Degrees of freedom

1.E+05

1.E+06

The first step in the generation of an FE model is to obtain a geometrical representation of the real system. Some models use a simplified representation of the geometrical shape of the tooth, using elliptical paraboloids or similar shapes to represent the root (Asmussen et al., 2005; Bourauel et al., 2000; Holmes et al., 1996) or cylindrical blocks to represent the alveolar bone (Barjau-Escribano et al., 2006; Holmes et al., 1996; Mezzomo et al., 2011). Other researchers have tried to represent the real geometry better by using anatomical data (Adanir & Belli, 2007; Lanza et al., 2005), X-ray images (Maceri et al., 2007) or CT data (Magne, 2007; Tajima et al., 2009). Authors using two-dimensional data sometimes make use of appropriate algorithms to obtain a three-dimensional model (Maceri et al., 2007). With advances in CAD software and 3D scanning methods, most of the recent works obtain the external geometry of a real representative tooth or a plaster model using 3D digitising scanners (Ausiello et al., 2001; Ferrari et al., 2008; Ichim et al., 2006) and import it into a 3D CAD software application. The geometrical representation of the different components of the restoration is obtained later in CAD using Boolean algebra. The authors recently used this procedure (Gonzalez-Lluch et al., 2009b) to obtain a realistic geometrical model of a maxillary central incisor, which is shown, in a sagittal section, in Fig. 3.

### **3.2 Components in the model**

The natural tooth is the reference for comparing the biomechanical behaviour of an endodontic restoration. Several works in the literature have modelled the natural tooth (Dejak et al., 2003; Middleton et al., 1996; Zarone et al., 2006). Most of modelled natural teeth included enamel, dentine, cortical and cancellous bone, pulp and ligament (Middleton et al., 1996; Rees & Jacobsen, 1997). Some of them did not include the ligament (Zarone et al., 2006). These studies used the model of the natural tooth to compare the calculated biomechanical response with the experimental results (Rees & Jacobsen, 1997), to evaluate the behaviour of a restored tooth against a natural tooth (Davy et al., 1981; Maceri et al., 2009; Soares et al., 2008b; Zarone et al., 2006) or both (Ferrari et al., 2008). Cementum is not considered in most of the models, due to its reduced thickness, although a recent work (Ren et al., 2010) has reported the importance of the cemento-dentinal junction and cementum in stress distribution in the root and the ligament.

Biomechanical Models of Endodontic Restorations 139

2011; Uddanwadiker et al., 2007), 2.5 mm (Coelho et al., 2009) , 1 mm (Toksavul et al., 2006). Ichim et al. (Ichim et al., 2006) prepared biomechanical models without a ferrule and with a ferrule of different heights (0.5, 1.0, 1.5, 2 mm) to study its effect on the strength of the

Mesh definition is a key point in FEA. The mechanical behaviour of a continuous domain with an infinite number of degrees of freedom is approximated in the model with the simplified mechanical response of a set of discrete finite elements with a limited number of degrees of freedom. This process occurs when a mesh of finite elements is defined for the original system. The accuracy of the results is highly dependent on the characteristics of this mesh. A coarse mesh will produce results in a short computation time, but the accuracy of these results will be compromised. A finer mesh may improve the validity of the results but with the cost of a higher processing time. A coarse mesh is affected by two main sources of error: firstly, an error is introduced in representing the boundaries of the real geometry; and secondly, the mechanical response of the continuous material in each finite element is represented by a simplified polynomial function, which is only an approximation of the actual response. In biomechanical applications, such as dentistry, the geometry of the real system is highly irregular, so the mesh density should be enough to obtain a good

Generation of a good mesh of FE for a restored tooth is always a challenge. The complicated shape of the tooth makes it difficult to produce a mesh manually. Moreover, some elements of a dental restoration present dimensions that are quite different from one another. For example, the luting cement used to bond the post to the root, or the PDL, have a thickness of a fraction of a millimetre, which is much smaller than the post dimensions, although its total length would be of several millimetres. To obtain a good mesh, a small element size has to be used in this thinner component, but such a size cannot be maintained for all the components, because, otherwise, the number of degrees of freedom would increase to a limit beyond the capacity of the computer or the computation time would increase excessively. A good meshing strategy should increase mesh density in areas where a greater expected stress gradient is expected or with thinner components, and increase the size of the elements in other parts of the model. Two different basic strategies can be used to improve finite element results from a first tentative model, namely the h-method or the p-method (Zienkiewicz & Taylor, 1989). In the h-method, the mesh is refined using elements of a smaller size, whereas in the p-method, the size of the elements is maintained but the order of the interpolation function is increased. A mixed strategy that combines both options can also be used. Most of the works in the literature about endodontic simulation use finite elements with linear or quadratic interpolation and refine the mesh using smaller elements

The first FE models used two-dimensional finite elements to represent the tooth and endodontic restoration (Craig & Farah, 1977; Davy et al., 1981), assuming the axisymmetric or plane strain hypothesis. In these planar models triangular or quadrangular elements with three or four nodes, respectively, are used to mesh the system. The size of the finite elements in these first models was relatively large owing to the limitations in computer resources. Davy et al. (Davy et al., 1981) used quadrangular elements for an incisor and tested different meshes to decide on the mesh density by comparing displacements of selected nodes. Finally the finer mesh tested was selected,

to improve the representation, so the h-method is prevalent.

restoration.

**3.3 Mesh definition** 

geometrical representation.

Fig. 3. Sagittal section of an endodontically restored maxillary central incisor

Previous works simulating post-core endodontic restorations have not always included all the actual components of the restoration. Some works assumed that bone deformation is negligible and did not include this component in the model (Boschian Pest et al., 2006; Davy et al., 1981; Maceri et al., 2009). Not all models in the literature include the bone. When included, sometimes it is modelled with uniform properties (Imanishi et al., 2003; Nakamura et al., 2006), although most of the models consider the cancellous bone and a layer of cortical bone near the bone surface (Asmussen et al., 2005; Gonzalez-Lluch et al., 2009b).

Thin structures, such as the PDL and cement are not always considered in the FE models. In natural teeth, the PDL thickness is approximately between 0.125 mm and 0.375 mm (Rees & Jacobsen, 1997). This range is covered in different biomechanical models (Asmussen et al., 2005; Boschian Pest et al., 2006; Ferrari et al., 2008; Ichim et al., 2006; Soares et al., 2010). However, some works did not include the ligament (Yaman et al., 1998; Zarone et al., 2006), despite it has been reported that stress distribution is affected to an important degree by this omission (Davy et al., 1981) or even by a geometrically simplified representation (Toms & Eberhardt, 2003). When cement layers are considered in the biomechanical model, they are usually represented with a constant small thickness (Asmussen et al., 2005; Maceri et al., 2009; Okamoto et al., 2008; Schmitter et al., 2010). However, some works simplify the model by obviating the cement in the interface between the core and the crown (Gonzalez-Lluch et al., 2009b; Lanza et al., 2005) or even in the interface between the post and the root (Adanir & Belli, 2007; Hsu et al., 2009).

Several types of endodontic posts can be found in the literature (Christensen, 1998; Fernandes et al., 2003). Originally, cast post-core systems were used as a single metal alloy unit. Subsequently, prefabricated posts made out of stainless steel, titanium or precious metals were used, while more recently prefabricated fibre posts were introduced. All these post types have been studied with FE models.

Some earlier laboratory studies maintain that preparing a ferrule might improve the resistance of post-core system (Assif & Gorfil, 1994; Aykent et al., 2006; Ichim et al., 2006; Tan et al., 2005; Zhi-Yue & Yu-Xing, 2003). Accordingly, many works in the literature prepared a ferrule of varying heights in the biomechanical model: 2 mm (Mezzomo et al., 2011; Uddanwadiker et al., 2007), 2.5 mm (Coelho et al., 2009) , 1 mm (Toksavul et al., 2006). Ichim et al. (Ichim et al., 2006) prepared biomechanical models without a ferrule and with a ferrule of different heights (0.5, 1.0, 1.5, 2 mm) to study its effect on the strength of the restoration.

### **3.3 Mesh definition**

138 Theoretical Biomechanics

Fig. 3. Sagittal section of an endodontically restored maxillary central incisor

bone surface (Asmussen et al., 2005; Gonzalez-Lluch et al., 2009b).

& Belli, 2007; Hsu et al., 2009).

post types have been studied with FE models.

Previous works simulating post-core endodontic restorations have not always included all the actual components of the restoration. Some works assumed that bone deformation is negligible and did not include this component in the model (Boschian Pest et al., 2006; Davy et al., 1981; Maceri et al., 2009). Not all models in the literature include the bone. When included, sometimes it is modelled with uniform properties (Imanishi et al., 2003; Nakamura et al., 2006), although most of the models consider the cancellous bone and a layer of cortical bone near the

Thin structures, such as the PDL and cement are not always considered in the FE models. In natural teeth, the PDL thickness is approximately between 0.125 mm and 0.375 mm (Rees & Jacobsen, 1997). This range is covered in different biomechanical models (Asmussen et al., 2005; Boschian Pest et al., 2006; Ferrari et al., 2008; Ichim et al., 2006; Soares et al., 2010). However, some works did not include the ligament (Yaman et al., 1998; Zarone et al., 2006), despite it has been reported that stress distribution is affected to an important degree by this omission (Davy et al., 1981) or even by a geometrically simplified representation (Toms & Eberhardt, 2003). When cement layers are considered in the biomechanical model, they are usually represented with a constant small thickness (Asmussen et al., 2005; Maceri et al., 2009; Okamoto et al., 2008; Schmitter et al., 2010). However, some works simplify the model by obviating the cement in the interface between the core and the crown (Gonzalez-Lluch et al., 2009b; Lanza et al., 2005) or even in the interface between the post and the root (Adanir

Several types of endodontic posts can be found in the literature (Christensen, 1998; Fernandes et al., 2003). Originally, cast post-core systems were used as a single metal alloy unit. Subsequently, prefabricated posts made out of stainless steel, titanium or precious metals were used, while more recently prefabricated fibre posts were introduced. All these

Some earlier laboratory studies maintain that preparing a ferrule might improve the resistance of post-core system (Assif & Gorfil, 1994; Aykent et al., 2006; Ichim et al., 2006; Tan et al., 2005; Zhi-Yue & Yu-Xing, 2003). Accordingly, many works in the literature prepared a ferrule of varying heights in the biomechanical model: 2 mm (Mezzomo et al., Mesh definition is a key point in FEA. The mechanical behaviour of a continuous domain with an infinite number of degrees of freedom is approximated in the model with the simplified mechanical response of a set of discrete finite elements with a limited number of degrees of freedom. This process occurs when a mesh of finite elements is defined for the original system. The accuracy of the results is highly dependent on the characteristics of this mesh. A coarse mesh will produce results in a short computation time, but the accuracy of these results will be compromised. A finer mesh may improve the validity of the results but with the cost of a higher processing time. A coarse mesh is affected by two main sources of error: firstly, an error is introduced in representing the boundaries of the real geometry; and secondly, the mechanical response of the continuous material in each finite element is represented by a simplified polynomial function, which is only an approximation of the actual response. In biomechanical applications, such as dentistry, the geometry of the real system is highly irregular, so the mesh density should be enough to obtain a good geometrical representation.

Generation of a good mesh of FE for a restored tooth is always a challenge. The complicated shape of the tooth makes it difficult to produce a mesh manually. Moreover, some elements of a dental restoration present dimensions that are quite different from one another. For example, the luting cement used to bond the post to the root, or the PDL, have a thickness of a fraction of a millimetre, which is much smaller than the post dimensions, although its total length would be of several millimetres. To obtain a good mesh, a small element size has to be used in this thinner component, but such a size cannot be maintained for all the components, because, otherwise, the number of degrees of freedom would increase to a limit beyond the capacity of the computer or the computation time would increase excessively. A good meshing strategy should increase mesh density in areas where a greater expected stress gradient is expected or with thinner components, and increase the size of the elements in other parts of the model. Two different basic strategies can be used to improve finite element results from a first tentative model, namely the h-method or the p-method (Zienkiewicz & Taylor, 1989). In the h-method, the mesh is refined using elements of a smaller size, whereas in the p-method, the size of the elements is maintained but the order of the interpolation function is increased. A mixed strategy that combines both options can also be used. Most of the works in the literature about endodontic simulation use finite elements with linear or quadratic interpolation and refine the mesh using smaller elements to improve the representation, so the h-method is prevalent.

The first FE models used two-dimensional finite elements to represent the tooth and endodontic restoration (Craig & Farah, 1977; Davy et al., 1981), assuming the axisymmetric or plane strain hypothesis. In these planar models triangular or quadrangular elements with three or four nodes, respectively, are used to mesh the system. The size of the finite elements in these first models was relatively large owing to the limitations in computer resources. Davy et al. (Davy et al., 1981) used quadrangular elements for an incisor and tested different meshes to decide on the mesh density by comparing displacements of selected nodes. Finally the finer mesh tested was selected,

Biomechanical Models of Endodontic Restorations 141

stressed area, such as the post-cement interface and the cemento-enamel junction (Garbin et al., 2010; Zarone et al., 2006). However, no explicit mention is made of the parameter used to test the convergence in most of the works. Hsu et al. (Hsu et al., 2009) based convergence on the total deformation and established a difference of 1% as the limit to consider convergence. The number of nodes and elements for some three-dimensional models of the endodontically restored tooth are presented in the Table 1. It can be observed that a lower number of elements are used when the mesh is composed of hexahedral elements. The average size of elements in most of these models is close to 0.2 mm or 0.3 mm (Gonzalez-Lluch et al., 2009b; Maceri et al., 2009). The authors have shown that a mesh control of 0.3 mm is a good compromise between accuracy and computation time

Fig. 4. Sagittal section of a typical mesh with tetrahedral elements for a restored tooth

Reference # Elements # Nodes Type of element (Zarone et al., 2006) 13272 15152 8-node hexahedral (Ferrari et al., 2008) 31240 35841 8-node hexahedral (Gonzalez-Lluch et al., 2009b) 399000 69000 4-node tetrahedral (Maceri et al., 2009) 130000 185000 10-node tetrahedral (Schmitter et al., 2010) 95000 100000 20-node hexahedral (Mezzomo et al., 2011) 23000-33000 30000-42000 4-node tetrahedral Table 1. Type and number of elements and number of nodes of selected FE models

(Gonzalez-Lluch et al., 2009a).

with elements of about 1 mm in the coronal-apical direction and with a dimension close to 0.3 mm in the lingual-vestibular direction in the cervical area. Middleton et al. (Middleton et al., 1996) used quadrangular elements with eight nodes to study the effect of the PDL in bone modelling. Although the mesh used for tooth and bone was relatively coarse, the ligament was modelled with a much finer mesh, as it was the part of interest in that particular study. More recently, Pegoretti et al. (Pegoretti et al., 2002) also used plain strain two-dimensional elements and modelled the endodontic restoration using about 4000 quadrilateral elements of four nodes. The mesh presented in that work employed elements with an aspect ratio close to one and smaller elements were used near the cement and for the ligament. The aspect ratio for a finite element is the ratio of the longest to the shortest side of the element and should be as close to one as possible in order to minimise possible inaccuracies in the model. In other recent two-dimensional model, convergence tests were used to recommend the element size of 0.1 mm (Coelho et al., 2009). These planar FE models are simplifications of the real system and hence limited. Although they can be explained by the computer limitations in the last few decades, they are no longer justified nowadays. Some authors present pseudo three-dimensional models to represent the tooth (Adanir & Belli, 2007; Li et al., 2006). They obtain a planar representation of the geometry from a sagittal section of the tooth and then a solid from an extrusion normal to this planar model, using a fictitious small thickness. This approach is doubtful because it is as assuming a planar stress hypothesis, which is difficult to justify with the geometry of the real tooth. Similarly, an assumption of planar strain or axisymmetry is difficult to justify if a three-dimensional model can be used.

