothetic convergence testing of mandible FE models meshed by 0.6 mm elements with no refinement (mesh 1) and with refinement levels 2, 3, 4 and 5 at a posterior implant region (meshes 2, 3, 4 and 5, respectively).

(Figure 7). Some authors considered that the convergence criterion between meshes refinement was a change of less than 5 (Li, Kong et al. 2009; Lin et al. 2010) or 6% (Huang, Hsu et al. 2008; Huang et al. 2009) in the maximum simulated stress of the bone-implant edge. Alternatively, a mesh can be considered converged when the rate of density change was less than 10-5 between subsequent iterations. (Hasan, Rahimi et al. 2011)

#### **6. Loading and boundary conditions**

90 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Fig. 6. (A) 2D Finite element model meshed with triangular elements. (B) 3D Finite element

To address this problem, researchers often devise strategies to minimize computational expense, such as taking advantage of symmetry when possible by modeling only half the structure, using a generally coarse mesh with finer elements only near regions of geometric complexity or high stress and strain, and/or using a 2D model when it suffices. (Richmond,

A convergence study of the model is always important to verify the mesh quality. A measurement of convergence is the degree of difference in the total strain energy between two successive mesh refinements. (Hasan, Rahimi et al. 2011) When the difference in energy is less than some tolerable limit specified by the user, the solution is considered converged

Fig. 7. Maximum principal stress (σmax) in hypothetic convergence testing of mandible FE models meshed by 0.6 mm elements with no refinement (mesh 1) and with refinement levels

2, 3, 4 and 5 at a posterior implant region (meshes 2, 3, 4 and 5, respectively).

model meshed with tetrahedral elements.

Wright et al. 2005)

A attempting to successfully replicate the clinical situation that an implant might encounter in the oral environment, it is also important to understand and correctly reproduce the natural forces that are exerted throughout the system. These forces are mainly the result of the masticatory muscles action, and are related to the amount, frequency and duration of the masticatory function.

Forces acting on dental implants possess both magnitude and direction, and are referred to as vector quantities. For accurate predictions on the implant-bone behavior, it is essential to determine realistic in vivo loading magnitudes and directions. However, at each specific bite point, bite forces can be generated in a wide range of directions. Also, although bite forces are generally act downward, toward the apex of the implant and thereby tending to compress the implant into the alveolar bone, tensile forces and bending moments may also be present depending on where the bite force is applied relative to the implant-supported prosthesis. This fact is even more important when the investigation aims to simulate multiple-implant treatment modalities, because of the geometric factors involving the restorations that are linking the implants, such as the existence of distal cantilevers. (Mericske-Stern et al. 1996; Fontijn-Tekamp et al. 1998) Nevertheless, although for implants used in single-tooth replacement simulations the in vivo forces ought to replicate the forces exerted on natural teeth, factors such as the width of the crown occlusal table, the height of the abutment above the bone level, and the angulation of the implant with respect to the occlusal plane will affect the value of the moment on the single-tooth implant.

A significant amount of investigations have assumed the direction of the load applied to the implant to be horizontal, vertical and oblique. However, the rationale for use of an oblique loading condition is based on finding the vertical (axial) forces directed to the implant system that are relatively low and well tolerated in comparison to oblique forces, which generate bending moments (Figure 8A). The combination of axial and transverse loading, termed as mixed loading, simulates practical conditions where the actual applied force may be inclined with respect to the implant axis and components can be solved in the longitudinal and transverse directions. (Shunmugasamy et al. 2011) These oblique forces have been considered more clinically realistic in FEA than vertical ones. (Chun et al. 2002; Pierrisnard et al. 2002)

Several investigators have tried to gain insight into implant loading magnitudes by performing tests using experimental, analytical, and computer-based simulations of various implant-supported prosthesis types. (Mericske-Stern, Piotti et al. 1996; Fontijn-Tekamp, Slagter et al. 1998; Duyck et al. 2000; Morneburg and Proschel 2002) Bite forces ranging from 50 to 400 N in the molar regions and 25 to 170 N in the incisor areas have been reported. These variations are influenced by patient's gender, muscle mass, exercise, diet, bite location, parafunction, number of teeth and implants, type of implant-supported prosthesis, physical status and age. (Duyck, Van Oosterwyck et al. 2000; Morneburg and Proschel 2002; Eser, Akca et al. 2009)

Critical Aspects for Mechanical Simulation in Dental Implantology 93

Shunmugasamy and coworkers (Shunmugasamy, Gupta et al. 2011) simulated different macro-geometries of unit implants and fixed the base of the outer sides of the cortical bone.

Fig. 9. (A) 3D CT-based model of an upper central incisor extraction socket and an implant model positioned inside of the alveolus. (B) Proximal view. (C) Final model: only the

The validation of the results is the final and most important step in FEA, depending on the degree to which the system reflects its biologic influence reality. (Richmond, Wright et al. 2005) For instance, a model would be precise but inaccurate if the mesh is exceedingly dense but the loading and boundary conditions are unrealistic. (Richmond, Wright et al. 2005)

Validation of FEA entails comparing the behavior of the model with *in vivo* or *in vitro* data gathered from parts of the modeled structures. A combination of *in vitro* and *in vivo* experimentation potentially offers the best validation. *In vitro* validation allows one to carefully control the loads and boundary conditions in order to assess the validity of the model's geometry and elastic properties. (Richmond, Wright et al. 2005) Studies of the consistency of numeric models and their agreement with biologic data are scarce (Mellal et al. 2004; Ozan et al. 2010) and the level of agreement and consistency between different engineering methods, especially those regarding quantified stress/strain, remains a concern.

The analysis criteria widely used in implant prosthodontics are: (1) von Mises stress (σvM) (Yokoyama, Wakabayashi et al. 2005; Assuncao, Tabata et al. 2008; Huang, Hsu et al. 2008; Lim, Chew et al. 2008; Rubo and Souza 2008; Schrotenboer et al. 2008; Assuncao et al. 2009; Assuncao et al. 2009; Chaichanasiri, Nanakorn et al. 2009; Ding et al. 2009; Fazel, Aalai et al. 2009; Gomes et al. 2009; Kong et al. 2009; Li, Kong et al. 2009; Qian et al. 2009; Abreu et al. 2010; Ao et al. 2010; Caglar et al. 2010; Eraslan and Inan 2010; Eser et al. 2010; Faegh and Muftu 2010; Gomes et al. 2010; Miyamoto et al. 2010; Okumura, Stegaroiu et al. 2010; Pessoa, Muraru et al. 2010; Sagat, Yalcin et al. 2010; Takahashi et al. 2010; Teixeira et al. 2010; Winter

relevant segment were included.

(Iplikcioglu et al. 2003)

**7. Validation and interpretation of FE modeling** 

**8. Analysis criteria involved in steps of the FEA** 

Fig. 8. (A**)** Transversal loading being applied on the lingual surface of the anterior crown. (B) Boundary conditions of the anterior maxilla at the upper and lateral sides. The green region represent the fixed support constrained of *x*, *y* and *z* directions (displacement = 0).

Ideally, the entire jawbone structure should be evaluated for its contribution to the force exerted onto the dental implant. However, since the simulation of the whole mandibular and maxillary bone is very elaborate, smaller models have been proposed. (Lin, Chang et al. 2007; Lin et al. 2008; Natali, Carniel et al. 2010; Pessoa et al. 2010). (Pierrisnard et al. 2003; Tada, Stegaroiu et al. 2003) It can be explained by the Saint Venant principles, the actual force system may be replaced by a equivalent load system and the distribution of stress and strains are only affected near the region of loading. (Ugural 2003) Thus, if the interest in the study is on the biomechanical peri-implant environment, the modeling of no more than the relevant segment of the bone is required (Figure 9). This procedure allows saving computing and modeling time. In addition, Teixeira et al. (Teixeira et al. 1998) demonstrated by a 3D FEA that modeling the mandible at a distance greater than 4.2 mm medially or distally from the implant did not result in any significant improvement in accuracy. Hence, besides the application of a proper implant loading, the determination of restrictions to the model displacement compatible with the anatomic segment to be simulated is advised. Obviously, if necessary, expanding the domain of the model could reduce the effect of inaccurate modeling of the boundary condition. (Zhou et al. 1999)

It has been common to apply fixed constraints to the uper region of the maxilla and to its lateral sides during the construction of the FE model to simulate the continuity of the bone (Figure 8B). (Van Staden, Guan et al. 2006) These fixed supports represent the constrained of *x*, *y* and *z* directions (displacement = 0). (Chaichanasiri et al. 2009; Hsu, Fuh et al. 2009; Faegh and Muftu 2010; Wu, Liao et al. 2010)

Ishigaki and coworkers (Ishigaki et al. 2003) determined the directions of displacement constrains, which were applied to the jawbone according to the angles of the closing pathways of chopping type and grinding type chewing patters, where the models were constrained at the base of the maxillary first molar to avoid sliding of the entire model.

Fig. 8. (A**)** Transversal loading being applied on the lingual surface of the anterior crown. (B) Boundary conditions of the anterior maxilla at the upper and lateral sides. The green region represent the fixed support constrained of *x*, *y* and *z* directions (displacement = 0).

Ideally, the entire jawbone structure should be evaluated for its contribution to the force exerted onto the dental implant. However, since the simulation of the whole mandibular and maxillary bone is very elaborate, smaller models have been proposed. (Lin, Chang et al. 2007; Lin et al. 2008; Natali, Carniel et al. 2010; Pessoa et al. 2010). (Pierrisnard et al. 2003; Tada, Stegaroiu et al. 2003) It can be explained by the Saint Venant principles, the actual force system may be replaced by a equivalent load system and the distribution of stress and strains are only affected near the region of loading. (Ugural 2003) Thus, if the interest in the study is on the biomechanical peri-implant environment, the modeling of no more than the relevant segment of the bone is required (Figure 9). This procedure allows saving computing and modeling time. In addition, Teixeira et al. (Teixeira et al. 1998) demonstrated by a 3D FEA that modeling the mandible at a distance greater than 4.2 mm medially or distally from the implant did not result in any significant improvement in accuracy. Hence, besides the application of a proper implant loading, the determination of restrictions to the model displacement compatible with the anatomic segment to be simulated is advised. Obviously, if necessary, expanding the domain of the model could reduce the effect of inaccurate

It has been common to apply fixed constraints to the uper region of the maxilla and to its lateral sides during the construction of the FE model to simulate the continuity of the bone (Figure 8B). (Van Staden, Guan et al. 2006) These fixed supports represent the constrained of *x*, *y* and *z* directions (displacement = 0). (Chaichanasiri et al. 2009; Hsu, Fuh et al. 2009;

Ishigaki and coworkers (Ishigaki et al. 2003) determined the directions of displacement constrains, which were applied to the jawbone according to the angles of the closing pathways of chopping type and grinding type chewing patters, where the models were constrained at the base of the maxillary first molar to avoid sliding of the entire model.

modeling of the boundary condition. (Zhou et al. 1999)

Faegh and Muftu 2010; Wu, Liao et al. 2010)

Shunmugasamy and coworkers (Shunmugasamy, Gupta et al. 2011) simulated different macro-geometries of unit implants and fixed the base of the outer sides of the cortical bone.

Fig. 9. (A) 3D CT-based model of an upper central incisor extraction socket and an implant model positioned inside of the alveolus. (B) Proximal view. (C) Final model: only the relevant segment were included.

#### **7. Validation and interpretation of FE modeling**

The validation of the results is the final and most important step in FEA, depending on the degree to which the system reflects its biologic influence reality. (Richmond, Wright et al. 2005) For instance, a model would be precise but inaccurate if the mesh is exceedingly dense but the loading and boundary conditions are unrealistic. (Richmond, Wright et al. 2005)

Validation of FEA entails comparing the behavior of the model with *in vivo* or *in vitro* data gathered from parts of the modeled structures. A combination of *in vitro* and *in vivo* experimentation potentially offers the best validation. *In vitro* validation allows one to carefully control the loads and boundary conditions in order to assess the validity of the model's geometry and elastic properties. (Richmond, Wright et al. 2005) Studies of the consistency of numeric models and their agreement with biologic data are scarce (Mellal et al. 2004; Ozan et al. 2010) and the level of agreement and consistency between different engineering methods, especially those regarding quantified stress/strain, remains a concern. (Iplikcioglu et al. 2003)

### **8. Analysis criteria involved in steps of the FEA**

The analysis criteria widely used in implant prosthodontics are: (1) von Mises stress (σvM) (Yokoyama, Wakabayashi et al. 2005; Assuncao, Tabata et al. 2008; Huang, Hsu et al. 2008; Lim, Chew et al. 2008; Rubo and Souza 2008; Schrotenboer et al. 2008; Assuncao et al. 2009; Assuncao et al. 2009; Chaichanasiri, Nanakorn et al. 2009; Ding et al. 2009; Fazel, Aalai et al. 2009; Gomes et al. 2009; Kong et al. 2009; Li, Kong et al. 2009; Qian et al. 2009; Abreu et al. 2010; Ao et al. 2010; Caglar et al. 2010; Eraslan and Inan 2010; Eser et al. 2010; Faegh and Muftu 2010; Gomes et al. 2010; Miyamoto et al. 2010; Okumura, Stegaroiu et al. 2010; Pessoa, Muraru et al. 2010; Sagat, Yalcin et al. 2010; Takahashi et al. 2010; Teixeira et al. 2010; Winter

Critical Aspects for Mechanical Simulation in Dental Implantology 95

component design, further improvement is achievable through better understanding the different dental implant treatment modalities' biomechanical behavior. While the field has gained key knowledge in model fabrication (experimental designing) and improvements in all steps here described concerning the basic FEA experimental design, it is expected that all steps will be further refined based on future gains in computer power and correlations made between modeling results and clinical observation, ultimately providing improved

Abreu, R. T., A. O. Spazzin, P. Y. Noritomi, et al. (2010). Influence of material of

Adell, R., B. Eriksson, U. Lekholm, et al. (1990). Long-term follow-up study of

Ao, J., T. Li, Y. Liu, et al. (2010). Optimal design of thread height and width on an

Assuncao, W. G., V. A. Barao, L. F. Tabata, et al. (2009). Biomechanics studies in dentistry:

Assuncao, W. G., L. F. Tabata, V. A. Barao, et al. (2008). Comparison of stress distribution

Bevilacqua, M., T. Tealdo, M. Menini, et al. (2011). The influence of cantilever length and

Bozini, T., H. Petridis&K. Garefis (2011). A meta-analysis of prosthodontic complication

Vol.20, No.4, pp. 1173-1177, 1536-3732 (Electronic) 1049-2275 (Linking). Assuncao, W. G., E. A. Gomes, V. A. Barao, et al. (2009). Stress analysis in simulation models

overdenture-retaining bar with vertical misfit on three-dimensional stress distribution, *Journal of prosthodontics: official journal of the American College of Prosthodontists*, Vol.19, No.6, pp. 425-431, 1532-849X (Electronic) 1059-941X

osseointegrated implants in the treatment of totally edentulous jaws, *The International journal of oral & maxillofacial implants*, Vol.5, No.4, pp. 347-359, 0882-

immediately loaded cylinder implant: a finite element analysis, *Computers in biology and medicine*, Vol.40, No.8, pp. 681-686, 1879-0534 (Electronic) 0010-4825 (Linking). Assuncao, W. G., V. A. Barao, L. F. Tabata, et al. (2009). Comparison between complete

denture and implant-retained overdenture: effect of different mucosa thickness and resiliency on stress distribution, *Gerodontology*, Vol.26, No.4, pp. 273-281, 1741-2358

bioengineering applied in oral implantology, *The Journal of craniofacial surgery*,

with or without implant threads representation, *The International journal of oral & maxillofacial implants*, Vol.24, No.6, pp. 1040-1044, 0882-2786 (Print) 0882-2786

between complete denture and implant-retained overdenture-2D FEA, *J Oral Rehabil*, Vol.35, No.10, pp. 766-774, 1365-2842 (Electronic) 0305-182X (Linking). Assuncao, W. G., L. F. Tabata, V. A. Barao, et al. (2008). Comparison of stress distribution

between complete denture and implant-retained overdenture-2D FEA, *Journal of oral rehabilitation*, Vol.35, No.10, pp. 766-774, 1365-2842 (Electronic) 0305-182X

implant inclination on stress distribution in maxillary implant-supported fixed dentures, *The Journal of prosthetic dentistry*, Vol.105, No.1, pp. 5-13, 1097-6841

rates of implant-supported fixed dental prostheses in edentulous patients after an

care for patients in need of oral rehabilitation.

2786 (Print) 0882-2786 (Linking).

(Electronic) 0734-0664 (Linking).

(Electronic) 0022-3913 (Linking).

**10. References** 

(Linking).

(Linking).

(Linking).

et al. 2010; Bevilacqua et al. 2011; Burak Ozcelik, Ersoy et al. 2011; de Almeida, Rocha et al. 2011; Okumura et al. 2011), (2) maximum and minimum principal elastic strain (Єmax and Єmin) (Saab et al. 2007; Qian, Todo et al. 2009; Chou, Muftu et al. 2010; Danza, Quaranta et al. 2010; Eser, Tonuk et al. 2010; Limbert et al. 2010; Okumura, Stegaroiu et al. 2010; Pessoa, Muraru et al. 2010; de Almeida, Rocha et al. 2011) and (3) maximum and minimum principal stress (σmax and σmin). (de Almeida, Rocha et al. 2010; Degerliyurt et al. 2010; Hudieb et al. 2010; Jofre et al. 2010; Wu, Liao et al. 2010; de Almeida, Rocha et al. 2011)

Although the majority of the studies have used the σvM for evaluation of bone interface stress, several studies suggest that the magnitudes of the concentrations should be presented in the σmax for evaluation of stress distribution in a brittle structure as bone, (de Almeida, Rocha et al. 2010; Degerliyurt, Simsek et al. 2010; Wu, Liao et al. 2010) as this criterion offers the possibility of making a distinction between tensile stress and compressive stress. (Degerliyurt, Simsek et al. 2010) In addition, displacement components of specific points may provide information about the deformation of the model and facilitate interpretation of the results. Principal stress values of fragile compact bone can be compared with its ultimate compressive strength and ultimate tensile strength values. (Ciftci and Canay 2000; Furmanski et al. 2009)

In fact, interfacial failure and bone resorption under different stress types are attributed to different mechanisms. Accordingly, it may be erroneous to emphasize the peak compressive or σvM without considering the risks of the tensile and the shear stresses at the interface. (Hudieb, Wakabayashi et al. 2010) However, when the titanium component is available as abutment, screw and implant, which are ductile structures, σvM is a recommended analysis criterion. (Cattaneo et al. 2005; Degerliyurt, Simsek et al. 2010; Wu, Liao et al. 2010)

Considering the FEA characteristics described above, the main advantage is the virtual simulation of real structures that are difficult to be clinically evaluated, i.e. stress and strain distribution on periimplant bone. Moreover, it is a low cost alternative in comparison to other *in vitro* methods since only a virtual model is used. Limitations of the FEA models include mainly the patient specific anatomy, and parameters such as the wet environment and damage accumulation under repetitive loading.

