journal of prosthodontic research 58 (2014) 92–101
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Review
Finite element contact analysis as a critical technique in dental biomechanics: A review Natsuko Murakami DDS, PhD, Noriyuki Wakabayashi DDS, PhD* Removable Partial Prosthodontics, Oral Health Sciences, Graduate School of Medical and Dental Sciences, Tokyo Medical and Dental University, Japan
article info
abstract
Article history:
Purpose: Nonlinear finite element contact analysis is used to mathematically estimate stress
Received 27 February 2014
and strain in a time- and status-dependent mechanical model. However, the benefits and
Accepted 10 March 2014
limitations of this method have not been thoroughly examined.
Available online 4 April 2014
Study selection: The current review summarizes the utility of contact analysis in character-
Keywords:
implant integration.
Finite element analysis
Results: Opposing tooth contact, friction, and sliding phenomena were simulated to esti-
izing individual stressors: (1) tooth-to-tooth contact, (2) restorative interface, and (3) bone–
Stress
mate stress distribution and assess the failure risk for tooth enamel, composite, and ceramic
Dental
restorations. Mechanical tests such as the flexural tests were simulated with the contact
Implant
analysis to determine the rationale underlying experimental findings. The tooth–restoration complex was modeled with interface contact elements that simulate imperfect bonding,
Nonlinear
and the normal and tangential stresses were calculated to predict failure progression. Previous studies have used a friction coefficient to represent osseointegration adjacent to dental implants, but the relationship between interface friction and the bone quality is unknown. An understanding of the local stress and strain may better predict loss of osseointegration, however, the effective stress as a critical contributor to bone degradation and formation has not been established. Conclusions: Contact analysis provides numerous benefits for dental and oral health sciences, however, a fundamental understanding and improved methodology are necessary. # 2014 Published by Elsevier Ireland on behalf of Japan Prosthodontic Society.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . Tooth-to-tooth contact . . Restorative interface . . . . Bone–implant integration
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* Corresponding author at: Removable Partial Prosthodontics, Oral Health Sciences, Graduate School of Medical and Dental Sciences, Tokyo Medical and Dental University, 1-5-45 Yushima, Bunkyo, Tokyo 113-8549, Japan. Tel.: +81 358034935. E-mail address: [email protected] (N. Wakabayashi). http://dx.doi.org/10.1016/j.jpor.2014.03.001 1883-1958/# 2014 Published by Elsevier Ireland on behalf of Japan Prosthodontic Society.
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journal of prosthodontic research 58 (2014) 92–101
4.1. Contact option at the osseointegration . 4.2. Coefficient of friction. . . . . . . . . . . . . . . 4.3. Effective stress and strain thresholds . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
The finite element (FE) method features a series of computational procedures that calculate the stress and strain within a structural model caused by external force, pressure, thermal change, magnetic field power, and other factors. The method is extremely useful in estimating the biomechanical characteristics of dental prostheses and supporting oral tissues that are difficult to measure in vivo. The stress and strain estimated through model structures can be analyzed using visualization software within the FE environment to evaluate a variety of physical parameters. Until recently, linear static models have been employed primarily in dental biomechanics. A constant elastic modulus representing the linear stress–strain relationship of each material or oral tissue may be entered into a FE program. Linear analyses are valid in model structures with a linear stress–strain relationship up to a stress level known as the proportional limit and a within a bonded single unit volume. However, most oral physical phenomena are not adequately simulated by linear static structures; oral tissues and biomaterials exhibit time-dependent and status-dependent characteristics under mechanical stress. A more realistic simulation generates nonlinearities especially in principal categories such as changing status. This structural behavior is commonly observed in intra-oral contacts such as between occluding antagonistic teeth, adjacent teeth, and frictional contact between the denture and supporting tissues. Stress and strain may change dramatically with the changing status of each contact. Therefore, accurate calculation of the mechanical output from model simulations is essential. Finite element contact analysis has recently appeared in numerous dental and prosthodontic studies. The method has become an increasingly powerful predictor of the realistic structural stress and strain that cannot be estimated in a linear static model. However, the benefits and limitations have not been thoroughly examined, particularly for data interpretation. The key elements required for the design and appropriate utilization of this method should be fully discussed. The present review describes the recent developments in the application of contact analysis to prosthodontics research of tooth-to-tooth contact, restorative interface, and bone–implant integration.
2.
