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Mechanobiological Analysis of Molar Teeth with Carious Lesions through the Finite Element Method

Mechanobiological Analysis of Molar Teeth with Carious Lesions through the Finite Element Method Hindawi Applied Bionics and Biomechanics Volume 2018, Article ID 1815830, 13 pages https://doi.org/10.1155/2018/1815830 Research Article Mechanobiological Analysis of Molar Teeth with Carious Lesions through the Finite Element Method R. A. Hernández-Vázquez , Betriz Romero-Ángeles, Guillermo Urriolagoitia-Sosa, Juan Alejandro Vázquez-Feijoo, Rodrigo Arturo Marquet-Rivera, and Guillermo Urriolagoitia-Calderón Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, Sección de Estudios de Posgrado e Investigación, Unidad Profesional Adolfo López Mateos “Zacatenco”, Avenida Instituto Politécnico Nacional s/n, Edificio 5, 2do. Piso, Col. Lindavista, C.P. 07320 Ciudad de Mexico, Mexico Correspondence should be addressed to R. A. Hernández-Vázquez; alyzia.hv@esimez.mx Received 18 May 2018; Revised 23 July 2018; Accepted 14 August 2018; Published 14 October 2018 Academic Editor: Jan Harm Koolstra Copyright © 2018 R. A. Hernández-Vázquez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The analysis of the distribution of stress in dental organs is a poorly studied area. That is why computational mechanobiological analysis at the tissue level using the finite element method is very useful to achieve a better understanding of the biomechanics and the behaviour of dental tissues in various pathologies. This knowledge will allow better diagnoses, customize treatment plans, and establish the basis for the development of better restoration materials. In the present work, through the use of high- fidelity biomodels, computational mechanobiological analyses were performed on four molar models affected with four different degrees of caries, which are subjected to masticatory forces. With the analyses performed, it is possible to observe that the masticatory forces that act on the enamel are not transmitted to the dentin and to the bone and periodontal ligament to protect the nerve, as it happens in a healthy dental organ. With the presence of decay, these forces are transmitted partly to the pulp. The reactions to the external loads on the dental organs depend on the advances of the carious lesion that they present, since the distribution of stresses is different in a healthy tooth. functional balance of the system, which in turn causes the 1. Introduction occlusion not only to be altered but also to be lost [4]. This Mechanobiology is an area of recent application in dentistry in turn causes alterations in its biomechanics. [1, 2]; this is dedicated to the analysis of stresses and The morphology of each dental organ is designed to per- deformations in tissues in living beings. Study tools used in form this function. The occlusal surface of the posterior teeth (molars and premolars) has cusps, depressions, grooves, and engineering are applied in the structures of living beings. The structures that make up the stomatognathic system have pits that coincide with those of the opposing tooth (Figure 1), a harmonious morphology with a high degree of specializa- as well as the incisal edges, cusps, and cingules in the case of tion. All the bony, neuromuscular, articular, periodontal, the anterior teeth (incisors and canines) [5]. Together with and occlusal components establish an equilibrium between the rest of the components of the system, they allow the teeth them that allow their complex physiology [3]. Even the finest to perform their function in a phenomenon called occlusion detail, such as a roughness, a groove, a cusp, or an orifice, [6], which is regulated by the masticatory forces acting on fulfills a specific function. In particular, when it comes to each of the teeth. the chewing function, the collusive components or dental Due to this, analysis of the distribution of stresses gener- organs are the protagonists. When the total or partial loss ated by occlusal or masticatory loads, in a biological system of any of these components occurs, there is a loss of the such as the teeth, is a complex problem [7], because of the 2 Applied Bionics and Biomechanics (a) (b) (c) Figure 1: Harmonious relationship of dental morphology. (a) Dental occlusion, (b) points where the upper teeth make contact with the lower teeth, according to their morphology, and (c) detail of occlusal assembly between premolars. (a) (b) Figure 2: Micrographs of dental enamel. (a) Distribution of enamel prisms and (b) hydroxyapatite prisms. nature of the tissues of the dental organ, such as nonhomoge- packed by a delicate network of organic material that neous materials and the geometric irregularities of their con- surrounds them (Figure 2). The prism is considered as the basic structural unit of the enamel, and the prismatic enamel tours and anatomical forms. In addition to this, the tooth in its structure is formed by enamel, dentine, and pulp, is the set of prisms that constitutes most of the mineralized whose mechanical properties differ from one another. The matrix [8]. distribution of stresses is also affected by health and patho- For its part, dentine is the one that provides support to logical states, which makes their analysis even more complex. enamel and transmits the forces it receives. Its microstructure The teeth concentrates the forces generated by the mus- is dominated by the presence of crystal structures in the cles in small areas such as contact surfaces and cusps or shape of an inverted cone called dentinal tubules. Between incisal edges. each tubule, there is a peritubular channel which contains Dental enamel is the tissue most exposed to high occlusal collagen I, hydroxyapatite crystals, and dentinal fluid. It forces, due to the processing of food during mastication in constitutes the organic or intertubular matrix [9]. the oral cavity and the occlusal contact between the teeth, Sometimes, nature uses processes like those that engi- for which it suffers attrition and fractures. It is a highly neering uses to improve a design; a clear example is what is specialized crystalline structure. The prisms that structure it known as residual stresses, which are used to reinforce are package of hydroxyapatite crystals ordered and densely materials, with a process called cold hardening. A similar Applied Bionics and Biomechanics 3 (a) (b) (c) (d) Figure 3: Stages of carious lesions. (a) First stages (enamel), (b) second stages (enamel and dentin), (c) third stages (affectation of the dental nerve), and (d) fourth stages (destruction of morphology). phenomenon occurs in the teeth. It has been found that the discomfort [14], being the second most common disease worldwide. It is a chronic multifactorial pathology [15] mineral particles of the dentin are precompressed, which prevents the propagation of cracks and increases the strength located in the hard tissues of the dental organ. It is character- of their biostructure. Dentin is a material similar to bone ized by the loss of the organic matrix due to the action of the formed from mineral nanoparticles, mainly hydroxyapatite. organic acid that is produced because of the bacterial metab- These mineral nanoparticles are embedded in collagen fibres, olism of the microbiota of the oral cavity. It takes place in a with which they are firmly interconnected. When the colla- progressively way, beginning with the demineralization of gen fibres contract, the adhered mineral particles get even the tissue at the ultrastructural level, until it reaches the total more compressed. This allows to prevent fissures, and the loss of the piece [16] (Figure 3). compression is carried out in such a way that the cracks This condition would cause a total imbalance of the cannot easily reach the internal part of the