**Biomechanical Models of Endodontic Restorations**

Antonio Pérez-González, Carmen González-Lluch, Joaquín L. Sancho-Bru, Pablo J. Rodríguez-Cervantes and José L. Iserte-Vilar *Universitat Jaume I Spain* 

### **1. Introduction**

132 Theoretical Biomechanics

Zhang, X., Jiang, G., Wu, C., Woo, S. L., (2008). A subject-specific finite element model of the

August 20-24, pp. 891-894.

anterior cruciate ligament. In *30th Annual International Conference of the IEEE Engineering in Medicine & Biology Society*. Vancouver, British Columbia, Canada,

> Endodontic treatment is one of the most widely used techniques in present-day odontology owing to the tendency to save teeth whenever possible. In endodontic therapy, the injured pulp of a tooth (located in the interior of the tooth and containing nerves and other vital tissues) is cleaned out and then the space is disinfected and subsequently filled with restorative material. This process is commonly known as root canal treatment. The devitalised tooth resulting from endodontics, has a different stiffness and resistance as compared to the original tooth and is less resistant as a consequence of the loss of tooth structure (Walton & Torabinejad, 2002). The use of intraradicular posts has extended as a technique to restore teeth that have lost a considerable amount of coronal tooth structure. After removing the pulp, the intraradicular post is introduced into the devitalised root. The post helps to support the final restoration and join it to the root (Christensen, 1998). Fig. 1a shows the typical structure of a tooth endodontically restored with a post. The post is inserted into the devitalised root canal, which has previously been obturated at its apical end with a biocompatible polymer called gutta-percha. Cement is used to bond the post to the root canal and a core is placed over the remaining dentine and the post. Finally, an artificial crown is used to achieve an external appearance that is similar to that of the original tooth. Nowadays most of the posts are prefabricated in a range of different materials and designs (Scotti & Ferrari, 2004). However, before prefabricated post became generalised, cast post and cores were used as a single metal alloy unit (Fig. 1b). Cast postcore systems take longer to make and involve an intermediate laboratory stage in which the retention system is created, which makes the whole process more costly. In comparison, prefabricated posts do not need this intermediate stage, which means that the whole restoration process can be performed in a single visit and is obviously easier and cheaper for the patient (Christensen, 1998). Nonetheless, the adaptation of the prefabricated posts to the root canal may be less accurate (Chan et al., 1993).

> As the endodontically restored tooth is composed of materials that are different to those of natural teeth, it is expected to have a different biomechanical response under oral loads. The deformation of the system under flexural, compressive or tension forces could be different and so its mechanical strength under static or fatigue loads. Ideally, it seems interesting that the biomechanical behaviour of the restored system should preferably resemble that of the original tooth as much as possible in order to avoid failure of the repaired tooth or its

Biomechanical Models of Endodontic Restorations 135

numerical biomechanical models has considerably increased considerably in recent years (Barjau-Escribano et al., 2006; Boschian Pest et al., 2006; Genovese et al., 2005; Pegoretti et al., 2002). These biomechanical models mainly use the finite element analysis (FEA) technique to obtain a numerical representation of the real system and to compute mechanical stresses and strains under simulated loads. In FEA, the system to be studied is divided into a set of small discrete elements (finite elements, FE) defined by a limited number of nodes. A simplified constitutive equation for the finite elements is defined to represent their mechanical response, as a relationship between nodal loads and nodal displacements, and is expressed as the stiffness matrix of the element. By forcing the nodes shared by adjacent elements to have the same displacements, the element stiffness matrices of the elements are combined to give the global stiffness matrix. Nodal displacements can be solved by imposing boundary conditions on this problem, i.e. applied loads and forced displacements and, subsequently, the strains and stresses are then calculated from them. As the size of the finite elements decreases (thus increasing the number of elements), the accuracy of the method increases, but at the same time the computational cost also increases significantly. The use of FEA for studying the restored tooth has some advantages with respect to the other alternatives cited above. Its main advantage is that it allows a highly controlled analysis of one or several specific parameters on a single tooth model. This results in a better comprehension of either the individual effect or that combined with different parameters. Moreover, numerical analysis with biomechanical models is faster and cheaper than *in vitro* experimentation and eliminates the ethical implications related with the collection of real tooth specimens for *in vitro* experiments or invasive *in vivo* measurements. In contrast, FEA also has some drawbacks namely, the clinical application of their results is conditioned by

the accuracy of the model and its previous validation based on experimental data.

pursued in this field in order to obtain more comprehensive and accurate models.