Three-dimensional models have been used extensively for simulating endodontic treatment since the beginning of the century (Asmussen et al., 2005; Ausiello et al., 2002; Barjau-Escribano et al., 2006; Boschian Pest et al., 2006; Genovese et al., 2005; Lanza et al., 2005). In recent years some of the most complete three-dimensional models ever developed for studying the biomechanics of endodontic restorations have been reported (Ferrari et al., 2008; Garbin et al., 2010; Gonzalez-Lluch et al., 2009b; Maceri et al., 2009; Mezzomo et al., 2011; Okamoto et al., 2008; Schmitter et al., 2010). In these three-dimensional models, tetrahedral or hexahedral finite elements are used. The mesh is created normally using the mesher of a commercial finite element package software, such as Ansys (Garbin et al., 2010; Hsu et al., 2009; Mezzomo et al., 2011), MSC-Nastran (Gonzalez-Lluch et al., 2009b; Maceri et al., 2009; Rodríguez-Cervantes et al., 2007), MSC-Marc (Okada et al., 2008; Pegoretti et al., 2002) or Cosmos (Ichim et al., 2006). In those cases, a tetrahedral mesh is the typical option because of the complicated geometries of the tooth. As an example, Fig. 4 shows a sagittal section of a mesh using tetrahedral elements for a restored incisor, generated by the authors using the Pro/Engineer FEM mesher. Some works, however, have used hexahedral elements (Ferrari et al., 2008; Okamoto et al., 2008; Schmitter et al., 2010; Zarone et al., 2006), which provide models with good results using fewer degrees of freedom, but at the cost of a more difficult mesh generation. In some models, elements with quadratic interpolation, i.e. 20-node hexahedral or 10-node tetrahedral, are used to improve the results (Maceri et al., 2009; Schmitter et al., 2010). Some attempts have been made in recent years to automate the process of creating high-quality meshes using hexahedral elements from anatomical CT data (Clement et al., 2004).

The use of convergence tests is the most commonly reported method for deciding mesh density in the majority of previous works (Ferrari et al., 2008; Maceri et al., 2009; Schmitter et al., 2010). Some authors report special attention to the convergence of results near the more

with elements of about 1 mm in the coronal-apical direction and with a dimension close to 0.3 mm in the lingual-vestibular direction in the cervical area. Middleton et al. (Middleton et al., 1996) used quadrangular elements with eight nodes to study the effect of the PDL in bone modelling. Although the mesh used for tooth and bone was relatively coarse, the ligament was modelled with a much finer mesh, as it was the part of interest in that particular study. More recently, Pegoretti et al. (Pegoretti et al., 2002) also used plain strain two-dimensional elements and modelled the endodontic restoration using about 4000 quadrilateral elements of four nodes. The mesh presented in that work employed elements with an aspect ratio close to one and smaller elements were used near the cement and for the ligament. The aspect ratio for a finite element is the ratio of the longest to the shortest side of the element and should be as close to one as possible in order to minimise possible inaccuracies in the model. In other recent two-dimensional model, convergence tests were used to recommend the element size of 0.1 mm (Coelho et al., 2009). These planar FE models are simplifications of the real system and hence limited. Although they can be explained by the computer limitations in the last few decades, they are no longer justified nowadays. Some authors present pseudo three-dimensional models to represent the tooth (Adanir & Belli, 2007; Li et al., 2006). They obtain a planar representation of the geometry from a sagittal section of the tooth and then a solid from an extrusion normal to this planar model, using a fictitious small thickness. This approach is doubtful because it is as assuming a planar stress hypothesis, which is difficult to justify with the geometry of the real tooth. Similarly, an assumption of planar strain or

axisymmetry is difficult to justify if a three-dimensional model can be used.

(Clement et al., 2004).

Three-dimensional models have been used extensively for simulating endodontic treatment since the beginning of the century (Asmussen et al., 2005; Ausiello et al., 2002; Barjau-Escribano et al., 2006; Boschian Pest et al., 2006; Genovese et al., 2005; Lanza et al., 2005). In recent years some of the most complete three-dimensional models ever developed for studying the biomechanics of endodontic restorations have been reported (Ferrari et al., 2008; Garbin et al., 2010; Gonzalez-Lluch et al., 2009b; Maceri et al., 2009; Mezzomo et al., 2011; Okamoto et al., 2008; Schmitter et al., 2010). In these three-dimensional models, tetrahedral or hexahedral finite elements are used. The mesh is created normally using the mesher of a commercial finite element package software, such as Ansys (Garbin et al., 2010; Hsu et al., 2009; Mezzomo et al., 2011), MSC-Nastran (Gonzalez-Lluch et al., 2009b; Maceri et al., 2009; Rodríguez-Cervantes et al., 2007), MSC-Marc (Okada et al., 2008; Pegoretti et al., 2002) or Cosmos (Ichim et al., 2006). In those cases, a tetrahedral mesh is the typical option because of the complicated geometries of the tooth. As an example, Fig. 4 shows a sagittal section of a mesh using tetrahedral elements for a restored incisor, generated by the authors using the Pro/Engineer FEM mesher. Some works, however, have used hexahedral elements (Ferrari et al., 2008; Okamoto et al., 2008; Schmitter et al., 2010; Zarone et al., 2006), which provide models with good results using fewer degrees of freedom, but at the cost of a more difficult mesh generation. In some models, elements with quadratic interpolation, i.e. 20-node hexahedral or 10-node tetrahedral, are used to improve the results (Maceri et al., 2009; Schmitter et al., 2010). Some attempts have been made in recent years to automate the process of creating high-quality meshes using hexahedral elements from anatomical CT data

The use of convergence tests is the most commonly reported method for deciding mesh density in the majority of previous works (Ferrari et al., 2008; Maceri et al., 2009; Schmitter et al., 2010). Some authors report special attention to the convergence of results near the more stressed area, such as the post-cement interface and the cemento-enamel junction (Garbin et al., 2010; Zarone et al., 2006). However, no explicit mention is made of the parameter used to test the convergence in most of the works. Hsu et al. (Hsu et al., 2009) based convergence on the total deformation and established a difference of 1% as the limit to consider convergence. The number of nodes and elements for some three-dimensional models of the endodontically restored tooth are presented in the Table 1. It can be observed that a lower number of elements are used when the mesh is composed of hexahedral elements. The average size of elements in most of these models is close to 0.2 mm or 0.3 mm (Gonzalez-Lluch et al., 2009b; Maceri et al., 2009). The authors have shown that a mesh control of 0.3 mm is a good compromise between accuracy and computation time (Gonzalez-Lluch et al., 2009a).

Fig. 4. Sagittal section of a typical mesh with tetrahedral elements for a restored tooth


Table 1. Type and number of elements and number of nodes of selected FE models

Biomechanical Models of Endodontic Restorations 143

technical problems associated to testing small specimens (Anusavice, 2003). Moreover, some data are not easily available, especially when considering commercial restorative materials. Table 2 shows the range of elastic properties used in different recent biomechanical models

Material Young's modulus (GPa) Poisson's ratio Dentine 15-18.6 0.30-0.32 Enamel 41-84.1 0.30-0.33 Cementum 13.7-18.7 0.30 Cancellous bone 0.345-1.37 0.22-0.31 Cortical bone 10.7-13.8 0.22-0.31 Periodontal ligament (PDL) 0.05-0.0689 (\*) 0.45-0.49 Resin composite 6-20 0.24-0.35 Cement 4.5-95 0.22-0.35 Glass fibre 20-45 0.25-0.30 Stainless steel 207-210 0.30 Gold alloy 93-120 0.33-0.44 Titanium alloy 107 0.34 Gutta-percha 0.00069(\*)-0.1 0.30-0.49 Porcelain 68.8-69 0.28 Amalgam 13.72 0.33 Zirconia 205-210 0.30-0.31 Ceramic 62-380 0.25-0.31

(\*) With respect to the value of 6.89·10-2 GPa for the elastic modulus of the PDL and the value of 6.9·10-4 GPa for that of gutta-percha, used in many works, a recent paper (Ruse, 2008) has argued that these data are erroneous and have been disseminated by hundreds of work using FEA in recent years as a consequence of a lack of rigour when it comes to citing and checking original papers. Following Ruse, correct values for the PDL are three orders of magnitude lower and for gutta-percha at least two orders

Table 2. Range of values for elastic properties for dental materials used in some FE models (Ferrari et al., 2008; Ferrari et al., 2008; Garbin et al., 2010; Gonzalez-Lluch et al., 2009b; Hsu et al., 2009; Maceri et al., 2009; Mezzomo et al., 2011; Okamoto et al., 2008; Schmitter et al.,

In almost all FE models in the literature, the materials of the natural tooth (dentine and enamel) are considered homogeneous, isotropic and elastic. However, in a critical review of the literature Kinney et al. (Kinney et al., 2003) showed that dentine exhibits an hexagonal anisotropy in the Young's modulus with a greater value (36 GPa) in the direction of the tubules than in the orthogonal direction (29 GPa). Later, Ferrari et al. considered this with an orthotropic model for dentine in the FE model (Ferrari et al., 2008). Kinney et al. (Kinney et al., 2003) also reported that dentine, at small deformations, displays a viscoelastic response, i.e. its deformation depends not only on the load that are applied but also on the time since application. Viscoelastic materials, under a constant stress, continue to deform with time, thereby presenting a time-dependence in the mechanical response. However, little is known about how to characterise this viscoelastic behaviour and in the range of frequencies of physiological interest (0.1-10 Hz) Kinney et al. concluded that dentine can be treated as a

in the literature.

of magnitude greater.

perfectly elastic solid.

2010; Suzuki et al., 2008; Toparli, 2003).

The assumption of perfect adherence between adjacent components is the rule of thumb for most of the FE models presented in the literature, i.e. nodes located on the contacting surfaces of two components are shared by these components. Post cement is normally supposed to be joined perfectly to the root dentine and to the post. However, some works represent some exceptions to this habitual modelling hypothesis. Ausiello et al. (Ausiello et al., 2002) modelled the adhesive layer in composite restorations of premolars using spring elements, which allow different normal and shear stiffness to be represented, and a good agreement with experimental data was obtained. This representation of cement was also repeated in later works by the same group (Ferrari et al., 2008; Maceri et al., 2009). Schmitter et al. (Schmitter et al., 2010) modelled different debonding states in the adhesive layers, by introducing gaps between components, to investigate possible failure modes and to compare with experimental results and a similar procedure was used by Ichim et al. (Ichim et al., 2006). Asmussen et al. (Asmussen et al., 2005) used contact finite elements to simulate non-bonding states. These contact elements allow a separation between contacting surfaces but not interpenetration between them and can simulate shear friction in the contact area. Finally, in one recent study (Garbin et al., 2010) the cement layer was modelled using shell elements in Ansys.

Different types of finite elements have been used to represent the PDL. Most of the works represent the ligament with solid isotropic elements filling the reduced thickness of the ligament, but other authors use spring elements (Maceri et al., 2009), beam elements (Freitas Junior et al., 2010) or shell elements (Garbin et al., 2010). Finally, in order to represent the viscoelastic behaviour of the ligament better, (Natali et al., 2004) proposed the use of a finite element with customised formulation, using a specific Helmholtz free-energy density. To the authors' knowledge, however, this model has not been used to simulate endodontic restorations.

### **3.4 Material properties**

Material properties are one of the key points in the definition of the model. Accuracy of the results will depend on a good representation of the actual material properties, which are not always well known. Most of the FE models assume the materials of the restored tooth are isotropic, homogeneous, elastic and linear. A material is considered homogenous if its composition can be considered as being the same at all the points, isotropic if the mechanical response is indifferent to the direction of the applied forces, elastic if it returns to the undeformed position after releasing the applied load, and linear if load and deformation are proportional. These four hypotheses are typical in a great part of all FE models in engineering, resulting in linear models for which the response (stress or strain) is proportional to the applied load. Although they allow the definition and interpretation of the model to be simplified to an important extent, these hypotheses can introduce errors in the representation of certain materials. In endodontic restoration, some materials such as bone, dentine, fibre posts or the PDL are known to be anisotropic and even not homogeneous. For example, van-Ruijven et al. (van Ruijven et al., 2006) reported that neglecting the non-homogeneity of the alveolar bone results in important changes in stresses and strains in the model. Moreover, the mechanical response of the PDL has been reported to be non-linear and viscoelastic (Pini et al., 2002). However, not only is the model that is used to represent material important, but also the numerical values of the mechanical properties, such as Young's modulus and Poisson's ratio, are of great importance to obtain an accurate model. Reported values from different sources vary markedly due to the

The assumption of perfect adherence between adjacent components is the rule of thumb for most of the FE models presented in the literature, i.e. nodes located on the contacting surfaces of two components are shared by these components. Post cement is normally supposed to be joined perfectly to the root dentine and to the post. However, some works represent some exceptions to this habitual modelling hypothesis. Ausiello et al. (Ausiello et al., 2002) modelled the adhesive layer in composite restorations of premolars using spring elements, which allow different normal and shear stiffness to be represented, and a good agreement with experimental data was obtained. This representation of cement was also repeated in later works by the same group (Ferrari et al., 2008; Maceri et al., 2009). Schmitter et al. (Schmitter et al., 2010) modelled different debonding states in the adhesive layers, by introducing gaps between components, to investigate possible failure modes and to compare with experimental results and a similar procedure was used by Ichim et al. (Ichim et al., 2006). Asmussen et al. (Asmussen et al., 2005) used contact finite elements to simulate non-bonding states. These contact elements allow a separation between contacting surfaces but not interpenetration between them and can simulate shear friction in the contact area. Finally, in one recent study (Garbin et al., 2010) the cement layer was modelled using shell

Different types of finite elements have been used to represent the PDL. Most of the works represent the ligament with solid isotropic elements filling the reduced thickness of the ligament, but other authors use spring elements (Maceri et al., 2009), beam elements (Freitas Junior et al., 2010) or shell elements (Garbin et al., 2010). Finally, in order to represent the viscoelastic behaviour of the ligament better, (Natali et al., 2004) proposed the use of a finite element with customised formulation, using a specific Helmholtz free-energy density. To the authors' knowledge, however, this model has not been used to simulate endodontic

Material properties are one of the key points in the definition of the model. Accuracy of the results will depend on a good representation of the actual material properties, which are not always well known. Most of the FE models assume the materials of the restored tooth are isotropic, homogeneous, elastic and linear. A material is considered homogenous if its composition can be considered as being the same at all the points, isotropic if the mechanical response is indifferent to the direction of the applied forces, elastic if it returns to the undeformed position after releasing the applied load, and linear if load and deformation are proportional. These four hypotheses are typical in a great part of all FE models in engineering, resulting in linear models for which the response (stress or strain) is proportional to the applied load. Although they allow the definition and interpretation of the model to be simplified to an important extent, these hypotheses can introduce errors in the representation of certain materials. In endodontic restoration, some materials such as bone, dentine, fibre posts or the PDL are known to be anisotropic and even not homogeneous. For example, van-Ruijven et al. (van Ruijven et al., 2006) reported that neglecting the non-homogeneity of the alveolar bone results in important changes in stresses and strains in the model. Moreover, the mechanical response of the PDL has been reported to be non-linear and viscoelastic (Pini et al., 2002). However, not only is the model that is used to represent material important, but also the numerical values of the mechanical properties, such as Young's modulus and Poisson's ratio, are of great importance to obtain an accurate model. Reported values from different sources vary markedly due to the

elements in Ansys.

restorations.