#### **9. Summary and final remarks**

The success of a dental implant depends on a variety of biomechanical factors including the design and position of the implant, implant-abutment connection, cantilever length, surface roughness, bone quality and type, depth of insertion, arch configuration, the nature of boneimplant interface, and occlusal conditions. (Randow, Glantz et al. 1986; Adell, Eriksson et al. 1990; van Steenberghe, Lekholm et al. 1990; Bozkaya and Muftu 2004; Bozkaya, Muftu et al. 2004; Misch et al. 2005; Yokoyama, Wakabayashi et al. 2005; De Smet et al. 2007; Abreu, Spazzin et al. 2010; Turkyilmaz 2011) All these biomechanical factors have been simulated by FEA in previous studies. While an increased number and quality of investigations has been published over the last decade, the results are often contradictory due to differences in model construction and meshing.

The biomechanical behavior of all components used in implant prosthodontics has been regarded as an important factor in determining the life expectancy of the restoration. Although substantial improvement has been made concerning implant/restorative component design, further improvement is achievable through better understanding the different dental implant treatment modalities' biomechanical behavior. While the field has gained key knowledge in model fabrication (experimental designing) and improvements in all steps here described concerning the basic FEA experimental design, it is expected that all steps will be further refined based on future gains in computer power and correlations made between modeling results and clinical observation, ultimately providing improved care for patients in need of oral rehabilitation.

#### **10. References**

94 Finite Element Analysis – From Biomedical Applications to Industrial Developments

et al. 2010; Bevilacqua et al. 2011; Burak Ozcelik, Ersoy et al. 2011; de Almeida, Rocha et al. 2011; Okumura et al. 2011), (2) maximum and minimum principal elastic strain (Єmax and Єmin) (Saab et al. 2007; Qian, Todo et al. 2009; Chou, Muftu et al. 2010; Danza, Quaranta et al. 2010; Eser, Tonuk et al. 2010; Limbert et al. 2010; Okumura, Stegaroiu et al. 2010; Pessoa, Muraru et al. 2010; de Almeida, Rocha et al. 2011) and (3) maximum and minimum principal stress (σmax and σmin). (de Almeida, Rocha et al. 2010; Degerliyurt et al. 2010; Hudieb et al.

Although the majority of the studies have used the σvM for evaluation of bone interface stress, several studies suggest that the magnitudes of the concentrations should be presented in the σmax for evaluation of stress distribution in a brittle structure as bone, (de Almeida, Rocha et al. 2010; Degerliyurt, Simsek et al. 2010; Wu, Liao et al. 2010) as this criterion offers the possibility of making a distinction between tensile stress and compressive stress. (Degerliyurt, Simsek et al. 2010) In addition, displacement components of specific points may provide information about the deformation of the model and facilitate interpretation of the results. Principal stress values of fragile compact bone can be compared with its ultimate compressive strength and ultimate tensile strength values. (Ciftci and

In fact, interfacial failure and bone resorption under different stress types are attributed to different mechanisms. Accordingly, it may be erroneous to emphasize the peak compressive or σvM without considering the risks of the tensile and the shear stresses at the interface. (Hudieb, Wakabayashi et al. 2010) However, when the titanium component is available as abutment, screw and implant, which are ductile structures, σvM is a recommended analysis

Considering the FEA characteristics described above, the main advantage is the virtual simulation of real structures that are difficult to be clinically evaluated, i.e. stress and strain distribution on periimplant bone. Moreover, it is a low cost alternative in comparison to other *in vitro* methods since only a virtual model is used. Limitations of the FEA models include mainly the patient specific anatomy, and parameters such as the wet environment

The success of a dental implant depends on a variety of biomechanical factors including the design and position of the implant, implant-abutment connection, cantilever length, surface roughness, bone quality and type, depth of insertion, arch configuration, the nature of boneimplant interface, and occlusal conditions. (Randow, Glantz et al. 1986; Adell, Eriksson et al. 1990; van Steenberghe, Lekholm et al. 1990; Bozkaya and Muftu 2004; Bozkaya, Muftu et al. 2004; Misch et al. 2005; Yokoyama, Wakabayashi et al. 2005; De Smet et al. 2007; Abreu, Spazzin et al. 2010; Turkyilmaz 2011) All these biomechanical factors have been simulated by FEA in previous studies. While an increased number and quality of investigations has been published over the last decade, the results are often contradictory due to differences in

The biomechanical behavior of all components used in implant prosthodontics has been regarded as an important factor in determining the life expectancy of the restoration. Although substantial improvement has been made concerning implant/restorative

criterion. (Cattaneo et al. 2005; Degerliyurt, Simsek et al. 2010; Wu, Liao et al. 2010)

2010; Jofre et al. 2010; Wu, Liao et al. 2010; de Almeida, Rocha et al. 2011)

Canay 2000; Furmanski et al. 2009)

and damage accumulation under repetitive loading.

**9. Summary and final remarks** 

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switching on crestal bone stress levels: a finite element analysis, *Journal of periodontology*, Vol.79, No.11, pp. 2166-2172, 0022-3492 (Print) 0022-3492 (Linking). Sevimay, M., F. Turhan, M. A. Kilicarslan, et al. (2005). Three-dimensional finite element

analysis of the effect of different bone quality on stress distribution in an implantsupported crown, *The Journal of prosthetic dentistry*, Vol.93, No.3, pp. 227-234, 0022-

factors on the immediate biomechanical surrounding for a series of dental implant designs, *J Biomech Eng*, Vol.133, No.3, pp. 031005, 1528-8951 (Electronic) 0148-0731

prostheses supported by four or six implants: a three-dimensional finite element analysis, *The International journal of oral & maxillofacial implants*, Vol.25, No.2, pp.

on implant placement: an anisotropic bone stress analysis, *Braz Dent J*, Vol.21, No.6,

strain adaptive bone modeling and remodeling, *The Journal of prosthetic dentistry*,


**5**

*Brazil* 

**Evaluation of Stress Distribution**

**in Implant-Supported Restoration**

**Under Different Simulated Loads** 

Paulo Roberto R. Ventura1, Isis Andréa V. P. Poiate1,

*1Federal Fluminense University, 2Pontifical Catholic University,* 

Edgard Poiate Junior2 and Adalberto Bastos de Vasconcellos1

Technical and scientific developments, in the form of osseointegrated implants, have made Restorative Dentistry evolve to become a highly successful form of treatment (Adell *et al*., 1981; Binon, 2000), considered as an additional safe tool oral surgeons have at their disposal to rehabilitate partially and totally edentulous patients. However, many negative factors can interfere in the treatment, causing clinical and functional complications that may culminate in the loss of osseointegration (Adell *et al.,* 1981; Jemt *et al.*, 1989; Naert *et al.*, 2001a, 2001b; Isidor, 2006). The attachment of the crown to the abutment and of the latter to the implant, the loosening of the fixation screw, and even its fracture may arise from a poor distribution of occlusal loads in the prosthesis-implant

The finite element method (FEM) has been applied to the prognosis of stress distribution in both the implant and its interface with the adjacent bone, for a comparison not only of several geometries and applied loads (Hansson, 1999; Bozkaya, Muftu, Muftu, 2004), but also of different clinical situations (Van Oostewyck *et al.,* 2002) and prosthesis designs (Papavasiliou *et al.,* 1996). The study of stresses using the FEM is basically a virtual simulation of two- or three-dimensional mathematical models, in which all biological and material structures involved can be discretized, that is, subdivided into smaller structures

Considering the physical alterations in components and the peri-implant tissue alterations to which they are subject under an unbalanced distribution of occlusal loads, it is fundamental to evaluate the distribution of the stresses generated by functional and parafunctional masticatory loads on implant-supported restorations to reach a better understanding of their possible biomechanical etiologic agents. The purpose of this study was to analyze the stress distribution in single implant-supported restorations, as well as in the peri-implant bone tissue, by means of a three-dimensional model, using the finite

that preserve individual anatomical and mechanical features.

**1. Introduction** 

set (Binon, 1994).

element method.


## **Evaluation of Stress Distribution in Implant-Supported Restoration Under Different Simulated Loads**

Paulo Roberto R. Ventura1, Isis Andréa V. P. Poiate1, Edgard Poiate Junior2 and Adalberto Bastos de Vasconcellos1 *1Federal Fluminense University, 2Pontifical Catholic University, Brazil* 

#### **1. Introduction**

106 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Winter, W., S. Mohrle, S. Holst, et al. (2010). Bone loading caused by different types of

Wu, T., W. Liao, N. Dai, et al. (2010). Design of a custom angled abutment for dental

Yang, H. S., L. A. Lang&D. A. Felton (1999). Finite element stress analysis on the effect of

Yang, H. S., L. A. Lang, A. Molina, et al. (2001). The effects of dowel design and load

Yokoyama, S., N. Wakabayashi, M. Shiota, et al. (2005). Stress analysis in edentulous

Zarone, F., A. Apicella, L. Nicolais, et al. (2003). Mandibular flexure and stress build-up in

Zhou, X., Z. Zhao, M. Zhao, et al. (1999). [The boundary design of mandibular model by

(Linking).

(Linking).

9290 (Linking).

3913 (Print) 0022-3913 (Linking).

0022-3913 (Print) 0022-3913 (Linking).

0882-2786 (Print) 0882-2786 (Linking).

29-32, 1000-1182 (Print) 1000-1182 (Linking).

misfits of implant-supported fixed dental prostheses: a three-dimensional finite element analysis based on experimental results, *The International journal of oral & maxillofacial implants*, Vol.25, No.5, pp. 947-952, 0882-2786 (Print) 0882-2786

implants using computer-aided design and nonlinear finite element analysis, *Journal of biomechanics*, Vol.43, No.10, pp. 1941-1946, 1873-2380 (Electronic) 0021-

splinting in fixed partial dentures, *J Prosthet Dent*, Vol.81, No.6, pp. 721-728, 0022-

direction on dowel-and-core restorations, *J Prosthet Dent*, Vol.85, No.6, pp. 558-567,

mandibular bone supporting implant-retained 1-piece or multiple superstructures, *The International journal of oral & maxillofacial implants*, Vol.20, No.4, pp. 578-583,

mandibular full-arch fixed prostheses supported by osseointegrated implants, *Clinical oral implants research*, Vol.14, No.1, pp. 103-114, 0905-7161 (Print) 0905-7161

means of the three-dimensional finite element method], *Hua xi kou qiang yi xue za zhi = Huaxi kouqiang yixue zazhi = West China journal of stomatology*, Vol.17, No.1, pp. Technical and scientific developments, in the form of osseointegrated implants, have made Restorative Dentistry evolve to become a highly successful form of treatment (Adell *et al*., 1981; Binon, 2000), considered as an additional safe tool oral surgeons have at their disposal to rehabilitate partially and totally edentulous patients. However, many negative factors can interfere in the treatment, causing clinical and functional complications that may culminate in the loss of osseointegration (Adell *et al.,* 1981; Jemt *et al.*, 1989; Naert *et al.*, 2001a, 2001b; Isidor, 2006). The attachment of the crown to the abutment and of the latter to the implant, the loosening of the fixation screw, and even its fracture may arise from a poor distribution of occlusal loads in the prosthesis-implant set (Binon, 1994).

The finite element method (FEM) has been applied to the prognosis of stress distribution in both the implant and its interface with the adjacent bone, for a comparison not only of several geometries and applied loads (Hansson, 1999; Bozkaya, Muftu, Muftu, 2004), but also of different clinical situations (Van Oostewyck *et al.,* 2002) and prosthesis designs (Papavasiliou *et al.,* 1996). The study of stresses using the FEM is basically a virtual simulation of two- or three-dimensional mathematical models, in which all biological and material structures involved can be discretized, that is, subdivided into smaller structures that preserve individual anatomical and mechanical features.

Considering the physical alterations in components and the peri-implant tissue alterations to which they are subject under an unbalanced distribution of occlusal loads, it is fundamental to evaluate the distribution of the stresses generated by functional and parafunctional masticatory loads on implant-supported restorations to reach a better understanding of their possible biomechanical etiologic agents. The purpose of this study was to analyze the stress distribution in single implant-supported restorations, as well as in the peri-implant bone tissue, by means of a three-dimensional model, using the finite element method.

Evaluation of Stress Distribution in

also favoring the same spatial positioning.

Implant-Supported Restoration Under Different Simulated Loads 109

for cemented prosthesis with a 2 mm cervical collar (sectioned at the occlusal end to obtain a

The implant was installed with the platform cervical boundary coinciding with the boundary of the bone crest, and the axial, mesio-distal, and labio-palatine positions equivalent to the root portion of the original sound tooth. Thus, the prosthetic metalloceramic crown could be constructed from a cervical prolongation of the sound tooth crown,

The model geometries were used to generate volumetric meshes using the tetrahedral element topology (Tet4), that is, a pyramidal element of four faces with six edges and one node on each edge. Elements with edges of 0.05 mm were used in regions of high curvature,

To create the volumetric meshes, it was necessary to proceed from the smaller or most internal structure to the larger or most external, that is, in this case the volumetric mesh was created first in the screw, and then the sequence adopted continued to the extremities, following the procedures in Poiate et al. (2008, 2009a, 2009b, 2011). This procedure assures perfect congruence in the FEM. The degree of discretization in the model derives from studies on the convergence of results and from the capacity of the computer used in the analyses, in order to assure adequate density in the finite element mesh for each model, describing the geometry of different components in a rather realistic way. The discretization detailed above corresponded to the maximum discretization established on a Pentium Core Duo 1.6 MHz computer, with 3.0 GB of RAM and a 160 GB hard disk. Thus, the model

The occlusal pattern adopted in this study was the cusp-marginal ridge (one tooth to two teeth), thus justifying the positioning of the simulated occlusal loads. Therefore, four load conditions were applied, with different inclinations and points of application for a total

a. load distributed among 38 node points, 19 of which on an area of 0.85 mm² of the vestibular cusp, and 19 on an area of 0.75 mm² of the lingual cusp (Kumagai *et al*., 1999), with an inclination of 45º, but with the resultant (291.36 N) parallel to the tooth long

final length of 3.55 mm); and one titanium fixation screw with a torque of 20 Ncm.

Fig. 1. Development of the model based on a sound tooth (Poiate, 2007).

small size, or regions of transition between structures of up to 0.3 mm.

discretization generated 164,848 node points and 1,011,727 elements.

axis, aiming at evaluating the effect of the axial force.

**2.1.2 Load conditions** 

static load of 291.36 N (Ferrario *et al.*, 2004):

#### **2. Material and methods**

The finite element method is a technique to solve a complex problem subdividing it into a set of smaller and simpler problems (elements), which can be solved using numerical techniques. In other words, it is a method in which a formulation of equations for each finite element combines elements to get a solution for the whole problem, rather than solving it in just one operation (Holmgren *et al*., 1998).

To divide the region of interest into elements, it is necessary to generate a mesh. The process to generate the mesh, the elements, their respective nodes, and the boundary conditions is referred to as the problem's own "discretization" (Geng et al., 2001).

The method of analysis with three-dimensional finite elements (3D FEA) was used. It is composed of three sequential and well-defined phases: pre-processing, processing, and post-processing, all described in detail below.

#### **2.1 Pre-processing**

In the pre-processing phase, a mathematical model of the object or structure under study is developed using computer-aided design (CAD). In the generation of the FEM, this geometry is discretized. Next, a physical phase is considered, in which properties equal to those of the materials or structures of the real model they represent are attributed to the elements of the mesh, and some hypotheses are formulated to make the analysis (linear elasticity) viable or just to address the lack of knowledge about the behavior of the material or structure represented (homogeneous and isotropic). Finally, the conditions of model fixation and the characteristics of force (load) are also applied.

#### **2.1.1 The development of a geometric model**

A model simulating a single implant-supported prosthesis was developed in the region of the second upper premolar. It is composed of a metallo-ceramic crown on a machinedsurface external hexagon implant of regular diameter (Master Screw, Conexão Sistemas de Prótese Ltda – SP, Brasil). The development of this prosthesis model and of the peri-implant support bone was carried out based on the sound tooth model constructed by Poiate (2007). The model was generated with the MSC/PATRAN 2005r2 software and the simulation was carried out with the MSC/NASTRAN 2005r1 software (The MacNeal-Schwendler Corporation - USA). To make the support structures, the anatomical dimensions of the cortical and cancelous bone were based on an image of a vestibulo-lingual cross-section of the upper premolar region presented by Berkovitz, Holland and Moxham (2004). This image was digitized with a high-resolution scanner. The vestibulo-lingual dimensions of the cortical and cancelous bone were taken by the DigXY 1.2 software, developed to digitize bitmap data, largely employed to digitize x,y coordinates from graphs. The model includes the geometries of cortical and cancelous bone, implant, screw, abutment, infrastructure, and of the crown's ceramic layer.

The dental structure of the original model was removed and models of abutment, fixation screw, and implant were imported, supplied by Conexão Sistemas de Prótese Ltda. The modeling carried out included components equivalent to one external hexagon implant Master Screw of 3.75 mm in diameter and 13 mm in length; one titanium straight abutment

The finite element method is a technique to solve a complex problem subdividing it into a set of smaller and simpler problems (elements), which can be solved using numerical techniques. In other words, it is a method in which a formulation of equations for each finite element combines elements to get a solution for the whole problem, rather than solving it in

To divide the region of interest into elements, it is necessary to generate a mesh. The process to generate the mesh, the elements, their respective nodes, and the boundary conditions is

The method of analysis with three-dimensional finite elements (3D FEA) was used. It is composed of three sequential and well-defined phases: pre-processing, processing, and

In the pre-processing phase, a mathematical model of the object or structure under study is developed using computer-aided design (CAD). In the generation of the FEM, this geometry is discretized. Next, a physical phase is considered, in which properties equal to those of the materials or structures of the real model they represent are attributed to the elements of the mesh, and some hypotheses are formulated to make the analysis (linear elasticity) viable or just to address the lack of knowledge about the behavior of the material or structure represented (homogeneous and isotropic). Finally, the conditions of model fixation and the

A model simulating a single implant-supported prosthesis was developed in the region of the second upper premolar. It is composed of a metallo-ceramic crown on a machinedsurface external hexagon implant of regular diameter (Master Screw, Conexão Sistemas de Prótese Ltda – SP, Brasil). The development of this prosthesis model and of the peri-implant support bone was carried out based on the sound tooth model constructed by Poiate (2007). The model was generated with the MSC/PATRAN 2005r2 software and the simulation was carried out with the MSC/NASTRAN 2005r1 software (The MacNeal-Schwendler Corporation - USA). To make the support structures, the anatomical dimensions of the cortical and cancelous bone were based on an image of a vestibulo-lingual cross-section of the upper premolar region presented by Berkovitz, Holland and Moxham (2004). This image was digitized with a high-resolution scanner. The vestibulo-lingual dimensions of the cortical and cancelous bone were taken by the DigXY 1.2 software, developed to digitize bitmap data, largely employed to digitize x,y coordinates from graphs. The model includes the geometries of cortical and cancelous bone, implant, screw, abutment, infrastructure, and

The dental structure of the original model was removed and models of abutment, fixation screw, and implant were imported, supplied by Conexão Sistemas de Prótese Ltda. The modeling carried out included components equivalent to one external hexagon implant Master Screw of 3.75 mm in diameter and 13 mm in length; one titanium straight abutment

referred to as the problem's own "discretization" (Geng et al., 2001).