**restorations** 

**2. Historical overview of biomechanical models for simulating endodontic** 

FEA was developed in the fifties of the last century in the aircraft industry and was not used in dentistry until the seventies. One of the first works on the subject was the doctoral dissertation of Farah, at the University of Michigan, in 1972, which dealt with the simulation of molar restoration using photoelasticity and FEA. Some later works were presented by this research group in following years (Craig & Farah, 1977; Farah et al., 1973; Farah & Craig, 1974). Other pioneering works in those years were presented by Thresher and Saito (Thresher & Saito, 1973) and Selna et al.(Selna et al., 1975). These first works were twodimensional and were conducted with programs developed by the researchers themselves. Later, the advances in computer resources and in commercial software for FEA extended the use of different commercial programs, allowing more accurate and predictive threedimensional models to be developed (Asmussen et al., 2005; Barjau-Escribano et al., 2006;

In this chapter, a comprehensive review of the state of the art of biomechanical models of endodontic restorations is presented. First, we describe a review of the evolution of these models throughout their brief history. Then, the different aspects related to the definition of the model are analysed with reference to the way they were treated in previous models in the literature. Finally, we provide an overview of the main conclusions reached in previous studies using biomechanical models about the effect of the different parameters involved in the endodontic restoration. Lastly, we present possible lines for future research that can be

antagonists or adjacent teeth. Nevertheless, this is only an ideal for endodontic therapy and actual restored teeth would differ from this ideal depending on the geometry of the remaining tooth structure, the material and geometry of the components used in the restoration and the technique used by the dentist.

Fig. 1. Endodontic restoration with prefabricated post (a) and cast post-core (b)

The interest in performing an in-depth analysis of the mechanical response of restored teeth becomes apparent from the above considerations and in last few decades a considerable amount of research has been devoted to this objective. Some works have compared this mechanical response with that of natural teeth and others have studied the effect of using different procedures or components for the restoration in order to obtain conclusions about which is preferable for clinical use. The conclusions from these investigations have contributed to the advance of the restorative techniques in Endodontics, that have allowed achieving higher survival rates.

With regard to the methodology used in this field of research, three main lines are distinguished: *in vitro* experimentation, retrospective clinical studies and studies with numerical biomechanical models. The first usually covers the effects of the specific parameters (material, post design, post length or post diameter) on the resistance or the retention of the restoration (Fokkinga et al., 2005; Gallo et al., 2002; Pereira et al., 2006). The majority of these works are static experiments and very few studies have been carried out under cyclic loading or fatigue (Isidor et al., 1999; Sahafi et al., 2005). Retrospective studies are less common given that they require longer times to be performed, often years, and they are based on the study of failures in restorations performed on real patients (Fox et al., 2004; Nothdurft & Pospiech, 2006; Torbjorner et al., 1995). Although working conditions are the current oral dynamic conditions in these studies, the results may be conditioned by loss of control over parameters that are beyond the scope of what is being studied owing to the variability that exists from one patient to another. Finally, the number of studies based on

antagonists or adjacent teeth. Nevertheless, this is only an ideal for endodontic therapy and actual restored teeth would differ from this ideal depending on the geometry of the remaining tooth structure, the material and geometry of the components used in the

> Cast post-core

> > b

Fig. 1. Endodontic restoration with prefabricated post (a) and cast post-core (b)

Bone

Periodontal ligament (PDL)

Crown

Gingiva

The interest in performing an in-depth analysis of the mechanical response of restored teeth becomes apparent from the above considerations and in last few decades a considerable amount of research has been devoted to this objective. Some works have compared this mechanical response with that of natural teeth and others have studied the effect of using different procedures or components for the restoration in order to obtain conclusions about which is preferable for clinical use. The conclusions from these investigations have contributed to the advance of the restorative techniques in Endodontics, that have allowed

With regard to the methodology used in this field of research, three main lines are distinguished: *in vitro* experimentation, retrospective clinical studies and studies with numerical biomechanical models. The first usually covers the effects of the specific parameters (material, post design, post length or post diameter) on the resistance or the retention of the restoration (Fokkinga et al., 2005; Gallo et al., 2002; Pereira et al., 2006). The majority of these works are static experiments and very few studies have been carried out under cyclic loading or fatigue (Isidor et al., 1999; Sahafi et al., 2005). Retrospective studies are less common given that they require longer times to be performed, often years, and they are based on the study of failures in restorations performed on real patients (Fox et al., 2004; Nothdurft & Pospiech, 2006; Torbjorner et al., 1995). Although working conditions are the current oral dynamic conditions in these studies, the results may be conditioned by loss of control over parameters that are beyond the scope of what is being studied owing to the variability that exists from one patient to another. Finally, the number of studies based on

restoration and the technique used by the dentist.

a

Core Post head

Post

Root (dentine)

Guttapercha

achieving higher survival rates.

