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Finite Element Analysis Application in Dentistry: Biomechanical Performance of Fixed Prosthodontics***

	Abstract 1. Introduction 2. FEA concept 3. Pre Processing 4. Modeling 5. Meshing 6.Processing 7.Discussion 8.Conclusion 9. References
Finite Element Analysis Application in Dentistry:
Biomechanical Performance of Fixed Prosthodontics


Abeer Aljareh ⁽¹⁾ Nabil Alhouri ⁽²⁾
Mohanad Kadhim ⁽³⁾Ayham Darwich^(4,5)
⁽¹⁾Postgraduate Student at The Department of Fixed Prosthodontics, Faculty of Dentistry, Damascus University, Syria
⁽²⁾Professor at The Department of Fixed Prosthodontics, Faculty of Dentistry, Damascus University, Syria
⁽³⁾Professor at The Department of periodontology, Faculty of Dentistry, Karbala University, Iraq
⁽⁴⁾Professor at Faculty of Biomedical Engineering, Al-Andalus University for Medical Sciences, Tartous, Syria
⁽⁵⁾Professor at Faculty of Technical Engineering, University of Tartous, Tartous, Syria
abeer12.jareh@damascusuniversity.edu.sy


Abstract
Background of the study: Finite element analysis (FEA) is an effective research tool that has been used widely in dentistry. Subsequent computational procedures are carried out to complete any FE analysis. FE method offers numerous advantages over in vitro studies since it can determine stress and strain in studied structures. Biomechanical behavior of different structures and materials can be also appraised by FEA studies. Many dental researches in the field of fixed prosthodontics have been accomplished using FE method.
Aim of the study: The present paper aims to clarify FEA concept and its steps. It also reviews FEA applications in dentistry, particularly in fixed prosthodontics.
Keywords: finite element analysis, stress, dentistry, fixed prosthodontics, biomechanical behavior.


Introduction:
There has been increasing interaction between different fields of science. For instance, the interaction between engineering and the medical fields which has been applied successfully in dentistry.1,2 The oral cavity has a unique, complicated environment that in vitro studies might not be able to accurately replicate.1,2 Furthermore, stresses are difficult to be studied directly. In other words, stresses and deformation cannot be studied clinically in anatomic structures and dental restorations.1 Numerous methods have been used to study stresses in dental structures. For example, brittle coatings analysis, strain gauges, photoelasticity, and finally finite element analysis.3
Finite element analysis (FEA) or method (FEM) is a numerical method to study stresses and strains in complex structures, including dental structures.3 Finite element analysis was first introduced in the airplane industry in 1956. It had been applied in space engineering only for many years before it entered the dental field in the 1970s when Thresher and Saito used FE analysis to assess stress in human teeth instead of using photoelasticity tests.4,5
Biomechanical behavior of anatomic structures and dental restorations has been a topic of interest since choosing the best material and the most appropriate design of preparation are still real challenges.4,6,7 FEA could help researchers to analyze stresses in such complex structures.8-11 Moreover, it could help manufacturers and dentists to promote appropriate designs of dental restorations from materials with high mechanical properties. It might also improve the strength of the tooth-restoration complex as it can predict the pattern of stress distribution inside this complex.4 Numerous studies have been implemented using FEA in almost all dental aspects.6,7,11-15 Therefore, this article aims at introducing the finite element analysis concept and reviewing FEA application in dentistry, particularly in fixed prosthodontics.


