In restored teeth, stresses at the tooth–restoration interface during masticatory processes may fracture the teeth or the restoration and cracks may grow and propagate. The aim was to apply numerical methodologies to simulate the behavior of a restored tooth and to evaluate fatigue lifetimes before crack failure.
Materials and methods
Using a CAD–FEM procedure and fatigue mechanic laws, the fatigue damage of a restored molar was numerically estimated. Tessellated surfaces of enamel and dentin were extracted by applying segmentation and classification algorithms, to sets of 2D image data. A user-friendly GUI, which enables selection and visualization of 3D tessellated surfaces, was developed in a MatLab ® environment. The tooth-boundary surfaces of enamel and dentin were then created by sweeping operations through cross-sections. A class II MOD cavity preparation was then added into the 3D model and tetrahedral mesh elements were generated. Fatigue simulation was performed by combining a preliminary static FEA simulation with classical fatigue mechanical laws.
Regions with the shortest fatigue-life were located around the fillets of the class II MOD cavity, where the static stress was highest.
The described method can be successfully adopted to generate detailed 3D-FE models of molar teeth, with different cavities and restorative materials. This method could be quickly implemented for other dental or biomechanical applications.
Resin-composite materials have been widely and increasingly used today in adhesive dental restorative procedures, particularly in posterior teeth . An important advantage over metallic filling materials is the well-known possibility of bonding the restoration to dental tissues by modern adhesives . Significant disadvantages of many of these materials are still the polymerization shrinkage and the disparity in mechanical characteristics with the hard tooth structures, enamel, and dentin . During polymerization, shrinkage stresses affect not only the adhesive interface but also the whole adhesively bonded restoration and the residual tooth structure. Also the mechanical mismatch leads to enhanced interfacial stress concentrations during mastication . Occlusal loading of adhesively restored class II MOD restorations does not invariably lead to equal fracture resistance values or to comparable fracture pattern . Furthermore, repeated functional loading will determine the fatigue response of the restored tooth and may finally result in failure . It appears crucial to understand the physical phenomena occurring in adhesive resin-composite posterior restorations to help prevent failure. Crack growth and propagation inside the restored teeth under occlusal fatigue loading have been already investigated in natural dental samples . Fracture patterns, anyway, in cohesive or adhesive failure are not always similar and it is still difficult to understand their causes and how to investigate where and when they happen. Modern CAD–FEM (Computer Aided Design and Finite Element Method) methodologies play an essential role in biomedical investigations of clinical and therapeutical situations in different dental fields . In fact, some dental and medical research may be expensive and ethically questionable when conducted on live subjects. The use of virtual models and simulation, on the contrary, can contribute to better performance of an investigation, reducing costs of in vitro and in vivo experiments and improving benefits. Thus, in adhesive and restorative dentistry there has arisen strong interest in CAD–FEM with studies on different loading conditions at the tooth–restoration interface , on transfer of occlusal forces through molars , on tooth deformation , and on residual shrinkage stresses within dental composite cavities .
2D and 3D Finite Element Analysis have been successfully introduced in the past for dental applications . In FEA the 3D tooth geometry is approximated by many small simple-shaped elements and deformation in term of stress–strain is evaluated for the single elements and finally calculated for the whole structure. Nevertheless, teeth are not simple geometric representations and external and internal data acquisition has often been obtained by different scanning procedures with the goal of representing real anatomical shapes and layered structures. Today, micro XMT firstly and micro-CT later are a more modern approach to achieve a detailed, realistic 3D finite element model in medicine and in dentistry . Over the years, several dental applications have been carried out in determining the responses of the restored tooth to different influences and stresses coming from outside, such as occlusal loading and tearing forces. Many authors have addressed their attention on fracture mechanism occurring into restored teeth. All these works are related to “in vivo” or “in vitro” experimental analyses. However, very few works focused on numerical evaluation (for example, through CAD–FEM approaches) of fatigue and fracture behavior of restored teeth.