teeth, which could occlusal relationships, since the morphology of the dental damage the pulp [10]. In this case, the residual stresses are organs is altered, up to the loss of the dental organ, which represented by the masticatory loads, which propitiate this causes the total loss of the interocclusal harmony. In turn, it phenomenon that prevents the propagation of faults, with generates collateral damages such as systemic diseases and residual reactions throughout the stomatognathic System, the simple fact of chewing. A detailed information on the distribution of stresses and which is well worth analysing. Therefore, their mechanobio- location of the critical load points is of vital importance, since logical analysis proposes new guidelines for the actions that the dental organs have as their main function the adequate can be applied to counteract their affections and to be more transmission of forces to grind the food. Discrete analysis certain that the treatments and restorative materials act properly. Likewise, it will establish the foundations of the of stress distribution is essential in these organs and tissues, because the main objective is to understand how these efforts future of odontology of prevention. are distributed and transmitted and find stress concentra- In general, there are no publications that have numerical tions in specific places due to damage to the tooth as a data on the distribution of efforts as the caries progresses in function of its growth, adaptation, and structural modifica- its different degrees. Therefore, the present work represents a novelty in this subject and field of study. There are some tion [11]. This type of analysis would allow, for example, the control of dental regeneration. Recent studies have docu- works that raise the possibility of performing analyses of this mented that there is some regenerative capacity of dental pathology, but it is not the main objective of them; therefore, enamel (as when it is subjected to chemical compounds such they do not present numerical results in relation to the dis- as fluoride [12]). tribution of efforts when a carious lesion occurs. In addition, the models that were made in these studies are theoretical One of the main pathologies that appear in the teeth is dental caries. This disease, according to the World Health models of reality; they are not models with the characteris- Organization, is a condition and is estimated that five million tics such as the ones we are presenting. The design of the people worldwide suffer from it [13]. In global terms, between cavities that they propose represents the caries; they are 60% and 90% of school-age children and nearly 100% of adults made with cuts to the model, not how the caries is presented in reality. have dental caries, often accompanied by pain or feeling of 4 Applied Bionics and Biomechanics (a) (b) (c) (d) Figure 4: Biomodels with the four degrees of decay, based on (a) first grade, (b) second grade, (c) third grade, and (d) fourth grade. The biomodels of the present work are the results of For the modelling, the methodology described in a previ- ous work was used [17]. As already mentioned, the images of radiographic and tomographic images of cases of real caries at different levels, which comply with the natural history of the molars were obtained through the use of 3D imaging, by the disease. In addition, no real case is equal to another. They means of a digital volumetric tomography (TVD) of the max- were obtained with a new technique of imaging (cone beam). illa and mandible with the cone beam computed tomography This system is used to obtain images in difficult to visualize system (CBTC), to obtain the DICOM files. This system is tissues; it is widely used in medicine and dentistry in the widely used for studies of the maxillofacial area, since it craniofacial region. It provides images with a resolution of allows a better visualization of difficult access tissues. submillimetres of high diagnostic quality and excellent The tomography used is Batex brand model EZ3D, which visualization. In addition to this, the methodology used to has a KvP of 90.0 mA (3.8 light beam intensity). A total of 477 generate them ensures biofidelity. For these same reasons, images or slices were obtained from each tomography, whose the numerical results would be difficult to compare with distance between them (slide thickness) is 0.5 mm. The space other works, given the differences of the models used in between the pixels (pixel spacing) is of a ratio of 0.3/0.3 mm. other works. The obtained models have elements of high order (tetrahe- dral with a total of 10 nodes per element) and consider three different materials that correspond to the tissues that struc- 2. Materials and Methods ture the dental organ (enamel, dentine, and pulp). The FEM analyses were carried out with the ANSYS® This study is aimed at carrying out the analysis of four case computer program. The number of elements and nodes of studies that address the problem of finding the stresses that each biomodel is the following: biomodel of the control case occur in a dental organ when it is subjected to the forces of 74,907 elements and 129,005 nodes. First-degree caries have 76,851 elements and 132,043 nodes. Seventy-degree caries occlusion and its structural integrity is compromised, by the presence of carious lesions in its four degrees. Addition- have 77,419 elements and 133,497 nodes. Third-degree caries ally, there is a control case in a healthy molar without decay. have 80,317 elements and 137,666 nodes. Fourth-degree For each of the cases, a high-fidelity biomodel of a lower left caries have 81,318 elements and 141,830 nodes. first molar, representing each stage of caries, was used, The structural analysis has an elastic, linear, and homo- obtaining a total of five biomodels represented as follows: geneous behaviour. The mechanical properties of the tissues control case: without decay; case 1: caries in enamel on the are described in Table 1 [18–20]. occlusal face, in the main sulcus; case 2: caries in enamel The convergences in the generation of discretized meshes were analysed. Discretization was carried out in a semicon- and dentin on the occlusal face, starting in the main sulcus; case 3: caries in enamel and dentin and with pulp communi- trolled way, since the geometries did not allow for a fully cation in the interproximal zone; and case 4: caries of enamel controlled discretization, in the same way that an automatic on the occlusal face, in the main sulcus that extends over one was not suitable. We chose the size of the Maya that much of the dentin, causing involvement of the pulp cham- would allow the resolution of the analysis and that the differ- ences between the results would be negligible or consistent. ber and dental pulp (Figure 4). Applied Bionics and Biomechanics 5 Oclusal Table 1: Mechanical properties used in the analysis. pressure Enamel Dentin Pulp 3 3 3 Density 0.25 g/cm 0.31 g/cm 0.1 g/cm Young’s module 70 GPa 18.3 GPa 2 GPa Dimensional Poisson ratio 0.30 0.30 0.45 In relation to the modelling of carious lesions, the Movement restriction following should be mentioned. The axes in tissue ablations UX = UY = UZ = 0 of enamel and dentine (caries) are real, where caries is an irregular area, the edges are angular, and therefore the Rot X = Rot Y = Rot = 0 singularities due to the stress concentrators are real and consistent. These irregularities are observed in the images provided in the radiographs and tomographies, so that the Figure 5: Applied loads and boundary conditions. resulting biomodels are real reproductions of decayed molars. The boundary conditions were established in the root zone, restricting the displacements and rotations in the of the stresses. That is, it is representative of the distribution directions of the X, Y, and Z axes. A load