numerical biomechanical models has considerably increased considerably in recent years (Barjau-Escribano et al., 2006; Boschian Pest et al., 2006; Genovese et al., 2005; Pegoretti et al., 2002). These biomechanical models mainly use the finite element analysis (FEA) technique to obtain a numerical representation of the real system and to compute mechanical stresses and strains under simulated loads. In FEA, the system to be studied is divided into a set of small discrete elements (finite elements, FE) defined by a limited number of nodes. A simplified constitutive equation for the finite elements is defined to represent their mechanical response, as a relationship between nodal loads and nodal displacements, and is expressed as the stiffness matrix of the element. By forcing the nodes shared by adjacent elements to have the same displacements, the element stiffness matrices of the elements are combined to give the global stiffness matrix. Nodal displacements can be solved by imposing boundary conditions on this problem, i.e. applied loads and forced displacements and, subsequently, the strains and stresses are then calculated from them. As the size of the finite elements decreases (thus increasing the number of elements), the accuracy of the method increases, but at the same time the computational cost also increases significantly.

The use of FEA for studying the restored tooth has some advantages with respect to the other alternatives cited above. Its main advantage is that it allows a highly controlled analysis of one or several specific parameters on a single tooth model. This results in a better comprehension of either the individual effect or that combined with different parameters. Moreover, numerical analysis with biomechanical models is faster and cheaper than *in vitro* experimentation and eliminates the ethical implications related with the collection of real tooth specimens for *in vitro* experiments or invasive *in vivo* measurements. In contrast, FEA also has some drawbacks namely, the clinical application of their results is conditioned by the accuracy of the model and its previous validation based on experimental data.

In this chapter, a comprehensive review of the state of the art of biomechanical models of endodontic restorations is presented. First, we describe a review of the evolution of these models throughout their brief history. Then, the different aspects related to the definition of the model are analysed with reference to the way they were treated in previous models in the literature. Finally, we provide an overview of the main conclusions reached in previous studies using biomechanical models about the effect of the different parameters involved in the endodontic restoration. Lastly, we present possible lines for future research that can be pursued in this field in order to obtain more comprehensive and accurate models.

### **2. Historical overview of biomechanical models for simulating endodontic restorations**

FEA was developed in the fifties of the last century in the aircraft industry and was not used in dentistry until the seventies. One of the first works on the subject was the doctoral dissertation of Farah, at the University of Michigan, in 1972, which dealt with the simulation of molar restoration using photoelasticity and FEA. Some later works were presented by this research group in following years (Craig & Farah, 1977; Farah et al., 1973; Farah & Craig, 1974). Other pioneering works in those years were presented by Thresher and Saito (Thresher & Saito, 1973) and Selna et al.(Selna et al., 1975). These first works were twodimensional and were conducted with programs developed by the researchers themselves. Later, the advances in computer resources and in commercial software for FEA extended the use of different commercial programs, allowing more accurate and predictive threedimensional models to be developed (Asmussen et al., 2005; Barjau-Escribano et al., 2006;

Biomechanical Models of Endodontic Restorations 137

the other hand, masticatory loads are variable, implying that the restoration is subject to fatigue, but very few fatigue simulations are found in the literature up until now (Huysmans & Van der Varst, 1993). The authors have recently presented a study on fatigue simulation of a

The validity of the conclusions reached from biomechanical models of the endodontic restoration will depend on the accuracy with which these models can represent the real system. A number of different factors related to the definition of the model conditions the quality of the results obtained from FEA, namely the accuracy of the geometric representation of the system, the components included in the model, the quality of the discretisation and the constitutive equations for finite elements, the material properties, or the boundary conditions. Moreover, processing and interpreting the results is a challenging task that can compromise

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

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

accuracy of the conclusions from FEA. All these questions are reviewed in this section.

maxillary central incisor, which is shown, in a sagittal section, in Fig. 3.

whole endodontic restoration (Sancho-Bru et al., 2009).

**3. Biomechanical model definition** 

**3.1 Geometric model** 

**3.2 Components in the model** 

stress distribution in the root and the ligament.

Boschian Pest et al., 2006; Ferrari et al., 2008; Genovese et al., 2005; Gonzalez-Lluch et al., 2009b; Maceri et al., 2009; Rodríguez-Cervantes et al., 2007; Sorrentino et al., 2007).

The number of publications dealing with FE simulation in endodontics has increased exponentially in recent years. For example, in 2011 a manual search in the SCOPUS database in 2011 for papers containing simultaneously the terms *finite element* and *endodontic* within journals on the subjects of Engineering and Medicine and Dentistry found 14 papers until 2000, 31 papers from 2000 to 2009 and 11 papers in the short period including 2010 and first two months of 2011.

The number of elements in the mesh has an influence on the number of degrees of freedom considered in the FE model. Using a mesh with a smaller element size is equivalent to increasing the number of elements and nodes of the model and consequently in the number of degrees of freedom of the problem. This number of degrees of freedom has also increased exponentially in recent years, as shown in Fig. 2.

Fig. 2. Number of degrees of freedom of FE models in different works from the literature.