**2. Material and methods** 

**2.1 Pre-processing** 

just one operation (Holmgren *et al*., 1998).

post-processing, all described in detail below.

characteristics of force (load) are also applied.

**2.1.1 The development of a geometric model** 

of the crown's ceramic layer.

for cemented prosthesis with a 2 mm cervical collar (sectioned at the occlusal end to obtain a final length of 3.55 mm); and one titanium fixation screw with a torque of 20 Ncm.

The implant was installed with the platform cervical boundary coinciding with the boundary of the bone crest, and the axial, mesio-distal, and labio-palatine positions equivalent to the root portion of the original sound tooth. Thus, the prosthetic metalloceramic crown could be constructed from a cervical prolongation of the sound tooth crown, also favoring the same spatial positioning.

Fig. 1. Development of the model based on a sound tooth (Poiate, 2007).

The model geometries were used to generate volumetric meshes using the tetrahedral element topology (Tet4), that is, a pyramidal element of four faces with six edges and one node on each edge. Elements with edges of 0.05 mm were used in regions of high curvature, small size, or regions of transition between structures of up to 0.3 mm.

To create the volumetric meshes, it was necessary to proceed from the smaller or most internal structure to the larger or most external, that is, in this case the volumetric mesh was created first in the screw, and then the sequence adopted continued to the extremities, following the procedures in Poiate et al. (2008, 2009a, 2009b, 2011). This procedure assures perfect congruence in the FEM. The degree of discretization in the model derives from studies on the convergence of results and from the capacity of the computer used in the analyses, in order to assure adequate density in the finite element mesh for each model, describing the geometry of different components in a rather realistic way. The discretization detailed above corresponded to the maximum discretization established on a Pentium Core Duo 1.6 MHz computer, with 3.0 GB of RAM and a 160 GB hard disk. Thus, the model discretization generated 164,848 node points and 1,011,727 elements.

#### **2.1.2 Load conditions**

The occlusal pattern adopted in this study was the cusp-marginal ridge (one tooth to two teeth), thus justifying the positioning of the simulated occlusal loads. Therefore, four load conditions were applied, with different inclinations and points of application for a total static load of 291.36 N (Ferrario *et al.*, 2004):

a. load distributed among 38 node points, 19 of which on an area of 0.85 mm² of the vestibular cusp, and 19 on an area of 0.75 mm² of the lingual cusp (Kumagai *et al*., 1999), with an inclination of 45º, but with the resultant (291.36 N) parallel to the tooth long axis, aiming at evaluating the effect of the axial force.

Evaluation of Stress Distribution in

**2.1.5 Formulated hypotheses** 

**2.2 Processing** 

**2.3 Post-processing** 

**2.3.1 Analysis of results** 

Titanium (implant,

the structures.

interrelation with the others (Çiftçi, Canay, 2000; Silva, 2005).

nodes according to the load applied (Holmgren *et al.,* 1998).

**(MPa)**

Table 2. Tensile and compressive strength and bibliographical references.

post-processing to visualize and evaluate results.

**Structure/Material Tensile Strength**

Implant-Supported Restoration Under Different Simulated Loads 111

As in any type of numerical analysis, some hypotheses needed to be formulated to make modeling and problem solving processes viable. It must be emphasized that the numerical tools available to analyze stresses are much more advanced than the knowledge about the mechanical properties of the structures involved. As in Poiate's studies (2005, 2006), all constant structures in the model behaved isotropically (mechanical properties did not vary according to the direction), homogeneously (properties were constant independent of location), and were linearly elastic (strains were directly proportional to the force applied), characterized by two material constants, the Elastic Modulus (*E*) and the Poisson's Ratio (�), with interfaces between structures presumed to be perfectly bonded. Therefore, the interface implant-bone was considered fully osseointegrated, and the cement layer regarded as negligible, since this study did not intend to analyze stresses on that structure nor its

The software used in the analysis was *MSC/NASTRAN* (The MacNeal-Schwendler Corporation – USA), version 2005r1, on a Pentium Dual-Core computer, with a 1.7 GHz processor, a 160 Gb hard disk, and 2 Gb of RAM. This software takes information from the previous phase (pre-processing) into account. Based on the contact relationship among mesh elements, it makes a series of mathematical calculations organized in an algorithm, that is, a sequence of instructions logically ordered to solve a problem in a finite number of phases (Silva, 2005). What is mathematically analyzed, in general, is the displacement of element

The software *MSC/PATRAN* 2005r2 employed in the pre-processing was also used in the

The principal stresses peak values were compared to the values of tensile and compressive strength in the model structures to analyze whether these loads were potentially harmful to

Bone 121 167 Tanaka *et al*. 2003 /

Cortical Bone - 173 Reilly and Burstein,

Cancelous Bone - 167 Çiftçi and Canay, 2000 Ni-Cr 790 - O'Brien, 2005

abutment and screw) 930 - O'Brien, 2005 Feldspathic Ceramic 37.2 150 O'Brien, 2005

**Compressive Strength (MPa)** **Bibliographical Reference** 

O'Brien, 2005

1975


Fig. 2. a) axial load, b) vestibular lever, c) proximal lever and d) torsion.

#### **2.1.3 Fixation conditions**

Contour conditions, also called fixation or bonding conditions, are those determined for the edges or extremities of modeled structures, so that they have some spatial support, with displacement and/or rotation constraint, to allow for the analysis under the applied loads (Bathe, 1996).

For the simulation, the following fixation conditions were applied: in the maxillary sinus, translation restricted to the directions x, y and z, and rotations restricted to the axes x, y and z; at the mesial and distal ends of the cortical and cancelous bone, translation restricted to the direction x and rotations restricted to the axes y and z.

#### **2.1.4 Definition of the mechanical properties of anatomic structures**

Mechanical properties were attributed to each element in the discretized model, such as the elastic modulus (*E*) and the Poisson's ratio (), considering the particularities of all anatomical structures and materials represented by the elements in the composition of the three-dimensional model. Table 1 shows the values applied to each of these properties, as well as their respective bibliographical references.


Table 1. Mechanical properties of structures and bibliographical references.

#### **2.1.5 Formulated hypotheses**

110 Finite Element Analysis – From Biomedical Applications to Industrial Developments

b. oblique load, with an inclination of 45º, distributed among 19 node points on an area of 0.85 mm² of the transverse ridge of the vestibular cusp, aiming at evaluating the

c. load with 0º of inclination distributed among 19 node points on an area of 0.80 mm² of

d. oblique load, with an inclination of 45º in the vestibular direction, distributed among 19 node points on an area of 0.80 mm² of the mesial marginal ridge, aiming at evaluating

the mesial marginal ridge, aiming at evaluating the proximal lever effect.

a) b) c) d)

Contour conditions, also called fixation or bonding conditions, are those determined for the edges or extremities of modeled structures, so that they have some spatial support, with displacement and/or rotation constraint, to allow for the analysis under the applied loads

For the simulation, the following fixation conditions were applied: in the maxillary sinus, translation restricted to the directions x, y and z, and rotations restricted to the axes x, y and z; at the mesial and distal ends of the cortical and cancelous bone, translation restricted to

Mechanical properties were attributed to each element in the discretized model, such as the elastic modulus (*E*) and the Poisson's ratio (), considering the particularities of all anatomical structures and materials represented by the elements in the composition of the three-dimensional model. Table 1 shows the values applied to each of these properties, as

Cortical bone 13.70 0.30 Ko *et al*. (1992) Cancelous bone 1.37 0.30 Ko *et al*. (1992) Ni-Cr 188.00 0.33 Vasconcellos (1999)

abutment and screw) 110.00 0.35 Iplikçioglu and Akça (2002) Feldspathic Ceramic 82.20 0.35 Peyton and Craig (1963)

**Poisson's Ratio** 

**() Bibliographical Reference** 

Fig. 2. a) axial load, b) vestibular lever, c) proximal lever and d) torsion.

**2.1.4 Definition of the mechanical properties of anatomic structures** 

**(GPa)** 

Table 1. Mechanical properties of structures and bibliographical references.

the direction x and rotations restricted to the axes y and z.

well as their respective bibliographical references.

**Structure/Material Elastic Modulus E** 

Titanium (implant,

vestibular lever effect.

**2.1.3 Fixation conditions** 

(Bathe, 1996).

the effect of torsion on the long axis.

As in any type of numerical analysis, some hypotheses needed to be formulated to make modeling and problem solving processes viable. It must be emphasized that the numerical tools available to analyze stresses are much more advanced than the knowledge about the mechanical properties of the structures involved. As in Poiate's studies (2005, 2006), all constant structures in the model behaved isotropically (mechanical properties did not vary according to the direction), homogeneously (properties were constant independent of location), and were linearly elastic (strains were directly proportional to the force applied), characterized by two material constants, the Elastic Modulus (*E*) and the Poisson's Ratio (�), with interfaces between structures presumed to be perfectly bonded. Therefore, the interface implant-bone was considered fully osseointegrated, and the cement layer regarded as negligible, since this study did not intend to analyze stresses on that structure nor its interrelation with the others (Çiftçi, Canay, 2000; Silva, 2005).

#### **2.2 Processing**

The software used in the analysis was *MSC/NASTRAN* (The MacNeal-Schwendler Corporation – USA), version 2005r1, on a Pentium Dual-Core computer, with a 1.7 GHz processor, a 160 Gb hard disk, and 2 Gb of RAM. This software takes information from the previous phase (pre-processing) into account. Based on the contact relationship among mesh elements, it makes a series of mathematical calculations organized in an algorithm, that is, a sequence of instructions logically ordered to solve a problem in a finite number of phases (Silva, 2005). What is mathematically analyzed, in general, is the displacement of element nodes according to the load applied (Holmgren *et al.,* 1998).

#### **2.3 Post-processing**

The software *MSC/PATRAN* 2005r2 employed in the pre-processing was also used in the post-processing to visualize and evaluate results.

#### **2.3.1 Analysis of results**

The principal stresses peak values were compared to the values of tensile and compressive strength in the model structures to analyze whether these loads were potentially harmful to the structures.


Table 2. Tensile and compressive strength and bibliographical references.

Evaluation of Stress Distribution in

ligament and tooth in the latter.

Implant-Supported Restoration Under Different Simulated Loads 113

Fig. 4. Occlusal view. Maximum principal stresses in model under longitudinal load.

value (37.2 MPa), suggesting a stronger tendency for micro-cracks in the ceramic.

contact with the implant platform, with a peak value of 8 MPa.

In Figure 4, it can be observed that the higher compressive stress values are located at points under the area where the load was applied (10 to 55 MPa, well below the tensile strength value of 150 MPa). On the other hand, the increasing tensile stress gradient, between 40 and 60 MPa, concentrated on the region of the central groove, is higher than the tensile strength

To compare and discuss the results obtained by Poiate (2007), the stress values found in the cortical and cancelous bone of a sound tooth model are twice lower than the values found in the implant model (between 5 and 10 MPa), in consequence of the absence of periodontal

In Figure 5, very low compressive stresses can be seen in the region of the cortical bone in

Fig. 5. Perspective view of the cortical bone. Maximum principal stresses in model under

Figure 6 shows that the concentration of compressive stresses on the region in contact with the implant is on a narrow strip on the external surface of the cortical bone. Next to the

longitudinal load. In A, disto-vestibular view; in B, mesio-palatine view.

cancelous bone, it shows a concentration of tensile stresses of 10 MPa.

The von Mises criterion, or theory of the maximum distortion energy, was also used in this study to analyze the results from the implant and its components. It is a rupture criterion to evaluate ductile materials, based on the determination of the maximum distortion energy of a structure, that is, of the energy related to changes in form (as opposed to the energy related to changes in the volume of material) (Beer and Johnston, 1995). The structure failure occurs when, at any point of the material, the distortion energy per unit of volume is higher than the yield strength value obtained for the material in a tensile test.

The scale of stresses (which appear in different colors in the figures) does not have equal intervals. This is a result of the stresses in action on each group of models (different types of load). Thus, a single scale was defined for all models (except for the axially loaded models) to make the comparison easier.

#### **3. Results and discussion**

#### **3.1 Axial load – Load with resultant parallel to the long axis**

In Figure 3 (perspective views), compressive stresses can be seen in the region of the abutment and in the cervical region of the implant, but not intense enough to cause harm (between 1 and 5 MPa).

Again in Figure 3 (internal views), it can be observed the result of the internal maximum principal stresses. The concentration of tensile stresses on the region of the central groove of the occlusal surface (40 to 60 MPa), seen in Figure 4, shows a narrow strip on the external surface, which indicates a stronger tendency for rupture or formation of small cracks in the ceramic surface only, since the decreasing gradient of stresses reach a peak value of only 20 MPa on the ceramic close to the infrastructure. These tensile stresses reach the infrastructure (Ni-Cr), but they are not harmful, given their low intensity, with a peak value of 20 MPa, well below the tensile strength of the material.

Fig. 3. Perspective and internal views of all structures. Maximum principal stresses in model under longitudinal load.

The von Mises criterion, or theory of the maximum distortion energy, was also used in this study to analyze the results from the implant and its components. It is a rupture criterion to evaluate ductile materials, based on the determination of the maximum distortion energy of a structure, that is, of the energy related to changes in form (as opposed to the energy related to changes in the volume of material) (Beer and Johnston, 1995). The structure failure occurs when, at any point of the material, the distortion energy per unit of volume is higher

The scale of stresses (which appear in different colors in the figures) does not have equal intervals. This is a result of the stresses in action on each group of models (different types of load). Thus, a single scale was defined for all models (except for the axially loaded models)

In Figure 3 (perspective views), compressive stresses can be seen in the region of the abutment and in the cervical region of the implant, but not intense enough to cause harm

Again in Figure 3 (internal views), it can be observed the result of the internal maximum principal stresses. The concentration of tensile stresses on the region of the central groove of the occlusal surface (40 to 60 MPa), seen in Figure 4, shows a narrow strip on the external surface, which indicates a stronger tendency for rupture or formation of small cracks in the ceramic surface only, since the decreasing gradient of stresses reach a peak value of only 20 MPa on the ceramic close to the infrastructure. These tensile stresses reach the infrastructure (Ni-Cr), but they are not harmful, given their low intensity, with a peak value of 20 MPa,

Fig. 3. Perspective and internal views of all structures. Maximum principal stresses in model

than the yield strength value obtained for the material in a tensile test.

**3.1 Axial load – Load with resultant parallel to the long axis** 

to make the comparison easier.

**3. Results and discussion** 

well below the tensile strength of the material.

(between 1 and 5 MPa).

under longitudinal load.

Fig. 4. Occlusal view. Maximum principal stresses in model under longitudinal load.

In Figure 4, it can be observed that the higher compressive stress values are located at points under the area where the load was applied (10 to 55 MPa, well below the tensile strength value of 150 MPa). On the other hand, the increasing tensile stress gradient, between 40 and 60 MPa, concentrated on the region of the central groove, is higher than the tensile strength value (37.2 MPa), suggesting a stronger tendency for micro-cracks in the ceramic.

To compare and discuss the results obtained by Poiate (2007), the stress values found in the cortical and cancelous bone of a sound tooth model are twice lower than the values found in the implant model (between 5 and 10 MPa), in consequence of the absence of periodontal ligament and tooth in the latter.

In Figure 5, very low compressive stresses can be seen in the region of the cortical bone in contact with the implant platform, with a peak value of 8 MPa.

Fig. 5. Perspective view of the cortical bone. Maximum principal stresses in model under longitudinal load. In A, disto-vestibular view; in B, mesio-palatine view.

Figure 6 shows that the concentration of compressive stresses on the region in contact with the implant is on a narrow strip on the external surface of the cortical bone. Next to the cancelous bone, it shows a concentration of tensile stresses of 10 MPa.

Evaluation of Stress Distribution in

under longitudinal load.

under longitudinal load.

Implant-Supported Restoration Under Different Simulated Loads 115

Fig. 8. Perspective and internal view of the implant. Maximum principal stresses in model

According to Dinato (2001), occlusal forces with axial resultant produce a vertical load and do not exert force on the screw nor cause screw loosening. In Figure 9 (perspectives A and B), it can be seen a prevalence of low-intensity tensile stresses on the screw, with higher values concentrated on the apex, on the distal surface (74 MPa). Again in Figure 9 (sections A and B), compressive stresses between 1 and 10 MPa are seen on the coronal third, in the region without spindle. Tensile stresses from 20 to 60 MPa are seen on the screw apex.

Fig. 9. Perspective and internal view of the screw. Maximum principal stresses in model

in the medium third and in the platform of the implant (Figure 8).

To obtain uniform stress distribution in implants, it is necessary a precise adjustment of the abutment, since the unit closest to the load will be subject to the greatest stresses (Rangert, Jemt and Jörneus, 1989). Contrary to this statement, when under axial load the highest stress values are below the cervical third of the abutment (in contact with the screw, see Figure 10),

Fig. 6. Internal view of the cortical bone. Maximum principal stresses in model under longitudinal load. In A, mesio-distal section; in B, vestibulo-palatine section.

In Figure 7, there is a concentration of bulb-shaped tensile stresses on the cancelous bone in contact with the implant, with values ranging from 0.5 to 2 MPa approximately. It is also possible to see that the compressive stresses in contact with the implant thread decrease towards the implant apex, even though the tensile stresses in contact with the implant thread are constant (2 to 4 MPa).

Fig. 7. Perspective and internal view of the cancelous bone. Maximum principal stresses in model under longitudinal load.