was applied on of efforts that is the main objective of this work. In addition, the occlusal surface in the four biomodels. The magnitude von Mises encompasses the shear and normal energy, which of the applied load is 150 N/mm corresponding to the bite can occur in these cases. The shear stresses have the same force between the two molars [21–25], which is distributed distribution, and the magnitudes obtained are very similar, locally in the application zone on the form of a pressure. So and for the objectives of the work, they do not provide more that the pressure is the same in all cases, the area of action information. In future jobs where the fracture or failure is is different because of the different degrees of caries they considered, they could be considered. represent. The load is uniform for optimization of the simu- The results obtained are shown from Tables 2 to 7 and lation, and this does not affect the results or the conclusions, from Figures 6 to 20. since it is not a comparison between the cases, so it is not The von Mises stresses of the control case are observed necessary to adjust to the same load (Figure 5). from Figures 6 to 8. The contacts between the tissues were considered for the The von Mises stresses of the case 1: 1st degree caries are analyses performed. The software used has a function called observed from Figures 9 to 11. Contact Manager®, through which the contact simulation The von Mises stresses of the case 2: 2nd degree caries was carried out. This tool presents a subfunction called are observed from Figures 12 to 14. Contact Wizard® in which information was given to establish The von Mises stresses of the case 3: 3rd degree caries are which material or tissue is the contact and which will be the observed from Figures 15 to 17. target. For the simulation, the tissue was established as a The von Mises stresses of the case 4: 4th degree caries contact with the enamel and as a target for the dentin. To are observed from Figures 18 to 20. establish their contact relationship, a coefficient of friction was also added, which is according to the literature that is 4. Discussion established between ceramic materials, since the crystalline nature of these tissues, as well as bone tissue, has been With regard to the unitary strain, it can be found that, considered as ceramic materials [26]. although the same load is applied in all the study cases, the resulting stress is greater as the degree of caries increases. This would lead to the expectation that greater deformation 3. Results should occur as the degree of caries increases. However, the Unitary deformation, directional deformation, nominal less deformation that occurs, the greater the degree of caries. stresses, shear stresses, principal stresses, and von Mises stress That is, a stiffening is generated (to name it somehow) which were analysed during the application of the load that simu- is due to the interaction of the mechanical properties that are lates dental bite or occlusion. It should be mentioned that established between the three dental tissues. The rigidity is Von Mises stresses are not considered here as a failure crite- assumed by the least deformation that was obtained. At this rion (which is mainly applicable to ductile materials), but as point, it is prudent to clarify that no real case is equal to a unique nondirectional value that allows to have a global another; however, aspects such as the pressure exercised are criterion on the load at each tooth point, since it is obtained the same in all cases. from the deformation energy. Several authors use it as a The geometric conditions cause that the direct compari- criterion to evaluate restorations [27–29]. son between the different degrees of decay can give the idea However, in the body of work, only the efforts of von that it is due to these differences in the geometry, the motive Mises for tissue are shown. This is due to the fact that the of the results, and the nature of the unit deformations. How- main efforts are obviated with those of von Mises, since it is ever, the effect of stiffening in all four models is consistent. In the total deformation energy that indicates the magnitude addition, to represent what actually happens with the natural Z 6 Applied Bionics and Biomechanics controlled and designed case that is uniform, the pulp Table 2: Unitary strain. can come to present greater load (it is not possible to do Control 1st degree 2nd degree 3rd degree 4th degree this in a real case because it would have to be done in a case caries caries caries caries healthy molar and that is allowed to get sick with caries, −5 −12 −16 −16 −16 3.66 × 10 3.06 × 10 3.80 × 10 5.04 × 10 6.07 × 10 without getting to treat it until it reaches a fourth grade, which is ethically complicated that can be done). So, to achieve this, it would be necessary to develop a theoretical Table 3: Control case: von Mises stresses results. model away from reality. Values Enamel Dentin Pulp It is also true that, in the present work, the pulp is modelled as a complete tissue, when in reality it is a Maximum 0.012 Pa 0.0027 Pa 0.0007 −8 −9 −7 vascular-nervous package. However, this condition is Minimum 3.11 × 10 Pa 2.99 × 10 Pa 2.34 × 10 Pa consistent in many publications that do so, since it is a very appropriate approximation model. It is possible to consider Table 4: 1st degree caries: von Mises stresses results. that this behaviour could only be proven in an experimental model, which would also be complicated, since the blood Values Enamel Dentin Pulp vessels and nerves that make it up are capillary. But there 6 6 6 Maximum 1.42 × 10 Pa 16.46 × 10 Pa 48.50 × 10 Pa are still no conditions to be able to perform in this way. 6 6 6 Minimum 1.02 × 10 Pa 1.31 × 10 Pa 3.77 × 10 Pa Nevertheless, as already mentioned, this does not change the way in which the effort is being distributed. This makes the biomodels more than adequate. This does not mean that, Table 5: 2nd degree caries: von Mises stresses results. in future investigations, optimizations or anisotropic or even Values Enamel Dentin Pulp orthotropic analyses cannot be made. 6 6 6 Specifically, with the von Mises stresses, it is possible to Maximum 49.29 × 10 Pa 10.33 × 10 Pa 42.13 × 10 Pa 6 6 6 observe that the values in comparison with the control case Minimum 3.76 × 10 Pa 1.14 × 10 Pa 3.28 × 10 Pa are increased. In the same way, this is related to the structural integrity of the molar for each case of caries; the geometry Table 6: 3rd degree caries: von Mises stresses results. modified by the degree of caries is a factor that affects these results, due to structural change. The tissue loss varies in each Values Enamel Dentin Pulp case. In addition, the occlusal forces that act (horizontal and 6 6 6 Maximum 43.37 × 10 Pa 15.92 × 10 Pa 2.63 × 10 Pa vertical) play an important role in these results, since the 6 6 6 Minimum 1.60 × 10 Pa 1.22 × 10 Pa 0.93 × 10 Pa disposition and depth of the lesion modify the resulting efforts and reactions. This could again suggest the idea that it is due to the way Table 7: 4th degree caries: von Mises stresses results. in which caries was generated in biomodels that singularities Values Enamel Dentin Pulp are presented as stress concentrators and intensifiers. In 6 6 6 Maximum 57.65 × 10 Pa 18.67 × 10 Pa 1.34 × 10 Pa clinical cases, this is a reality. Caries produces irregularities 6 6 6 in the geometry of dental morphology. These irregularities Minimum 2.57 × 10 Pa 0.41 × 10 Pa 0.13 × 10 Pa actually produce stress concentrators; omitting them in the generation of the biomodel would modify the results of what history of caries disease, the biomodels used are not a contin- actually occurs in the molar with decay. uous growth of the same caries and are not an elaborate design to control, but are results of intakes radiographic 5. Conclusions and