FEA Concept:
One of three methods for resolving engineering problems within complex materials and structures is finite element analysis (FEA).16 These methods are numerical, analytical, and experimental. FEA, which is a numerical method, contains multiple computational procedures to determine stress and strain in each element in the structure under study.4 Any structure contains an infinite number of points which makes it impossible to solve problems in this structure. The examined structure, also known as the problem domain, is divided into finite points, which are referred to as elements, in order to solve the problem. These elements are combined together by nodes. Each node connects to another node by elements. The elements and nodes form a complex system called mesh, which is essential to perform the FEA as it is programmed to include the structure properties.4,17 Thus, mathematical equations are calculated on a finite number of points and the answers are then inserted for the entire domain to solve various engineering problems in many kinds of complex geometries under varied circumstances.4,17

Depending on the models, there are two types of analysis; 2D (two-dimensional) and 3D (3-dimensional) modeling. The 2D modeling, by which FEA had started, is simple but its results might not be accurate. Also, it uses triangular or quadrilateral elements. Most of the FEA studies today utilize 3D modeling, which is more accurate and can reflect actual geometry. 3D modeling also uses quadrilateral or polyhedral elements with up to 14 faces.4
Finite element analysis consists of three main steps; pre-processing, processing, and post-processing.1,4

Pre-Processing:
The first step of any FEA study contains three sub-steps; modeling, meshing, and The last sub-step is the determination of materials properties, boundary conditions, and loading conditions.1

Modeling:
The geometry, which may be 2D or 3D, is acquired in this step. While accurate 2D models might only provide an exact solution for some aspects of the problem, accurate 3D models enable researchers to solve more complex problems with great deal precision.18 Reproducing a 3D model of a complex geometry, particularly anatomic structures such as teeth and periodontal tissues, may be the most sophisticated and time-consuming step in an FEA study.19
Dental models have been created using many methods and software systems. For example, Tribst et al., 2021 and Souza et al., 2015 used Rhinoceros software,20,21 while Houdaifa et al., 2022 and Neto et al., 2021 utilized SolidWorks to build the dental structures as well as dental restorations and prosthodontics.15,22 Using a laser scanner, other researchers first scanned the tooth before preparation and then did so again after preparation.23,24 More information about anatomic structures has been gained recently thanks to the advance of medical imaging technology,.19 The most trustworthy source for dental models today is medical images, such as computed tomographic images CT,19,25,26 and cone-beam computed tomographic images CBCT.6,7 CT and CBCT images provide DICOM files (Digital Imaging and Communications in Medicine) which can be modified by interactive software for medical images.6,7,19,25,26 3D models obtained by real teeth scanning or imaging, referred to as reverse engineering technique, are more acceptable since they help researches to reproduce dental geometry with more appropriate shape and much more accurate dimensions.25
Whatever the method to obtain investigated structures was, they should be represented and simplified by a number of elements, lines, or splines rather than their intricate geometry.17 At this step, the size, number, and shape of the elements are decided. The desired accuracy of the analysis and the available efficiency of the computer are also taken into consideration while choosing these details. Moreover, some geometrical details can be ignored while others still need to be represented despite being complex.17 This stage may contain errors as the operator might oversimplify the model. Consequently, the results would not be exact.19 Although this step plays an essential role in the entire process, it depends on the researcher's skills and their knowledge.17

Meshing:
Structures should be divided into a finite number of elements and nodes, as was previously described. As a result, these structures could be studied and used to calculate predictable results within them. By meshing, The calculation would be performed on each node to find the solution of the whole structure. It is thought that increasing the number of elements and nodes leads to more precise results with fewer errors.4 However, a convergence test could be implemented to estimate the required refinement of the mesh, in which increasing the number of elements and nodes does not affect the results.20
Even though triangular, quadrilateral, or polyhedral elements are typically employed in medical and dental studies, more specified elements may be used in engineering researches like mass, spring, and damper elements.4 The majority of programs now enable automatic or semi-automatic meshing which might make this important procedure less time-consuming.4