The aim of this work was to create a 3D model of a restored molar tooth and then to generate a FE model, by applying linear FEA analysis and fatigue mechanical laws, to numerically estimate the lifecycle before significant fatigue damage of the restored tooth occurred.
The paper describes a general approach to numerical prediction of fatigue lifetime before crack failure in composite-restored teeth.
Materials and methods
Fig. 1 shows the general work-flow adopted to perform the fatigue numerical simulation of a restored human molar tooth.
Image data were processed in MatLab ® 2007a (Mathworks, University of Waterloo). Using image segmentation and classification procedures a 3D tessellated model of an un-treated tooth was created. Tessellated models (into ASCII .stl format) were then converted into surfaces by fitting cross-sections. This operation was performed into SolidWorks ® 2008sp3 CAD system (Dassault Systèmes, USA), where a class II MOD cavity was added. Finally, the FE model was generated and the fatigue simulation was performed in Comsol Multiphysics ® 3.5a (COMSOL AB, Sweden) environment. Here, the “Live-Link” import module was used to directly import CAD geometry from SolidWorks ® .
CAD model generation and cavity preparation
An un-treated human posterior molar was digitized with a high resolution micro-CT scanner system (1072, SkyScan, Belgium). 471 slices were made with a voxel dimension of 19.47 μm. The use of micro-CT images to generate FE models results from the facility of getting the outer contours very close to the original scanned structure. In the present research, the main aim is to identify the macro-structure of the tooth. Therefore, there is no need of slices which includes micro structural changes such as mineral density differences into the same tooth-tissue. Finally, based on these explanations, only 91 slices were accounted in the following.
Iso-surfaces were detected by using a clustering algorithm: the “ K -Means” clustering method was here implemented . This method operates by grouping image pixels, defined with their gray scale, into K groups/clusters. A constant pixel value (the centroid of the cluster) is associated with each cluster. By using this classification, for the i th slice, logical matrices (pixel mask) of pixels are introduced: the value of the t th pixel is equal to “1” if it belongs to the k th cluster, otherwise it is assumed equal to “0”. In the present application, the number of clusters was two: enamel and dentin regions. Once clustering classification had been performed for all image slices, 3D tessellated surfaces were created by using the marching-cube algorithm . This numerical procedure allows to connect pixels having the same value on different slices. The resulting model was a tessellated surface made of triangles connecting 3D points. The “isosurface” function available in MatLab ® was used for this purpose.
Fig. 2 shows tessellated surfaces both of enamel and dentin.
Segmentation and classification algorithms were implemented within a user-friendly MatLab ® Graphical User Interface (GUI), which prompted the user: (i) to load images (in .bmp or .jpg formats), (ii) to apply automatic filters and segmenting algorithms, and (iii) to visualize 3D models and to export tessellated surfaces into standard .stl format. Furthermore, the Region of Interest (ROI) could be selected for processing and then 3D visualization. Fig. 3 shows the MatLab ® ’s GUI interface.
Tessellated models were then imported into SolidWorks ® CAD environment, where the “SCANto3D” add-in module was used to process and manage imported tessellated surfaces. Starting from cross-sections (created as intersection curves between cut planes and the same tessellated surface), sweeping surfaces were generated.
At this stage, the tooth model consisted of three tessellated surfaces: the external enamel surface ( Fig. 4 (a) ), the interfacial surface between enamel and dentin ( Fig. 4 (b)) and the interfacial surface between dentin and the pulp cavity ( Fig. 4 (c)).
Once enamel, dentin and pulp 3D volumes were generated, Boolean operations were used to ensure congruence between the related interfacial surfaces. For instance, the dentin volume was created by subtracting the pulp cavity volume. Then, the enamel 3D volume was obtained by subtracting the external enamel and the dentin volumes.
Given the 3D CAD model of the un-restored tooth, a class II MOD cavity preparation ( Fig. 4 (d)) was introduced into the model. The shape and the dimensions of the MOD cavity were taken from the literature .