**3.4 Material properties** 

technical problems associated to testing small specimens (Anusavice, 2003). Moreover, some data are not easily available, especially when considering commercial restorative materials. Table 2 shows the range of elastic properties used in different recent biomechanical models in the literature.


(\*) With respect to the value of 6.89·10-2 GPa for the elastic modulus of the PDL and the value of 6.9·10-4 GPa for that of gutta-percha, used in many works, a recent paper (Ruse, 2008) has argued that these data are erroneous and have been disseminated by hundreds of work using FEA in recent years as a consequence of a lack of rigour when it comes to citing and checking original papers. Following Ruse, correct values for the PDL are three orders of magnitude lower and for gutta-percha at least two orders of magnitude greater.

Table 2. Range of values for elastic properties for dental materials used in some FE models (Ferrari et al., 2008; Ferrari et al., 2008; Garbin et al., 2010; Gonzalez-Lluch et al., 2009b; Hsu et al., 2009; Maceri et al., 2009; Mezzomo et al., 2011; Okamoto et al., 2008; Schmitter et al., 2010; Suzuki et al., 2008; Toparli, 2003).

In almost all FE models in the literature, the materials of the natural tooth (dentine and enamel) are considered homogeneous, isotropic and elastic. However, in a critical review of the literature Kinney et al. (Kinney et al., 2003) showed that dentine exhibits an hexagonal anisotropy in the Young's modulus with a greater value (36 GPa) in the direction of the tubules than in the orthogonal direction (29 GPa). Later, Ferrari et al. considered this with an orthotropic model for dentine in the FE model (Ferrari et al., 2008). Kinney et al. (Kinney et al., 2003) also reported that dentine, at small deformations, displays a viscoelastic response, i.e. its deformation depends not only on the load that are applied but also on the time since application. Viscoelastic materials, under a constant stress, continue to deform with time, thereby presenting a time-dependence in the mechanical response. However, little is known about how to characterise this viscoelastic behaviour and in the range of frequencies of physiological interest (0.1-10 Hz) Kinney et al. concluded that dentine can be treated as a perfectly elastic solid.

Biomechanical Models of Endodontic Restorations 145

limitations to the movement of some parts of the model boundary, which are necessary to obtain a unique solution for the displacement of the different points of the model. A clear definition of the boundary conditions that are employed should be included in the description of any FE model, as any changes in these conditions could introduce important differences in the results. Boundary conditions should represent the actual loads and constraints imposed on the real system as well as possible. In this section, we present a review of how these boundary conditions are defined in the biomechanical models

Most of the models presented in the literature limit the analysis to only one tooth. Some works include a portion of the alveolar bone surrounding that tooth. In these cases, restrictions are imposed on the external surfaces of the solid representing the bone (Gonzalez-Lluch et al., 2009b; Hsu et al., 2009). Other works do not include the bone and consider the outer surface of the PDL to be fixed (Ichim et al., 2006; Maceri et al., 2009). Finally, some models do not include the PDL and constrain the system in the external nodes of the root (Lanza et al., 2005; Sorrentino et al., 2007; Zarone et al., 2006). However, this last solution is not recommended because several works have shown that omission of the PDL implies significant changes in stress distribution (Cattaneo et al., 2005; Davy et al., 1981; Qian et al., 2001). Some studies try to reproduce *in vitro* experiments on endodontically restored teeth using FE models. As the rule of thumb in the experimental setup is to substitute the bone by a resin block, the models used in those studies introduce the constraints in the outline of the block of resin, which is attached to the testing machine in the

To reduce the size of the model or in order to have an easier way to show stresses and strains in the sagittal section of the restored tooth, some works include only half of the tooth in the model by eliminating the mesial or distal half (Ferrari et al., 2008; Schmitter et al., 2010). In those cases, a symmetry boundary condition is imposed on the nodes of the cutting plane, thus preventing its movement out of this plane. Although this simplification could be an interesting way to reduce computational time, the simulation of the total tooth is more

A considerable number of works in the literature make no explicit reference to the constraints in the paper (Asmussen et al., 2005; Holmes et al., 1996; Nakamura et al., 2006). The second type of boundary conditions correspond to forces applied to the system. Oral loads can be variable and several simulations can be performed over the model to simulate different oral situations such as occlusion, grinding, mastication of sticky food, impacts on the tooth, bruxism, and so forth. Most FE models of endodontic restorations simulate a maxillary incisor and introduce a load inclined at 45º to the root axis applied to the palatal surface of the tooth in the vestibular direction to represent the direction of oral loads in type I occlusion (Coelho et al., 2009; Hsu et al., 2009; Mezzomo et al., 2011; Okamoto et al., 2008). Other angles close to this value of 45º, ranging from 30º to 60º, have been used in other works (Barjau-Escribano et al., 2006; Ferrari et al., 2008; Garbin et al., 2010; Ichim et al., 2006; Lanza et al., 2005; Zarone et al., 2006). Apart from masticatory loads, other oral conditions have also been simulated, including impacts, using an angle of 90º to the long axis of the tooth, or bruxism, using an angle of 0º (Genovese et al., 2005; Pegoretti et al., 2002). As far as the position of the applied load is concerned, it is typically applied on the palatal surface, below the incisal edge, near the junction of the incisal third and the medial third (Hsu et al., 2009; Li et al., 2006), distal to the cingulum (Ichim et al., 2006). It is important to note that few details are given in the papers about the size of the area over which the force is applied,

described in the literature.

experiment (Schmitter et al., 2010; Soares et al., 2008b).

realistic, as the tooth is not geometrically symmetric.

The PDL is a component that surrounds the root of the tooth and connects it to the bone. It contributes to load distribution and damping from the tooth to the bone. Its mechanical behaviour is anisotropic as a consequence of its composition, with collagen fibres that run in different directions (Zhurov et al., 2007). Between the collagen fibres there are tissue fluids that provide a damping behaviour, which enables the ligament to offer a viscous and timedependent response (Komatsu, 2010). Some works have shown that the elastic response of the PDL is nonlinear (Pini et al., 2002). The elastic modulus is lower for small strains and increases suddenly for strains greater than near 20%-30% in both tension and compression (Pini et al., 2002). However, some models of the ligament in endodontic simulation simplify this characteristic using an isotropic, linear and perfectly elastic material (Garbin et al., 2010; Mezzomo et al., 2011; Okamoto et al., 2008). This simplification avoids the iterative procedure associated to a non-linear model that entails an important increase in computation time. Rees and Jacobsen (Rees & Jacobsen, 1997) analysed the optimal value of the Young's modulus of the PDL (when using an isotropic and linear constitutive model) which provides a good correlation between experimental results and computations of the FE model. They found an optimal value near 50 MPa. This value is used in several works (Davy et al., 1981; Dejak et al., 2005; Garbin et al., 2010) and a similar value of 68.9 MPa in others (Asmussen et al., 2005; Gonzalez-Lluch et al., 2009b; Holmes et al., 1996; Pegoretti et al., 2002). Some recent works consider the non-linear mechanical response of the ligament using a non-linear stress-strain relationship for the finite elements representing the PDL (Okada et al., 2008; Suzuki et al., 2008; Uddanwadiker et al., 2007). Maceri et al. (Maceri et al., 2009) used a set of different spring elements connecting dentine to the bone to represent the non-linear response of the ligament. In an attempt to represent orthodontic movements better, Qian et al. (Qian et al., 2001) proposed the use of special finite elements including fibres that connect two opposing faces of the element. In a further step, some recent works (Ferrari et al., 2008; Sorrentino et al., 2009) considered a more complicated model that takes into account the nonlinearity and the time-dependent response of the ligament, using different stress-strain curves for different values of strain rate. Including viscoelasticity in the model is important to be able to simulate the response under dynamical loads correctly, but the fact that most of the models are used in static simulation explains its absence in the majority of the works. The inclusion of the non-linearity and viscoelasticity of the ligament in the constitutive formulation of customised finite elements has been investigated in recent years (Natali et al., 2004; Zhurov et al., 2007), but to the authors' knowledge it has not yet been implemented in a simulation of endodontic restorations. Maceri et al. (Maceri et al., 2010) proposed recently a method to model the PDL and other materials composed of soft collagen tissues, by means of a nanoscale model and a two-step micro–macro homogenisation technique.

Composite posts, such as glass-fibre or carbon-fibre, exhibit anisotropic behaviour due to the presence of reinforcing fibres oriented in the direction of the post axis. An orthotropic model, with different elastic moduli in the direction of the fibres and in the transverse direction, has been used by several authors to represent this material in FE models of the restored tooth (Barjau-Escribano et al., 2006; Garbin et al., 2010; Maceri et al., 2009; Schmitter et al., 2010; Sorrentino et al., 2007).

### **3.5 Boundary conditions**

Boundary conditions in FEA define the known constraints and forces acting on the system. In endodontic simulation, forces represent the loads acting on the tooth and constraints are

The PDL is a component that surrounds the root of the tooth and connects it to the bone. It contributes to load distribution and damping from the tooth to the bone. Its mechanical behaviour is anisotropic as a consequence of its composition, with collagen fibres that run in different directions (Zhurov et al., 2007). Between the collagen fibres there are tissue fluids that provide a damping behaviour, which enables the ligament to offer a viscous and timedependent response (Komatsu, 2010). Some works have shown that the elastic response of the PDL is nonlinear (Pini et al., 2002). The elastic modulus is lower for small strains and increases suddenly for strains greater than near 20%-30% in both tension and compression (Pini et al., 2002). However, some models of the ligament in endodontic simulation simplify this characteristic using an isotropic, linear and perfectly elastic material (Garbin et al., 2010; Mezzomo et al., 2011; Okamoto et al., 2008). This simplification avoids the iterative procedure associated to a non-linear model that entails an important increase in computation time. Rees and Jacobsen (Rees & Jacobsen, 1997) analysed the optimal value of the Young's modulus of the PDL (when using an isotropic and linear constitutive model) which provides a good correlation between experimental results and computations of the FE model. They found an optimal value near 50 MPa. This value is used in several works (Davy et al., 1981; Dejak et al., 2005; Garbin et al., 2010) and a similar value of 68.9 MPa in others (Asmussen et al., 2005; Gonzalez-Lluch et al., 2009b; Holmes et al., 1996; Pegoretti et al., 2002). Some recent works consider the non-linear mechanical response of the ligament using a non-linear stress-strain relationship for the finite elements representing the PDL (Okada et al., 2008; Suzuki et al., 2008; Uddanwadiker et al., 2007). Maceri et al. (Maceri et al., 2009) used a set of different spring elements connecting dentine to the bone to represent the non-linear response of the ligament. In an attempt to represent orthodontic movements better, Qian et al. (Qian et al., 2001) proposed the use of special finite elements including fibres that connect two opposing faces of the element. In a further step, some recent works (Ferrari et al., 2008; Sorrentino et al., 2009) considered a more complicated model that takes into account the nonlinearity and the time-dependent response of the ligament, using different stress-strain curves for different values of strain rate. Including viscoelasticity in the model is important to be able to simulate the response under dynamical loads correctly, but the fact that most of the models are used in static simulation explains its absence in the majority of the works. The inclusion of the non-linearity and viscoelasticity of the ligament in the constitutive formulation of customised finite elements has been investigated in recent years (Natali et al., 2004; Zhurov et al., 2007), but to the authors' knowledge it has not yet been implemented in a simulation of endodontic restorations. Maceri et al. (Maceri et al., 2010) proposed recently a method to model the PDL and other materials composed of soft collagen tissues, by means of a nano-

scale model and a two-step micro–macro homogenisation technique.

et al., 2010; Sorrentino et al., 2007).

**3.5 Boundary conditions** 

Composite posts, such as glass-fibre or carbon-fibre, exhibit anisotropic behaviour due to the presence of reinforcing fibres oriented in the direction of the post axis. An orthotropic model, with different elastic moduli in the direction of the fibres and in the transverse direction, has been used by several authors to represent this material in FE models of the restored tooth (Barjau-Escribano et al., 2006; Garbin et al., 2010; Maceri et al., 2009; Schmitter

Boundary conditions in FEA define the known constraints and forces acting on the system. In endodontic simulation, forces represent the loads acting on the tooth and constraints are limitations to the movement of some parts of the model boundary, which are necessary to obtain a unique solution for the displacement of the different points of the model. A clear definition of the boundary conditions that are employed should be included in the description of any FE model, as any changes in these conditions could introduce important differences in the results. Boundary conditions should represent the actual loads and constraints imposed on the real system as well as possible. In this section, we present a review of how these boundary conditions are defined in the biomechanical models described in the literature.

Most of the models presented in the literature limit the analysis to only one tooth. Some works include a portion of the alveolar bone surrounding that tooth. In these cases, restrictions are imposed on the external surfaces of the solid representing the bone (Gonzalez-Lluch et al., 2009b; Hsu et al., 2009). Other works do not include the bone and consider the outer surface of the PDL to be fixed (Ichim et al., 2006; Maceri et al., 2009). Finally, some models do not include the PDL and constrain the system in the external nodes of the root (Lanza et al., 2005; Sorrentino et al., 2007; Zarone et al., 2006). However, this last solution is not recommended because several works have shown that omission of the PDL implies significant changes in stress distribution (Cattaneo et al., 2005; Davy et al., 1981; Qian et al., 2001). Some studies try to reproduce *in vitro* experiments on endodontically restored teeth using FE models. As the rule of thumb in the experimental setup is to substitute the bone by a resin block, the models used in those studies introduce the constraints in the outline of the block of resin, which is attached to the testing machine in the experiment (Schmitter et al., 2010; Soares et al., 2008b).