According to Lehmann and Elias (2008), to minimize this higher concentration of stresses, new implant forms should be employed, among them platform switching and micro-threads in the most cervical region of the implant.

The implant and its components withstood the stress and did not reach the tensile strength of the material (Figure 8). Tensile stresses between 10 and 20 MPa can be seen in the most cervical region of the implant. The apical third of the implant shows lower tensile stresses (up to 1 MPa) as a result of its anatomy, with no threads in portions of each quadrant, thus minimizing stress concentration. Again in Figure 8, there are compressive stresses on the apex (between 1 and 10 MPa), as well as on the medium third of the implant, a region corresponding to the screw apex (between 1 and 10 MPa). Low tensile stresses are found in the region in contact with the apical third of the screw, in the medium third of the implant (between 10 and 60 MPa), which might result in the loss of stability of the fixation screw.

Fig. 6. Internal view of the cortical bone. Maximum principal stresses in model under

In Figure 7, there is a concentration of bulb-shaped tensile stresses on the cancelous bone in contact with the implant, with values ranging from 0.5 to 2 MPa approximately. It is also possible to see that the compressive stresses in contact with the implant thread decrease towards the implant apex, even though the tensile stresses in contact with the implant

 Fig. 7. Perspective and internal view of the cancelous bone. Maximum principal stresses in

According to Lehmann and Elias (2008), to minimize this higher concentration of stresses, new implant forms should be employed, among them platform switching and micro-threads

The implant and its components withstood the stress and did not reach the tensile strength of the material (Figure 8). Tensile stresses between 10 and 20 MPa can be seen in the most cervical region of the implant. The apical third of the implant shows lower tensile stresses (up to 1 MPa) as a result of its anatomy, with no threads in portions of each quadrant, thus minimizing stress concentration. Again in Figure 8, there are compressive stresses on the apex (between 1 and 10 MPa), as well as on the medium third of the implant, a region corresponding to the screw apex (between 1 and 10 MPa). Low tensile stresses are found in the region in contact with the apical third of the screw, in the medium third of the implant (between 10 and 60 MPa), which might result in the loss of stability of the fixation screw.

longitudinal load. In A, mesio-distal section; in B, vestibulo-palatine section.

thread are constant (2 to 4 MPa).

model under longitudinal load.

in the most cervical region of the implant.

Fig. 8. Perspective and internal view of the implant. Maximum principal stresses in model under longitudinal load.

According to Dinato (2001), occlusal forces with axial resultant produce a vertical load and do not exert force on the screw nor cause screw loosening. In Figure 9 (perspectives A and B), it can be seen a prevalence of low-intensity tensile stresses on the screw, with higher values concentrated on the apex, on the distal surface (74 MPa). Again in Figure 9 (sections A and B), compressive stresses between 1 and 10 MPa are seen on the coronal third, in the region without spindle. Tensile stresses from 20 to 60 MPa are seen on the screw apex.

Fig. 9. Perspective and internal view of the screw. Maximum principal stresses in model under longitudinal load.

To obtain uniform stress distribution in implants, it is necessary a precise adjustment of the abutment, since the unit closest to the load will be subject to the greatest stresses (Rangert, Jemt and Jörneus, 1989). Contrary to this statement, when under axial load the highest stress values are below the cervical third of the abutment (in contact with the screw, see Figure 10), in the medium third and in the platform of the implant (Figure 8).

Evaluation of Stress Distribution in

vestibular cusp.

cracks in the ceramic.

on the screw apex, in the medium third of the implant.

suggesting that there will be a fracture in the cortical bone.

Implant-Supported Restoration Under Different Simulated Loads 117

seen on the implant and on the abutment (Figure 11, section B). It was observed that stress concentration under the place where the load is applied is punctual and well located. In Figure 11 (section A), there is an accumulation of tensile stresses ranging from 50 to 100 MPa

Fig. 12. Occlusal view. Maximum principal stresses in model with a load of 45 on the

In Figure 12, it can be seen that the highest compressive stress values are under the area of application of a load of approximately 156 MPa, rather higher than the values found in the model under axial load (between 10 and 55 MPa, see Figure 4), higher than the strength value (150 MPa). Also, as it occurred under axial load, the tensile stress values adjacent to the area of load application exceeded its strength value, suggesting the possibility of micro-

A study by Faulkner *et al.* (1998) showed, as this one does, that implants loaded with forces that are distant from their axis bear considerable stresses on the bone crest. In Figure 13, there are tensile stresses concentrated in the region of contact with the implant on the palatine surface, with a peak value of 300 MPa, higher than the bone tensile strength,

Fig. 13. Perspective view of the cortical bone. Maximum principal stresses in model with a load of 45 on the vestibular cusp. In A, disto-vestibular view; in B, mesio-palatine view.

Fig. 10. Perspective and internal view of the abutment. Maximum principal stresses in model under longitudinal load.

The results of the longitudinal load revealed that there is a concentration of von Mises stresses, with a peak value of 80 MPa, on the implant head in contact with the cortical bone and on the abutment collar in contact with the implant, as well as on the screw apex, with a peak value of 130 MPa.

#### **3.2 Vestibular lever effect – Load of 45˚ on the vestibular cusp**

Mastication produces mainly vertical forces, but also transverse forces originating from the horizontal movement of the jaw and from the inclination of the cusps. These forces are transferred to the implant through the prosthesis, transforming occlusal forces into bone stresses (Rangert, Jemt e Jörneus, 1989).

Fig. 11. Perspective and internal view of all structures. Maximum principal stresses in model with a load of 45 on the vestibular cusp.

Applying load on the vestibular cusp (Figure 11) produces compression on the vestibular cusp, more concentrated on the region of contact with the cortical bone, which acts as a fulcrum. The tensile stress concentrates and reaches its maximum on the opposite side, the palatine. In Figure 11 (perspective B), tensile stresses between 300 and 740 MPa are seen in the abutment neck, but they are lower than the titanium tensile strength of 930 MPa. When analyzing the ceramic, a concentration of tensile stresses with peak value of 740 MPa can be

Fig. 10. Perspective and internal view of the abutment. Maximum principal stresses in

**3.2 Vestibular lever effect – Load of 45˚ on the vestibular cusp** 

The results of the longitudinal load revealed that there is a concentration of von Mises stresses, with a peak value of 80 MPa, on the implant head in contact with the cortical bone and on the abutment collar in contact with the implant, as well as on the screw apex, with a

Mastication produces mainly vertical forces, but also transverse forces originating from the horizontal movement of the jaw and from the inclination of the cusps. These forces are transferred to the implant through the prosthesis, transforming occlusal forces into bone

Fig. 11. Perspective and internal view of all structures. Maximum principal stresses in model

Applying load on the vestibular cusp (Figure 11) produces compression on the vestibular cusp, more concentrated on the region of contact with the cortical bone, which acts as a fulcrum. The tensile stress concentrates and reaches its maximum on the opposite side, the palatine. In Figure 11 (perspective B), tensile stresses between 300 and 740 MPa are seen in the abutment neck, but they are lower than the titanium tensile strength of 930 MPa. When analyzing the ceramic, a concentration of tensile stresses with peak value of 740 MPa can be

model under longitudinal load.

stresses (Rangert, Jemt e Jörneus, 1989).

with a load of 45 on the vestibular cusp.

peak value of 130 MPa.

seen on the implant and on the abutment (Figure 11, section B). It was observed that stress concentration under the place where the load is applied is punctual and well located. In Figure 11 (section A), there is an accumulation of tensile stresses ranging from 50 to 100 MPa on the screw apex, in the medium third of the implant.

Fig. 12. Occlusal view. Maximum principal stresses in model with a load of 45 on the vestibular cusp.

In Figure 12, it can be seen that the highest compressive stress values are under the area of application of a load of approximately 156 MPa, rather higher than the values found in the model under axial load (between 10 and 55 MPa, see Figure 4), higher than the strength value (150 MPa). Also, as it occurred under axial load, the tensile stress values adjacent to the area of load application exceeded its strength value, suggesting the possibility of microcracks in the ceramic.

A study by Faulkner *et al.* (1998) showed, as this one does, that implants loaded with forces that are distant from their axis bear considerable stresses on the bone crest. In Figure 13, there are tensile stresses concentrated in the region of contact with the implant on the palatine surface, with a peak value of 300 MPa, higher than the bone tensile strength, suggesting that there will be a fracture in the cortical bone.

Fig. 13. Perspective view of the cortical bone. Maximum principal stresses in model with a load of 45 on the vestibular cusp. In A, disto-vestibular view; in B, mesio-palatine view.

Evaluation of Stress Distribution in

with a load of 45 on the vestibular cusp.

with a load of 45 on the vestibular cusp.

bone).

Implant-Supported Restoration Under Different Simulated Loads 119

In Figure 16 (perspective B), there are tensile stresses on the medium and cervical third of the implant, which was expected, since the load was applied to the vestibular cusp. The implant is being bent, with a fulcrum in the region close to the cortical bone, where there is a decreasing dissipation of tensile stresses on the palatine surface. Again in Figure 16 (internal view), there are tensile stresses on the cervical third of the implant, on the lingual surface, which was expected as well, since the load was applied to the vestibular cusp, with a peak

Fig. 16. Perspective and internal view of the implant. Maximum principal stresses in model

In Figure 17, there is a prevalence of tensile stresses on the cervical third of the screw, between the beginning of the thread and the head, on the palatine surface, which was already expected, since the load was applied to the vestibular cusp and the screw was being bent, with a fulcrum in the cervical region in contact with the implant (above the cortical

Fig. 17. Perspective and internal view of the screw. Maximum principal stresses in model

Understanding this pattern of stress distribution in the screw is important to explain the problems and complications encountered, such as screw fractures, as described by Zarb and

value of 300 MPa, rather lower than the tensile strength of the material (930 MPa).

Figure 14 shows that only a narrow strip, close to the implant, exceeds the cortical bone tensile strength. However, the incidence of excess stresses on the cortical bone produces micro-fractures and consequent resorption (Burr *et al.,* 1985; Papavasiliou *et al.,* 1996; Holmgren *et al.*, 1998).

Fig. 14. Internal view of the cortical bone. Maximum principal stresses in model with a load of 45 on the vestibular cusp. In A, mesio-distal section; in B, vestibulo-palatine section.

The maximum principal stresses on the cancelous bone are shown in Figure 15.

Fig. 15. Perspective and internal view of the cancelous bone. Maximum principal stresses in model with a load of 45 on the vestibular cusp.

Comparing the results from models with axial load and vestibular lever effect, it could be observed that the result from the internal maximum principal stresses on the cancelous bone differs in intensity and in the pattern of stress distribution. There is no concentration of bulb-shaped tensile stresses on the cancelous bone in contact with the implant, as occurred in the previous model. Moreover, the tensile stress peak value in the model with vestibular lever effect is 56.9 times higher. It is also possible to see that the compressive stresses in contact with the implant thread appear only on the vestibular surface, which was already expected, since the load was applied to the vestibular cusp, where the implant is being bent. It can be seen that stresses on the cancelous bone in contact with thread pitches or between them reach high values, especially in the coronal third (between 300 and 740 MPa), exceeding the bone tensile strength (121 MPa, Table 2) and suggesting the formation of micro-fractures and consequent resorption (Papavasiliou *et al.,* 1996; Holmgren *et al.*, 1998; De Tolla *et al.,* 2000).

Figure 14 shows that only a narrow strip, close to the implant, exceeds the cortical bone tensile strength. However, the incidence of excess stresses on the cortical bone produces micro-fractures and consequent resorption (Burr *et al.,* 1985; Papavasiliou *et al.,* 1996;

Fig. 14. Internal view of the cortical bone. Maximum principal stresses in model with a load of 45 on the vestibular cusp. In A, mesio-distal section; in B, vestibulo-palatine section.

Fig. 15. Perspective and internal view of the cancelous bone. Maximum principal stresses in

Comparing the results from models with axial load and vestibular lever effect, it could be observed that the result from the internal maximum principal stresses on the cancelous bone differs in intensity and in the pattern of stress distribution. There is no concentration of bulb-shaped tensile stresses on the cancelous bone in contact with the implant, as occurred in the previous model. Moreover, the tensile stress peak value in the model with vestibular lever effect is 56.9 times higher. It is also possible to see that the compressive stresses in contact with the implant thread appear only on the vestibular surface, which was already expected, since the load was applied to the vestibular cusp, where the implant is being bent. It can be seen that stresses on the cancelous bone in contact with thread pitches or between them reach high values, especially in the coronal third (between 300 and 740 MPa), exceeding the bone tensile strength (121 MPa, Table 2) and suggesting the formation of micro-fractures and consequent resorption (Papavasiliou *et al.,* 1996; Holmgren *et al.*, 1998;

model with a load of 45 on the vestibular cusp.

De Tolla *et al.,* 2000).

The maximum principal stresses on the cancelous bone are shown in Figure 15.

Holmgren *et al.*, 1998).

In Figure 16 (perspective B), there are tensile stresses on the medium and cervical third of the implant, which was expected, since the load was applied to the vestibular cusp. The implant is being bent, with a fulcrum in the region close to the cortical bone, where there is a decreasing dissipation of tensile stresses on the palatine surface. Again in Figure 16 (internal view), there are tensile stresses on the cervical third of the implant, on the lingual surface, which was expected as well, since the load was applied to the vestibular cusp, with a peak value of 300 MPa, rather lower than the tensile strength of the material (930 MPa).

Fig. 16. Perspective and internal view of the implant. Maximum principal stresses in model with a load of 45 on the vestibular cusp.

In Figure 17, there is a prevalence of tensile stresses on the cervical third of the screw, between the beginning of the thread and the head, on the palatine surface, which was already expected, since the load was applied to the vestibular cusp and the screw was being bent, with a fulcrum in the cervical region in contact with the implant (above the cortical bone).

Fig. 17. Perspective and internal view of the screw. Maximum principal stresses in model with a load of 45 on the vestibular cusp.

Understanding this pattern of stress distribution in the screw is important to explain the problems and complications encountered, such as screw fractures, as described by Zarb and

Evaluation of Stress Distribution in

with a load of 0 on the mesial marginal ridge.

section, in the occlusal distal region (5 to 10 MPa).

peak value of 5 MPa).

marginal ridge.

Implant-Supported Restoration Under Different Simulated Loads 121

Fig. 19. Perspective and internal view of all structures. Maximum principal stresses in model

According to Gross (2001), neither axial nor non-axial load generate stress concentration on the implant apex. Under the load that generates the proximal lever effect, these results are confirmed in this study. We could observe that the region of the apex does not concentrate compressive stresses, which are dissipated throughout the apical third of the model (with a

However, there is a concentration of tensile stresses on the screw apex, with a peak value of 50 MPa (medium third of implant, Figure 19, section A), and tensile stresses on the distal surface of the implant and on the apical third of the abutment, with values between 50 and 100 MPa. Although the tensile stress values found in the implant and its components are lower than their tensile strength value (Table 2), its presence is important, so that we understand complications, such as screw loss and/or loosening, caused by the micromovements generated between these two surfaces. Again in Figure 19, it is possible to see a concentration of low-intensity tensile stresses only on the cortical bone, in a mesio-distal

Fig. 20. Occlusal view. Maximum principal stresses in model with a load of 0 on the mesial

Schmitt (1990), although they seem slightly probable when we compare the stress values (150 MPa) with the amount necessary for a rupture (930 MPa).

The results revealed in Figure 18 showed a narrow strip of tensile stresses between 300 and 740 MPa on the external surface of the abutment neck, with dissipation to the palatine and internal surface, but not exceeding its tensile strength. There are tensile stress values (650 MPa in the abutment collar in contact with the implant) rather higher than those in the model under longitudinal load (peak value of 80 MPa).

Fig. 18. Perspective and internal view of the abutment. Maximum principal stresses in model with a load of 45 on the vestibular cusp.

The results from the distribution of von Mises stresses in the implant and its components showed that higher stress values are found also in the implant head, when compared with the axial model.

#### **3.3 Proximal lever effect – Load of 0˚ on the mesial marginal ridge**

In Figure 19, there are tensile stresses between 150 and 300 MPa on the cortical bone in contact with the implant. They were already expected, given the direction of the force on the crown, which pulled the implant neck in this area. It is also possible to see tensile stresses (5 MPa) on the line between the abutment and the ceramic, which could be explained by compression on materials of different rigidity.

In Figure 19, it can also be seen that the highest compressive stress values are under the application area of a load ranging from approximately 60 to 120 MPa, slightly lower than the values found in the model with vestibular lever effect (150 MPa), and below the ceramic compressive strength (150 MPa, Table 2). It shows the place of load application with a welllocated and punctual stress concentration; however, this stress concentration does not transfer to the vestibular surface (Figure 19, perspective A) and transfers only slightly to the palatine surface (Figure 19, perspective B). On the other hand, there are tensile stresses close to the load application point, with values between 50 and 100 MPa, higher than the ceramic tensile strength, suggesting a possible formation of micro-cracks.

Schmitt (1990), although they seem slightly probable when we compare the stress values

The results revealed in Figure 18 showed a narrow strip of tensile stresses between 300 and 740 MPa on the external surface of the abutment neck, with dissipation to the palatine and internal surface, but not exceeding its tensile strength. There are tensile stress values (650 MPa in the abutment collar in contact with the implant) rather higher than those in the

Fig. 18. Perspective and internal view of the abutment. Maximum principal stresses in

**3.3 Proximal lever effect – Load of 0˚ on the mesial marginal ridge** 

tensile strength, suggesting a possible formation of micro-cracks.

The results from the distribution of von Mises stresses in the implant and its components showed that higher stress values are found also in the implant head, when compared with

In Figure 19, there are tensile stresses between 150 and 300 MPa on the cortical bone in contact with the implant. They were already expected, given the direction of the force on the crown, which pulled the implant neck in this area. It is also possible to see tensile stresses (5 MPa) on the line between the abutment and the ceramic, which could be explained by

In Figure 19, it can also be seen that the highest compressive stress values are under the application area of a load ranging from approximately 60 to 120 MPa, slightly lower than the values found in the model with vestibular lever effect (150 MPa), and below the ceramic compressive strength (150 MPa, Table 2). It shows the place of load application with a welllocated and punctual stress concentration; however, this stress concentration does not transfer to the vestibular surface (Figure 19, perspective A) and transfers only slightly to the palatine surface (Figure 19, perspective B). On the other hand, there are tensile stresses close to the load application point, with values between 50 and 100 MPa, higher than the ceramic

(150 MPa) with the amount necessary for a rupture (930 MPa).

model under longitudinal load (peak value of 80 MPa).

model with a load of 45 on the vestibular cusp.

compression on materials of different rigidity.

the axial model.