tomographic of cases of real caries at different levels, which comply with the natural history of the disease, which With the present work, it was possible to demonstrate that is why many aspects are not directly comparable. the presence of caries causes mainly three phenomena: This could suggest that the number of variables that can be presented would compromise the results of the analyses. (1) There is an overload in the contact of the ameloden- However, such control would distance work from reality. tinaria junction. As already mentioned, it is not the intention to compare (2) This demonstrates the traditional empirical knowl- one case with another since; to make a comparison, it would edge of dentistry that the dental organ can fail due have to grow the same decay. Each caries is different, and to the loss of elasticity provided by dentin. therefore, the efforts are not comparable. What is tried to demonstrate is that, in all cases, there are first efforts in the (3) The simple fact of presenting a carious lesion area of the amelodentinaria union and that, as the degree of causes the pulp to be affected from a mechanical caries increases, the pulp begins to load, unlike the control point of view. case where the dental organ is healthy and the pulp is not reached by any kind of effort. In an investigation published prior to the present work At the moment that the caries appears, begin to [17], it was found that there are stress concentrations in the present efforts. It is possible to come to think that in a amelodentinous junction (located in the anatomical neck of Occlusal Applied Bionics and Biomechanics 7 Lingual Vestibular Distal Mesial Distal Vestibular Occlusal Y Z (a) (b) -8 3.11 × 10 0.002 0.005 0.008 0.011 0.001 0.004 0.006 0.009 0.012 (Pa) Figure 6: Control case: von Mises stresses in enamel: (a) occlusal view (b) cervical view. Lingual Vestibular Z X -9 2.99 × 10 0.0006 0.0012 0.0018 0.0024 0.0003 0.0009 0.0015 0.0021 0.0027 (Pa) Figure 7: Control case: von Mises stresses in dentin. Vestibular Occlusal -7 2.34 × 10 0.0001 0.0003 0.00048 0.0006 -5 0.0002 0.0004 0.0005 0.007 (Pa) 8.13 × 10 Figure 8: Control case: von Mises stresses in pulp. the teeth, it is the junction or articulation zone of enamel but also when the dental organ presents caries, one of the and dentin) in healthy molars due to the simple fact of first consequences is that in the amelodentinous junction mastication. Not only does this work corroborate this fact, greater points of concentration of efforts are manifested. 8 Applied Bionics and Biomechanics Lingual Vestibular Lingual Distal Mesial Mesial Distal Y Z Vestibular Lingual Vestibular (a) (b) 1.02 9.46 18.92 28.37 37.83 4.73 14.19 23.65 33.10 42.56 × 10 (Pa) Figure 9: 1st degree caries: von Mises stresses in enamel: (a) occlusal view (b) cervical view. Occlusal Vestibular Lingual Mesial Z X Distal 1.31 3.65 7.31 10.97 14.63 1.82 5.48 9.14 12.80 16.46 × 10 (Pa) Figure 10: 1st degree caries: von Mises stresses in dentin. Occlusal Lingual Cervical Vestibular Mesial Distal 3.77 10.78 21.15 32.33 43.11 5.39 16.17 26.94 37.72 48.50 × 10 (Pa) Figure 11: 1st degree caries: von Mises stresses in pulp. Applied Bionics and Biomechanics 9 Lingual Vestibular Lingual Distal Mesial Distal Mesial Vestibular Lingual Y Z Vestibular (a) (b) 3.76 10.98 21.93 32.87 43.82 5.51 16.45 27.40 38.35 49.29 × 10 (Pa) Figure 12: 2nd degree caries: von Mises stresses in enamel: (a) occlusal view (b) cervical view. Occlusal Vestibular Lingual Mesial Z X Distal 1.148 2.29 4.59 6.89 9.18 1.14 3.44 5.74 8.03 10.33 × 10 (Pa) Figure 13: 2nd degree caries: von Mises stresses in dentin. Occlusal Lingual Cervical Vestibular Distal Mesial 3.28 9.36 18.72 28.08 37.45 4.68 14.04 23.40 32.76 42.13 × 10 (Pa) Figure 14: 2nd degree caries: von Mises stresses in pulp. 10 Applied Bionics and Biomechanics Lingual Lingual Vestibular Distal Mesial Distal Mesial Vestibular Lingual Vestibular Y Z (a) (b) 1.60 9.64 19.27 28.91 38.54 4.83 14.46 24.09 33.72 43.37 × 10 (Pa) Figure 15: 3rd degree caries: von Mises stresses in enamel: (a) occlusal view (b) cervical view. Occlusal Vestibular Lingual Mesial Z X Distal 1.22 3.53 7.07 10.61 14.15 1.76 5.30 8.46 12.38 15.92 × 10 (Pa) Figure 16: 3rd degree caries: von Mises stresses in dentin. Lingual Vestibular Occlusal Distal 0.93 0.58 1.17 1.75 2.34 0.36 0.87 1.46 2.05 2.63 × 10 (Pa) Figure 17: 3rd degree caries: von Mises stresses in pulp. Applied Bionics and Biomechanics 11 Lingual Lingual Vestibular Distal Mesial Distal Mesial Vestibular Lingual Z Vestibular (a) (b) 2.57 12.83 25.63 38.44 51.24 6.42 19.23 32.03 44.84 57.65 × 10 (Pa) Figure 18: 4th degree caries: von Mises stresses in enamel: (a) occlusal view (b) cervical view. Z X Mesial Lingual Vestibular Occlusal Distal 0.41 4.15 8.30 12.45 16.60 2.07 18.67 × 10 6.22 10.37 14.52 (Pa) Figure 19: 4th degree caries: von Mises stresses in dentin. Distal Lingual Vestibular Occlusal Mesial Z X 0.13 0.29 0.59 0.89 1.19 0.14 0.44 0.74 1.04 1.34 × 10 (Pa) Figure 20: 4th degree caries: von Mises stresses in pulp. 12 Applied Bionics and Biomechanics recovery. It has been found that, as already mentioned, as Establishing with this are areas with propensity to present lesions known as noncarious, before functional loads and the mechanical stimulation of the dentine produces its not for functional loads, as it usually happens as a prece- regeneration, then it would be desirable that the restorative materials could fulfill that function of stimulation that would dent of this type of injuries. On the other hand, when the caries causes the dentin to lead to the regeneration of the damaged tissue. decrease, the elasticity of the dental organ tissue system is lost as well, since this property is provided by this tissue. There- Data Availability fore, there is a predisposition to failure, mainly the enamel, which is consistent with the existing empirical knowledge The data used to support the findings of this study are in dentistry, which states that a dental organ that suffers from included within the article. caries has a greater risk of fracture by force of normal bite, than a healthy tooth. It is common to hear from dentists that Conflicts of Interest the simple fact of presenting decay makes the tooth more fragile and the greater the degree of decay, the greater the risk The authors declare that they have no competing interests. of that organ ruptures. With this, it is possible to establish that, in addition to Acknowledgments biological and chemical factors that cause that there is greater predisposition to the failure of a dental organ with caries, The authors gratefully acknowledge the Instituto Politécnico there are factors of a mechanical nature that cause this fail- Nacional and the Consejo Nacional de Ciencia y Tecnología. ure. In this same way, not only are biological and chemical factors cause irritation in the pulp that in the end will trigger inflammation, pain, and even necrosis. When losing the References dentine or seeing its elastic property diminished, its function [1] A. Buganza Tepole, “Computational systems mechanobiology of dissipating the masticatory forces decreases, with which of wound healing,” Computer Methods in Applied Mechanics the pulp is involved in the distribution of stresses that do and Engineering, vol. 314, pp. 46–70, 2017. not occur in a healthy tooth. [2] J. C. Vanegas-Acosta, N. S. Landinez P., D. A. Garzón- With this it can be deduced that regardless of the size Alvarado, and M. C. Casale R., “A finite element method or depth of the caries, the pulp will be affected as long as approach for the mechanobiological modeling of the the dentin loses its elastic function. This should cause osseointegration of a dental implant,” Computer Methods reflection that, once the decay is eliminated and the mate- and Programs in Biomedicine, vol. 101, no. 3, pp. 297–314, rials to restore the remaining tissues are selected, they must restore the lost elastic functions of the system, in such [3] T. Gedrange, C. Kunert-Keil, F. Heinemann, and a way that the pula is no longer involved and it is possible M. Dominiak, “Tissue engineering and oral rehabilitation in to reestablish the health and functioning of the dental organ the stomatognathic system,” BioMed Research International, and its tissues. vol. 2017, Article ID 4519568, 2 pages, 2017. 