To reduce the size of the model or in order to have an easier way to show stresses and strains in the sagittal section of the restored tooth, some works include only half of the tooth in the model by eliminating the mesial or distal half (Ferrari et al., 2008; Schmitter et al., 2010). In those cases, a symmetry boundary condition is imposed on the nodes of the cutting plane, thus preventing its movement out of this plane. Although this simplification could be an interesting way to reduce computational time, the simulation of the total tooth is more realistic, as the tooth is not geometrically symmetric.

A considerable number of works in the literature make no explicit reference to the constraints in the paper (Asmussen et al., 2005; Holmes et al., 1996; Nakamura et al., 2006).

The second type of boundary conditions correspond to forces applied to the system. Oral loads can be variable and several simulations can be performed over the model to simulate different oral situations such as occlusion, grinding, mastication of sticky food, impacts on the tooth, bruxism, and so forth. Most FE models of endodontic restorations simulate a maxillary incisor and introduce a load inclined at 45º to the root axis applied to the palatal surface of the tooth in the vestibular direction to represent the direction of oral loads in type I occlusion (Coelho et al., 2009; Hsu et al., 2009; Mezzomo et al., 2011; Okamoto et al., 2008). Other angles close to this value of 45º, ranging from 30º to 60º, have been used in other works (Barjau-Escribano et al., 2006; Ferrari et al., 2008; Garbin et al., 2010; Ichim et al., 2006; Lanza et al., 2005; Zarone et al., 2006). Apart from masticatory loads, other oral conditions have also been simulated, including impacts, using an angle of 90º to the long axis of the tooth, or bruxism, using an angle of 0º (Genovese et al., 2005; Pegoretti et al., 2002). As far as the position of the applied load is concerned, it is typically applied on the palatal surface, below the incisal edge, near the junction of the incisal third and the medial third (Hsu et al., 2009; Li et al., 2006), distal to the cingulum (Ichim et al., 2006). It is important to note that few details are given in the papers about the size of the area over which the force is applied,

Biomechanical Models of Endodontic Restorations 147

properties of different materials. However, the second and third conditions are not met in most of the previous works, due to the lack of consistent and complete data about the strength of some dental materials. More especially, however, the interpretation of results is hampered by the absence of a post-hoc analysis of failure in each component of the model or

To meet the second condition, information about compressive and tensile strengths is needed for each material in the restoration. The compressive strength is usually obtained experimentally by a compressive test using cylindrical specimens. The tensile strength is obtained by applying an axial force over specimens with a cylindrical or rectangular crosssection and is a typical test for metals and other ductile materials. This type of test, however, is rarely used for brittle materials. Technical problems related with gripping and aligning the brittle specimens are often cited as an explanation for not measuring the tensile strengths (Ban & Anusavice, 1990; Xie et al., 2000). Alternatively, the diametral tensile test (DTT) is commonly used to obtain the diametral tensile strength (DTS) (Probster et al., 1997; Xie et al., 2000) because of its simplicity and reproducibility (Coelho Santos et al., 2004). The DTT is performed by compressing a cylindrical specimen with its axis perpendicular to the load direction. Tensile strength can also be obtained indirectly as a flexural strength (FS) with three- or four-point flexural tests (FT) (Probster et al., 1997; Xie et al., 2000). However, DTS and FS are obtained in loading states that are not uniaxial and the results of these tests are not equivalent, as numerous previous works have shown for different dental materials (Ban & Anusavice, 1990; Probster et al., 1997; Xie et al., 2000). Despite this, DTS and FS have been used interchangeably in recent works as a reference to compare with computed maximal stresses in finite element models of dental restorations (De Jager et al., 2006; Imanishi et al., 2003). On analysing the stress state in both tests, FS seems preferable as a reference for the tensile strength of a ductile material because the stress state is uniaxial at the failure point, and this can only be said of a DTT if a plain stress state is assumed, which

Finally, the third condition for a correct interpretation of the FE results is related to the selection of the proper failure criterion for comparing actual stress with the admissible stress limits of different materials present in the restoration. A stress criterion or failure theory combines principal stresses at a point in a solid with the compressive strength and tensile strength of the material to obtain a safety factor at this point. Safety factor values lower than unity indicate that the material is prone to mechanical failure at this point and values greater than unity indicate a safe condition at this point. Different criteria such as Von Mises (VM), Rankine (R), Coulomb-Mohr (CM) or Modified Mohr (MM), are usually reported in mechanical engineering manuals for isotropic materials (Shigley & Mischke, 2002). Some of these criteria are better suited for ductile materials, whereas others are more accurate for predicting the failure of brittle materials. In a recent study, Christensen (Christensen, 2006) proposed the use of a novel failure criterion that is valid for ductile and brittle materials. To date, little research has been devoted to the interpretation of FEM results in endodontic restorations. Most previous work analyzed the results of FE simulations from Von Mises maximal stresses (Asmussen et al., 2005; Boschian Pest et al., 2006; Hsu et al., 2009; Pegoretti et al., 2002; Sorrentino et al., 2007), implicitly assuming the validity of the VM criterion for all the materials used in the restoration. However, it is known that the VM criterion is only valid for ductile materials with equal compressive and tensile strength (De Groot et al., 1987), but materials exhibiting brittle behaviour such as ceramics, cements or resin composites are frequently used in endodontic restorations. Even dentine presents reported

the use of the Von Mises equivalent strength as the reference value for this analysis.

if far from being the actual situation in real tests.

although this factor can introduce local changes in stresses and strains near the application zone. Therefore, the authors recommend reporting this information in future works. Two different approaches are used in the selection of the magnitude of the loads applied to the model. Since many of the models are linear, an arbitrary small load of 1 N (Davy et al., 1981) or 10 N (Sorrentino et al., 2007) can be used to analyse the system and to obtain information about the stress distribution, which can be translated to greater force scenarios by a simple scaling operation. This approach is valid to compare stresses among different restorative designs or with respect to the natural tooth. However, other works use a force that represents the expected oral forces in order to obtain stress values that are directly comparable to the stress limits for restorative materials. In these cases, forces of 100 N (Gurbuz et al., 2008; Mezzomo et al., 2011), 200 N (Nakamura et al., 2006; Uddanwadiker et al., 2007) or 300 N (Gonzalez-Lluch et al., 2009b; Hsu et al., 2009) are typical.

For premolars, forces at 0º or 45º were used on the crown (Maceri et al., 2009; Toparli, 2003). Ausiello et al. (Ausiello et al., 2002) distributed an axial load over two different points on the crown using two rigid bars. Okada et al. (Okada et al., 2008) used the data obtained experimentally with a device to measure masticatory forces in vivo to decide the magnitude and direction of the loads in a model of a first premolar. During the experiment a piece of beef jerky was masticated and the resulting load was predominantly directed in the apicocoronal direction (164.3 N) with lower and similar components in the mesial-distal direction and bucco-palatal direction (-28.9 N and -23.9 N respectively). For molars, Imanishi (Imanishi et al., 2003) used three different forces of 225 N to simulate masticatory loads with angles of 0º, 45º and 90º with respect to the radicular axis, and applied it in outer incline of the buccal cusps.

Restored teeth are also subject to thermomechanical loading, because of the transient changes in the temperature of the different components of the restoration when hot or cold foods or liquids are introduced into the mouth. To date very few works included these types of boundary conditions in FE models. Gungor et al. (Gungor et al., 2004) presented a simulation that analysed stresses induced in a first premolar with different all-ceramic crown materials as a consequence of thermal loading.

### **4. Interpreting FEA results**

A correct interpretation of FEA results should be based on the stress and strength of each component in the system. To obtain accurate conclusions from this interpretation, three conditions must be fulfilled: (1) stress values must be reliable, i.e., the FE model should be an adequate representation of the real system, (2) strengths of the different materials present in the model must be known and (3) a suitable failure criterion must be used to compare values of computed stresses, which are bi-axial or tri-axial, with material strengths frequently obtained under conditions of a uniaxial stress state (tension or compression).

The first condition is progressively closer to being fulfilled with the development of more accurate three-dimensional models, with finer meshes and with more components represented in the system. However, as has been commented above, the accuracy of most of the models is still questionable. Many models consider that all materials display linear isotropic behaviour, although this is a simplification for components such as dentine, PDL or fibre posts. A good representation of bonded interfaces is difficult due to their reduced thickness and different ways to model them have been tested, as has been commented in previous sections of this chapter. Moreover, some uncertainty exists about the mechanical

although this factor can introduce local changes in stresses and strains near the application zone. Therefore, the authors recommend reporting this information in future works. Two different approaches are used in the selection of the magnitude of the loads applied to the model. Since many of the models are linear, an arbitrary small load of 1 N (Davy et al., 1981) or 10 N (Sorrentino et al., 2007) can be used to analyse the system and to obtain information about the stress distribution, which can be translated to greater force scenarios by a simple scaling operation. This approach is valid to compare stresses among different restorative designs or with respect to the natural tooth. However, other works use a force that represents the expected oral forces in order to obtain stress values that are directly comparable to the stress limits for restorative materials. In these cases, forces of 100 N (Gurbuz et al., 2008; Mezzomo et al., 2011), 200 N (Nakamura et al., 2006; Uddanwadiker et

For premolars, forces at 0º or 45º were used on the crown (Maceri et al., 2009; Toparli, 2003). Ausiello et al. (Ausiello et al., 2002) distributed an axial load over two different points on the crown using two rigid bars. Okada et al. (Okada et al., 2008) used the data obtained experimentally with a device to measure masticatory forces in vivo to decide the magnitude and direction of the loads in a model of a first premolar. During the experiment a piece of beef jerky was masticated and the resulting load was predominantly directed in the apicocoronal direction (164.3 N) with lower and similar components in the mesial-distal direction and bucco-palatal direction (-28.9 N and -23.9 N respectively). For molars, Imanishi (Imanishi et al., 2003) used three different forces of 225 N to simulate masticatory loads with angles of 0º, 45º and 90º with respect to the radicular axis, and applied it in outer incline of

Restored teeth are also subject to thermomechanical loading, because of the transient changes in the temperature of the different components of the restoration when hot or cold foods or liquids are introduced into the mouth. To date very few works included these types of boundary conditions in FE models. Gungor et al. (Gungor et al., 2004) presented a simulation that analysed stresses induced in a first premolar with different all-ceramic

A correct interpretation of FEA results should be based on the stress and strength of each component in the system. To obtain accurate conclusions from this interpretation, three conditions must be fulfilled: (1) stress values must be reliable, i.e., the FE model should be an adequate representation of the real system, (2) strengths of the different materials present in the model must be known and (3) a suitable failure criterion must be used to compare values of computed stresses, which are bi-axial or tri-axial, with material strengths frequently obtained under conditions of a uniaxial stress state (tension or compression). The first condition is progressively closer to being fulfilled with the development of more accurate three-dimensional models, with finer meshes and with more components represented in the system. However, as has been commented above, the accuracy of most of the models is still questionable. Many models consider that all materials display linear isotropic behaviour, although this is a simplification for components such as dentine, PDL or fibre posts. A good representation of bonded interfaces is difficult due to their reduced thickness and different ways to model them have been tested, as has been commented in previous sections of this chapter. Moreover, some uncertainty exists about the mechanical

al., 2007) or 300 N (Gonzalez-Lluch et al., 2009b; Hsu et al., 2009) are typical.

crown materials as a consequence of thermal loading.

the buccal cusps.

**4. Interpreting FEA results** 

properties of different materials. However, the second and third conditions are not met in most of the previous works, due to the lack of consistent and complete data about the strength of some dental materials. More especially, however, the interpretation of results is hampered by the absence of a post-hoc analysis of failure in each component of the model or the use of the Von Mises equivalent strength as the reference value for this analysis.

To meet the second condition, information about compressive and tensile strengths is needed for each material in the restoration. The compressive strength is usually obtained experimentally by a compressive test using cylindrical specimens. The tensile strength is obtained by applying an axial force over specimens with a cylindrical or rectangular crosssection and is a typical test for metals and other ductile materials. This type of test, however, is rarely used for brittle materials. Technical problems related with gripping and aligning the brittle specimens are often cited as an explanation for not measuring the tensile strengths (Ban & Anusavice, 1990; Xie et al., 2000). Alternatively, the diametral tensile test (DTT) is commonly used to obtain the diametral tensile strength (DTS) (Probster et al., 1997; Xie et al., 2000) because of its simplicity and reproducibility (Coelho Santos et al., 2004). The DTT is performed by compressing a cylindrical specimen with its axis perpendicular to the load direction. Tensile strength can also be obtained indirectly as a flexural strength (FS) with three- or four-point flexural tests (FT) (Probster et al., 1997; Xie et al., 2000). However, DTS and FS are obtained in loading states that are not uniaxial and the results of these tests are not equivalent, as numerous previous works have shown for different dental materials (Ban & Anusavice, 1990; Probster et al., 1997; Xie et al., 2000). Despite this, DTS and FS have been used interchangeably in recent works as a reference to compare with computed maximal stresses in finite element models of dental restorations (De Jager et al., 2006; Imanishi et al., 2003). On analysing the stress state in both tests, FS seems preferable as a reference for the tensile strength of a ductile material because the stress state is uniaxial at the failure point, and this can only be said of a DTT if a plain stress state is assumed, which if far from being the actual situation in real tests.

Finally, the third condition for a correct interpretation of the FE results is related to the selection of the proper failure criterion for comparing actual stress with the admissible stress limits of different materials present in the restoration. A stress criterion or failure theory combines principal stresses at a point in a solid with the compressive strength and tensile strength of the material to obtain a safety factor at this point. Safety factor values lower than unity indicate that the material is prone to mechanical failure at this point and values greater than unity indicate a safe condition at this point. Different criteria such as Von Mises (VM), Rankine (R), Coulomb-Mohr (CM) or Modified Mohr (MM), are usually reported in mechanical engineering manuals for isotropic materials (Shigley & Mischke, 2002). Some of these criteria are better suited for ductile materials, whereas others are more accurate for predicting the failure of brittle materials. In a recent study, Christensen (Christensen, 2006) proposed the use of a novel failure criterion that is valid for ductile and brittle materials.