Fig. 19. Perspective and internal view of all structures. Maximum principal stresses in model with a load of 0 on the mesial marginal ridge.

According to Gross (2001), neither axial nor non-axial load generate stress concentration on the implant apex. Under the load that generates the proximal lever effect, these results are confirmed in this study. We could observe that the region of the apex does not concentrate compressive stresses, which are dissipated throughout the apical third of the model (with a peak value of 5 MPa).

However, there is a concentration of tensile stresses on the screw apex, with a peak value of 50 MPa (medium third of implant, Figure 19, section A), and tensile stresses on the distal surface of the implant and on the apical third of the abutment, with values between 50 and 100 MPa. Although the tensile stress values found in the implant and its components are lower than their tensile strength value (Table 2), its presence is important, so that we understand complications, such as screw loss and/or loosening, caused by the micromovements generated between these two surfaces. Again in Figure 19, it is possible to see a concentration of low-intensity tensile stresses only on the cortical bone, in a mesio-distal section, in the occlusal distal region (5 to 10 MPa).

Fig. 20. Occlusal view. Maximum principal stresses in model with a load of 0 on the mesial marginal ridge.

Evaluation of Stress Distribution in

Implant-Supported Restoration Under Different Simulated Loads 123

layer of the more elastic cancelous bone. Natural teeth themselves under load generate

 Fig. 23. Perspective and internal view of the cancelous bone. Maximum principal stresses in

Also in Figure 23, it can be observed that, with a force with inclination of 0º to the axial axis of the set, there was a higher concentration of tensile stresses on the side opposite to the applied force, but with stresses limited to the screw threads of the component, with a peak value of 300 MPa, exceeding the bone tensile strength (121 MPa, Table 2), which suggests

In Figure 24, there is a concentration of tensile stresses only on the cervical third of the implant (with a peak value of 300 MPa). Comparing the models with vestibular and proximal lever effect, it can be considered that the area of tensile stress concentration is larger than in the model that produces a vestibular lever effect, suggesting a stronger possibility of osseointegration failure. The clinical success derived from osseointegration proves that implants resist firmly to the masticatory load; however, the concentration of

Again in Figure 24 (internal view), there are tensile stresses located on the cervical third of the implant, on the distal surface, which was expected, since the load was applied to the mesial marginal ridge, with a peak value of 150 MPa, rather lower than the tensile strength of the material (930 MPa) and 50% lower than the peak value of tensile stresses found in the

Fig. 24. Perspective and internal view of the implant. Maximum principal stresses in model

higher stresses next to the cortical bone (Caputo and Standlee, 1987).

model with a load of 0 on the mesial marginal ridge.

the formation of micro-fractures and consequent resorption.

stresses can result in the loss of osseointegration (Adell *et al.,* 1981).

implant of the model that produces a vestibular lever effect.

with a load of 0 on the mesial marginal ridge.

In Figure 21, low-intensity tensile stresses can be seen in the distal cervical region. However, the internal surface of the cortical bone in contact with the implant shows tensile stresses between 50 and 150 MPa, exceeding the bone tensile strength (121 MPa, Table 2), which suggests the formation of micro-fractures and consequent resorption.

Fig. 21. View of the cortical bone. Maximum principal stresses in model with a load of 0 on the mesial marginal ridge. In A, disto-vestibular view; in B, mesio-palatine view.

In Figure 22, it is clear the presence of tensile stresses with intensity between 50 and 150 MPa on the cortical bone in contact with the implant, on the distal surface (Figure 22B), exceeding the bone tensile strength (Table 2). In implant-supported restorations, stresses are close to the bone crest (Misch *et al*., 2001), altering the existing process of bone crest remodeling.

Fig. 22. Internal view of the cortical bone. Maximum principal stresses in model with a load of 0 on the mesial marginal ridge. In A, mesio-distal section; in B, vestibulo-palatine section.

In Figure 23, it can be observed that there is a dissipation of low-intensity tensile stresses in the distal cervical region from the cortical to the cancelous bone, as well as a concentration of stresses in contact with the implant. The bone anatomy itself leads to the concentration of local forces, given the existence of an external layer of rigid cortical bone and of an internal

In Figure 21, low-intensity tensile stresses can be seen in the distal cervical region. However, the internal surface of the cortical bone in contact with the implant shows tensile stresses between 50 and 150 MPa, exceeding the bone tensile strength (121 MPa, Table 2), which

Fig. 21. View of the cortical bone. Maximum principal stresses in model with a load of 0 on

In Figure 22, it is clear the presence of tensile stresses with intensity between 50 and 150 MPa on the cortical bone in contact with the implant, on the distal surface (Figure 22B), exceeding the bone tensile strength (Table 2). In implant-supported restorations, stresses are close to the bone crest (Misch *et al*., 2001), altering the existing process of bone crest

Fig. 22. Internal view of the cortical bone. Maximum principal stresses in model with a load of 0 on the mesial marginal ridge. In A, mesio-distal section; in B, vestibulo-palatine section.

In Figure 23, it can be observed that there is a dissipation of low-intensity tensile stresses in the distal cervical region from the cortical to the cancelous bone, as well as a concentration of stresses in contact with the implant. The bone anatomy itself leads to the concentration of local forces, given the existence of an external layer of rigid cortical bone and of an internal

the mesial marginal ridge. In A, disto-vestibular view; in B, mesio-palatine view.

remodeling.

suggests the formation of micro-fractures and consequent resorption.

layer of the more elastic cancelous bone. Natural teeth themselves under load generate higher stresses next to the cortical bone (Caputo and Standlee, 1987).

Fig. 23. Perspective and internal view of the cancelous bone. Maximum principal stresses in model with a load of 0 on the mesial marginal ridge.

Also in Figure 23, it can be observed that, with a force with inclination of 0º to the axial axis of the set, there was a higher concentration of tensile stresses on the side opposite to the applied force, but with stresses limited to the screw threads of the component, with a peak value of 300 MPa, exceeding the bone tensile strength (121 MPa, Table 2), which suggests the formation of micro-fractures and consequent resorption.

In Figure 24, there is a concentration of tensile stresses only on the cervical third of the implant (with a peak value of 300 MPa). Comparing the models with vestibular and proximal lever effect, it can be considered that the area of tensile stress concentration is larger than in the model that produces a vestibular lever effect, suggesting a stronger possibility of osseointegration failure. The clinical success derived from osseointegration proves that implants resist firmly to the masticatory load; however, the concentration of stresses can result in the loss of osseointegration (Adell *et al.,* 1981).

Again in Figure 24 (internal view), there are tensile stresses located on the cervical third of the implant, on the distal surface, which was expected, since the load was applied to the mesial marginal ridge, with a peak value of 150 MPa, rather lower than the tensile strength of the material (930 MPa) and 50% lower than the peak value of tensile stresses found in the implant of the model that produces a vestibular lever effect.

Fig. 24. Perspective and internal view of the implant. Maximum principal stresses in model with a load of 0 on the mesial marginal ridge.

Evaluation of Stress Distribution in

Implant-Supported Restoration Under Different Simulated Loads 125

Single implant-supported restorations can also be subject to rotational or torsion forces, whenever they are clinically demanded through the functional contacts cusp/marginal

Figure 27 shows tensile stresses between 50 and 300 MPa on the abutment, which was already expected, in view of the direction of the force on the crown, which pulls the implant in this area. Again in Figure 27 (section B), it can be seen that there is a concentration of tensile stresses on the cervical and medium third of the implant (peak value of 300 MPa) on

In Figure 28, it can be seen that the highest compressive stress values are under the area where the load was applied (10 to 80 MPa, lower than the strength value of 150 MPa), which is a stress concentration point with no transmission to the vestibular surface. However, the existence of tensile stresses close to the application of load (50 to 100 MPa) exceeds the

Fig. 27. Perspective and internal view of all structures. Maximum principal stresses in model

Fig. 28. Occlusal view of all structures. Maximum principal stresses in model with a load of

**3.4 Torsion effect – Load of 45˚ on the mesial marginal ridge** 

ridge, that is, in a one tooth to two teeth relation (Cohen *et al*., 1995).

ceramic strength (37.2 MPa) and suggests a localized micro-crack.

the palatine surface, lower than the mechanical strength.

with a load of 45 on the mesial marginal ridge.

45 on the mesial marginal ridge.

In Figure 25, there is a prevalence of tensile stresses on the cervical third of the screw, between the beginning of the thread and its head or platform, on the distal side, which was already expected, since the load was applied to the mesial marginal ridge and the screw is being bent, with a fulcrum in the cervical region, in contact with the implant (above the cortical bone).

Fig. 25. Perspective and internal view of the screw. Maximum principal stresses in model with a load of 0 on the mesial marginal ridge.

Sealing, obtained by the precise adjustment of the abutment surface to the implant, would prevent problems of a peri-implant nature and minimize the development of tangential forces harmful to the interface implant-bone tissue, which could lead to osseointegration failure (Adell *et al*., 1981). The maximum principal stresses on the abutment are shown in Figure 26.

Fig. 26. Perspective and internal view of the abutment. Maximum principal stresses in model with a load of 0 on the mesial marginal ridge.

The results revealed values of von Mises stresses (peak value of 300 MPa in the abutment collar in contact with the implant) lower than those found in the model that produces the vestibular lever effect (peak value of 650 MPa), but higher than those found in the model under longitudinal load (peak value of 80 MPa). Higher values of von Mises stresses are also found in the implant platform, when compared to the axial model.

In Figure 25, there is a prevalence of tensile stresses on the cervical third of the screw, between the beginning of the thread and its head or platform, on the distal side, which was already expected, since the load was applied to the mesial marginal ridge and the screw is being bent, with a fulcrum in the cervical region, in contact with the implant (above the

Fig. 25. Perspective and internal view of the screw. Maximum principal stresses in model

Sealing, obtained by the precise adjustment of the abutment surface to the implant, would prevent problems of a peri-implant nature and minimize the development of tangential forces harmful to the interface implant-bone tissue, which could lead to osseointegration failure (Adell *et al*., 1981). The maximum principal stresses on the abutment are shown in

Fig. 26. Perspective and internal view of the abutment. Maximum principal stresses in

The results revealed values of von Mises stresses (peak value of 300 MPa in the abutment collar in contact with the implant) lower than those found in the model that produces the vestibular lever effect (peak value of 650 MPa), but higher than those found in the model under longitudinal load (peak value of 80 MPa). Higher values of von Mises stresses are also

with a load of 0 on the mesial marginal ridge.

model with a load of 0 on the mesial marginal ridge.

found in the implant platform, when compared to the axial model.

cortical bone).

Figure 26.

#### **3.4 Torsion effect – Load of 45˚ on the mesial marginal ridge**

Single implant-supported restorations can also be subject to rotational or torsion forces, whenever they are clinically demanded through the functional contacts cusp/marginal ridge, that is, in a one tooth to two teeth relation (Cohen *et al*., 1995).

Figure 27 shows tensile stresses between 50 and 300 MPa on the abutment, which was already expected, in view of the direction of the force on the crown, which pulls the implant in this area. Again in Figure 27 (section B), it can be seen that there is a concentration of tensile stresses on the cervical and medium third of the implant (peak value of 300 MPa) on the palatine surface, lower than the mechanical strength.

In Figure 28, it can be seen that the highest compressive stress values are under the area where the load was applied (10 to 80 MPa, lower than the strength value of 150 MPa), which is a stress concentration point with no transmission to the vestibular surface. However, the existence of tensile stresses close to the application of load (50 to 100 MPa) exceeds the ceramic strength (37.2 MPa) and suggests a localized micro-crack.

Fig. 27. Perspective and internal view of all structures. Maximum principal stresses in model with a load of 45 on the mesial marginal ridge.

Fig. 28. Occlusal view of all structures. Maximum principal stresses in model with a load of 45 on the mesial marginal ridge.

Evaluation of Stress Distribution in

Implant-Supported Restoration Under Different Simulated Loads 127

In Figure 31, there are tensile stresses on the cortical bone (on the palatine surface) in contact with the implant that exceed the tensile strength value and overload the implant. According to Rangert *et al.* (1995), in a retrospective analysis, this overload induces bone resorption,

Also in Figure 31, there are distribution of low-intensity tensile stresses (5 to 10 MPa) and

The concentration of tensile stresses on the cancelous bone in contact with the implant (especially on the threads) reaches the cervical (higher intensity) and medium (lower intensity) thirds on their palatine surface. This was expected, as a result of the direction of the load application. According to Rangert *et al.* (1989), threads reduce the shear stress on

Tensile stresses between 50 and 100 MPa (Figure 32) on the implant platform are five times higher than the stresses found in the axial model (10 to 20 MPa). In Figure 32, it can also be observed that there is a concentration of tensile stresses on the medium and cervical third of the implant, as a consequence of load application, in addition to the torsion effect. Moreover, the implant is being bent, with a fulcrum in the region close to the cortical bone, where there is dissipation of decreasing tensile stresses on the palatine surface. Tensile stresses are also seen in the medium and cervical third of the implant, on the distal and lingual surface, a region corresponding to the fulcrum in the cortical bone and to the screw thread (between 50 and 150 MPa). Tensile stresses are found in the region in contact with the apical third of the screw, medium third of the implant. Although the tensile stress value at the end of the screw thread is lower than the mechanical strength of the material, it might

 Fig. 32. Perspective view of the implant. Maximum principal stresses in model with a load of

In a disto-vestibular view, Figure 33 shows compressive stresses between approximately 5 and 20 MPa, concentrated on the coronal third of the screw, in the region without spindle. Tensile stresses can also be seen in the apex (between 50 and 150 MPa) and in the screw thread (5 to 10 MPa), in a mesio-lingual view, suggesting the possibility of screw loosening in this region. In Figure 33 (sections A and B), there is a prevalence of tensile stresses between 50 and 150 MPa, but concentrated on the medium-coronal third of the screw, a

region without thread, and not exceeding the tensile strength of the material.

which seems to precede and contribute to the fracture of implant components.

stress concentration on the palatine surface and around the implant.

the implant-bone interface when it is under axial load.

contribute to the loss of stability in the fixation screw.

45 on the mesial marginal ridge.

In Figure 29, a dissipation of low-intensity compressive stresses can be seen in the vestibular surface of the cortical bone (5 to 20 MPa), and there are tensile stresses on the palatine surface (peak value of 10 MPa). However, high-intensity tensile stresses are present on the cortical bone in contact with the implant (palatine). This is better shown in Figure 30.

Fig. 29. Perspective view of the cortical bone. Maximum principal stresses in model with a load of 45 on the mesial marginal ridge. In A, disto-vestibular view; in B, mesio-palatine view.

Fig. 30. Internal view of the cortical bone. Maximum principal stresses in model with a load of 45 on the mesial marginal ridge. In A, mesio-distal section; in B, vestibulo-lingual section.

Fig. 31. Perspective and internal view of the cancelous bone. Maximum principal stresses in model with a load of 45 on the mesial marginal ridge.

In Figure 29, a dissipation of low-intensity compressive stresses can be seen in the vestibular surface of the cortical bone (5 to 20 MPa), and there are tensile stresses on the palatine surface (peak value of 10 MPa). However, high-intensity tensile stresses are present on the

cortical bone in contact with the implant (palatine). This is better shown in Figure 30.

Fig. 29. Perspective view of the cortical bone. Maximum principal stresses in model with a load of 45 on the mesial marginal ridge. In A, disto-vestibular view; in B, mesio-palatine

Fig. 30. Internal view of the cortical bone. Maximum principal stresses in model with a load of 45 on the mesial marginal ridge. In A, mesio-distal section; in B, vestibulo-lingual

 Fig. 31. Perspective and internal view of the cancelous bone. Maximum principal stresses in

model with a load of 45 on the mesial marginal ridge.

view.

section.

In Figure 31, there are tensile stresses on the cortical bone (on the palatine surface) in contact with the implant that exceed the tensile strength value and overload the implant. According to Rangert *et al.* (1995), in a retrospective analysis, this overload induces bone resorption, which seems to precede and contribute to the fracture of implant components.

Also in Figure 31, there are distribution of low-intensity tensile stresses (5 to 10 MPa) and stress concentration on the palatine surface and around the implant.

The concentration of tensile stresses on the cancelous bone in contact with the implant (especially on the threads) reaches the cervical (higher intensity) and medium (lower intensity) thirds on their palatine surface. This was expected, as a result of the direction of the load application. According to Rangert *et al.* (1989), threads reduce the shear stress on the implant-bone interface when it is under axial load.

Tensile stresses between 50 and 100 MPa (Figure 32) on the implant platform are five times higher than the stresses found in the axial model (10 to 20 MPa). In Figure 32, it can also be observed that there is a concentration of tensile stresses on the medium and cervical third of the implant, as a consequence of load application, in addition to the torsion effect. Moreover, the implant is being bent, with a fulcrum in the region close to the cortical bone, where there is dissipation of decreasing tensile stresses on the palatine surface. Tensile stresses are also seen in the medium and cervical third of the implant, on the distal and lingual surface, a region corresponding to the fulcrum in the cortical bone and to the screw thread (between 50 and 150 MPa). Tensile stresses are found in the region in contact with the apical third of the screw, medium third of the implant. Although the tensile stress value at the end of the screw thread is lower than the mechanical strength of the material, it might contribute to the loss of stability in the fixation screw.

Fig. 32. Perspective view of the implant. Maximum principal stresses in model with a load of 45 on the mesial marginal ridge.

In a disto-vestibular view, Figure 33 shows compressive stresses between approximately 5 and 20 MPa, concentrated on the coronal third of the screw, in the region without spindle. Tensile stresses can also be seen in the apex (between 50 and 150 MPa) and in the screw thread (5 to 10 MPa), in a mesio-lingual view, suggesting the possibility of screw loosening in this region. In Figure 33 (sections A and B), there is a prevalence of tensile stresses between 50 and 150 MPa, but concentrated on the medium-coronal third of the screw, a region without thread, and not exceeding the tensile strength of the material.

Fig. 33. Perspective view of the screw. Maximum principal stresses in model with a load of 45 on the mesial marginal ridge.

In Figure 34, the prevalence of tensile stresses can be seen on the abutment external surface (it shows only a narrow strip of low-intensity compressive stresses, of up to 5 MPa, on the vestibular surface). This prevalence of tensile stresses on the abutment external surface is important to explain complications such as the loss and/or loosening of screws, since these stresses are directly related to the abutment-implant interface, as a result of the creation of micro-movements between the two surfaces when a non-axial load is applied to them.

Fig. 34. Perspective and internal view of the abutment. Maximum principal stresses in model with a load of 45 on the mesial marginal ridge.