Material Properties:
The mechanical properties of any structure or material, which control how it responds to different circumstances, are entered into the FEA software and programmed in the mesh.4
Poisson's ratio and modulus of elasticity must be determined for solid structures. They are used in almost all FEA dental studies.24-27 However, more properties might be required according to the aims of the analysis and the failure theories used in it. For instance, thermal parameters including the coefficient of thermal expansion and thermal conductivity should be identified when the influence of thermal change is studied.27 Ultimate tensile and compressive strengths are required when modified von Mises theory is used.24 They are also needed when calculating Mohr–Coulomb ratio.7,28 Furthermore, S-N curves, which represent alternating stress versus the number of cycles, should be set in the FEA software to investigate the fatigue behavior of studied materials and analyze stresses inside them under cyclic loading.11,29
Although the determination of material properties is a main step, it may be difficult to find the required properties of all studied structures.17 S-N curves, for example, are not available for all dental restorative materials and some tissues.29



Boundary & Loading Conditions:
To get appropriate results, it is crucial to exactly determine the support structures and loading conditions according to the problem.17 The fixed or support area should represent the actual geometry or anatomy.11,13 In order to prevent any interference with the results, it should be defined as the farthest surface from the location of loading. 15 Many types of loading, ranging from normal forces to parafunction forces, could be applied in various magnitudes and directions.6,15,24,27

Processing:
It is also called the solution step. It is completely done by computer software which calculates all the equations and performs the matrix formulations to give the final solution.4

Post-Processing:
The results are shown numerically and by visual maps. Then, the researcher should think of ways to improve the design if necessary. Verification is also involved in this step to make sure that the results are accurate and reliable for the problem.4