To date, little research has been devoted to the interpretation of FEM results in endodontic restorations. Most previous work analyzed the results of FE simulations from Von Mises maximal stresses (Asmussen et al., 2005; Boschian Pest et al., 2006; Hsu et al., 2009; Pegoretti et al., 2002; Sorrentino et al., 2007), implicitly assuming the validity of the VM criterion for all the materials used in the restoration. However, it is known that the VM criterion is only valid for ductile materials with equal compressive and tensile strength (De Groot et al., 1987), but materials exhibiting brittle behaviour such as ceramics, cements or resin composites are frequently used in endodontic restorations. Even dentine presents reported

Biomechanical Models of Endodontic Restorations 149

representations of some projection of the model using a coloured scale to represent a particular scalar result. Fringe plots allow the distribution of stress or strain in some part of the model to be seen in one picture. They are the usual way to represent the FE results in most works today (Mezzomo et al., 2011; Okamoto et al., 2008; Silva et al., 2009), although they are better suited for qualitative analysis than for quantitative ones. Maceri et al. (Maceri et al., 2009) employed an averaging function to obtain several mean stress values at each section of the tooth in the coronal-apical direction and considered these functions as indicators of the risk of failure. With this procedure, they can easily compare different restoration methods in a plot with simple curves along the axis of the tooth, although the reliability of this customised parameter as risk

Statistical tests have been used in some recent works to compare results from different FE models. Hsu et al. used t-tests over sets of points in the areas of interest to compare different restorative solutions (Hsu et al., 2009). The authors have recently presented a similar approach for using ANOVA tests as a method to assess the significance of different restorative designs (Pérez-González et al., 2010). This option for the interpretation of FE

Validation of biomechanical models employed in numerical simulations should be carried out in order to ensure the validity of the results that are obtained. In other biomechanical ambits, model validation is considered to be of major importance (Dalstra et al., 1995; Gupta et al., 2004). Despite this, very few of the works that make use of FEA for endodontic restoration research include such validations of their models. Moreover, little work has been

Generally speaking, two stages should be accomplished in order to validate the FE models. First, some sort of convergence tests have to be performed, where, according to the FEM theory (Zienkiewicz & Taylor, 1989), subsequent refinements of the mesh should make the results to converge. Second, the numerical results have to be compared with those obtained from experimental tests. This comparison can be achieved in different ways that can be found in the scientific literature. Some works only perform the convergence test for validation (Garbin et al., 2010; Sorrentino et al., 2007) and even in theses cases no detailed

One way to accomplish validation of the model is to compare the relation between the load applied and the displacement obtained from both the numerical and the mechanical tests. In some cases (Ausiello et al., 2001), mechanical tests can be undertaken using a testing machine with a constant rate of displacement, which allows two continuous curves to be compared. In other cases, the comparison is achieved only for some discrete values of load and displacements values (Rappelli et al., 2005). In other works the variables that are compared are the load and the strain values (Magne & Tan, 2008; Tajima et al., 2009) using strain gauges, although the process may be found to be more difficult and tedious. In the work by Tajima et al., the utilisation of the strain gauge technique allows simultaneous comparison of the strain values obtained from FE calculation and from experimental tests at four different points, which can be seen as a more rigorous validation. Furthermore, six

An indirect form of experimental validation may be established when the numerical results of stress distribution obtained by the FE model agree with the results obtained by fracture

results should be investigated in more detail in the future.

carried out on validation procedures in this particular field.

occlusal points are concurrently loaded during experiments.

**5. Validation of biomechanical models** 

information about this stage is offered.

indicator is questionable.

values of compressive strength that are significantly greater than its tensile strength (Craig & Powers, 2002). Some authors suggest the use of the Rankine or Maximum Normal Stress criterion to evaluate the failure in dentine, using the maximum principal stress to analyse the results (Ichim et al., 2006; Maceri et al., 2009; Nakamura et al., 2006). Others analyse the results of shear stress at the post-dentine interface, and indicate that this value should be compared to the reported shear bond strengths in order to evaluate the risk of loosing retention (Asmussen et al., 2005; Maceri et al., 2009; Pegoretti et al., 2002). Wakabayashi et al. (Wakabayashi et al., 2008) also underline the importance of shear stresses to anticipate the probability of failure in the adhesive joints of the restoration. Okamoto et al. (Okamoto et al., 2008) used an iterative procedure to account for debonding in the cement layer. Their method involved reducing the elastic modulus of the finite elements of the cement with shear or normal stresses beyond their corresponding strength to a value close to zero and then recalculating the model until they reached a final solution with a reduced adhesion surface in the cement layer.

DeGroot et al. (De Groot et al., 1987) compared three criteria to analyse FEM results in composite resin: Von Mises, a modified Von Mises criterion presented by Williams (Williams, 1973), and the Drüker-Prager criterion. From their results, they concluded that the Drüker-Prager criterion is more suitable for describing the failure of this material. Recently, Christensen (Christensen, 2006) proposed a unified failure criterion for ductile and brittle materials, which is equivalent to the modified Von Mises criterion proposed by Williams with an additional modification for brittle materials. The same author also demonstrated some unrealistic behaviour of the Drüker-Prager and Coulomb-Mohr criteria under certain important stress states.

The interpretation of failure probability in anisotropic materials is even more difficult and few researchers address it. Dejak et al. (Dejak et al., 2007) applied the Tsai-Wu criterion for anisotropic materials to dentine, enamel and resin composites in molars with ceramic inlays. From the above discussion, it becomes apparent to the authors that more research is needed to clarify the question of how to interpret the results of FE models correctly. In the authors' opinion, DTS should not be used as a reference for the tensile strength in a failure criterion, and FS is a better option. As regards the failure criterion, a critical review is needed about the extensive use of the Von Mises criterion and the use of the novel Christensen criterion should be taken into consideration as an interesting alternative (Pérez-González et al., 2011). One of the problems related to the interpretation of FE results is the huge quantity of numerical results obtained by the program, namely the displacements and stresses in all the nodes for every dimension considered (two for planar models and three for three-dimensional models). As there may be several thousands of nodes in current FE models, it is not feasible to present the results in a table or a list, at least not for all the nodes. For a good interpretation of the results, one has to know what to look for and where to look for it. It is clear that maximal stresses are the first option to search for points where the system is prone to failure, although the absolute maximum may not be the initial failure point because stress has to be compared to strength in the different components. For a better understanding of the stress distribution in the tooth, some works present the results by giving a list of stresses for selected points or a plot representing the stress values as a function of point location in some trajectory inside the tooth. For example, Davy et al. (Davy et al., 1981) examined the stresses at selected points located near the cervical with different positions in the coronal-apical direction or in the buccal-labial direction. Posterior works (Hsu et al., 2009; Ichim et al., 2006; Pegoretti et al., 2002) also maintain this method, although complemented with fringe plots. Fringe plots are colour

values of compressive strength that are significantly greater than its tensile strength (Craig & Powers, 2002). Some authors suggest the use of the Rankine or Maximum Normal Stress criterion to evaluate the failure in dentine, using the maximum principal stress to analyse the results (Ichim et al., 2006; Maceri et al., 2009; Nakamura et al., 2006). Others analyse the results of shear stress at the post-dentine interface, and indicate that this value should be compared to the reported shear bond strengths in order to evaluate the risk of loosing retention (Asmussen et al., 2005; Maceri et al., 2009; Pegoretti et al., 2002). Wakabayashi et al. (Wakabayashi et al., 2008) also underline the importance of shear stresses to anticipate the probability of failure in the adhesive joints of the restoration. Okamoto et al. (Okamoto et al., 2008) used an iterative procedure to account for debonding in the cement layer. Their method involved reducing the elastic modulus of the finite elements of the cement with shear or normal stresses beyond their corresponding strength to a value close to zero and then recalculating the model until they reached a final solution with a reduced adhesion

DeGroot et al. (De Groot et al., 1987) compared three criteria to analyse FEM results in composite resin: Von Mises, a modified Von Mises criterion presented by Williams (Williams, 1973), and the Drüker-Prager criterion. From their results, they concluded that the Drüker-Prager criterion is more suitable for describing the failure of this material. Recently, Christensen (Christensen, 2006) proposed a unified failure criterion for ductile and brittle materials, which is equivalent to the modified Von Mises criterion proposed by Williams with an additional modification for brittle materials. The same author also demonstrated some unrealistic behaviour of the Drüker-Prager and Coulomb-Mohr criteria

The interpretation of failure probability in anisotropic materials is even more difficult and few researchers address it. Dejak et al. (Dejak et al., 2007) applied the Tsai-Wu criterion for anisotropic materials to dentine, enamel and resin composites in molars with ceramic inlays. From the above discussion, it becomes apparent to the authors that more research is needed to clarify the question of how to interpret the results of FE models correctly. In the authors' opinion, DTS should not be used as a reference for the tensile strength in a failure criterion, and FS is a better option. As regards the failure criterion, a critical review is needed about the extensive use of the Von Mises criterion and the use of the novel Christensen criterion should be taken into consideration as an interesting alternative (Pérez-González et al., 2011). One of the problems related to the interpretation of FE results is the huge quantity of numerical results obtained by the program, namely the displacements and stresses in all the nodes for every dimension considered (two for planar models and three for three-dimensional models). As there may be several thousands of nodes in current FE models, it is not feasible to present the results in a table or a list, at least not for all the nodes. For a good interpretation of the results, one has to know what to look for and where to look for it. It is clear that maximal stresses are the first option to search for points where the system is prone to failure, although the absolute maximum may not be the initial failure point because stress has to be compared to strength in the different components. For a better understanding of the stress distribution in the tooth, some works present the results by giving a list of stresses for selected points or a plot representing the stress values as a function of point location in some trajectory inside the tooth. For example, Davy et al. (Davy et al., 1981) examined the stresses at selected points located near the cervical with different positions in the coronal-apical direction or in the buccal-labial direction. Posterior works (Hsu et al., 2009; Ichim et al., 2006; Pegoretti et al., 2002) also maintain this method, although complemented with fringe plots. Fringe plots are colour

surface in the cement layer.

under certain important stress states.

representations of some projection of the model using a coloured scale to represent a particular scalar result. Fringe plots allow the distribution of stress or strain in some part of the model to be seen in one picture. They are the usual way to represent the FE results in most works today (Mezzomo et al., 2011; Okamoto et al., 2008; Silva et al., 2009), although they are better suited for qualitative analysis than for quantitative ones. Maceri et al. (Maceri et al., 2009) employed an averaging function to obtain several mean stress values at each section of the tooth in the coronal-apical direction and considered these functions as indicators of the risk of failure. With this procedure, they can easily compare different restoration methods in a plot with simple curves along the axis of the tooth, although the reliability of this customised parameter as risk indicator is questionable.

Statistical tests have been used in some recent works to compare results from different FE models. Hsu et al. used t-tests over sets of points in the areas of interest to compare different restorative solutions (Hsu et al., 2009). The authors have recently presented a similar approach for using ANOVA tests as a method to assess the significance of different restorative designs (Pérez-González et al., 2010). This option for the interpretation of FE results should be investigated in more detail in the future.

### **5. Validation of biomechanical models**

Validation of biomechanical models employed in numerical simulations should be carried out in order to ensure the validity of the results that are obtained. In other biomechanical ambits, model validation is considered to be of major importance (Dalstra et al., 1995; Gupta et al., 2004). Despite this, very few of the works that make use of FEA for endodontic restoration research include such validations of their models. Moreover, little work has been carried out on validation procedures in this particular field.

Generally speaking, two stages should be accomplished in order to validate the FE models. First, some sort of convergence tests have to be performed, where, according to the FEM theory (Zienkiewicz & Taylor, 1989), subsequent refinements of the mesh should make the results to converge. Second, the numerical results have to be compared with those obtained from experimental tests. This comparison can be achieved in different ways that can be found in the scientific literature. Some works only perform the convergence test for validation (Garbin et al., 2010; Sorrentino et al., 2007) and even in theses cases no detailed information about this stage is offered.

One way to accomplish validation of the model is to compare the relation between the load applied and the displacement obtained from both the numerical and the mechanical tests. In some cases (Ausiello et al., 2001), mechanical tests can be undertaken using a testing machine with a constant rate of displacement, which allows two continuous curves to be compared. In other cases, the comparison is achieved only for some discrete values of load and displacements values (Rappelli et al., 2005). In other works the variables that are compared are the load and the strain values (Magne & Tan, 2008; Tajima et al., 2009) using strain gauges, although the process may be found to be more difficult and tedious. In the work by Tajima et al., the utilisation of the strain gauge technique allows simultaneous comparison of the strain values obtained from FE calculation and from experimental tests at four different points, which can be seen as a more rigorous validation. Furthermore, six occlusal points are concurrently loaded during experiments.

An indirect form of experimental validation may be established when the numerical results of stress distribution obtained by the FE model agree with the results obtained by fracture

Biomechanical Models of Endodontic Restorations 151

work, Sancho-Bru et al. (Sancho-Bru et al., 2009) recently proposed a way to use finite element results for a fatigue analysis of dental restorations. From that work, it was concluded that, although restorations using glass fibre posts are able to bear higher static loads, both stainless steel and glass fibre post systems would have a similar life under dynamic loads, although this conclusion should be confirmed using more detailed models

The effect that post diameter has on the stress distribution over the tooth is not the same for all post materials. Several works (Boschian Pest et al., 2006; Nakamura et al., 2006; Rodríguez-Cervantes et al., 2007) have indicated that the effect of diameter is greater for metallic posts than for fibre posts. This could be explained by the fact that the elastic modulus of fibre posts is more comparable with that of dentine, thus producing a more homogenous restoration that is consequently less affected by a change in the diameter of the tooth. In a work by the authors (Rodríguez-Cervantes et al., 2007), we confirmed with *in vitro* tests and FE models that increases in post diameter for metallic posts tend to reduce the failure loads for the restored tooth without the crown. However, in a later work, it was observed that the inclusion of the crown meant that the post diameter did not influence the biomechanical performance of the post systems no matter what post material was used (Gonzalez-Lluch et al., 2009b). For fibre posts, as has been said, the importance of post diameter is lower and the conclusions of different works vary. Boschian-Pest et al. (Boschian Pest et al., 2006), for example, recommended using small diameters to avoid weakening the root, whereas Okamoto et al. (Okamoto et al., 2008) recommended the use of a large

Two of the earlier FE works (Davy et al., 1981; Holmes et al., 1996) studied metallic post systems by means of two-dimensional finite element models and reached opposing conclusions. The first predicted minor changes in the stress patterns in dentine for the length variations considered; the only effect of post length was a change in the location of the stress concentrations that occurred at the post apex. In contrast, the second predicted higher shear stresses as the length of the metallic post decreased. This controversy seems to continue in two later works that compared the effect of post length for fibre posts. On the one hand, Boschian Pest et al. (Boschian Pest et al., 2006) suggested using a post inserted as deeply as possible, because shorter posts would increase stresses in the luting materials, although minor effects were recognised in the root. Ferrari et al.(Ferrari et al., 2008), on the other hand, concluded that post length does not influence the biomechanics of restored teeth. The same conclusion about the non-significance of the post length was obtained by the authors in a study with *in vitro* tests and with FEA for both metallic and fibre posts (Rodríguez-Cervantes et al., 2007). More recently Hsu et al. (Hsu et al., 2009) concluded that when a metal post is used, the post should be as long as possible, while the biomechanical performance of a glass-fibre post combined with a composite resin core was less sensitive to post length. Others authors, like Chuang et al. (Chuang et al., 2010), using an FE model supported by an *in vitro* study, concluded that post length is more decisive for root fracture resistance in teeth restored with metal posts than in those restored with fibre posts, since long metal posts give rise to stress concentration

in the future.