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Hindawi Applied Bionics and Biomechanics Volume 2018, Article ID 1815830, 13 pages https://doi.org/10.1155/2018/1815830 Research Article Mechanobiological Analysis of Molar Teeth with Carious Lesions through the Finite Element Method R. A. Hernández-Vázquez , Betriz Romero-Ángeles, Guillermo Urriolagoitia-Sosa, Juan Alejandro Vázquez-Feijoo, Rodrigo Arturo Marquet-Rivera, and Guillermo Urriolagoitia-Calderón Instituto Politécnico Nacional, Escuela Superior de Ingeniería Mecánica y Eléctrica, Sección de Estudios de Posgrado e Investigación, Unidad Profesional Adolfo López Mateos “Zacatenco”, Avenida Instituto Politécnico Nacional s/n, Edificio 5, 2do. Piso, Col. Lindavista, C.P. 07320 Ciudad de Mexico, Mexico Correspondence should be addressed to R. A. Hernández-Vázquez; alyzia.hv@esimez.mx Received 18 May 2018; Revised 23 July 2018; Accepted 14 August 2018; Published 14 October 2018 Academic Editor: Jan Harm Koolstra Copyright © 2018 R. A. Hernández-Vázquez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The analysis of the distribution of stress in dental organs is a poorly studied area. That is why computational mechanobiological analysis at the tissue level using the finite element method is very useful to achieve a better understanding of the biomechanics and the behaviour of dental tissues in various pathologies. This knowledge will allow better diagnoses, customize treatment plans, and establish the basis for the development of better restoration materials. In the present work, through the use of high- fidelity biomodels, computational mechanobiological analyses were performed on four molar models affected with four different degrees of caries, which are subjected to masticatory forces. With the analyses performed, it is possible to observe that the masticatory forces that act on the enamel are not transmitted to the dentin and to the bone and periodontal ligament to protect the nerve, as it happens in a healthy dental organ. With the presence of decay, these forces are transmitted partly to the pulp. The reactions to the external loads on the dental organs depend on the advances of the carious lesion that they present, since the distribution of stresses is different in a healthy tooth. functional balance of the system, which in turn causes the 1. Introduction occlusion not only to be altered but also to be lost [4]. This Mechanobiology is an area of recent application in dentistry in turn causes alterations in its biomechanics. [1, 2]; this is dedicated to the analysis of stresses and The morphology of each dental organ is designed to per- deformations in tissues in living beings. Study tools used in form this function. The occlusal surface of the posterior teeth (molars and premolars) has cusps, depressions, grooves, and engineering are applied in the structures of living beings. The structures that make up the stomatognathic system have pits that coincide with those of the opposing tooth (Figure 1), a harmonious morphology with a high degree of specializa- as well as the incisal edges, cusps, and cingules in the case of tion. All the bony, neuromuscular, articular, periodontal, the anterior teeth (incisors and canines) [5]. Together with and occlusal components establish an equilibrium between the rest of the components of the system, they allow the teeth them that allow their complex physiology [3]. Even the finest to perform their function in a phenomenon called occlusion detail, such as a roughness, a groove, a cusp, or an orifice, [6], which is regulated by the masticatory forces acting on fulfills a specific function. In particular, when it comes to each of the teeth. the chewing function, the collusive components or dental Due to this, analysis of the distribution of stresses gener- organs are the protagonists. When the total or partial loss ated by occlusal or masticatory loads, in a biological system of any of these components occurs, there is a loss of the such as the teeth, is a complex problem [7], because of the 2 Applied Bionics and Biomechanics (a) (b) (c) Figure 1: Harmonious relationship of dental morphology. (a) Dental occlusion, (b) points where the upper teeth make contact with the lower teeth, according to their morphology, and (c) detail of occlusal assembly between premolars. (a) (b) Figure 2: Micrographs of dental enamel. (a) Distribution of enamel prisms and (b) hydroxyapatite prisms. nature of the tissues of the dental organ, such as nonhomoge- packed by a delicate network of organic material that neous materials and the geometric irregularities of their con- surrounds them (Figure 2). The prism is considered as the basic structural unit of the enamel, and the prismatic enamel tours and anatomical forms. In addition to this, the tooth in its structure is formed by enamel, dentine, and pulp, is the set of prisms that constitutes most of the mineralized whose mechanical properties differ from one another. The matrix [8]. distribution of stresses is also affected by health and patho- For its part, dentine is the one that provides support to logical states, which makes their analysis even more complex. enamel and transmits the forces it receives. Its microstructure The teeth concentrates the forces generated by the mus- is dominated by the presence of crystal structures in the cles in small areas such as contact surfaces and cusps or shape of an inverted cone called dentinal tubules. Between incisal edges. each tubule, there is a peritubular channel which contains Dental enamel is the tissue most exposed to high occlusal collagen I, hydroxyapatite crystals, and dentinal fluid. It forces, due to the processing of food during mastication in constitutes the organic or intertubular matrix [9]. the oral cavity and the occlusal contact between the teeth, Sometimes, nature uses processes like those that engi- for which it suffers attrition and fractures. It is a highly neering uses to improve a design; a clear example is what is specialized crystalline structure. The prisms that structure it known as residual stresses, which are used to reinforce are package of hydroxyapatite crystals ordered and densely materials, with a process called cold hardening. A similar Applied Bionics and Biomechanics 3 (a) (b) (c) (d) Figure 3: Stages of carious lesions. (a) First stages (enamel), (b) second stages (enamel and dentin), (c) third stages (affectation of the dental nerve), and (d) fourth stages (destruction of morphology). phenomenon occurs in the teeth. It has been found that the discomfort [14], being the second most common disease worldwide. It is a chronic multifactorial pathology [15] mineral particles of the dentin are precompressed, which prevents the propagation of cracks and increases the strength located in the hard tissues of the dental organ. It is character- of their biostructure. Dentin is a material similar to bone ized by the loss of the organic matrix due to the action of the formed from mineral nanoparticles, mainly hydroxyapatite. organic acid that is produced because of the bacterial metab- These mineral nanoparticles are embedded in collagen fibres, olism of the microbiota of the oral cavity. It takes place in a with which they are firmly interconnected. When the colla- progressively way, beginning with the demineralization of gen fibres contract, the adhered mineral particles get even the tissue