The result of von Mises stresses revealed tensile stresses values in the abutment collar in contact with the implant (peak value of 600 MPa) that are higher than those found in the proximal lever model (peak value of 300 MPa) and close to those found in the model that produces a vestibular lever effect (peak value of 650 MPa), and considerably higher than those found in the model under longitudinal load (peak value of 80 MPa).

Evaluation of Stress Distribution in

**5. Acknowledgements** 

10, pp. 387-416.

pp. 48-63.

350-57.

15, No. 4, pp. 5715-82.

**6. References** 

important to achieve success in this therapy.

(LEFO), Fluminense Federal University, for their support.

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Implant-Supported Restoration Under Different Simulated Loads 129

load with a lever effect, no load was able to fracture the components of the implant system simulated here. However, they may suggest loosening of the screw, micro-cracks in the ceramic, and bone micro-fractures (resorptions), except for the model under axial load, which proved to be the least harmful to the stability of the rehabilitating system under consideration. It can be stated that a careful observation of the criteria for rehabilitation using implant-supported restorations, as regards the direction of occlusal loads, is crucially

This study was partially based on a thesis submitted to The Fluminense Federal University, in fulfillment of the requirements for the degree of Master of Science. The authors are grateful to the State of Rio de Janeiro Research Foundation (FAPERJ), the Conexão Sistemas de Prótese Ltda and to the Finite Element Laboratory in Dentistry

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#### **4. Conclusions**

Taking into account that the results produced in this study, for all models, revealed a higher concentration of forces on the cervical region and higher stress values in the models under load with a lever effect, no load was able to fracture the components of the implant system simulated here. However, they may suggest loosening of the screw, micro-cracks in the ceramic, and bone micro-fractures (resorptions), except for the model under axial load, which proved to be the least harmful to the stability of the rehabilitating system under consideration. It can be stated that a careful observation of the criteria for rehabilitation using implant-supported restorations, as regards the direction of occlusal loads, is crucially important to achieve success in this therapy.

### **5. Acknowledgements**

This study was partially based on a thesis submitted to The Fluminense Federal University, in fulfillment of the requirements for the degree of Master of Science. The authors are grateful to the State of Rio de Janeiro Research Foundation (FAPERJ), the Conexão Sistemas de Prótese Ltda and to the Finite Element Laboratory in Dentistry (LEFO), Fluminense Federal University, for their support.

#### **6. References**

128 Finite Element Analysis – From Biomedical Applications to Industrial Developments

 Fig. 33. Perspective view of the screw. Maximum principal stresses in model with a load of

In Figure 34, the prevalence of tensile stresses can be seen on the abutment external surface (it shows only a narrow strip of low-intensity compressive stresses, of up to 5 MPa, on the vestibular surface). This prevalence of tensile stresses on the abutment external surface is important to explain complications such as the loss and/or loosening of screws, since these stresses are directly related to the abutment-implant interface, as a result of the creation of micro-movements between the two surfaces when a non-axial load is applied to them.

Fig. 34. Perspective and internal view of the abutment. Maximum principal stresses in

those found in the model under longitudinal load (peak value of 80 MPa).

The result of von Mises stresses revealed tensile stresses values in the abutment collar in contact with the implant (peak value of 600 MPa) that are higher than those found in the proximal lever model (peak value of 300 MPa) and close to those found in the model that produces a vestibular lever effect (peak value of 650 MPa), and considerably higher than

Taking into account that the results produced in this study, for all models, revealed a higher concentration of forces on the cervical region and higher stress values in the models under

model with a load of 45 on the mesial marginal ridge.

**4. Conclusions** 

45 on the mesial marginal ridge.


Evaluation of Stress Distribution in

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**6**

 *Brazil* 

**Biomechanical Analysis of Restored**

For the prosthetic reconstruction of endodontically treated teeth with large loss of tooth structure a lot of times become indispensable to obtain retention by the use of post and core systems. The retention loss and the dental fractures are the two failures more commonly

The dental fracture tends to happen longitudinally with the end below the alveolar bone crest, what constitutes a no restorative failure and leads to the tooth's loss. This failure type is attributed mainly to the use of posts with length and/or diameter incorrect and

The intra-radicular post is used to provide retention to a core, but would also have the function of distributing the functional load in a larger area of the remaining coronary structure and dental root. However, many authors (Assif & Gofil, 1994; Martinez-Insua et al., 1999; Rundquist & Versluis, 2006; Whitworth et al., 2002) have been demonstrating that this reconstruction method doesn't restore the original strength of a vital tooth, what is attributed to the wedge effect that this restoration type causes and to the stiffness difference

This effect appears when the load induces the post intrusion inside the dental root. When being pushed, the wedge tends to increase the perimeter of the remnant transversal section due to the tensile stress guided parallel to the circular outlines of the tooth's transversal section. These tangent stresses can cause vertical fracture (Rundquist & Versluis, 2006).

An attempt to increase the root strength front to the physiologic load is the ferrule making that is a metallic necklace of 360° that surrounds the axial walls of the remnant dentine and can be propitiated by the core or by the crown, and tends to produce the encirclement of the root. This "ferrule effect" would protect the pulpless tooth against fracture by counteracting spreading forces generated by the post. The core should extend apical to the shoulder of the

described in this restoration type (Ferrari & Mannocci, 2000).

deficiencies in dental structure preservation1.

between the post and the tooth.

**1. Introduction** 

**Teeth with Cast Intra-Radicular** 

Isis Andréa Venturini Pola Poiate1,

*1Federal Fluminense University, 2Pontifical Catholic University, 3University of São Paulo,* 

**Retainer with and Without Ferrule**

Edgard Poiate Junior2 and Rafael Yagϋe Ballester3

Zarb, GA; Schmitt, A. (1990). The longitudinal clinical effectiveness of osseointegrated dental implants: Toronto study. Part III: Problems and complications encountered. *J Posthet Dent*, Vol. 64, No. 2, pp. 185-94.

### **Biomechanical Analysis of Restored Teeth with Cast Intra-Radicular Retainer with and Without Ferrule**

```
Isis Andréa Venturini Pola Poiate1, 
Edgard Poiate Junior2 and Rafael Yagϋe Ballester3
                         1Federal Fluminense University, 
                           2Pontifical Catholic University, 
                                3University of São Paulo, 
                                                   Brazil
```
#### **1. Introduction**

132 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Zarb, GA; Schmitt, A. (1990). The longitudinal clinical effectiveness of osseointegrated

*Posthet Dent*, Vol. 64, No. 2, pp. 185-94.

dental implants: Toronto study. Part III: Problems and complications encountered. *J* 

For the prosthetic reconstruction of endodontically treated teeth with large loss of tooth structure a lot of times become indispensable to obtain retention by the use of post and core systems. The retention loss and the dental fractures are the two failures more commonly described in this restoration type (Ferrari & Mannocci, 2000).

The dental fracture tends to happen longitudinally with the end below the alveolar bone crest, what constitutes a no restorative failure and leads to the tooth's loss. This failure type is attributed mainly to the use of posts with length and/or diameter incorrect and deficiencies in dental structure preservation1.

The intra-radicular post is used to provide retention to a core, but would also have the function of distributing the functional load in a larger area of the remaining coronary structure and dental root. However, many authors (Assif & Gofil, 1994; Martinez-Insua et al., 1999; Rundquist & Versluis, 2006; Whitworth et al., 2002) have been demonstrating that this reconstruction method doesn't restore the original strength of a vital tooth, what is attributed to the wedge effect that this restoration type causes and to the stiffness difference between the post and the tooth.

This effect appears when the load induces the post intrusion inside the dental root. When being pushed, the wedge tends to increase the perimeter of the remnant transversal section due to the tensile stress guided parallel to the circular outlines of the tooth's transversal section. These tangent stresses can cause vertical fracture (Rundquist & Versluis, 2006).

An attempt to increase the root strength front to the physiologic load is the ferrule making that is a metallic necklace of 360° that surrounds the axial walls of the remnant dentine and can be propitiated by the core or by the crown, and tends to produce the encirclement of the root. This "ferrule effect" would protect the pulpless tooth against fracture by counteracting spreading forces generated by the post. The core should extend apical to the shoulder of the

Biomechanical Analysis of Restored Teeth with

(Fig. 2).

Cast Intra-Radicular Retainer with and Without Ferrule 135

In four models with intra-radicular retainer was varied the ending line of the core preparation (with ferrule geometry given by the core) and another model was simulated

The width and the height of the ferrule were determined as a proportion of the thickness of radicular dentine found in the cementoenamel junction region on the lingual side (1.85 mm), separated for three thirds. For other words, the ferrule with x height varying of 0.62 x 0.62

All of the models with intra-radicular retainer presented the same characteristics in relation to the supporting structures, porcelain-faced crown, cement thickness and apical seal

with simple core (without ferrule), staying the same coronary restoration.

Fig. 1. Ferrule geometries of the radicular dentine in a premolar root.

With the measures of the thickness in each one of the structures (pulp, dentin, enamel) in each face (mesial, distal, lingual, vestibular) a coordinates system was generated with the origin in the pulpal apex and all the values of thickness of the structures were transformed in coordinates. As many coordinates existed to generate the points in the Finite Element (FE) program, a routine in PCL (Patran Command Language) was created to read the spreadsheet and to generate the points. After the points generation (Fig. 3a), the curves were generated and, soon afterwards, the surfaces. Starting from the dental structures surfaces built, the superficial meshes were generated with triangular linear elements (Tri3). After

(T1H1), 0.62 x 1.34 (T1H2), 1.34 x 0.62 (T2H1) and 1.34 x 1.34 mm (T2H2), (Fig. 1).

preparation to provide a 1.5 to 3.0 mm ferrule of the intact tooth structure (Tan et al., 2005; Pereira et al., 2006; Morgano, 1996; Morgano & Bracket, 1999).

According to Loney et al (1990), the ferrule supplied by the core contributes for the most balanced stress distribution in the dental root and it could compress the remnant structure.

The understanding of the biomechanical principles and restorations applicability is important to design restorations to provide larger strength and retention. The aim of this study was to evaluate the ferrule geometry formed by core on the stress developed in the dental root when a second superior premolar is submitted to four different load conditions.

#### **2. Materials and methods**

A natural healthy tooth model (H) was generated in the MSC/PATRAN 2005 (MSC Software Corporation, Santa Ana, CA, USA). On that model modifications were accomplished for the creation of five new models with retainer intra-radicular.

The 3D model of the maxillary second premolar was built based on data presented by Shillinburg & Grace (1973), regarding the measures of the dentin thickness in the mesiodistal and vestibular-lingual axis in four horizontal slices (with 3.5 mm interval) accomplished along the main axis of the dental root starting from the cementoenamel junction. The dentin's thickness in the remnant of the root was built by the interpolation with a spline curve starting from the existent data, following the root anatomy. The pulp dimensions were built based on the data presented by Green & Brooklyn (1960), that it supplies the average diameter of the apical foramen and Shillinburg & Grace (1973), that it supplies the average dimensions in the cervical area.

For the making of the crown geometry the enamel thickness and coronary dentin were based on the data presented by Shillinburg & Grace (1973), through slices of interval of 1 mm, by Cantisano at al. (1987) and Ueti at al. (1997). The height of the vestibular cusp was considered larger than the lingual cusp in 0.9 mm (Shillinburg et al., 1972).

The lamina dura was represented with mechanical properties equal of the cortical bone and as a uniform layer of 0.25 mm thickness as well as the periodontal ligament (Lee et al., 2000).

The models that represented intra-radicular retainer were built from the natural healthy tooth model already. Specify modifications were accomplished in the area corresponding to the pulp and to the dentin, in the places corresponding to the core and porcelain-faced crown.

The dimension of the pulp was enlarged to reproduce the endodontic access stage (Cohen & Hargreaves 2005), the biomechanical prepare and the filling of the root canal.

The porcelain-faced crown was represented with the same outline of the enamel, being average thickness of 1.5 mm in the vestibular face, 1.2 mm in the lingual face, 1.0 mm in the proximal face and 2.0 mm in the occlusal face, with an end in chamfer around the core. The minimum thickness of the metal (NiCr) in the restoration porcelain-faced was 0.3 mm (Yamamoto, 1985).

The zinc phosphate cement was select because is usually used for cementation of the crown and of intra-radicular retainer with thickness from 50 to 100 m (Anusavice, 2003) and the gutta-percha height was of 5 mm (Morgano, 1996).

preparation to provide a 1.5 to 3.0 mm ferrule of the intact tooth structure (Tan et al., 2005;

According to Loney et al (1990), the ferrule supplied by the core contributes for the most balanced stress distribution in the dental root and it could compress the remnant structure. The understanding of the biomechanical principles and restorations applicability is important to design restorations to provide larger strength and retention. The aim of this study was to evaluate the ferrule geometry formed by core on the stress developed in the dental root when a second superior premolar is submitted to four different load conditions.

A natural healthy tooth model (H) was generated in the MSC/PATRAN 2005 (MSC Software Corporation, Santa Ana, CA, USA). On that model modifications were

The 3D model of the maxillary second premolar was built based on data presented by Shillinburg & Grace (1973), regarding the measures of the dentin thickness in the mesiodistal and vestibular-lingual axis in four horizontal slices (with 3.5 mm interval) accomplished along the main axis of the dental root starting from the cementoenamel junction. The dentin's thickness in the remnant of the root was built by the interpolation with a spline curve starting from the existent data, following the root anatomy. The pulp dimensions were built based on the data presented by Green & Brooklyn (1960), that it supplies the average diameter of the apical foramen and Shillinburg & Grace (1973), that it

For the making of the crown geometry the enamel thickness and coronary dentin were based on the data presented by Shillinburg & Grace (1973), through slices of interval of 1 mm, by Cantisano at al. (1987) and Ueti at al. (1997). The height of the vestibular cusp was

The lamina dura was represented with mechanical properties equal of the cortical bone and as a uniform layer of 0.25 mm thickness as well as the periodontal ligament (Lee et al., 2000). The models that represented intra-radicular retainer were built from the natural healthy tooth model already. Specify modifications were accomplished in the area corresponding to the pulp

The dimension of the pulp was enlarged to reproduce the endodontic access stage (Cohen &

The porcelain-faced crown was represented with the same outline of the enamel, being average thickness of 1.5 mm in the vestibular face, 1.2 mm in the lingual face, 1.0 mm in the proximal face and 2.0 mm in the occlusal face, with an end in chamfer around the core. The minimum thickness of the metal (NiCr) in the restoration porcelain-faced was 0.3 mm

The zinc phosphate cement was select because is usually used for cementation of the crown and of intra-radicular retainer with thickness from 50 to 100 m (Anusavice, 2003) and the

accomplished for the creation of five new models with retainer intra-radicular.

considered larger than the lingual cusp in 0.9 mm (Shillinburg et al., 1972).

and to the dentin, in the places corresponding to the core and porcelain-faced crown.

Hargreaves 2005), the biomechanical prepare and the filling of the root canal.

Pereira et al., 2006; Morgano, 1996; Morgano & Bracket, 1999).

supplies the average dimensions in the cervical area.

gutta-percha height was of 5 mm (Morgano, 1996).

**2. Materials and methods** 

(Yamamoto, 1985).

In four models with intra-radicular retainer was varied the ending line of the core preparation (with ferrule geometry given by the core) and another model was simulated with simple core (without ferrule), staying the same coronary restoration.

The width and the height of the ferrule were determined as a proportion of the thickness of radicular dentine found in the cementoenamel junction region on the lingual side (1.85 mm), separated for three thirds. For other words, the ferrule with x height varying of 0.62 x 0.62 (T1H1), 0.62 x 1.34 (T1H2), 1.34 x 0.62 (T2H1) and 1.34 x 1.34 mm (T2H2), (Fig. 1).

All of the models with intra-radicular retainer presented the same characteristics in relation to the supporting structures, porcelain-faced crown, cement thickness and apical seal (Fig. 2).

Fig. 1. Ferrule geometries of the radicular dentine in a premolar root.

With the measures of the thickness in each one of the structures (pulp, dentin, enamel) in each face (mesial, distal, lingual, vestibular) a coordinates system was generated with the origin in the pulpal apex and all the values of thickness of the structures were transformed in coordinates. As many coordinates existed to generate the points in the Finite Element (FE) program, a routine in PCL (Patran Command Language) was created to read the spreadsheet and to generate the points. After the points generation (Fig. 3a), the curves were generated and, soon afterwards, the surfaces. Starting from the dental structures surfaces built, the superficial meshes were generated with triangular linear elements (Tri3). After

Biomechanical Analysis of Restored Teeth with

T1H1 1 x 1

Structure / Material Young´s Modulus

Table 1. Models and mesh size.

simplification was justified.

Cast radicular retainer

Cast Intra-Radicular Retainer with and Without Ferrule 137

H Natural healthy tooth 179403 1109929

T1H2 179430 1142542 T2H1 189438 1200778 T2H2 180720 1148853 T0H0 184750 1175175

It was assumed that all the structures in the models were, homogeneous, isotropic and linearly elastic behavior as characterized by two physical properties: Young´s Modulus (E) and Poisson's Ratio (), Table 2. The interfaces between the structures were presumed to be perfectly united, because the aim was to provide a comparison of our approach, that this

Phosphate of Zinc Cement 13.00 0.35 Powers at al. (1976) Feldspathic Ceramic 82.80 0.35 Peyton & Craig (1963)

(ILOR56: gold-alloy post) 93.00 0.33 Pegoretti et al. (2002)

Finally, boundary conditions or the model constraint and the loads are also applied. The constraint conditions applied were: in the maxillary sinus, translation in x, y and z directions and rotations in x, y and z axis, fully anchor; in the mesial and distal extremities of the cortical and spongy bone, translation in x direction (perpendicular to this faces) and rotations in y and z axis were anchor. Four load cases were built varying the site, inclination and the application

Initially, all of the models received the resultant load intensity of 291.36 N parallel to the tooth's long axis, with the aim to evaluate the wedge effect (Fig. 4). The load was applied on the lingual and vestibular cusp above the occlusal surface, at 45o in relation to the tooth's long axis (Holmes et al. 1996), distributed in 19 nodal points of 0.85 mm2 area in vestibular cusp and in 19 nodal points of 0.75 mm2 area of in lingual cusp (Kumugai et al., 1999).

area of a 291.36 N total static load, maximum masticatory force (Ferrario et al., 2004).

Table 2. Physical properties of the anatomical structures and materials.