**6.2 About the effect of post diameter** 

**6.3 About the effect of post length** 

diameter, even assuming that changes in stresses are slight.

experiments (Barjau-Escribano et al., 2006; Genovese et al., 2005; Huysmans & Van der Varst, 1993). In this sense, model zones with higher stresses should correspond to a higher probability of reaching the failure at these zones during fracture tests when the same external load is applied. Hence, the fracture modes observed during experiments can be compared with the stress distributions patterns obtained by FEA. In other cases, although both FEA and mechanical tests are implemented (Lang et al., 2001; Schmitter et al., 2010; Soares et al., 2008a), the results are considered complementary and no direct validation of the model is pursued.

When FEA is just a part of a more complex numerical model (Maceri et al., 2007), validation for the whole model has to be performed. In this particular case, validation was carried out by comparison with *in vivo* experimental data found in the literature.

### **6. Conclusions from previous research with biomechanical models**

Conclusions from comprehensive three-dimensional models, that capable of simulating the highly irregular shape of real teeth and the real 3D mastication forces acting on them, are summarized in the following section. These models have mainly contributed to the assessment of the effects of the material and the dimensions of the prefabricated posts on the static biomechanical performance of restored teeth, although other parameters have also been analysed.

### **6.1 About the effect of post material**

A conclusion that can be drawn from different works is that stresses distributed differently in natural teeth than in restored teeth and for the case of restored teeth, the distribution is affected in an important way by the material used for the post (Adanir & Belli, 2007; Coelho et al., 2009). The origin of this different distribution lies in the difference in the elastic modulus of the material (near 18 GPa for dentine, 30 GPa for glass fibre posts, and 210 GPa for stainless steel). Stress concentrates where non-homogeneous material distributions are present, just like interfaces. Interfaces of materials with different moduli of elasticity represent the weak link of restorative systems, as the toughness/stiffness mismatch influences the stress distribution (Barjau-Escribano et al., 2006; Sorrentino et al., 2007; Zarone et al., 2006). FE models that consider teeth under flexural-compressive loads concluded that the high elastic modulus of the metal posts caused the stress to be concentrated at the post-dentine junction (Genovese et al., 2005; Hsu et al., 2009; Okamoto et al., 2008; Pegoretti et al., 2002). Consequently, stresses on the root dentine-cortical bone area were weaker than those of the fibre-post group. The results of the FE model of Barjau-Escribano et al. (Barjau-Escribano et al., 2006) made it possible to identify the difference in the elastic moduli between the post and the dentine and core as the origin of stress concentrations at the post-core-cement interface that weakened the restored tooth when stainless steel posts were used. Other works are consistent with this finding (Boschian Pest et al., 2006; Pegoretti et al., 2002). Most of the studies found the highest stresses in restorations with endodontic posts located near the cervical region (Maceri et al., 2009; Mezzomo et al., 2011; Pierrisnard et al., 2002; Sorrentino et al., 2007), especially for fibre posts, whereas peak stresses tend to move apically when more rigid posts are used (Okamoto et al., 2008).

In the FE models from the literature, a monotonic static load was considered, which does not represent the clinical situation, where a dynamic load is characteristic. In a pioneering

experiments (Barjau-Escribano et al., 2006; Genovese et al., 2005; Huysmans & Van der Varst, 1993). In this sense, model zones with higher stresses should correspond to a higher probability of reaching the failure at these zones during fracture tests when the same external load is applied. Hence, the fracture modes observed during experiments can be compared with the stress distributions patterns obtained by FEA. In other cases, although both FEA and mechanical tests are implemented (Lang et al., 2001; Schmitter et al., 2010; Soares et al., 2008a), the results are considered complementary and no direct validation of

When FEA is just a part of a more complex numerical model (Maceri et al., 2007), validation for the whole model has to be performed. In this particular case, validation was carried out

Conclusions from comprehensive three-dimensional models, that capable of simulating the highly irregular shape of real teeth and the real 3D mastication forces acting on them, are summarized in the following section. These models have mainly contributed to the assessment of the effects of the material and the dimensions of the prefabricated posts on the static biomechanical performance of restored teeth, although other parameters have also

A conclusion that can be drawn from different works is that stresses distributed differently in natural teeth than in restored teeth and for the case of restored teeth, the distribution is affected in an important way by the material used for the post (Adanir & Belli, 2007; Coelho et al., 2009). The origin of this different distribution lies in the difference in the elastic modulus of the material (near 18 GPa for dentine, 30 GPa for glass fibre posts, and 210 GPa for stainless steel). Stress concentrates where non-homogeneous material distributions are present, just like interfaces. Interfaces of materials with different moduli of elasticity represent the weak link of restorative systems, as the toughness/stiffness mismatch influences the stress distribution (Barjau-Escribano et al., 2006; Sorrentino et al., 2007; Zarone et al., 2006). FE models that consider teeth under flexural-compressive loads concluded that the high elastic modulus of the metal posts caused the stress to be concentrated at the post-dentine junction (Genovese et al., 2005; Hsu et al., 2009; Okamoto et al., 2008; Pegoretti et al., 2002). Consequently, stresses on the root dentine-cortical bone area were weaker than those of the fibre-post group. The results of the FE model of Barjau-Escribano et al. (Barjau-Escribano et al., 2006) made it possible to identify the difference in the elastic moduli between the post and the dentine and core as the origin of stress concentrations at the post-core-cement interface that weakened the restored tooth when stainless steel posts were used. Other works are consistent with this finding (Boschian Pest et al., 2006; Pegoretti et al., 2002). Most of the studies found the highest stresses in restorations with endodontic posts located near the cervical region (Maceri et al., 2009; Mezzomo et al., 2011; Pierrisnard et al., 2002; Sorrentino et al., 2007), especially for fibre posts, whereas peak stresses tend to move apically when more rigid posts are used

In the FE models from the literature, a monotonic static load was considered, which does not represent the clinical situation, where a dynamic load is characteristic. In a pioneering

by comparison with *in vivo* experimental data found in the literature.

**6. Conclusions from previous research with biomechanical models** 

the model is pursued.

been analysed.

(Okamoto et al., 2008).

**6.1 About the effect of post material** 

work, Sancho-Bru et al. (Sancho-Bru et al., 2009) recently proposed a way to use finite element results for a fatigue analysis of dental restorations. From that work, it was concluded that, although restorations using glass fibre posts are able to bear higher static loads, both stainless steel and glass fibre post systems would have a similar life under dynamic loads, although this conclusion should be confirmed using more detailed models in the future.

### **6.2 About the effect of post diameter**

The effect that post diameter has on the stress distribution over the tooth is not the same for all post materials. Several works (Boschian Pest et al., 2006; Nakamura et al., 2006; Rodríguez-Cervantes et al., 2007) have indicated that the effect of diameter is greater for metallic posts than for fibre posts. This could be explained by the fact that the elastic modulus of fibre posts is more comparable with that of dentine, thus producing a more homogenous restoration that is consequently less affected by a change in the diameter of the tooth. In a work by the authors (Rodríguez-Cervantes et al., 2007), we confirmed with *in vitro* tests and FE models that increases in post diameter for metallic posts tend to reduce the failure loads for the restored tooth without the crown. However, in a later work, it was observed that the inclusion of the crown meant that the post diameter did not influence the biomechanical performance of the post systems no matter what post material was used (Gonzalez-Lluch et al., 2009b). For fibre posts, as has been said, the importance of post diameter is lower and the conclusions of different works vary. Boschian-Pest et al. (Boschian Pest et al., 2006), for example, recommended using small diameters to avoid weakening the root, whereas Okamoto et al. (Okamoto et al., 2008) recommended the use of a large diameter, even assuming that changes in stresses are slight.

### **6.3 About the effect of post length**

Two of the earlier FE works (Davy et al., 1981; Holmes et al., 1996) studied metallic post systems by means of two-dimensional finite element models and reached opposing conclusions. The first predicted minor changes in the stress patterns in dentine for the length variations considered; the only effect of post length was a change in the location of the stress concentrations that occurred at the post apex. In contrast, the second predicted higher shear stresses as the length of the metallic post decreased. This controversy seems to continue in two later works that compared the effect of post length for fibre posts. On the one hand, Boschian Pest et al. (Boschian Pest et al., 2006) suggested using a post inserted as deeply as possible, because shorter posts would increase stresses in the luting materials, although minor effects were recognised in the root. Ferrari et al.(Ferrari et al., 2008), on the other hand, concluded that post length does not influence the biomechanics of restored teeth. The same conclusion about the non-significance of the post length was obtained by the authors in a study with *in vitro* tests and with FEA for both metallic and fibre posts (Rodríguez-Cervantes et al., 2007). More recently Hsu et al. (Hsu et al., 2009) concluded that when a metal post is used, the post should be as long as possible, while the biomechanical performance of a glass-fibre post combined with a composite resin core was less sensitive to post length. Others authors, like Chuang et al. (Chuang et al., 2010), using an FE model supported by an *in vitro* study, concluded that post length is more decisive for root fracture resistance in teeth restored with metal posts than in those restored with fibre posts, since long metal posts give rise to stress concentration

Biomechanical Models of Endodontic Restorations 153

• Proper validation methods for FE models should be established and shared among researchers. As validation should be based on well-tested and documented experimental results, it is important to increment the quantity and quality of experimental data. To do so, information such as detailed geometrical data about the restored teeth (with different restorative solutions and subject to different loading situations) and experimental measures of strains or displacements should be available to researchers in order to test their numerical models. A common protocol to promote

• Interpretation of FE results is a key point for the future development and reliability of FE models. Of course, the best options to present the results of the models should be investigated, but more research is also needed to establish correct and validated failure

Adanir, N., Belli, S. (2007). Stress analysis of a maxillary central incisor restored with

Anusavice, K. J. (2003). *Phillips' Science of Dental Materials.* Saunders, ISBN: 13: 978-0-7216-

Asmussen, E., Peutzfeldt, A. & Sahafi, A. (2005). Finite element analysis of stresses in

Assif, D., Gorfil, C. (1994). Biomechanical considerations in restoring endodontically treated teeth. *The Journal of Prosthetic Dentistry,* Vol. 71, No.6, pp. 565-567, ISSN 0022-3913. Ausiello, P., Apicella, A. & Davidson, C. L. (2002). Effect of adhesive layer properties on

Ausiello, P., Apicella, A., Davidson, C. L. & Rengo, S. (2001). 3D-finite element analyses of

Ban, S., Anusavice, K. J. (1990). Influence of test method on failure stress of brittle dental

Barjau-Escribano, A., Sancho-Bru, J. L., Forner-Navarro, L., Rodríguez-Cervantes, P. J.,

Boschian Pest, L., Guidotti, S., Pietrabissa, R. & Gagliani, M. (2006). Stress distribution in a

Bourauel, C., Vollmer, D. & Jager, A. (2000). Application of bone remodeling theories in the

different posts. *European Journal of Dentistry,* Vol. 1, No.2, pp. 67-71, ISSN 1305-7456.

endodontically treated, dowel-restored teeth. *The Journal of Prosthetic Dentistry,* Vol.

stress distribution in composite restorations - A 3D finite element analysis. *Dental* 

cusp movements in a human upper premolar, restored with adhesive resin-based composites. *Journal of Biomechanics,* Vol. 34, No.10, pp. 1269-1277, ISSN 0021-9290. Aykent, F., Kalkan, M., Yucel, M. T. & Ozyesil, A. G. (2006). Effect of dentin bonding and

ferrule preparation on the fracture strength of crowned teeth restored with dowels and amalgam cores. *The Journal of Prosthetic Dentistry,* Vol. 95, No.4, pp. 297-301,

materials. *Journal of Dental Research,* Vol. 69, No.12, pp. 1791-1799, ISSN 0022-0345;

Perez-Gonzalez, A. & Sanchez-Marin, F. T. (2006). Influence of prefabricated post material on restored teeth: fracture strength and stress distribution. *Operative* 

post-restored tooth using the three-dimensional finite element method. *Journal of* 

simulation of orthodontic tooth movements. *Journal of Orofacial Orthopedics =* 

sharing of these data over the Internet would be an important advance.

criteria for the different materials and especially for bonded interfaces.

9387-3, United States of America.

ISSN 0022-3913.

0022-0345.

94, No.4, pp. 321-329, ISSN 0022-3913.

*Materials,* Vol. 18, No.4, pp. 295-303, ISSN 01095641.

*Dentistry,* Vol. 31, No.1, pp. 47-54, ISSN 0361-7734.

*Oral Rehabilitation,* Vol. 33, No.9, pp. 690-697, ISSN 0305-182X.

**8. References** 

in the apical portion of the root. They also concluded that post material has a greater effect on the location of peak stress, and on the resulting fracture pattern, than post length does.

### **6.4 About the effect of other restoration parameters**

FE models have corroborated other experimental results and recommendations, e.g., teeth prepared with a ferrule preparation tend to fail in a more favourable mode and exhibit greater mechanical resistance (Ichim et al., 2006) or crowns with small ferrule heights should be resin-bonded instead of using conventional cements (Schmitter et al., 2010). In a recent work, aimed at trying to find the optimal combination of crown material and luting agent, Suzuki et al. (Suzuki et al., 2008) concluded that polymer-based restorative material for the crown and composite cement were preferable to other restorative alternatives. Sorrentino et al. (Sorrentino et al., 2007) also found that changes in the crown and core materials affected stress distribution and that the stress concentrations in post-dentine interface moved apically when more rigid materials were used. The effect of the crown on stress distribution was also studied by the authors and compared with a restoration without the crown (Gonzalez-Lluch et al., 2009b). We confirmed the conclusion reached by Sorrentino et al., although it was also found that the addition of the crown did not affect the final strength of the restoration to any significant extent for either of the post systems considered (stainless steel and glass fibre). As for the effect of the cement properties, Li et al. (Li et al., 2006) found that the elastic modulus of the cement is an important parameter for stress distribution and concluded that cements with an elastic modulus similar to that of dentine should be used in weakened roots.