at the ultrastructural level, until it reaches the total more compressed. This allows to prevent fissures, and the loss of the piece [16] (Figure 3). compression is carried out in such a way that the cracks This condition would cause a total imbalance of the cannot easily reach the internal part of the teeth, which could occlusal relationships, since the morphology of the dental damage the pulp [10]. In this case, the residual stresses are organs is altered, up to the loss of the dental organ, which represented by the masticatory loads, which propitiate this causes the total loss of the interocclusal harmony. In turn, it phenomenon that prevents the propagation of faults, with generates collateral damages such as systemic diseases and residual reactions throughout the stomatognathic System, the simple fact of chewing. A detailed information on the distribution of stresses and which is well worth analysing. Therefore, their mechanobio- location of the critical load points is of vital importance, since logical analysis proposes new guidelines for the actions that the dental organs have as their main function the adequate can be applied to counteract their affections and to be more transmission of forces to grind the food. Discrete analysis certain that the treatments and restorative materials act properly. Likewise, it will establish the foundations of the of stress distribution is essential in these organs and tissues, because the main objective is to understand how these efforts future of odontology of prevention. are distributed and transmitted and find stress concentra- In general, there are no publications that have numerical tions in specific places due to damage to the tooth as a data on the distribution of efforts as the caries progresses in function of its growth, adaptation, and structural modifica- its different degrees. Therefore, the present work represents a novelty in this subject and field of study. There are some tion [11]. This type of analysis would allow, for example, the control of dental regeneration. Recent studies have docu- works that raise the possibility of performing analyses of this mented that there is some regenerative capacity of dental pathology, but it is not the main objective of them; therefore, enamel (as when it is subjected to chemical compounds such they do not present numerical results in relation to the dis- as fluoride [12]). tribution of efforts when a carious lesion occurs. In addition, the models that were made in these studies are theoretical One of the main pathologies that appear in the teeth is dental caries. This disease, according to the World Health models of reality; they are not models with the characteris- Organization, is a condition and is estimated that five million tics such as the ones we are presenting. The design of the people worldwide suffer from it [13]. In global terms, between cavities that they propose represents the caries; they are 60% and 90% of school-age children and nearly 100% of adults made with cuts to the model, not how the caries is presented in reality. have dental caries, often accompanied by pain or feeling of 4 Applied Bionics and Biomechanics (a) (b) (c) (d) Figure 4: Biomodels with the four degrees of decay, based on (a) first grade, (b) second grade, (c) third grade, and (d) fourth grade. The biomodels of the present work are the results of For the modelling, the methodology described in a previ- ous work was used [17]. As already mentioned, the images of radiographic and tomographic images of cases of real caries at different levels, which comply with the natural history of the molars were obtained through the use of 3D imaging, by the disease. In addition, no real case is equal to another. They means of a digital volumetric tomography (TVD) of the max- were obtained with a new technique of imaging (cone beam). illa and mandible with the cone beam computed tomography This system is used to obtain images in difficult to visualize system (CBTC), to obtain the DICOM files. This system is tissues; it is widely used in medicine and dentistry in the widely used for studies of the maxillofacial area, since it craniofacial region. It provides images with a resolution of allows a better visualization of difficult access tissues. submillimetres of high diagnostic quality and excellent The tomography used is Batex brand model EZ3D, which visualization. In addition to this, the methodology used to has a KvP of 90.0 mA (3.8 light beam intensity). A total of 477 generate them ensures biofidelity. For these same reasons, images or slices were obtained from each tomography, whose the numerical results would be difficult to compare with distance between them (slide thickness) is 0.5 mm. The space other works, given the differences of the models used in between the pixels (pixel spacing) is of a ratio of 0.3/0.3 mm. other works. The obtained models have elements of high order (tetrahe- dral with a total of 10 nodes per element) and consider three different materials that correspond to the tissues that struc- 2. Materials and Methods ture the dental organ (enamel, dentine, and pulp). The FEM analyses were carried out with the ANSYS® This study is aimed at carrying out the analysis of four case computer program. The number of elements and nodes of studies that address the problem of finding the stresses that each biomodel is the following: biomodel of the control case occur in a dental organ when it is subjected to the forces of 74,907 elements and 129,005 nodes. First-degree caries have 76,851 elements and 132,043 nodes. Seventy-degree caries occlusion and its structural integrity is compromised, by the presence of carious lesions in its four degrees. Addition- have 77,419 elements and 133,497 nodes. Third-degree caries ally, there is a control case in a healthy molar without decay. have 80,317 elements and 137,666 nodes. Fourth-degree For each of the cases, a high-fidelity biomodel of a lower left caries have 81,318 elements and 141,830 nodes. first molar, representing each stage of caries, was used, The structural analysis has an elastic, linear, and homo- obtaining a total of five biomodels represented as follows: geneous behaviour. The mechanical properties of the tissues control case: without decay; case 1: caries in enamel on the are described in Table 1 [18–20]. occlusal face, in the main sulcus; case 2: caries in enamel The convergences in the generation of discretized meshes were analysed. Discretization was carried out in a semicon- and dentin on the occlusal face, starting in the main sulcus; case 3: caries in enamel and dentin and with pulp communi- trolled way, since the geometries did not allow for a fully cation in the interproximal zone; and case 4: caries of enamel controlled discretization, in the same way that an automatic on the occlusal face, in the main sulcus that extends over one was not suitable. We chose the size of the Maya that much of the dentin, causing involvement of the pulp cham- would allow the resolution of the analysis and that the differ- ences between the results would be negligible or consistent. ber and dental pulp (Figure 4). Applied Bionics and Biomechanics 5 Oclusal Table 1: Mechanical properties used in the analysis. pressure Enamel Dentin Pulp 3 3 3 Density 0.25 g/cm 0.31 g/cm 0.1 g/cm Young’s module 70 GPa 18.3 GPa 2 GPa Dimensional Poisson ratio 0.30 0.30 0.45 In relation to the modelling of carious lesions, the Movement restriction following should be mentioned. The axes in tissue ablations UX = UY = UZ = 0 of enamel and dentine (caries) are real, where caries is an irregular area, the edges are angular, and therefore the Rot X = Rot Y = Rot = 0 singularities due to the stress concentrators are real and consistent. These irregularities are observed in the images provided in the radiographs and tomographies, so that the Figure 5: Applied