Most of the studies about endodontic restorations using FEA have dealt with incisor teeth, followed by premolars, molars and canines. In the majority of those works, the teeth were analysed under masticatory loads, although other load types, such as bruxism or accidental loads, were also simulated. The development of computers and simulation software has allowed more detailed models. Although an increasing number of studies consider all the components of the restored tooth in the model, other studies still simplify the model by eliminating some of the components. Sometimes the final crown is not included for the sake of simplification (Rodríguez-Cervantes et al., 2007), while in other cases the periodontal ligament (PDL), the luting cement, or both, are left out to simplify the model (Adanir & Belli, 2007; Asmussen et al., 2005; Hsu et al., 2009; Yaman et al., 2004; Zarone et al., 2006). To date, most of the FE models have been linear and static. In linear static models, the stress-

strain curve representing the material behaviour has a constant slope and the materials undergo deformation below their proportional limit. However, in some realistic situations this analysis is not accurate. Wakabayashi et al. (Wakabayashi et al., 2008) cited three main causes of non-linearity: (1) material non-linearities, like the response of the PDL, (2) the changing interrelation of objects, as when loads introduce contacts between components, and (3) geometric nonlinearities, as a consequence of large deformations in some components. In recent years, some works have introduced non-linear FE models as a better way to represent the response of endodontic restorations (Sorrentino et al., 2009; Uddanwadiker et al., 2007). On the other hand, masticatory loads are variable, implying that the restoration is subject to fatigue, but very few fatigue simulations are found in the literature up until now (Huysmans & Van der Varst, 1993). The authors have recently presented a study on fatigue simulation of a whole endodontic restoration (Sancho-Bru et al., 2009).

## **3. Biomechanical model definition**

The validity of the conclusions reached from biomechanical models of the endodontic restoration will depend on the accuracy with which these models can represent the real system. A number of different factors related to the definition of the model conditions the quality of the results obtained from FEA, namely the accuracy of the geometric representation of the system, the components included in the model, the quality of the discretisation and the constitutive equations for finite elements, the material properties, or the boundary conditions. Moreover, processing and interpreting the results is a challenging task that can compromise accuracy of the conclusions from FEA. All these questions are reviewed in this section.

### **3.1 Geometric model**

136 Theoretical Biomechanics

Boschian Pest et al., 2006; Ferrari et al., 2008; Genovese et al., 2005; Gonzalez-Lluch et al.,

The number of publications dealing with FE simulation in endodontics has increased exponentially in recent years. For example, in 2011 a manual search in the SCOPUS database in 2011 for papers containing simultaneously the terms *finite element* and *endodontic* within journals on the subjects of Engineering and Medicine and Dentistry found 14 papers until 2000, 31 papers from 2000 to 2009 and 11 papers in the short period including 2010 and first

The number of elements in the mesh has an influence on the number of degrees of freedom considered in the FE model. Using a mesh with a smaller element size is equivalent to increasing the number of elements and nodes of the model and consequently in the number of degrees of freedom of the problem. This number of degrees of freedom has also increased

> 1990 1995 2000 2005 2010 2015 Year

Fig. 2. Number of degrees of freedom of FE models in different works from the literature. Most of the studies about endodontic restorations using FEA have dealt with incisor teeth, followed by premolars, molars and canines. In the majority of those works, the teeth were analysed under masticatory loads, although other load types, such as bruxism or accidental loads, were also simulated. The development of computers and simulation software has allowed more detailed models. Although an increasing number of studies consider all the components of the restored tooth in the model, other studies still simplify the model by eliminating some of the components. Sometimes the final crown is not included for the sake of simplification (Rodríguez-Cervantes et al., 2007), while in other cases the periodontal ligament (PDL), the luting cement, or both, are left out to simplify the model (Adanir & Belli, 2007; Asmussen et al., 2005; Hsu et al., 2009; Yaman et al., 2004; Zarone et al., 2006). To date, most of the FE models have been linear and static. In linear static models, the stressstrain curve representing the material behaviour has a constant slope and the materials undergo deformation below their proportional limit. However, in some realistic situations this analysis is not accurate. Wakabayashi et al. (Wakabayashi et al., 2008) cited three main causes of non-linearity: (1) material non-linearities, like the response of the PDL, (2) the changing interrelation of objects, as when loads introduce contacts between components, and (3) geometric nonlinearities, as a consequence of large deformations in some components. In recent years, some works have introduced non-linear FE models as a better way to represent the response of endodontic restorations (Sorrentino et al., 2009; Uddanwadiker et al., 2007). On

2009b; Maceri et al., 2009; Rodríguez-Cervantes et al., 2007; Sorrentino et al., 2007).

two months of 2011.

exponentially in recent years, as shown in Fig. 2.

1.E+02

1.E+03

1.E+04