FEA Studies in Fixed Prosthodontics:
Stress in an upper incisor that had been restored with different prefabricated and custom-made post and core systems was the subject of a study by Kharboutly et al. in 2023. The posts were from gold, nickel-chrome, zirconia, and glass fiber whereas crowns were from zirconia and lithium disilicate ceramic. von Mises stress was analyzed after an oblique loading of 100 N was applied. The findings indicated that great stress concentration at the buccal side of all posts, notably in the middle third of the post, while the glass fiber post extended the stresses to the apical third of it. The last post demonstrated a homogeneous pattern of stress distribution in the tooth-restoration complex but it also increased the values of stress in dentin (16.71 to 16.74 MPa). Furthermore, zirconia and metal posts concentrated more stresses within them when compared to other posts (70.66 to 71.29 MPa and 68.92- 69.19 MPa, respectively). The highest stress was observed at the cervical and middle areas of the roots whatever the material of the crown and post was. Zirconia crown models recorded higher values of stress (21.79 to 36.51 MPa in crown, 14.04 to 71.29 MPa in posts and 15.13- 15.80 MPa in dentin) than lithium disilicate models (18.19 to 28.62 MPa in crown, 37.18 to 70.66 MPa in post, and 14.99 to 15.81 MPa in dentin). The authors found that crown material did not affect stress distribution in radicular dentin. Additionally, stress values decreased when the elastic modulus of the post increased. Furthermore, the behavior of zirconia posts was similar to custom-made metal posts which might make zirconia post and core a suitable choice for esthetic purposes.12
de Moura Martins et al., 2021 studied the effect of crowns made of two different materials with two occlusal thicknesses on stress distribution in molars. A nano resin ceramic material (Lava Ultimate) and lithium disilicate ceramic (e.max CAD) with an occlusal thickness of 1 mm or 2 mm were compared. The analysis was carried out when a 200 N was applied in two directions; vertical and oblique. They found that the lower the occlusal thickness, the higher the stress concentrated inside the crown. Additionally, 2 mm occlusal thickness of the crown reduced the stress transmission to the dental tissues. Furthermore, the nano resin ceramic caused higher stress concentration in the dental structures than the other studied material whatever its thickness and the loading direction were.30
Full crown and endocrown made of zirconia-reinforced glass ceramic or hybrid ceramic with different ferrules of 1 mm or 2 mm were examined for stress distribution in Tribst et al., 2021 investigation. Axial and oblique loadings of 300 N were applied on a first maxillary molar, and maximum principal stress was analyzed. Cohesive failure and adhesive failure in the cement layer were also calculated. Stresses were lower in endocrown (6.75 to 13.94 MPa) than in full crown (13.05 to 24.24 MPa) under axial loading whatever the material and the ferrule were. Stresses in the tooth were also lower in endocrown models when compared to full crown models (1.31 MPa and 5.09 MPa, respectively). Under oblique loading, the highest tensile stress was in glass ceramic endocrown with 1 mm ferrule (20.44 MPa), while the lowest values were in the glass ceramic full crown with 2 mm ferrule (8.82 MPa) and hybrid ceramic full crowns whatever the ferrule was (8.67 to 8.98 MPa). The adhesive failure risk of the cement layer was lower than the cohesive failure risk regardless of the restoration and its material. Moreover, the oblique loading increased the failure risk and the peak of stresses. Hybrid ceramic endocrown with 1 mm ferrule showed the greatest failure risk of the cement layer (0.34 MPa and 0.55 MPa under axial and oblique loadings, respectively), whereas glass ceramic full crown with 2 mm ferrule recorded the lowest failure risk (0.16 MPa and 0.31 MPa under axial and oblique loadings, respectively). The authors concluded that full crowns show lower tensile stress under oblique loading but endocrowns show less stress under axial loading. They also found that using hybrid ceramic may reduce stresses in the restoration.20
Zheng et al., 2021 investigated the biomechanical behavior of endocrown restorations made from several CAD-CAM materials; lithium disilicate ceramic, zirconia-reinforced lithium silicate ceramic, polymer infiltrated ceramic (PICN), and composite resin. The analysis was performed on a mandibular molar when axial and oblique loadings of 200 N were applied. Von Mises and maximum principal stress criteria were used to calculate stresses. Von Mises stress concentrated in the loading area of the endocrown. PICN and composite resin endocrowns increased von Mises stresses in enamel but they reduced stress in the pulp chamber. The greatest value of maximum principal stress in both enamel and endocrown was seen in the composite resin model (6.73 MPa and 45.10 MPa, respectively) while values of stress in dentin were similar in all models. The zirconia-reinforced lithium silicate ceramic model had the lowest values of maximum principal stress (4.53 MPa in enamel and 33.87 MPa in endocrown). According to the findings of this study, the composite resin endocrown could provide the best biomechanical behavior and stress distribution in the tooth-endocrown complex.31
Jafari et al., 2021 studied stress distribution in maxillary incisors restored with different posts. The tooth was restored with a monolithic zirconia crown and a post from Ni-Cr casting, glass fiber, titanium, or zirconia. The ferrule height was either 0 mm or 2 mm, with a 1 mm rounded shoulder finish line. Von Mises stresses were evaluated under a static loading of 100 N. The force was applied with an angle of 135° to the long axis of the tooth. Whatever the material of the post, the highest stresses were concentrated in the middle third of the post. More stresses were also seen in the coronal third of casting, titanium, and zirconia posts. The cervical region of dentin experienced the most stresses in the models with no ferrule. The authors found that although the pattern of stress distribution was more homogenous in the fiber post model than other models, it revealed greater stresses in the cervical area of dentin, especially when ferrule height was 0 mm.32
The biomechanical behavior of endodontically treated molars restored with endocrowns from different materials was evaluated in FEA and in vitro study by Dartora et al. in 2021. The examined materials of the endocrown were leucite-based glass ceramic, lithium disilicate-based glass ceramic, glass ceramic based on zirconia-reinforced lithium silicate, and monolithic zirconia. A static axial loading of 200 N was applied at 3 points on the occlusal surface of the endocrown. The stresses in all models were assessed using von Mises criterion. Color maps showed higher stresses in the loading points and the linear angle between the pulp chamber and axial walls. Additionally, stress values fell within the same range in all models (631 MPa in lithium disilicate and zirconia-reinforced ceramic models and 626 MPa, 636 MPa in zirconia and leucite endocrown models, respectively). According to the experimental and numerical findings, the authors concluded that monolithic zirconia might be better than other studied materials in terms of biomechanical performance despite its higher rate of catastrophic failure.33