### **7. Conclusions and proposals for future work**

In the last decades, biomechanical models of endodontic restorations have been developed increasingly using FEA. FE simulations carried out over this time have made it possible to gain a better understanding of how the restored tooth deforms and what stresses it is subject to under simulated loads. These investigations and others to be expected in the future with more comprehensive models could make a valuable contribution to the development of better restorative solutions in this area. With a view to this objective, future works should concentrate on improving current models in order to eliminate remaining weak points. In the authors' opinion, some of these possible future lines of research are:


### **8. References**

152 Theoretical Biomechanics

in the apical portion of the root. They also concluded that post material has a greater effect on the location of peak stress, and on the resulting fracture pattern, than post length

FE models have corroborated other experimental results and recommendations, e.g., teeth prepared with a ferrule preparation tend to fail in a more favourable mode and exhibit greater mechanical resistance (Ichim et al., 2006) or crowns with small ferrule heights should be resin-bonded instead of using conventional cements (Schmitter et al., 2010). In a recent work, aimed at trying to find the optimal combination of crown material and luting agent, Suzuki et al. (Suzuki et al., 2008) concluded that polymer-based restorative material for the crown and composite cement were preferable to other restorative alternatives. Sorrentino et al. (Sorrentino et al., 2007) also found that changes in the crown and core materials affected stress distribution and that the stress concentrations in post-dentine interface moved apically when more rigid materials were used. The effect of the crown on stress distribution was also studied by the authors and compared with a restoration without the crown (Gonzalez-Lluch et al., 2009b). We confirmed the conclusion reached by Sorrentino et al., although it was also found that the addition of the crown did not affect the final strength of the restoration to any significant extent for either of the post systems considered (stainless steel and glass fibre). As for the effect of the cement properties, Li et al. (Li et al., 2006) found that the elastic modulus of the cement is an important parameter for stress distribution and concluded that cements with an elastic

In the last decades, biomechanical models of endodontic restorations have been developed increasingly using FEA. FE simulations carried out over this time have made it possible to gain a better understanding of how the restored tooth deforms and what stresses it is subject to under simulated loads. These investigations and others to be expected in the future with more comprehensive models could make a valuable contribution to the development of better restorative solutions in this area. With a view to this objective, future works should concentrate on improving current models in order to eliminate remaining weak points. In

• Reliable data about the mechanical properties of the different materials used in clinical endodontic restorations are necessary to be able to have good models and interpret results correctly. The stress-strain curves of all these materials, under both tensile and compressive loads, as well as the failure limits, should be clearly established in the literature. Furthermore, common procedures should also be promoted to obtain these

• More comprehensive mechanical models will have to be developed in the future. Of course, they should be three-dimensional and represent all the components present in the restoration, but the model should also consider the possible anisotropy of materials, such as bone, dentine or PDL. Additionally, efforts have to be directed towards developing models that represent the nonlinear response of the restored tooth in a suitable manner, because of the nonlinearity of some components, such as the PDL, or

**6.4 About the effect of other restoration parameters** 

modulus similar to that of dentine should be used in weakened roots.

the authors' opinion, some of these possible future lines of research are:

data, and communicate and share them among researchers.

due to the appearance of contacts between components.

**7. Conclusions and proposals for future work** 

does.


Biomechanical Models of Endodontic Restorations 155

Dejak, B., Mlotkowski, A. & Romanowicz, M. (2005). Finite element analysis of mechanism

Dejak, B., Mlotkowski, A. & Romanowicz, M. (2003). Finite element analysis of stresses in

Farah, J. W., Craig, R. G. (1974). Finite element stress analysis of a restored axisymmetric first molar. *Journal of Dental Research,* Vol. 53, No.4, pp. 859-866, ISSN 0022-0345. Farah, J. W., Craig, R. G. & Sikarskie, D. L. (1973). Photoelastic and finite element stress

Fernandes, A. S., Shetty, S. & Coutinho, I. (2003). Factors determining post selection: a

Ferrari, M., Sorrentino, R., Zarone, F., Apicella, D., Aversa, R. & Apicella, A. (2008). Non-

Fokkinga, W. A., Le Bell, A. M., Kreulen, C. M., Lassila, L. V., Vallittu, P. K. & Creugers, N.

Fox, K., Wood, D. J. & Youngson, C. C. (2004). A clinical report of 85 fractured metallic post-

Freitas Junior, A. C., Rocha, E. P., Santos, P. H., Ko, C. C., Martin Junior, M. & de Almeida,

Gallo, J. R.,3rd, Miller, T., Xu, X. & Burgess, J. O. (2002). In vitro evaluation of the retention

Genovese, K., Lamberti, L. & Pappalettere, C. (2005). Finite element analysis of a new

Gonzalez-Lluch C, Pérez-González A, Sancho-Bru JL, Rodríguez-Cervantes PJ. Influencia

Gonzalez-Lluch, C., Rodríguez-Cervantes, P. J., Sancho-Bru, J. L., Perez-Gonzalez, A.,

*Biomedical Engineering,* Vol. 13, No.5, pp. 515-521, ISSN 1476-8259.

*Journal,* Vol. 43, No.12, pp. 1098-1107, ISSN 01432885.

maxilar restaurado endodónticamente. *CIBIM9*. 2009a.

*Biomechanics,* Vol. 38, No.12, pp. 2375-2389.

alveolar bone. *Dental Materials*, Vol. 27, No.4, pp. 485-498.

0022-3913.

0022-3913.

0143-2885.

No.6, pp. 591-597, ISSN 0022-3913.

pp. 511-514, IN9, 515-520, ISSN 0021-9290.

Vol. 38, No.4, pp. 230-237, ISSN 0143-2885.

of cervical lesion formation in simulated molars during mastication and parafunction. *The Journal of Prosthetic Dentistry,* Vol. 94, No.6, pp. 520-529, ISSN

molars during clenching and mastication. *The Journal of Prosthetic Dentistry,* Vol. 90,

analysis of a restored axisymmetric first molar. *Journal of Biomechanics,* Vol. 6, No.5,

literature review. *The Journal of Prosthetic Dentistry,* Vol. 90, No.6, pp. 556-562, ISSN

linear viscoelastic finite element analysis of the effect of the length of glass fiber posts on the biomechanical behaviour of directly restored incisors and surrounding

H. (2005). Ex vivo fracture resistance of direct resin composite complete crowns with and without posts on maxillary premolars. *International Endodontic Journal,*

retained crowns. *International Endodontic Journal,* Vol. 37, No.8, pp. 561-573, ISSN

E. O. (2010). Mechanics of the maxillary central incisor. Influence of the periodontal ligament represented by beam elements. *Computer Methods in Biomechanics and* 

of composite fiber and stainless steel posts. *Journal of Prosthodontics : Official Journal of the American College of Prosthodontists,* Vol. 11, No.1, pp. 25-29, ISSN 1059-941X. Garbin, C. A., Spazzin, A. O., Meira-Júnior, A. D., Loretto, S. C., Lyra, A. M. V. C. & Braz, R.

(2010). Biomechanical behaviour of a fractured maxillary incisor restored with direct composite resin only or with different post systems. *International Endodontic* 

customized composite post system for endodontically treated teeth. *Journal of* 

del mallado en la simulación por elementos finitos de un diente incisivo central

Barjau-Escribano, A., Vergara-Monedero, M. & Forner-Navarro, L. (2009b). Influence of material and diameter of pre-fabricated posts on maxillary central

*Fortschritte Der Kieferorthopadie : Organ/official Journal Deutsche Gesellschaft Fur Kieferorthopadie,* Vol. 61, No.4, pp. 266-279, ISSN 1434-5293.


Cattaneo, P. M., Dalstra, M. & Melsen, B. (2005). The finite element method: a tool to study

Chan, F. W., Harcourt, J. K. & Brockhurst, P. J. (1993). The effect of post adaptation in the

Christensen, G. J. (1998). Posts and cores: state of the art. *Journal of the American Dental* 

Christensen, R. M. (2006). A comparative evaluation of three isotropic, two property failure theories. *Journal of Applied Mechanics,* Vol. 73, No.5, pp. 852-859, ISSN 0021-8936. Chuang, S., Yaman, P., Herrero, A., Dennison, J. & Chang, C. (2010). Influence of post

Clement, R., Schneider, J., Brambs, H. -., Wunderlich, A., Geiger, M. & Sander, F. G. (2004).

Coelho Santos, G.,Jr, El-Mowafy, O. & Rubo, J. H. (2004). Diametral tensile strength of a

Coelho, C. S., Biffi, J. C., Silva, G. R., Abrahao, A., Campos, R. E. & Soares, C. J. (2009). Finite

Craig, R. G., Powers, J. M. (2002). *Restorative dental materials.* Mosby, 0-323-01442-9, United

Dalstra, M., Huiskes, R. & van Erning, L. (1995). Development and validation of a three-

Davy, D. T., Dilley, G. L. & Krejci, R. F. (1981). Determination of stress patterns in root-filled

De Groot, R., Peters, M. C., De Haan, Y. M., Dop, G. J. & Plasschaert, A. J. (1987). Failure

De Jager, N., de Kler, M. & van der Zel, J. M. (2006). The influence of different core material

Dejak, B., Mlotkowski, A. & Romanowicz, M. (2007). Strength estimation of different designs

*Journal of Prosthetic Dentistry,* Vol. 98, No.2, pp. 89-100, ISSN 0022-3913.

*Materials Journal,* Vol. 28, No.6, pp. 671-678, ISSN 0287-4547; 0287-4547. Craig, R. G., Farah, J. W. (1977). Stress analysis and design of single restorations and fixed

bridges. *Oral Sciences Reviews,* Vol. 10, pp. 45-74, ISSN 0300-4759.

*Engineering,* Vol. 117, No.3, pp. 272-278, ISSN 0148-0731; 0148-0731.

*Kieferorthopadie,* Vol. 61, No.4, pp. 266-279, ISSN 1434-5293.

*Association (1939),* Vol. 129, No.1, pp. 96-97, ISSN 0002-8177.

*Journal,* Vol. 38, No.1, pp. 39-45, ISSN 0045-0421.

73, No.2, pp. 135-144, ISSN 0169-2607.

ISSN 0022-0345; 0022-0345.

0022-3913.

States of America.

pp. 1301-1310, ISSN 0022-0345.

1748-1752, ISSN 0022-0345.

0109-5641.

*Fortschritte Der Kieferorthopadie : Organ/official Journal Deutsche Gesellschaft Fur* 

orthodontic tooth movement. *Journal of Dental Research,* Vol. 84, No.5, pp. 428-433,

root canal on retention of posts cemented with various cements. *Australian Dental* 

material and length on endodontically treated incisors: an in vitro and finite element study. *The Journal of Prosthetic Dentistry,* Vol. 104, No.6, pp. 379-388, ISSN

Quasi-automatic 3D finite element model generation for individual single-rooted teeth and periodontal ligament. *Computer Methods and Programs in Biomedicine,* Vol.

resin composite core with nonmetallic prefabricated posts: an in vitro study. *The Journal of Prosthetic Dentistry,* Vol. 91, No.4, pp. 335-341, ISSN 0022-3913; 0022-3913.

element analysis of weakened roots restored with composite resin and posts. *Dental* 

dimensional finite element model of the pelvic bone. *Journal of Biomechanical* 

teeth incorporating various dowel designs. *Journal of Dental Research,* Vol. 60, No.7,

stress criteria for composite resin. *Journal of Dental Research,* Vol. 66, No.12, pp.

on the FEA-determined stress distribution in dental crowns. *Dental Materials : Official Publication of the Academy of Dental Materials,* Vol. 22, No.3, pp. 234-242, ISSN

of ceramic inlays and onlays in molars based on the Tsai-Wu failure criterion. *The* 


Biomechanical Models of Endodontic Restorations 157

Li, L. L., Wang, Z. Y., Bai, Z. C., Mao, Y., Gao, B., Xin, H. T., Zhou, B., Zhang, Y. & Liu, B.

Maceri, F., Marino, M. & Vairo, G. (2010). A unified multiscale mechanical model for soft

Maceri, F., Martignoni, M. & Vairo, G. (2009). Optimal mechanical design of anatomical

Maceri, F., Martignoni, M. & Vairo, G. (2007). Mechanical behaviour of endodontic

Magne, P., Tan, D. T. (2008). Incisor compliance following operative procedures: a rapid 3-D

Magne, P. (2007). Efficient 3D finite element analysis of dental restorative procedures using micro-CT data. *Dental Materials,* Vol. 23, No.5, pp. 539-548, ISSN 0109-5641. Mezzomo, L. A., Corso, L., Marczak, R. J. & Rivaldo, E. G. (2011). Three-dimensional FEA of

Middleton, J., Jones, M. & Wilson, A. (1996). The role of the periodontal ligament in bone

Nakamura, T., Ohyama, T., Waki, T., Kinuta, S., Wakabayashi, K., Mutobe, Y., Takano, N. &

Natali, A., Pavan, P., Carniel, E. & Dorow, C. (2004). Viscoelastic response of the periodontal

Nothdurft, F. P., Pospiech, P. R. (2006). Clinical evaluation of pulpless teeth restored with

Okada, D., Miura, H., Suzuki, C., Komada, W., Shin, C., Yamamoto, M. & Masuoka, D. (2008).

*Biomedical Engineering,* Vol. 12, No.1, pp. 59-71, ISSN 10255842.

*Biomechanics,* Vol. 40, No.11, pp. 2386-2398, ISSN 0021-9290.

*of Orthodontics,* Vol. 109, No.2, pp. 155-162, ISSN 08895406.

Vol. 119, No.4, pp. 305-311, ISSN 0366-6999.

10, No.1, pp. 49-56, ISSN 1461-5185.

pp. 120-129, ISSN 1059941X.

150, ISSN 0287-4547.

No.4-5, pp. 222-230, ISSN 0300-8207.

*Dentistry,* Vol. 95, No.4, pp. 311-314.

Vol. 27, No.1, pp. 49-55, ISSN 0287-4547.

5641.

No.2, pp. 355-363.

*Publication of the Academy of Dental Materials,* Vol. 21, No.8, pp. 709-715, ISSN 0109-

(2006). Three-dimensional finite element analysis of weakened roots restored with different cements in combination with titanium alloy posts. *Chinese Medical Journal,*

collagenous tissues with regular fiber arrangement. *Journal of Biomechanics,* Vol. 43,

post-systems for endodontic restoration. *Computer Methods in Biomechanics and* 

restorations with multiple prefabricated posts: A finite-element approach. *Journal of* 

finite element analysis using micro-CT data. *The Journal of Adhesive Dentistry,* Vol.

effects of two dowel-and-core approaches and effects of canal flaring on stress distribution in endodontically treated teeth. *Journal of Prosthodontics,* Vol. 20, No.2,

modeling: the initial development of a time-dependent finite element model. *American Journal of Orthodontics and Dentofacial Orthopedics : Official Publication of the American Association of Orthodontists, its Constituent Societies, and the American Board* 

Yatani, H. (2006). Stress analysis of endodontically treated anterior teeth restored with different types of post material. *Dental Materials Journal,* Vol. 25, No.1, pp. 145-

ligament: an experimental-numerical analysis. *Connective Tissue Research,* Vol. 45,

conventionally cemented zirconia posts: A pilot study. *The Journal of Prosthetic* 

Stress distribution in roots restored with different types of post systems with composite resin. *Dental Materials Journal,* Vol. 27, No.4, pp. 605-611, ISSN 0287-4547. Okamoto, K., Ino, T., Iwase, N., Shimizu, E., Suzuki, M., Satoh, G., Ohkawa, S. & Fujisawa,

M. (2008). Three-dimensional finite element analysis of stress distribution in composite resin cores with fiber posts of varying diameters. *Dental Materials Journal,*

incisors restored with crown. *Journal of Oral Rehabilitation,* Vol. 36, No.10, pp. 737- 747, ISSN 1365-2842.