The CAD model was then imported in the Comsol Multiphysics ® environment by using the live connection available with SolidWorks ® .
FE model and simulation
In order to perform a comparative analysis in terms of fatigue lifetime, a class II MOD restored tooth (MOD model – see Fig. 5 (a) ) and the class II MOD cavity preparation (CAV model – see Fig. 5 (b)) was investigated.
Generally speaking, when a fatigue failure occurs, the process may be divided into three steps. After a large number of load cycles, damage is accumulated on a micromechanical scale and a crack is formed. With further load cycles the crack grows. Finally, when the crack reaches a critical dimension, the component catastrophically fails . The crack growth and propagation are studied in fracture mechanics. Fatigue analysis, however, focuses on damage accumulation and allows a prediction to be made concerning where the crack will grow and propagate.
In the present paper, the fatigue simulation was performed in two consecutive steps: firstly, the occlusal load was statically applied. The results of the first simulation were then post-processed by using linear fatigue mechanic laws .
The static FEM model was based on the following hypotheses :
enamel and dentin were assumed to exhibit linear and isotropic behavior,
an indirect class II MOD preparation was assumed: one part of the “adhesive area” was modeled to be the “adhesive layer”, contacting the dental walls. The other part was modeled to be the “cement”, contacting from one side this adhesive layer and from the other side the filling material, and
thickness of the adhesive area was assumed constant (see also Table 1 ).Table 1
Young’s modulus (GPa) Poisson’s ratio Thickness (μm) σ f (MPa) b Enamel 48.0 0.30 – 310.0 −0.111 Dentin 18.0 0.23 – 247.0 −0.111 Adhesive layer 4.5 0.30 10 – – Cement layer 6.0 0.30 70 – – Composite filling material 25.0 0.24 – – –
For the MOD restoration, a resin-composite with Young’s Modulus equal to 25.0 GPa was assumed, together with an adhesive of 4.5 GPa modulus, applied on the cavity walls, and a resin cement composite with 6.0 GPa modulus as cement layer (see Table 1 ).
The 3D domains (enamel, dentin and filling material) were meshed with linear tetrahedral elements. Shell elements, instead, with membrane and bending behavior, were used to model the adhesive area (cement layer + adhesive layer). In order to reach more accuracy where necessary, mesh size was strongly reduced (mean mesh size about 0.1 mm) on the fillets of the cavity. Fig. 5 (a) and (b) shows so-created mesh models of MOD preparation and CAV preparation, respectively. Moreover, identity pairs were introduced to assure the congruence at interfacial surfaces (enamel–dentin and filling material-sound tooth). For instance, identity pairs make the displacement fields equal among the interfacial boundary surfaces. All this allows good simulation of the physical correspondencies between enamel and dentin and the adhesive area with the filling material.
Fixed zero-displacements were assigned to the nodes belonging to the bottom surface of the dentin (see Fig. 5 (c)).
The occlusal load was set equal to 600 N. This load was applied as depicted into Fig. 5 (c): a 35° angle with respect to the longitudinal axis of the tooth and perpendicular to the palatal cusps. The adopted load condition corresponds to the experimental setup, suggested into .
Once the preliminary static analysis was performed, fatigue simulation was accomplished by using the Comsol Multiphysics ® function “lcfmultiaxlin”. This function allows to calculate number of cycles to fatigue failure for multi-axial fatigue loading based (i) on a linear elastic analysis and (ii) one-axial stress-life cycle (S–N) curves of adopted materials. Principal stresses from the linear-elastic analysis are then transformed to cycles to fatigue-failure using an approximate method developed by Hoffman–Seeger .
As proposed in , enamel and dentin have a “metal-like” stress-life cycle behavior, with an apparent fatigue limit. Experimental results have shown how the stress-life cycle (S–N) curve both for enamel and dentin may be mathematically expressed by the Basquin’s formulation, as stated into relationship (1) :