The influence of preparation depth and design on stress distribution in incisors restored with three ceramic veneers was appraised by Tsouknidas et al., 2020. Minimal preparation was butt joint and feather-edge with two depths; thin and thick. Thin preparation was represented by 0.3 mm preparation in the cervical and middle thirds and 0.4 mm in the incisal third while deep preparation was 0.4 mm in the cervical third and 0.9 mm in the middle and incisal thirds. Veneers were from three systems; feldspathic porcelain, heat-pressed IPS Empress, and heat-pressed IPS e.max Press. Analysis was done according to von Mises stresses under oblique loading of 200 N. The findings showed that the greatest stresses were at the cervical margins of the veneers. Regardless of the preparation design, feldspathic veneers models were associated with the highest values of stresses in the dental structures (352.17 MPa and 238.92 MPa in enamel and dentin, respectively). The greatest values of stress in veneer were seen in IPS Empress models (186.51 MPa). Minimal preparation had higher stresses in both dentin and veneers than deep preparation. Besides, the feather-edge design caused greater stresses in veneers and dental tissues compared to butt joint models. It is found that increasing the depth of preparation and using lithium disilicate ceramic might be preferable in terms of stress distribution in veneers and dental structures.34
Tribst et al., 2018 studied the effect of remaining coronal tissues (1.5, 3, or 4.5 mm) on stress distribution in a maxillary molar restored with lithium disilicate and leucite ceramic endocrown. Utilizing non-linear FEA, a 300 N was applied axially and maximum principal stress was calculated. The results revealed that the maximum values of stress in endocrown (15.8 MPa to 19.6 MPa) were lower than the fracture resistance of studied materials. Additionally, leucite ceramic endocrown showed a better pattern of stress distribution than lithium disilicate ceramic. Moreover, the stresses in the cement layer decreased as the amount of remaining dental structures increased. This study came to the conclusion that the greater the dental remnant, the higher stresses concentrate in the restoration which could protect the dental structures since they must be always preserved.35
The effect of finish line design on stress distribution in bi-layered and monolithic zirconia crowns on premolars under static loading was examined by Miura et al., 2018. They used cylindrical models as crown abutments from dentin and brass with three different types of the finish line; shoulder, rounded shoulder, and deep chamfer. Maximum principal stress was calculated under 1 N loading which was applied perpendicularly to the occlusal surface. They found that the monolithic zirconia crown with rounded shoulder and deep chamfer margins showed the lowest values of maximum stresses as they recorded 0.028 MPa and 0.031 MPa in dentin abutments, respectively. Therefore, rounded shoulder and deep chamfer margins could lower the risk of ceramic fracture in clinical practice.36
Gulec & Ulusoy, 2017 evaluated stress distribution in maxillary premolars with two designs of endocrown. The endocrown was with an extension to the pulp chamber only or with 3 mm intraradicular extensions. Three CAD/CAM materials were studied; felspathic ceramic, polymer infiltrated ceramic, and nanoceramic resin. Following the application of a 100 N loading, von Mises and maximum principal stress criteria were used to assess the stresses. The highest values of von Mises stress in enamel were seen in the PICN endocrown (24.54 MPa), whereas the lowest values were observed in the feldspathic endocrown with extension (9.66 MPa). Furthermore, the greatest maximum principal stress values in enamel were in resin endocrown (15.88 MPa) while the lowest values in enamel were seen in feldspathic endocrown with extension (5.78 MPa). In contrast, the feldspathic endocrown model showed the highest values of stresses in dentin while the least stresses in dentin were seen in resin endocrown models. This study's conclusion found that the endocrown with intraradicular extensions could protect teeth more effectively than the conventional endocrown. Polymer-infiltrated ceramic and CAD/CAM feldspathic ceramic could be suitable for such restorations.9
Tripathi et al, 2014 investigated the influence of preparation taper, height, and marginal design on stress in the luting cement. A CT scan of upper second premolar and molar was taken to develop the 3D model. Teeth were prepared with a shoulder margin on the buccal side and chamfer margin on the palatal side with a taper of 10˚ or 30˚ and a height of 3 mm or 5 mm. Teeth were restored with porcelain fused to metal crown and glass ionomer cement was represented with a thickness of 24 µm. Maximum shear stress and von Mises stress were calculated when a 100 N loading was applied horizontally and axially on a point. A further distributed axial loading was also applied at many points. The maximum shear stresses ranged from 1.70 to 3.93 MPa under horizontal point loading, 0.66 to 3.04 MPa under vertical point loading, and from 0.38 to 0.87 MPa under distributed loading. The preparation with 5 mm height and 10˚ taper showed the lowest stresses, whereas the preparation with the taper of 30˚ and the same height had the highest stresses. Distributed axial loading was associated with the most favorable stress distribution and the lowest stresses. The authors concluded that a smaller preparation taper is better in terms of stress concentration especially when the height of preparation increased. Moreover, because of the high stress concentration in chamfer margins, the cement margins might experience microfracture when these margins are used.37
In 2014, Oyar et al. carried out a 2-deminional FEA to appraise the influence of crown material and preparation on stress distribution. In order to examine the effect of two crown materials (In-Ceram, Empress Esthetic), and two occlusal preparations (anatomic, nonanatomic), a 200 N loading was applied on a second mandibular molar. The lowest stress was recorded in the core of the Empress Esthetic crown with nonanatomic occlusal reduction (40 to 80 MPa in the Empress Esthetic model and 80 to 140 MPa in the In-Ceram model). Stress was also seen in the lingual area of dentin in all models (23 MPa to 36 MPa). This investigation found that preparation design did not affect the pattern of stress distribution and the values of stresses. Moreover, the ceramic material with high elastic modulus increased the values of stresses within it and decreased stresses in the occlusal surface of dentin. It also concluded that nonanatomic reduction could be favorable for Empress Esthetic crowns.38