Pulp 0.02 0.45 Farah & Craig (1974) Dentin 18.60 0.31 Ko et al*.* (1992) Enamel 41.00 0.30 Ko et al. (1992) Periodontal ligament 0.0689 0.45 Weinstein et al. (1980) Cortical bone 13.70 0.30 Ko et al*.* (1992) Spongy bone 1.37 0.30 Ko et al. (1992) Gutta-percha 0.00069 0.45 Friedman et al.(1975) NiCr 188.00 0.33 Black & Hastings (1998)

nodes

(GPa) Poisson´s Ratio Reference

193034 1226486

Number of elements

Code Ferrule Thickness x Height Number of

that, volumetric meshes (Fig. 3d) with tetrahedral linear elements (Tet4) were generated, following the procedures in Poiate et al. (2008, 2009a, 2009b, 2011).

Fig. 2. Structures of the models with intra-radicular retainer.

Fig. 3. Sequence of procedures for 3D natural healthy tooth model generation, points generation, dental structures surfaces, superficial and volumetric mesh, respectively.

The degree of discretization of the FE models (Table 1) was established from convergence studies of the results in computer modeling (Pentium 4 3.2 GHz computer with 2.0 Gb RAM memory) to ensure that a proper FE model mesh density was generated and in this way model a realistically anatomic geometry.


Table 1. Models and mesh size.

136 Finite Element Analysis – From Biomedical Applications to Industrial Developments

that, volumetric meshes (Fig. 3d) with tetrahedral linear elements (Tet4) were generated,

following the procedures in Poiate et al. (2008, 2009a, 2009b, 2011).

Fig. 2. Structures of the models with intra-radicular retainer.

Fig. 3. Sequence of procedures for 3D natural healthy tooth model generation, points generation, dental structures surfaces, superficial and volumetric mesh, respectively.

model a realistically anatomic geometry.

The degree of discretization of the FE models (Table 1) was established from convergence studies of the results in computer modeling (Pentium 4 3.2 GHz computer with 2.0 Gb RAM memory) to ensure that a proper FE model mesh density was generated and in this way It was assumed that all the structures in the models were, homogeneous, isotropic and linearly elastic behavior as characterized by two physical properties: Young´s Modulus (E) and Poisson's Ratio (), Table 2. The interfaces between the structures were presumed to be perfectly united, because the aim was to provide a comparison of our approach, that this simplification was justified.


Table 2. Physical properties of the anatomical structures and materials.

Finally, boundary conditions or the model constraint and the loads are also applied. The constraint conditions applied were: in the maxillary sinus, translation in x, y and z directions and rotations in x, y and z axis, fully anchor; in the mesial and distal extremities of the cortical and spongy bone, translation in x direction (perpendicular to this faces) and rotations in y and z axis were anchor. Four load cases were built varying the site, inclination and the application area of a 291.36 N total static load, maximum masticatory force (Ferrario et al., 2004).

Initially, all of the models received the resultant load intensity of 291.36 N parallel to the tooth's long axis, with the aim to evaluate the wedge effect (Fig. 4). The load was applied on the lingual and vestibular cusp above the occlusal surface, at 45o in relation to the tooth's long axis (Holmes et al. 1996), distributed in 19 nodal points of 0.85 mm2 area in vestibular cusp and in 19 nodal points of 0.75 mm2 area of in lingual cusp (Kumugai et al., 1999).

Biomechanical Analysis of Restored Teeth with

Cast Intra-Radicular Retainer with and Without Ferrule 139

Fig. 5. MPS in all models loaded longitudinally with Vestibular (V)-Lingual (L) and Mesial

(M)–Distal (D) slices.

After that, the model with ferrule that best minimized the wedge effect, as well as the models that represent natural healthy tooth and restored with simple core (T0H0, without ferrule, Fig. 1) were submitted to the following loads (Fig. 4):


Fig. 4. Place, area and orientation of the load application, a)wedge effect, b)vestibular lever effect, c)proximal lever effect and d)torsion effect.

The processing stage or the solution analysis was performed using MSC/NASTRAN 2005 software (MSC Software Corporation, Santa Ana, CA, USA). The MSC/PATRAN 2005 software, used in the pre-processing, was also used for the post-processing, visualization, and evaluation of the results.

In this study, the maximum principal stress was used as the stress criterion to present the stress patterns distribution in the analyzed models. The cement and dentin's tensile strength of 8.3 MPa (Powers at al., 1976) and 103 MPa (Tanaka et al., 2003), respectively, as well as dentin´s compressive strength of 282 MPa (Tanaka et al., 2003), respectively, will serve as reference for results comparison to evaluate if the loads would be potentially harmful to the studied structures.

#### **3. Results**

#### **3.1 Wedge effect**

Under load with parallel resultant to the long axis, all of the models answered equally. The differences were just in the dentin area in contact with the ferrule (Fig. 5). The results

After that, the model with ferrule that best minimized the wedge effect, as well as the models that represent natural healthy tooth and restored with simple core (T0H0, without

a. Oblique Load with 45o in relation to the tooth's long axis distributed in 19 nodal points of 0.85 mm2 area in the vestibular cusp, with the aim to evaluate the vestibular lever

b. Load parallel to the tooth's long axis distributed in 19 nodal points of 0.80 mm2 area in the mesial marginal ridge, with the aim to evaluate the proximal lever effect; c. Oblique Load with 45o in relation to the tooth's long axis pointed to vestibular cusp, distributed in 19 nodal points of 0.80 mm2 area in the mesial marginal ridge, with the

Fig. 4. Place, area and orientation of the load application, a)wedge effect, b)vestibular lever

The processing stage or the solution analysis was performed using MSC/NASTRAN 2005 software (MSC Software Corporation, Santa Ana, CA, USA). The MSC/PATRAN 2005 software, used in the pre-processing, was also used for the post-processing, visualization,

In this study, the maximum principal stress was used as the stress criterion to present the stress patterns distribution in the analyzed models. The cement and dentin's tensile strength of 8.3 MPa (Powers at al., 1976) and 103 MPa (Tanaka et al., 2003), respectively, as well as dentin´s compressive strength of 282 MPa (Tanaka et al., 2003), respectively, will serve as reference for results comparison to evaluate if the loads would be potentially harmful to the

Under load with parallel resultant to the long axis, all of the models answered equally. The differences were just in the dentin area in contact with the ferrule (Fig. 5). The results

ferrule, Fig. 1) were submitted to the following loads (Fig. 4):

aim to evaluate the torsion effect.

effect, c)proximal lever effect and d)torsion effect.

and evaluation of the results.

studied structures.

**3.1 Wedge effect** 

**3. Results** 

effect;

Fig. 5. MPS in all models loaded longitudinally with Vestibular (V)-Lingual (L) and Mesial (M)–Distal (D) slices.

Biomechanical Analysis of Restored Teeth with

failure (8.3 MPa).

effect.

**3.2 Vestibular lever effect** 

models, respectively.

Cast Intra-Radicular Retainer with and Without Ferrule 141

obtained with models T1H1 and T2H1 presented similar pattern of stress distribution in Maximum Principal Stress (MPS), but different from models T1H2 and T2H2 (that present low tensile stress under the ferrule). In the model H, the dentin area between the bone and the enamel presents some tensile stress, but the Fig. 6 show that the stress vector direction is not totally tangential to the transversal section and it wouldn't be deleterious, nor for the intensity nor for the direction. Similar stress appears in all the models with retainer immediately for apical of the ferrule and cannot are responsible for eventual root fracture. The Fig. 7 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin. The results obtained with models T2H1, T0H0 and T1H1 presented a similar pattern of stress distribution in cement below the ferrula, but different from the model T1H2, which shows higher tensile stresses (up to 4 MPa), which by direction (Fig. 6) suggests greater tendency to loosen or break the cement layer in the region, because the stresses are significantly higher, but the failure of the cement seems unlikely when comparing the absolute values of stress with the stress required to the cement cohesive

Fig. 7. Perspective view of the cement between the retainer and dentin, MPS in the wedge

Under load with inclination of 45o in the vestibular cusp the T2H2, T0H0 and H models answered equally (Fig. 8), except in the radicular dentin that is being doubled and it is observed compressive stress in the vestibular, concentrated in the contact area with the cortical bone that acts as fulcrum (Fig. 9). The tensile stress concentrates and reaches the maximum on the opposite side, lingual, but, unexpectedly, it presents a larger extension in the model with ferrule, with maximum of 104, 110 and 109 MPa for H, T0H0 and T2H2

Fig. 6. MPS vectors in the surfaces of the radicular dentine in the wedge effect in M-D and V-L slices.

Fig. 6. MPS vectors in the surfaces of the radicular dentine in the wedge effect in M-D and

V-L slices.

obtained with models T1H1 and T2H1 presented similar pattern of stress distribution in Maximum Principal Stress (MPS), but different from models T1H2 and T2H2 (that present low tensile stress under the ferrule). In the model H, the dentin area between the bone and the enamel presents some tensile stress, but the Fig. 6 show that the stress vector direction is not totally tangential to the transversal section and it wouldn't be deleterious, nor for the intensity nor for the direction. Similar stress appears in all the models with retainer immediately for apical of the ferrule and cannot are responsible for eventual root fracture.

The Fig. 7 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin. The results obtained with models T2H1, T0H0 and T1H1 presented a similar pattern of stress distribution in cement below the ferrula, but different from the model T1H2, which shows higher tensile stresses (up to 4 MPa), which by direction (Fig. 6) suggests greater tendency to loosen or break the cement layer in the region, because the stresses are significantly higher, but the failure of the cement seems unlikely when comparing the absolute values of stress with the stress required to the cement cohesive failure (8.3 MPa).

Fig. 7. Perspective view of the cement between the retainer and dentin, MPS in the wedge effect.

#### **3.2 Vestibular lever effect**

Under load with inclination of 45o in the vestibular cusp the T2H2, T0H0 and H models answered equally (Fig. 8), except in the radicular dentin that is being doubled and it is observed compressive stress in the vestibular, concentrated in the contact area with the cortical bone that acts as fulcrum (Fig. 9). The tensile stress concentrates and reaches the maximum on the opposite side, lingual, but, unexpectedly, it presents a larger extension in the model with ferrule, with maximum of 104, 110 and 109 MPa for H, T0H0 and T2H2 models, respectively.

Biomechanical Analysis of Restored Teeth with

Cast Intra-Radicular Retainer with and Without Ferrule 143

The Fig. 11 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin and the Fig. 10 shows the MPS vectors in the internal and external surfaces of the radicular dentine at the intersections with in M-D and V-L planes.

Fig. 10. Perspective view of the cement between the retainer and dentin, MPS in the

vestibular lever effect.

Fig. 8. Perspective view of all structures, MPS in the vestibular lever effect.

Fig. 9. Perspective view of the dentin, MPS in the vestibular lever effect.

Fig. 8. Perspective view of all structures, MPS in the vestibular lever effect.

Fig. 9. Perspective view of the dentin, MPS in the vestibular lever effect.

The Fig. 11 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin and the Fig. 10 shows the MPS vectors in the internal and external surfaces of the radicular dentine at the intersections with in M-D and V-L planes.

Fig. 10. Perspective view of the cement between the retainer and dentin, MPS in the vestibular lever effect.

Biomechanical Analysis of Restored Teeth with

**3.3 Proximal lever effect** 

Cast Intra-Radicular Retainer with and Without Ferrule 145

The Fig. 12 shows the results under load parallel to the tooth's long axis in the mesial marginal ridge. The maximum tensile stress in dentin (Fig. 13) at the distal edge are also

much seemed (68, 68 and 67 MPa), smaller than in the vestibular lever effect.

Fig. 12. Perspective view of all structures, MPS in the proximal lever effect.

Fig. 11. MPS vectors in the surfaces of the radicular dentine in the vestibular lever effect in M-D and V-L slices.

#### **3.3 Proximal lever effect**

144 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Fig. 11. MPS vectors in the surfaces of the radicular dentine in the vestibular lever effect in

M-D and V-L slices.

The Fig. 12 shows the results under load parallel to the tooth's long axis in the mesial marginal ridge. The maximum tensile stress in dentin (Fig. 13) at the distal edge are also much seemed (68, 68 and 67 MPa), smaller than in the vestibular lever effect.

Fig. 12. Perspective view of all structures, MPS in the proximal lever effect.

Biomechanical Analysis of Restored Teeth with

Cast Intra-Radicular Retainer with and Without Ferrule 147

Fig. 15. MPS vectors in the surfaces of the radicular dentine in the proximal lever effect in

M-D and V-L slices.

Fig. 13. Perspective view of the dentin, MPS in the proximal lever effect.

The Fig. 14 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin and the Fig.15 shows the MPS vectors in the internal and external surfaces of the radicular dentine at the intersections with in M-D and V-L planes. The stress orientation in all of the cases is similar.

Fig. 14. Perspective view of the cement between the retainer and dentin, MPS in the proximal lever effect.

Biomechanical Analysis of Restored Teeth with Cast Intra-Radicular Retainer with and Without Ferrule 147

146 Finite Element Analysis – From Biomedical Applications to Industrial Developments

Fig. 13. Perspective view of the dentin, MPS in the proximal lever effect.

The stress orientation in all of the cases is similar.

proximal lever effect.

The Fig. 14 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin and the Fig.15 shows the MPS vectors in the internal and external surfaces of the radicular dentine at the intersections with in M-D and V-L planes.

Fig. 14. Perspective view of the cement between the retainer and dentin, MPS in the

Fig. 15. MPS vectors in the surfaces of the radicular dentine in the proximal lever effect in M-D and V-L slices.

Biomechanical Analysis of Restored Teeth with

Fig. 17. Perspective view of the dentin, MPS in the torsion effect.

effect.

Fig. 18. Perspective view of the cement between the retainer and dentin, MPS in the torsion

Cast Intra-Radicular Retainer with and Without Ferrule 149

#### **3.4 Torsion effect**

The Fig. 16 shows the MPS results in all structures under oblique load with 45o in relation to the tooth's long axis pointed to vestibular cusp in the mesial marginal ridge.

Fig. 16. Perspective view of all structures, MPS in the torsion effect.

The MPS in dentin are present in Fig. 17. The Fig. 18 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin and the Fig.19 shows the MPS vectors in the internal and external surfaces of the radicular dentine at the intersections with in M-D and V-L planes.

The Fig. 16 shows the MPS results in all structures under oblique load with 45o in relation to

the tooth's long axis pointed to vestibular cusp in the mesial marginal ridge.

Fig. 16. Perspective view of all structures, MPS in the torsion effect.

intersections with in M-D and V-L planes.

The MPS in dentin are present in Fig. 17. The Fig. 18 represents a perspective view of the distribution of the MPS in the cement between the retainer and dentin and the Fig.19 shows the MPS vectors in the internal and external surfaces of the radicular dentine at the

**3.4 Torsion effect** 

Fig. 17. Perspective view of the dentin, MPS in the torsion effect.

Fig. 18. Perspective view of the cement between the retainer and dentin, MPS in the torsion effect.

Biomechanical Analysis of Restored Teeth with

(Distance/Total length),.

cervical direction.

increase up to 5 MPa.

Cast Intra-Radicular Retainer with and Without Ferrule 151

The Figs 20 to 28 shows the MPS in a ridge A to B, indicated in the drawing corresponding to each graph, in the surface of root dentin (interface with periodontal ligament), depending on the relative position of the node under every load, due to the abcissa axis is adimensional

The Fig. 20 shows the outside edge of the lingual surface. It is observed that the models that generate the wedge or proximal lever present in the lingual edge, very low stresses, all of the same order of magnitude. Since the models that generate vestibular lever and torsion

**3.5 Stress plot in wedge, vestibular lever, proximal lever and torsion effect** 

have high tensile stresses in this edge, but below the dentin´s strenght limit.

Fig. 20. MPS at the edge of the root dentin on the lingual surface, from the apex to the

The Fig. 21 shows the outside edge of the vestibular surface. It is observed that the models that generate proximal wedge and proximal lever have the same order of magnitude of stresses acting on the vestibular edge with inflection behavior (tensile to compressive) from the apex to the cervical direction. In models that generate proximal lever, the tensile stresses near the apex are close to the wedge effect, however, near the neck, the stress tensile

Fig. 19. MPS vectors in the surfaces of the radicular dentine in the torsion effect in M-D and V-L slices.

Fig. 19. MPS vectors in the surfaces of the radicular dentine in the torsion effect in M-D and

V-L slices.

#### **3.5 Stress plot in wedge, vestibular lever, proximal lever and torsion effect**

The Figs 20 to 28 shows the MPS in a ridge A to B, indicated in the drawing corresponding to each graph, in the surface of root dentin (interface with periodontal ligament), depending on the relative position of the node under every load, due to the abcissa axis is adimensional (Distance/Total length),.

The Fig. 20 shows the outside edge of the lingual surface. It is observed that the models that generate the wedge or proximal lever present in the lingual edge, very low stresses, all of the same order of magnitude. Since the models that generate vestibular lever and torsion have high tensile stresses in this edge, but below the dentin´s strenght limit.

Fig. 20. MPS at the edge of the root dentin on the lingual surface, from the apex to the cervical direction.

The Fig. 21 shows the outside edge of the vestibular surface. It is observed that the models that generate proximal wedge and proximal lever have the same order of magnitude of stresses acting on the vestibular edge with inflection behavior (tensile to compressive) from the apex to the cervical direction. In models that generate proximal lever, the tensile stresses near the apex are close to the wedge effect, however, near the neck, the stress tensile increase up to 5 MPa.

Biomechanical Analysis of Restored Teeth with

Cast Intra-Radicular Retainer with and Without Ferrule 153

Fig. 22. MPS at the edge of the root dentin, the distal surface, from the apex to the cervical,

The Fig. 23 shows the outside edge of the mesial surface. We observed lowest stresses in wedge models. The inversion of tensile to compression stress occurs so evident in the

proximal lever models and so milder in torsion models.

under all loads

Fig. 21. MPS at the edge of the root dentin on the vestibular surface, from the apex to the cervical direction.

The Fig. 22 shows the outside edge of the distal surface. We observed lowest stresses in wedge models, followed by vestibular lever models, torsion models and reaching the maximum stresses in proximal lever models. In models of proximal lever and torsion effect are generated stress peaks on the same nodes of the same models; the lower stress in torsion models could be due to the lower longitudinal component of the load in this case.

Fig. 21. MPS at the edge of the root dentin on the vestibular surface, from the apex to the

models could be due to the lower longitudinal component of the load in this case.

The Fig. 22 shows the outside edge of the distal surface. We observed lowest stresses in wedge models, followed by vestibular lever models, torsion models and reaching the maximum stresses in proximal lever models. In models of proximal lever and torsion effect are generated stress peaks on the same nodes of the same models; the lower stress in torsion

cervical direction.

The Fig. 23 shows the outside edge of the mesial surface. We observed lowest stresses in wedge models. The inversion of tensile to compression stress occurs so evident in the proximal lever models and so milder in torsion models.