loads and boundary conditions. resulting biomodels are real reproductions of decayed molars. The boundary conditions were established in the root zone, restricting the displacements and rotations in the of the stresses. That is, it is representative of the distribution directions of the X, Y, and Z axes. A load was applied on of efforts that is the main objective of this work. In addition, the occlusal surface in the four biomodels. The magnitude von Mises encompasses the shear and normal energy, which of the applied load is 150 N/mm corresponding to the bite can occur in these cases. The shear stresses have the same force between the two molars [21–25], which is distributed distribution, and the magnitudes obtained are very similar, locally in the application zone on the form of a pressure. So and for the objectives of the work, they do not provide more that the pressure is the same in all cases, the area of action information. In future jobs where the fracture or failure is is different because of the different degrees of caries they considered, they could be considered. represent. The load is uniform for optimization of the simu- The results obtained are shown from Tables 2 to 7 and lation, and this does not affect the results or the conclusions, from Figures 6 to 20. since it is not a comparison between the cases, so it is not The von Mises stresses of the control case are observed necessary to adjust to the same load (Figure 5). from Figures 6 to 8. The contacts between the tissues were considered for the The von Mises stresses of the case 1: 1st degree caries are analyses performed. The software used has a function called observed from Figures 9 to 11. Contact Manager®, through which the contact simulation The von Mises stresses of the case 2: 2nd degree caries was carried out. This tool presents a subfunction called are observed from Figures 12 to 14. Contact Wizard® in which information was given to establish The von Mises stresses of the case 3: 3rd degree caries are which material or tissue is the contact and which will be the observed from Figures 15 to 17. target. For the simulation, the tissue was established as a The von Mises stresses of the case 4: 4th degree caries contact with the enamel and as a target for the dentin. To are observed from Figures 18 to 20. establish their contact relationship, a coefficient of friction was also added, which is according to the literature that is 4. Discussion established between ceramic materials, since the crystalline nature of these tissues, as well as bone tissue, has been With regard to the unitary strain, it can be found that, considered as ceramic materials [26]. although the same load is applied in all the study cases, the resulting stress is greater as the degree of caries increases. This would lead to the expectation that greater deformation 3. Results should occur as the degree of caries increases. However, the Unitary deformation, directional deformation, nominal less deformation that occurs, the greater the degree of caries. stresses, shear stresses, principal stresses, and von Mises stress That is, a stiffening is generated (to name it somehow) which were analysed during the application of the load that simu- is due to the interaction of the mechanical properties that are lates dental bite or occlusion. It should be mentioned that established between the three dental tissues. The rigidity is Von Mises stresses are not considered here as a failure crite- assumed by the least deformation that was obtained. At this rion (which is mainly applicable to ductile materials), but as point, it is prudent to clarify that no real case is equal to a unique nondirectional value that allows to have a global another; however, aspects such as the pressure exercised are criterion on the load at each tooth point, since it is obtained the same in all cases. from the deformation energy. Several authors use it as a The geometric conditions cause that the direct compari- criterion to evaluate restorations [27–29]. son between the different degrees of decay can give the idea However, in the body of work, only the efforts of von that it is due to these differences in the geometry, the motive Mises for tissue are shown. This is due to the fact that the of the results, and the nature of the unit deformations. How- main efforts are obviated with those of von Mises, since it is ever, the effect of stiffening in all four models is consistent. In the total deformation energy that indicates the magnitude addition, to represent what actually happens with the natural Z 6 Applied Bionics and Biomechanics controlled and designed case that is uniform, the pulp Table 2: Unitary strain. can come to present greater load (it is not possible to do Control 1st degree 2nd degree 3rd degree 4th degree this in a real case because it would have to be done in a case caries caries caries caries healthy molar and that is allowed to get sick with caries, −5 −12 −16 −16 −16 3.66 × 10 3.06 × 10 3.80 × 10 5.04 × 10 6.07 × 10 without getting to treat it until it reaches a fourth grade, which is ethically complicated that can be done). So, to achieve this, it would be necessary to develop a theoretical Table 3: Control case: von Mises stresses results. model away from reality. Values Enamel Dentin Pulp It is also true that, in the present work, the pulp is modelled as a complete tissue, when in reality it is a Maximum 0.012 Pa 0.0027 Pa 0.0007 −8 −9 −7 vascular-nervous package. However, this condition is Minimum 3.11 × 10 Pa 2.99 × 10 Pa 2.34 × 10 Pa consistent in many publications that do so, since it is a very appropriate approximation model. It is possible to consider Table 4: 1st degree caries: von Mises stresses results. that this behaviour could only be proven in an experimental model, which would also be complicated, since the blood Values Enamel Dentin Pulp vessels and nerves that make it up are capillary. But there 6 6 6 Maximum 1.42 × 10 Pa 16.46 × 10 Pa 48.50 × 10 Pa are still no conditions to be able to perform in this way. 6 6 6 Minimum 1.02 × 10 Pa 1.31 × 10 Pa 3.77 × 10 Pa Nevertheless, as already mentioned, this does not change the way in which the effort is being distributed. This makes the biomodels more than adequate. This does not mean that, Table 5: 2nd degree caries: von Mises stresses results. in future investigations, optimizations or anisotropic or even Values Enamel Dentin Pulp orthotropic analyses cannot be made. 6 6 6 Specifically, with the von Mises stresses, it is possible to Maximum 49.29 × 10 Pa 10.33 × 10 Pa 42.13 × 10 Pa 6 6 6 observe that the values in comparison with the control case Minimum 3.76 × 10 Pa 1.14 × 10 Pa 3.28 × 10 Pa are increased. In the same way, this is related to the structural integrity of the molar for each case of caries; the geometry Table 6: 3rd degree caries: von Mises stresses results. modified by the degree of caries is a factor that affects these results, due to structural change. The tissue loss varies in each Values Enamel Dentin Pulp case. In addition, the occlusal forces that act (horizontal and 6 6 6 Maximum 43.37 × 10 Pa 15.92 × 10 Pa 2.63 × 10 Pa vertical) play an important role in these results, since the 6 6 6 Minimum 1.60 × 10 Pa 1.22 × 10 Pa 0.93 × 10 Pa disposition and depth of the lesion modify the resulting efforts and reactions. This could again suggest the idea that it is due to the way Table 7: 4th degree caries: von Mises stresses results. in which caries was generated in biomodels that singularities Values Enamel Dentin Pulp are presented as stress concentrators and intensifiers. In 6 6 6 Maximum 57.65 × 10 Pa 18.67 × 10 Pa 1.34 × 10 Pa clinical cases, this is a reality. Caries produces irregularities 6 6 6 in the geometry of dental morphology. These irregularities Minimum 2.57 × 10 Pa 0.41 × 10 Pa 0.13 × 10 Pa actually produce stress concentrators; omitting them