Discussion:
Finite element method, which is based on mathematical modeling, is an effective tool in dental studies. It has been used widely to understand how various designs and materials of dental restorations may affect stress distribution in the tooth-restoration complex.9 FEA has many benefits over mechanical methods.3 This method reduces the experimental requirements. It is also less time-consuming and more cost-effective than in vitro studies. It does not need human materials although this might be less realistic.4,19 Besides, it determines stress distribution in biological tissues.4 The same models could be used to analyze numerous variables to investigate all potential solutions of the studied problem under static or cyclic conditions.4,39 Using this method, innovative ideas can be studied and evaluated before implementing them in clinical trials.1 For both simple and complex geometries, the FE method offers quick, effective, and accurate solutions.39 Nonetheless, the accuracy of results depends on the operator's skills and knowledge. For instance, inserting materials properties and determining loading conditions can be changed dramatically according to the accuracy and expertise of the operator.39 Thus, a well-trained operator is required.1 Furthermore the type, shape, position, and number of elements as well as the type of the mesh all affect how accurate the solution is. The operator may have an impact on the results explanation.39 Also, it might be hard to predict the type of failure in complex models which consist of various materials with complex geometries.4 Unfortunately, FEA cannot stimulate the exact biologic condition such as the dynamics of teeth and periodontal tissues.40 It cannot also replace mechanical and experimental methods completely.4


Conclusion:
Finite element analysis offers several advantages for researchers and it can be used effectually in the fields of dentistry such as fixed prosthodontics. However, precise care should be taken at every step from the early stage of obtaining the models until reading the results and explaining them objectively.


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Finite Element Analysis Application in Dentistry: Biomechanical Performance of Fixed Prosthodontics***