Biomechanical Analysis of Restored Teeth with

cervical, under all loads

cortical bone.

Cast Intra-Radicular Retainer with and Without Ferrule 155

Fig. 24. MPS at the edge of the root dentin on the lingual surface, from the apex to the

The Fig. 25 shows the inner edge of the vestibular surface. Tensile stresses are observed up to 27 MPa in the region near the retainer apex in the models that generate vestibular lever, followed by models that generate torsion, but in both types of loading, the stresses are smaller in the healthy tooth models. Figure 11 shows that the direction of the peak tensile is radial and therefore not likely to cause longitudinal fracture but delamination in the dentin and detachment of the retainer apex. In fact, presents a direction parallel to the tensile stress on the edge of the external vestibular root, which appears to be motivated by the apical ligament region, to oppose the rotation of the tooth that is supported by the vestibular

Fig. 23. MPS at the edge of the root dentin in the mesial side, from the apex to the cervical, under all loads.

The Fig. 24 shows the inner edge of the lingual surface. We observed lowest stresses in the models of wedge and tensile stresses up to 45 MPa in the models of vestibular lever, followed by models of torsion effect, but in both types of load, in models of healthy tooth the stresses are smaller. It seems important to note that the order of values is much lower than those in Fig. 20, corresponding to the outer surface, which leads to think that the fracture is expected to start on the outside, apparently motivated by folding of the tooth as a whole. Discard the possibility of fracture driven by stress concentration on the inside face, which could be motivated by the wedging effect promoted by the retainer.

The Fig. 20 shows that the highest stress on the outside edge was achieved by the healthy tooth model. However, it is unlikely that the healthy tooth had a greater tendency to fracture: this is not what is observed in practice.

On the other hand Fig. 24 shows the stress almost doubles in inner edge, for the cases of vestibular lever with retainer, compared to the healthy tooth. This suggests that the failure criterion should not be simply the value of the maximum principal stress, but must be influenced by the stress gradient.

Fig. 23. MPS at the edge of the root dentin in the mesial side, from the apex to the cervical,

The Fig. 24 shows the inner edge of the lingual surface. We observed lowest stresses in the models of wedge and tensile stresses up to 45 MPa in the models of vestibular lever, followed by models of torsion effect, but in both types of load, in models of healthy tooth the stresses are smaller. It seems important to note that the order of values is much lower than those in Fig. 20, corresponding to the outer surface, which leads to think that the fracture is expected to start on the outside, apparently motivated by folding of the tooth as a whole. Discard the possibility of fracture driven by stress concentration on the inside face,

The Fig. 20 shows that the highest stress on the outside edge was achieved by the healthy tooth model. However, it is unlikely that the healthy tooth had a greater tendency to

On the other hand Fig. 24 shows the stress almost doubles in inner edge, for the cases of vestibular lever with retainer, compared to the healthy tooth. This suggests that the failure criterion should not be simply the value of the maximum principal stress, but must be

which could be motivated by the wedging effect promoted by the retainer.

fracture: this is not what is observed in practice.

influenced by the stress gradient.

under all loads.

The Fig. 25 shows the inner edge of the vestibular surface. Tensile stresses are observed up to 27 MPa in the region near the retainer apex in the models that generate vestibular lever, followed by models that generate torsion, but in both types of loading, the stresses are smaller in the healthy tooth models. Figure 11 shows that the direction of the peak tensile is radial and therefore not likely to cause longitudinal fracture but delamination in the dentin and detachment of the retainer apex. In fact, presents a direction parallel to the tensile stress on the edge of the external vestibular root, which appears to be motivated by the apical ligament region, to oppose the rotation of the tooth that is supported by the vestibular cortical bone.

Biomechanical Analysis of Restored Teeth with

Cast Intra-Radicular Retainer with and Without Ferrule 157

Fig. 26. MPS at the edge of the root dentin, the distal surface, from the apex to the cervical,

The Fig. 27 shows the inner edge of the mesial surface. We observed lowest stresses in wedge models. The highest tensile stress occur to the torsion effect, regardless of the presence or not of the ferrula, which also does not help to reduce the stress on the lever

under all loads.

vestibular.

Fig. 25. MPS at the edge of the root dentin on the vestibular surface, from the apex to the cervical, under all loads

The Fig. 26 shows the inner edge of the distal surface. We observed lowest stresses in wedge models. Except the loading wedge models, all have tensile stresses, which shows that practically the entire thickness of the wall works in traction. Moreover, none of the models exceeded the stress developed by the model of healthy tooth with proximal lever load, which leads to the hypothesis that in no case has reached the stress level compatible with the fracture.

Fig. 25. MPS at the edge of the root dentin on the vestibular surface, from the apex to the

The Fig. 26 shows the inner edge of the distal surface. We observed lowest stresses in wedge models. Except the loading wedge models, all have tensile stresses, which shows that practically the entire thickness of the wall works in traction. Moreover, none of the models exceeded the stress developed by the model of healthy tooth with proximal lever load, which leads to the hypothesis that in no case has reached the stress level compatible with

cervical, under all loads

the fracture.

Fig. 26. MPS at the edge of the root dentin, the distal surface, from the apex to the cervical, under all loads.

The Fig. 27 shows the inner edge of the mesial surface. We observed lowest stresses in wedge models. The highest tensile stress occur to the torsion effect, regardless of the presence or not of the ferrula, which also does not help to reduce the stress on the lever vestibular.

Biomechanical Analysis of Restored Teeth with

retainer under all loads.

**4. Discussion** 

Nicholls 1995).

justify the fracture.

Cast Intra-Radicular Retainer with and Without Ferrule 159

Fig. 28. MPS shown in the circular edge of the inner dentin (VMLDV) around the apex of the

Several studies (Pierrisnard et al., 2002; Zhi-Yue & Yu-Xing, 2003) show that the ferrule creates a positive effect in the reduction of the stress concentration in the dentin-core junction and helps to maintain the integrity of the cement seal in the crown (Libman &

However, in spite of some authors recommend a coronary minimum height of ferrule (Tan et al., 2005; Pereira et al., 2006; Morgano, 1996; Morgano & Bracket, 1999; Aykent, 2006) to increase the fracture strength values, in this study the ferrule didn't influence on the stress distribution. An important observation should be made: for the perfect adhesion among the

Under load with parallel resultant to the long axis, all of the models present compression stress in the root apex, but with very low intensity when compared with the compression strength of dentin. The light stress concentration found around of the post apex cannot are responsible by longitudinal fracture, because the orientation and the intensity would not

structures, it was not possible to notice a wedge effect appreciable.

Fig. 27. MPS at the edge of the root dentin in the mesial side, from the apex to the cervical, under all loads.

In Figure 28 shows the circular edge of the inner dentin around the apex of the retainer. The abscissa axis is adimensional (Distance/Total length), where the 0 and 1 represents the vestibular, 0.25 the mesial, 0.5 the lingual and 0.75 the distal aspect. We observed lowest stresses in wedge models, followed by models of proximal lever, torsion and vestibular lever models. The region that remains the most tensioned is the vestibular.

Fig. 27. MPS at the edge of the root dentin in the mesial side, from the apex to the cervical,

In Figure 28 shows the circular edge of the inner dentin around the apex of the retainer. The abscissa axis is adimensional (Distance/Total length), where the 0 and 1 represents the vestibular, 0.25 the mesial, 0.5 the lingual and 0.75 the distal aspect. We observed lowest stresses in wedge models, followed by models of proximal lever, torsion and vestibular

lever models. The region that remains the most tensioned is the vestibular.

under all loads.

Fig. 28. MPS shown in the circular edge of the inner dentin (VMLDV) around the apex of the retainer under all loads.

#### **4. Discussion**

Several studies (Pierrisnard et al., 2002; Zhi-Yue & Yu-Xing, 2003) show that the ferrule creates a positive effect in the reduction of the stress concentration in the dentin-core junction and helps to maintain the integrity of the cement seal in the crown (Libman & Nicholls 1995).

However, in spite of some authors recommend a coronary minimum height of ferrule (Tan et al., 2005; Pereira et al., 2006; Morgano, 1996; Morgano & Bracket, 1999; Aykent, 2006) to increase the fracture strength values, in this study the ferrule didn't influence on the stress distribution. An important observation should be made: for the perfect adhesion among the structures, it was not possible to notice a wedge effect appreciable.

Under load with parallel resultant to the long axis, all of the models present compression stress in the root apex, but with very low intensity when compared with the compression strength of dentin. The light stress concentration found around of the post apex cannot are responsible by longitudinal fracture, because the orientation and the intensity would not justify the fracture.

Biomechanical Analysis of Restored Teeth with

relationship to two teeth (Hemmings et al., 1991).

situation of mechanical effort what the root is submitted.

apical, what increases the lever arm progressively (Cohen et al., 1993).

Within the limitations of this study, the following conclusions were drawn:

load, in that presents discreet beneficial effect;

except for the case of longitudinal load;

propitiate the root fracture.

**5. Conclusions** 

it acts as a fulcrum;

reality;

Cast Intra-Radicular Retainer with and Without Ferrule 161

ones for cohesive fracture. After the cement failure the stress distribution change and could

The intra-radicular retainers can be subject also to rotation force or torsion, whenever requested clinically through contacts functional cusp in the ridge, in other words, tooth

The occlusal contacts in the marginal ridge of the premolar could produce a torsion loads that tends to rotate the root along the tooth's axis. For this reason is extremely important have in mind the use of retainers that offer larger safety to the radicular remnant in any

Under oblique load with 45o in relation to the tooth's long axis pointed to vestibular cusp in the mesial marginal ridge, the maximum stress and orientation evidences again that cement layer is prone to failure and that the stress concentration is more serious in the ferrule case. When the cement around of the most coronary portion deteriorates, the fulcrum migrates to

The modeling in this study considered all the components without setting out eventual contact problems, in other words, without cement failure, what couldn't happen in the clinic reality.

Regarding the clinical significance of the results, it seems that the best way to protect the radicular remnant when restored with post would be to guarantee the occlusal adjustment, that don't happen loads different from the longitudinal. The most vulnerable part of the whole restorative system is the cement layer, that doesn't resist to the tensile stresses result from the loads application different from the longitudinal load. The post perfectly adhered doesn't seem lead to the occurrence of longitudinal fracture (below the bone crest), but probably to fracture that would begin in the vestibular in the height of the bone crest.

1. In spite of none of the simulated cases to evidence the wedge effect, the longitudinal load produced stresses that don't justify the rupture nor of the dentine nor of the cement layer and the stresses are very inferior to developed by all the other load types; 2. The ferrule isn't necessary to improve the stresses distribution, except for longitudinal

3. In the vestibular load were found, in the lingual face, tensile stresses guided parallel to the longitudinal axis and magnitude enough to induce vertical fractures above the bone crest. These stresses are associated to the tooth's bend, leaning in the cortical bone that

4. In the cement layer were found stresses that are prone to fracture in all of the loads,

5. A concerted effort to develop other FE models to improve the results may provide more reliable data about the non-linearly of the structures and the cement failures in the clinic

6. The analyses help the field better understand the biomechanical analysis of restored

teeth with cast intra-radicular retainer with and without ferrule.

The results of models T1H1 and T2H1 show that the increase in the width of ferrule, keeping the height, it was important to the root protection with the ferrule effect, because the compressive stress increased four times. The model T2H2 presented compressive stress in the radicular dentin on the ferrule (0.5 to 2.0 MPa) and in the radicular dentin above the periodontal ligament (0.5 to 20 MPa), mesial and distal face (Fig. 4).

On the other hand, the results of models T1H2 and T2H2 show that tensile stresses disappears in the dentine in contact with the ferrule, what can be attributed to the increase in the ferrule width, keeping the height. This characteristic confers to the ferrule design some superiority, because it can be inferred that it will be more difficult than the cement is unstuck in that area, since the tensile stress isn't submitted. The model T2H2 seems also to minimize the tensile stress in the cervical dentine (compressive stress from 0.5 to 2.0 MPa) and in the radicular dentine near to the periodontal ligament, following by the model T2H1, T1H1 and T1H2.

Posteriors teeth can be subject also to vestibular lever effects whenever requested eccentric efforts in lateral excursion. The contact oclusal on the work side in lateral excursion can reach the vestibular cusp in posteriors teeth, generating lever force on the involved roots that serve as guide for those movements.

Under load with inclination of 45o in the vestibular cusp, the maximum stress value is compatible with the fracture occurrence. If the fracture begins at that place can follow a perpendicular plan to the tensile vector and to spread tending the uprighting that suggests a vertical radicular fracture would pass exactly in the bone crest limit.

Fractures vertical appear as a result of stresses generated inside the root canal (Lertchirakarn et al., 2003), but the models showed tensile stress concentrated in the post apex, although of smaller magnitude that in the height of the bone crest.

In all of the models, the root canal stress in the mesial-distal slice presented tensile stress (except for the area near to the post apex) and near to the bone crest presents change in the vector orientation due to the rotation of the same ones, that it seems related with the presence of cortical bone. Therefore, the fracture area not just depends on the post and ferrule, but also of location of the tooth in the alveolus.

An interesting finding was the relatively small difference among the three models that could be attributed to the fact of the interfaces between the structures of the models was simulated with perfect adhesion. The perfect adhesion can losing in the course of time, what would explain that the failures didn't use to happen in retainer recently cemented. On the other hand, the models showed high tensile stresses concentration in more than half of the cement layer, which could be responsible for cohesive fracture.

Under the load area the tensile stresses on the ferrule generates 40 MPa, exceeding the cement's tensile strength of 8.3 MPa and associated with the stress orientation (Fig. 11) suggests the tendency to the cement layer failure in this area.

Under load parallel to the tooth's long axis in the mesial marginal ridge, the tooth restored with post would present a tendency to the rupture similar to the one of the natural, what is not supported by the clinical observations. The explanation for this discrepancy could be again in the fact that the cement layer presented stresses much larger than the necessary

The results of models T1H1 and T2H1 show that the increase in the width of ferrule, keeping the height, it was important to the root protection with the ferrule effect, because the compressive stress increased four times. The model T2H2 presented compressive stress in the radicular dentin on the ferrule (0.5 to 2.0 MPa) and in the radicular dentin above the

On the other hand, the results of models T1H2 and T2H2 show that tensile stresses disappears in the dentine in contact with the ferrule, what can be attributed to the increase in the ferrule width, keeping the height. This characteristic confers to the ferrule design some superiority, because it can be inferred that it will be more difficult than the cement is unstuck in that area, since the tensile stress isn't submitted. The model T2H2 seems also to minimize the tensile stress in the cervical dentine (compressive stress from 0.5 to 2.0 MPa) and in the radicular dentine near to the periodontal ligament, following by the model T2H1,

Posteriors teeth can be subject also to vestibular lever effects whenever requested eccentric efforts in lateral excursion. The contact oclusal on the work side in lateral excursion can reach the vestibular cusp in posteriors teeth, generating lever force on the involved roots

Under load with inclination of 45o in the vestibular cusp, the maximum stress value is compatible with the fracture occurrence. If the fracture begins at that place can follow a perpendicular plan to the tensile vector and to spread tending the uprighting that suggests a

Fractures vertical appear as a result of stresses generated inside the root canal (Lertchirakarn et al., 2003), but the models showed tensile stress concentrated in the post apex, although of

In all of the models, the root canal stress in the mesial-distal slice presented tensile stress (except for the area near to the post apex) and near to the bone crest presents change in the vector orientation due to the rotation of the same ones, that it seems related with the presence of cortical bone. Therefore, the fracture area not just depends on the post and

An interesting finding was the relatively small difference among the three models that could be attributed to the fact of the interfaces between the structures of the models was simulated with perfect adhesion. The perfect adhesion can losing in the course of time, what would explain that the failures didn't use to happen in retainer recently cemented. On the other hand, the models showed high tensile stresses concentration in more than half of the cement

Under the load area the tensile stresses on the ferrule generates 40 MPa, exceeding the cement's tensile strength of 8.3 MPa and associated with the stress orientation (Fig. 11)

Under load parallel to the tooth's long axis in the mesial marginal ridge, the tooth restored with post would present a tendency to the rupture similar to the one of the natural, what is not supported by the clinical observations. The explanation for this discrepancy could be again in the fact that the cement layer presented stresses much larger than the necessary

periodontal ligament (0.5 to 20 MPa), mesial and distal face (Fig. 4).

vertical radicular fracture would pass exactly in the bone crest limit.

smaller magnitude that in the height of the bone crest.

ferrule, but also of location of the tooth in the alveolus.

layer, which could be responsible for cohesive fracture.

suggests the tendency to the cement layer failure in this area.

T1H1 and T1H2.

that serve as guide for those movements.

ones for cohesive fracture. After the cement failure the stress distribution change and could propitiate the root fracture.

The intra-radicular retainers can be subject also to rotation force or torsion, whenever requested clinically through contacts functional cusp in the ridge, in other words, tooth relationship to two teeth (Hemmings et al., 1991).

The occlusal contacts in the marginal ridge of the premolar could produce a torsion loads that tends to rotate the root along the tooth's axis. For this reason is extremely important have in mind the use of retainers that offer larger safety to the radicular remnant in any situation of mechanical effort what the root is submitted.

Under oblique load with 45o in relation to the tooth's long axis pointed to vestibular cusp in the mesial marginal ridge, the maximum stress and orientation evidences again that cement layer is prone to failure and that the stress concentration is more serious in the ferrule case. When the cement around of the most coronary portion deteriorates, the fulcrum migrates to apical, what increases the lever arm progressively (Cohen et al., 1993).

The modeling in this study considered all the components without setting out eventual contact problems, in other words, without cement failure, what couldn't happen in the clinic reality.

Regarding the clinical significance of the results, it seems that the best way to protect the radicular remnant when restored with post would be to guarantee the occlusal adjustment, that don't happen loads different from the longitudinal. The most vulnerable part of the whole restorative system is the cement layer, that doesn't resist to the tensile stresses result from the loads application different from the longitudinal load. The post perfectly adhered doesn't seem lead to the occurrence of longitudinal fracture (below the bone crest), but probably to fracture that would begin in the vestibular in the height of the bone crest.

### **5. Conclusions**

Within the limitations of this study, the following conclusions were drawn:


Biomechanical Analysis of Restored Teeth with

12.

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#### **6. Acknowledgements**

This study was partially based on a thesis submitted to the University of São Paulo, in fulfillment of the requirements for PhD degree. The authors are grateful to the CAPES (Coordenação de Aperfeicoamento de Pessoal de Nível Superior, Coordination for the Improvement of Higher Education Personnel) that supported this research project.

#### **7. References**


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**7. References** 


**Part 2** 

**Cardiovascular and Skeletal Systems** 