in the generation of the biomodel would modify the results of what history of caries disease, the biomodels used are not a contin- actually occurs in the molar with decay. uous growth of the same caries and are not an elaborate design to control, but are results of intakes radiographic 5. Conclusions and tomographic of cases of real caries at different levels, which comply with the natural history of the disease, which With the present work, it was possible to demonstrate that is why many aspects are not directly comparable. the presence of caries causes mainly three phenomena: This could suggest that the number of variables that can be presented would compromise the results of the analyses. (1) There is an overload in the contact of the ameloden- However, such control would distance work from reality. tinaria junction. As already mentioned, it is not the intention to compare (2) This demonstrates the traditional empirical knowl- one case with another since; to make a comparison, it would edge of dentistry that the dental organ can fail due have to grow the same decay. Each caries is different, and to the loss of elasticity provided by dentin. therefore, the efforts are not comparable. What is tried to demonstrate is that, in all cases, there are first efforts in the (3) The simple fact of presenting a carious lesion area of the amelodentinaria union and that, as the degree of causes the pulp to be affected from a mechanical caries increases, the pulp begins to load, unlike the control point of view. case where the dental organ is healthy and the pulp is not reached by any kind of effort. In an investigation published prior to the present work At the moment that the caries appears, begin to [17], it was found that there are stress concentrations in the present efforts. It is possible to come to think that in a amelodentinous junction (located in the anatomical neck of Occlusal Applied Bionics and Biomechanics 7 Lingual Vestibular Distal Mesial Distal Vestibular Occlusal Y Z (a) (b) -8 3.11 × 10 0.002 0.005 0.008 0.011 0.001 0.004 0.006 0.009 0.012 (Pa) Figure 6: Control case: von Mises stresses in enamel: (a) occlusal view (b) cervical view. Lingual Vestibular Z X -9 2.99 × 10 0.0006 0.0012 0.0018 0.0024 0.0003 0.0009 0.0015 0.0021 0.0027 (Pa) Figure 7: Control case: von Mises stresses in dentin. Vestibular Occlusal -7 2.34 × 10 0.0001 0.0003 0.00048 0.0006 -5 0.0002 0.0004 0.0005 0.007 (Pa) 8.13 × 10 Figure 8: Control case: von Mises stresses in pulp. the teeth, it is the junction or articulation zone of enamel but also when the dental organ presents caries, one of the and dentin) in healthy molars due to the simple fact of first consequences is that in the amelodentinous junction mastication. Not only does this work corroborate this fact, greater points of concentration of efforts are manifested. 8 Applied Bionics and Biomechanics Lingual Vestibular Lingual Distal Mesial Mesial Distal Y Z Vestibular Lingual Vestibular (a) (b) 1.02 9.46 18.92 28.37 37.83 4.73 14.19 23.65 33.10 42.56 × 10 (Pa) Figure 9: 1st degree caries: von Mises stresses in enamel: (a) occlusal view (b) cervical view. Occlusal Vestibular Lingual Mesial Z X Distal 1.31 3.65 7.31 10.97 14.63 1.82 5.48 9.14 12.80 16.46 × 10 (Pa) Figure 10: 1st degree caries: von Mises stresses in dentin. Occlusal Lingual Cervical Vestibular Mesial Distal 3.77 10.78 21.15 32.33 43.11 5.39 16.17 26.94 37.72 48.50 × 10 (Pa) Figure 11: 1st degree caries: von Mises stresses in pulp. Applied Bionics and Biomechanics 9 Lingual Vestibular Lingual Distal Mesial Distal Mesial Vestibular Lingual Y Z Vestibular (a) (b) 3.76 10.98 21.93 32.87 43.82 5.51 16.45 27.40 38.35 49.29 × 10 (Pa) Figure 12: 2nd degree caries: von Mises stresses in enamel: (a) occlusal view (b) cervical view. Occlusal Vestibular Lingual Mesial Z X Distal 1.148 2.29 4.59 6.89 9.18 1.14 3.44 5.74 8.03 10.33 × 10 (Pa) Figure 13: 2nd degree caries: von Mises stresses in dentin. Occlusal Lingual Cervical Vestibular Distal Mesial 3.28 9.36 18.72 28.08 37.45 4.68 14.04 23.40 32.76 42.13 × 10 (Pa) Figure 14: 2nd degree caries: von Mises stresses in pulp. 10 Applied Bionics and Biomechanics Lingual Lingual Vestibular Distal Mesial Distal Mesial Vestibular Lingual Vestibular Y Z (a) (b) 1.60 9.64 19.27 28.91 38.54 4.83 14.46 24.09 33.72 43.37 × 10 (Pa) Figure 15: 3rd degree caries: von Mises stresses in enamel: (a) occlusal view (b) cervical view. Occlusal Vestibular Lingual Mesial Z X Distal 1.22 3.53 7.07 10.61 14.15 1.76 5.30 8.46 12.38 15.92 × 10 (Pa) Figure 16: 3rd degree caries: von Mises stresses in dentin. Lingual Vestibular Occlusal Distal 0.93 0.58 1.17 1.75 2.34 0.36 0.87 1.46 2.05 2.63 × 10 (Pa) Figure 17: 3rd degree caries: von Mises stresses in pulp. Applied Bionics and Biomechanics 11 Lingual Lingual Vestibular Distal Mesial Distal Mesial Vestibular Lingual Z Vestibular (a) (b) 2.57 12.83 25.63 38.44 51.24 6.42 19.23 32.03 44.84 57.65 × 10 (Pa) Figure 18: 4th degree caries: von Mises stresses in enamel: (a) occlusal view (b) cervical view. Z X Mesial Lingual Vestibular Occlusal Distal 0.41 4.15 8.30 12.45 16.60 2.07 18.67 × 10 6.22 10.37 14.52 (Pa) Figure 19: 4th degree caries: von Mises stresses in dentin. Distal Lingual Vestibular Occlusal Mesial Z X 0.13 0.29 0.59 0.89 1.19 0.14 0.44 0.74 1.04 1.34 × 10 (Pa) Figure 20: 4th degree caries: von Mises stresses in pulp. 12 Applied Bionics and Biomechanics recovery. It has been found that, as already mentioned, as Establishing with this are areas with propensity to present lesions known as noncarious, before functional loads and the mechanical stimulation of the dentine produces its not for functional loads, as it usually happens as a prece- regeneration, then it would be desirable that the restorative materials could fulfill that function of stimulation that would dent of this type of injuries. On the other hand, when the caries causes the dentin to lead to the regeneration of the damaged tissue. decrease, the elasticity of the dental organ tissue system is lost as well, since this property is provided by this tissue. There- Data Availability fore, there is a predisposition to failure, mainly the enamel, which is consistent with the existing empirical knowledge The data used to support the findings of this study are in dentistry, which states that a dental organ that suffers from included within the article. caries has a greater risk of fracture by force of normal bite, than a healthy tooth. It is common to hear from dentists that Conflicts of Interest the simple fact of presenting decay makes the tooth more fragile and the greater the degree of decay, the greater the risk The authors declare that they have no competing interests. of that organ ruptures. With this, it is possible to establish that, in addition to Acknowledgments biological and chemical factors that cause that there is greater predisposition to the failure of a dental organ with caries, The authors gratefully acknowledge the Instituto Politécnico there are factors of a mechanical nature that cause this fail- Nacional and the Consejo Nacional de Ciencia y Tecnología. ure. In this same way, not only are biological and chemical factors cause irritation in the pulp that in the end will trigger inflammation, pain, and even necrosis. When losing the References dentine or seeing its elastic property diminished, its function [1] A. Buganza Tepole, “Computational systems mechanobiology of dissipating the masticatory forces decreases, with which of wound healing,” Computer Methods in Applied Mechanics the pulp is involved in the distribution of stresses that do and Engineering, vol. 314, pp. 46–70, 2017. not occur in a healthy tooth. [2] J. C. Vanegas-Acosta, N. S. 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Applied Bionics and BiomechanicsHindawi Publishing Corporation

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