Gungor, M. A., Kucuk, M., Dundar, M., Karaoglu, C. & Artunc, C. (2004). Effect of

Gupta, S., van der Helm, F. C., Sterk, J. C., van Keulen, F. & Kaptein, B. L. (2004).

Gurbuz, T., Sengul, F. & Altun, C. (2008). Finite element stress analysis of short-post core

Holmes, D. C., Diaz-Arnold, A. M. & Leary, J. M. (1996). Influence of post dimension on

Hsu, M. L., Chen, C. S., Chen, B. J., Huang, H. H. & Chang, C. L. (2009). Effects of post

Huysmans, M. C. D. N. J. M., Van der Varst, P. G. T. (1993). Finite element analysis of

Ichim, I., Kuzmanovic, D. V. & Love, R. M. (2006). A finite element analysis of ferrule design

Imanishi, A., Nakamura, T., Ohyama, T. & Nakamura, T. (2003). 3-D Finite element analysis of all-ceramic posterior crowns. *Journal of Oral Rehabilitation,* Vol. 30, pp. 818-822. Isidor, F., Brondum, K. & Ravnholt, G. (1999). The influence of post length and crown

Kinney, J. H., Marshall, S. J. & Marshall, G. W. (2003). The mechanical properties of human

Komatsu, K. (2010). Mechanical strength and viscoelastic response of the periodontal

Lang, L. A., Wang, R. F., Kang, B. & White, S. N. (2001). Validation of finite element analysis

Lanza, A., Aversa, R., Rengo, S., Apicella, D. & Apicella, A. (2005). 3D FEA of cemented

*Endodontic Journal,* Vol. 39, No.6, pp. 443-452, ISSN 0143-2885.

*Biologists,* Vol. 14, No.1, pp. 13-29, ISSN 1544-1113; 1045-4411.

502318. Epub 2009 Dec 15, ISSN 1758-7360.

*Journal,* Vol. 27, No.4, pp. 499-507, ISSN 0287-4547.

No.11, pp. 821-830, ISSN 1365-2842; 0305-182X.

747, ISSN 1365-2842.

172-178, ISSN 0305-182X.

140-147, ISSN 0022-3913.

pp. 57-64, ISSN 0300-5712.

ISSN 0893-2174.

654, ISSN 0022-3913.

0954-4119.

incisors restored with crown. *Journal of Oral Rehabilitation,* Vol. 36, No.10, pp. 737-

temperature and stress distribution on all-ceramic restorations by using a threedimensional finite element analysis. *Journal of Oral Rehabilitation,* Vol. 31, No.2, pp.

Development and experimental validation of a three-dimensional finite element model of the human scapula. *Proceedings of the Institution of Mechanical Engineers.Part H, Journal of Engineering in Medicine,* Vol. 218, No.2, pp. 127-142, ISSN

and over restorations prepared with different restorative materials. *Dental Materials* 

stress distribution in dentin. *The Journal of Prosthetic Dentistry,* Vol. 75, No.2, pp.

materials and length on the stress distribution of endodontically treated maxillary central incisors: a 3D finite element analysis. *Journal of Oral Rehabilitation,* Vol. 36,

quasistatic and fatigue failure of post and cores. *Journal of Dentistry,* Vol. 21, No.1,

on restoration resistance and distribution of stress within a root. *International* 

ferrule length on the resistance to cyclic loading of bovine teeth with prefabricated titanium posts. *The International Journal of Prosthodontics,* Vol. 12, No.1, pp. 78-82,

dentin: a critical review and re-evaluation of the dental literature. *Critical Reviews in Oral Biology and Medicine : An Official Publication of the American Association of Oral* 

ligament in relation to structure. *Journal of Dental Biomechanics,* Vol. 2010, pp.

in dental ceramics research. *The Journal of Prosthetic Dentistry,* Vol. 86, No.6, pp. 650-

steel, glass and carbon posts in a maxillary incisor. *Dental Materials : Official* 

*Publication of the Academy of Dental Materials,* Vol. 21, No.8, pp. 709-715, ISSN 0109- 5641.


Biomechanical Models of Endodontic Restorations 159

Schmitter, M., Rammelsberg, P., Lenz, J., Scheuber, S., Schweizerhof, K. & Rues, S. (2010).

Selna, L. G., Shillingburg, H. T.,Jr & Kerr, P. A. (1975). Finite element analysis of dental

Shigley.J.E. & Mischke, CR. (2002) *Mechanical Engineering Design*. 5th ed. McGraw-Hill, ISBN

Silva, N. R., Castro, C. G., Santos-Filho, P. C., Silva, G. R., Campos, R. E., Soares, P. V. &

Soares, C. J., Raposo, L. H. A., Soares, P. V., Santos-Filho, P. C. F., Menezes, M. S., Soares, P.

Soares, P. V., Santos-Filho, P. C., Gomide, H. A., Araujo, C. A., Martins, L. R. & Soares, C. J.

Soares, P. V., Santos-Filho, P. C. F., Queiroz, E. C., Araújo, T. C., Campos, R. E., Araújo, C. A.

Sorrentino, R., Apicella, D., Riccio, C., Gherlone, E., Zarone, F., Aversa, R., Garcia-Godoy, F.,

Suzuki, C., Miura, H., Okada, D. & Komada, W. (2008). Investigation of stress distribution in

Tajima, K., Chen, K. K., Takahashi, N., Noda, N., Nagamatsu, Y. & Kakigawa, H. (2009).

*Dental Materials,* Vol. 23, No.8, pp. 983-993, ISSN 0109-5641.

*Journal,* Vol. 27, No.2, pp. 229-236, ISSN 0287-4547.

*Dentistry,* Vol. 93, No.4, pp. 331-336, ISSN 0022-3913.

*Journal of Prosthodontics,* Vol. 19, No.2, pp. 130-137, ISSN 1059941X.

analysis. *Acta Biomaterialia,* Vol. 6, pp. 3747-3754, ISSN 1742-7061. Scotti R, Ferrari M. (2004) *Pernos de fibra*. Barcelona: Masson, ISBN 84-458-1358-7

*Materials Research,* Vol. 9, No.2, pp. 237-252, ISSN 0021-9304.

0072832096.

3913.

pp. 153-158, ISSN 1998-3603.

Vol. 17, No.2, pp. 114-119, ISSN 1059941X.

Teeth restored using fiber-reinforced posts: In vitro fracture tests and finite element

structures--axisymmetric and plane stress idealizations. *Journal of Biomedical* 

Soares, C. J. (2009). Influence of different post design and composition on stress distribution in maxillary central incisor: Finite element analysis. *Indian Journal of Dental Research : Official Publication of Indian Society for Dental Research,* Vol. 20, No.2,

B. F. & Magalhães, D. (2010). Effect of different cements on the biomechanical behavior of teeth restored with cast dowel-and-cores - in vitro and FEA analysis.

(2008a). Influence of restorative technique on the biomechanical behavior of endodontically treated maxillary premolars. Part II: strain measurement and stress distribution. *The Journal of Prosthetic Dentistry,* Vol. 99, No.2, pp. 114-122, ISSN 0022-

& Soares, C. J. (2008b). Fracture resistance and stress distribution in endodontically treated maxillary premolars restored with composite resin. *Journal of Prosthodontics,*

Ferrari, M. & Apicella, A. (2009). Nonlinear visco-elastic finite element analysis of different porcelain veneers configuration. *Journal of Biomedical Materials Research.Part B, Applied Biomaterials,* Vol. 91, No.2, pp. 727-736, ISSN 1552-4981. Sorrentino, R., Aversa, R., Ferro, V., Auriemma, T., Zarone, F., Ferrari, M. & Apicella, A.

(2007). Three-dimensional finite element analysis of strain and stress distributions in endodontically treated maxillary central incisors restored with different post, core and crown materials. *Dental Materials : Official Publication of the Academy of* 

roots restored with different crown materials and luting agents. *Dental Materials* 

Three-dimensional finite element modeling from CT images of tooth and its validation. *Dental Materials Journal,* Vol. 28, No.2, pp. 219-226, ISSN 0287-4547. Tan, P. L., Aquilino, S. A., Gratton, D. G., Stanford, C. M., Tan, S. C., Johnson, W. T. &

Dawson, D. (2005). In vitro fracture resistance of endodontically treated central incisors with varying ferrule heights and configurations. *The Journal of Prosthetic* 


Pegoretti, A., Fambri, L., Zappini, G. & Bianchetti, M. (2002). Finite element analysis of a

Pereira, J. R., de Ornelas, F., Conti, P. C. & do Valle, A. L. (2006). Effect of a crown ferrule on

Pérez-González, A., Iserte-Vilar, J.L., González-Lluch, C. (2011). Interpreting finite element

Pierrisnard, L., Bohin, F., Renault, P. & Barquins, M. (2002). Corono-radicular reconstruction

Pini, M., Wiskott, H. W., Scherrer, S. S., Botsis, J. & Belser, U. C. (2002). Mechanical

Probster, L., Geis-Gerstorfer, J., Kirchner, E. & Kanjantra, P. (1997). In vitro evaluation of a

Qian, H., Chen, J. & Katona, T. R. (2001). The influence of PDL principal fibers in a 3-

Rappelli, G., Scalise, L., Procaccini, M. & Tomasini, E. P. (2005). Stress distribution in fiber-

Rees, J. S., Jacobsen, P. H. (1997). Elastic modulus of the periodontal ligament. *Biomaterials,*

Ren, L. M., Wang, W. X., Takao, Y. & Chen, Z. X. (2010). Effects of cementum–dentine

Rodríguez-Cervantes, P. J., Sancho-Bru, J. L., Barjau-Escribano, A., Forner-Navarro, L.,

Ruse, N. D. (2008). Propagation of erroneous data for the modulus of elasticity of

Sancho-Bru, J. L., Rodríguez-Cervantes, P. J., Pérez-González, A. & González-Lluch, C.

links. *Dental Materials,* Vol. 24, No.12, pp. 1717-1719, ISSN 0109-5641. Sahafi, A., Peutzfeldt, A., Ravnholt, G., Asmussen, E. & Gotfredsen, K. (2005). Resistance to

*and Modelling, ASM 2009*, ISBN 9780889868083, Palma de Mallorca.

*Journal of Dentistry,* Vol. 38, No.11, pp. 882-891, ISSN 0300-5712.

*Prosthetic Dentistry,* Vol. 88, No.4, pp. 442-448, ISSN 0022-3913.

2667-2682, ISSN 0142-9612.

*Engineering 2010*, Valencia (Spain).

37, No.4, pp. 237-244, ISSN 0022-3484.

Vol. 93, No.5, pp. 425-432, ISSN 0022-3913.

Vol. 18, No.14, pp. 995-999, ISSN 0142-9612.

34, No.2, pp. 141-152, ISSN 0305-182X.

pp. 84-90, ISSN 1432-6981.

636-645, ISSN 0305-182X.

2011, 10:44.

glass fibre reinforced composite endodontic post. *Biomaterials,* Vol. 23, No.13, pp.

the fracture resistance of endodontically treated teeth restored with prefabricated posts. *The Journal of Prosthetic Dentistry,* Vol. 95, No.1, pp. 50-54, ISSN 0022-3913. Pérez-González, A., González-Lluch, C., Sancho-Bru, J. L., Rodríguez-Cervantes, P. J. &

Iserte-Vilar, J. L. (2010). FEM and statistical analysis for studying the optimal design of endodontic posts. *Computer Methods in Biomechanics and Biomedical* 

results for brittle materials in endodontic restorations. *Biomedical Engineering Online*

of pulpless teeth: A mechanical study using finite element analysis. *The Journal of* 

characterization of bovine periodontal ligament. *Journal of Periodontal Research,* Vol.

glass-ceramic restorative material. *Journal of Oral Rehabilitation,* Vol. 24, No.9, pp.

dimensional analysis of orthodontic tooth movement. *American Journal of Orthodontics and Dentofacial Orthopedics,* Vol. 120, No.3, pp. 272-279, ISSN 0889-5406.

reinforced composite inlay fixed partial dentures. *The Journal of Prosthetic Dentistry,*

junction and cementum on the mechanical response of tooth supporting structure.

Pérez-González, A. & Sánchez-Marín, F. T. (2007). Influence of prefabricated post dimensions on restored maxillary central incisors. *Journal of Oral Rehabilitation,* Vol.

periodontal ligament and gutta percha in FEM/FEA papers: A story of broken

cyclic loading of teeth restored with posts. *Clinical Oral Investigations,* Vol. 9, No.2,

(2009). Using fatigue analyses from FEA to study the biomechanics of restored teeth with intraradicular posts. *18th IASTED International Conference on Applied Simulation* 


**8** 

*USA* 

**Development and Validation of a** 

*Department of Industrial Engineering, University of Miami* 

One of the important applications of the computer modelling of human body is the area of joint replacement where a validated model can be used for surgery planning. It is known that the evolution of total knee and total hip replacement has been influenced to a great extent by the knowledge obtained from gait analysis studies (Andriacchi and Hurwitz, 1997). Many of the mechanical problems associated with these devices have been evaluated in terms of the mechanics of walking where the magnitude and pattern of the forces at the hip and knee joints obtained from gait analysis studies have been used as design criteria of

Gait analysis provides a unique opportunity to obtain objective information that cannot be obtained through other clinical means (Andriacchi and Hurwitz, 1997). For instance, several investigators have advocated the use of gait analysis for planning surgery and therapy treatments for children with cerebral palsy (Lofterod, et al., 2007; Kay, et al., 2000; Molenaers, et al., 2006). Improvement in gait after multi-level surgery using kinematic data has been documented, while kinematics provides information on dynamic joint motion kinetics is essential for differentiating between primary deformities and secondary

The potential benefits of gait analysis are improved treatment decision making, so that surgery and other treatments result in improved walking capability. Also, the information generated from the gait analysis of patients with total joint replacements has been utilized as a tool for assessing recovery following these procedures, where the key to the analysis of functionality following joint replacement is the ability to identify the adaptations

It is very difficult to determine muscle force/power output from multiple muscles simultaneously without affecting the pattern of normal movements (Naganoa, et al., 2005). Fortunately, computer modeling can provide useful insights for human biomechanics. Most in-vivo experiments only reveal the forces in the joint and not the surrounding muscle forces or their point of application. It is also known that finding the internal forces in the body by in-vivo experiments alone is difficult and sometime impossible. Because of the inherited redundancy in the musculoskeletal system (Crowninshield and Brand, 1981b) a desired motion can be achieved by an infinite

**1. Introduction** 

responses.

both total hip and total knee replacements.

corresponding to the joint design features.

**Model of the Lower Extremity** 

Shihab Asfour and Moataz Eltoukhy

**Three-Dimensional Biomechanical** 