Tooth-to-tooth contact
The fracture risk of the enamel and ceramic restorations is determined by calculating the stress and strain distributions associated with tooth-to-tooth contact under occlusal loading. To estimate occlusal surface stresses using a linear elastic model, force is applied onto a node or an element, or the pressure upon an area of the occlusal surface is estimated to
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96 96 97 99 99
simulate occlusal loading. The resultant stress may be erroneous because this assumed loading condition is likely to excessively concentrate stress peripherally around the loading site, which is far from the reality. These problems can be partially resolved by using contact analysis, which estimates sliding and friction phenomena during mastication using contact elements to simulate the contacting maxillary and mandibular teeth. However, the method is highly nonlinear and difficult to solve due to several limitations. First, the contact regions are unknown until a sequence of the problem has been solved. Depending upon the load, material, and environmental factors, the surfaces can move in and out of contact with each other unpredictably. Secondly, friction should be considered in most contact analyses. The frictional responses can be chaotic and make solution convergence difficult. In addition, contact simulations often require significant computer resources to perform. Contact analyses enhance the estimated accuracy of the Hertzian damage experienced by the enamel and dental ceramics. Fig. 1 demonstrates the stress distributions created by a simple point loading on the occlusal surface of a linear static model (left) and a nonlinear contact simulation using the opposing tooth cusp (right). The linear analysis predicted an unrealistically high stress concentration and a tensile area surrounding the loaded point that does not reflect a real condition. The contact analysis by contrast predicted a compressive area from the contact cusp. Contact analyses of tooth surfaces appeared within a twodimensional model devised by Magne and Belser [1]. They demonstrated the stress distribution in maxillary and mandibular molar teeth during working, non-working, and vertical closure of the jaw. The same group later comprised computerized tomography (CT)-based FE models using standard triangle language (STL) and Boolean operations (volume addition, intersection, or subtraction) [2–4]. The method recently estimated the fracture risk of ultra-thin composite resin occlusal veneers [5]. The forces acting upon the molar teeth during mastication constantly change direction, magnitude, and location, depending on the specific contact between opposing tooth surfaces. Dejak et al. conducted a nonlinear contact simulation of the interface between molar occlusal surfaces and morsels, then analyzed the induced stresses on a mandibular molar during clenching and chewing of morsels of various elastic moduli [6]. The same group later assessed the mechanism underlying cervical lesion formation [7]. Damaged elements were removed from the computer tooth model based on the Tsai–Wu criterion. In addition, the strength of mandibular molars restored with composite resin inlays was compared to those restored with ceramic inlays, according to the Mohr–Coulomb failure criterion [8]. Recently, the strength of thin-walled molar crowns comprising various materials was analyzed under simulated mastication [9].
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journal of prosthodontic research 58 (2014) 92–101
Fig. 1 – Linear static and nonlinear contact simulations of occlusal tooth contact. A static load (red arrow) was directed onto a surface spot of a linear static model (upper left) and a nonlinear analysis, conducted by contact simulation of opposing tooth cusps with a friction coefficient of 0.3 (upper right). The maximum first principal stress distributions of the functional cusp are illustrated using stress contour graphics, with the surface area of the highest stress level (tensile) colored red and that of the lowest level (compressive) colored blue (color bar by stress level). The linear static model shows an unrealistically sharp concentration of tension surrounding the loading spot (lower left). In contrast, the nonlinear model realistically represents an area of compressive contact (lower right).
Patient-specific or patient-based models have been used to assess the occlusal surface stress under mechanical function. The surface contacts between antagonistic maxillary and mandibular teeth during the power stroke has been kinetically analyzed using Occlusal Fingerprint Analyzer (OFA) software, which identified the occlusal contact areas in human or cadaver-based 3D FE models [10–12]. The combination of the FE method and OFA revealed the changing force distributions during occlusion created by interproximal wear facets. The interproximal contacts between natural teeth and implants were simulated to enable precise calculation of bone strain at the implant–bone interface [13]. The strain distributions based on FE model were compared to those determined experimentally using a full-field surface strain measurement technique (digital image correlation). A nonlinear frictionless condition was assumed, allowing free sliding and gap
formation between their relative displacements. The authors concluded that surface strain measurement effectively validated the FE models of dental and implant restorations. Contact analysis was also employed in a model of tooth enamel fracture with graded elastic modulus and toughness [14,15]. Another model analyzed variable wear using contact analysis by focusing on the relationship between tooth wear and food properties [16]. The model simulated a hemispherical shell (enamel) on a compliant interior (dentin) and replicated occlusal loading with contact to a flat or curved and hard or soft indenter. The effect of enamel microstructure prism orientation on surface stresses was also explored in yet another study [17]. Prism orientation is location-dependent, and the friction coefficient and wear resistance change accordingly. The study found greater wear resistance at the intercuspal region but less at the lateral enamel [17].
journal of prosthodontic research 58 (2014) 92–101
Contact analyses have been used extensively to evaluate dental ceramic failure mechanisms. Fracture strengths and failure modes obtained from indentation tests of dental ceramics have been interpreted using tailored FE models [18–20]. The stress distributions from the nonlinear calculations were frequently implicated in the failure mechanics. Contact-induced plastic deformation and cracking in yttriastabilized zirconia samples were demonstrated using both an experimental and FE simulated Hertzian contact model; the findings suggested that cyclic loading caused significant longterm strength degradation [21]. Contact analysis findings have been compared to those of SEM and EDX analyzing the veneered zirconia-based crowns fracture resistance of with either luted or fused veneer assessed by fractography [22]. Contact analysis has been used to simulate basic mechanical tests such as three-point and four-point flexural tests, which evaluate the rectangular beam specimen and loading pins contact surfaces [23], passive gripping of micro tensile devices and the gripping jig [24], and the contact area between the piston and dental ceramic disk in the piston-on-three-ball and piston-on-ring tests [25]. The contact analysis is essential to explain material failure and justify and verify in vitro experiments. If a three-point bending test is simulated by linear static model, the specimen beam is fixed by two columnar supports. This condition may create a high tensile stress on the lower beam not only centrally, but also near the supports. Yet, the tensile stress at the supports is unrealistic; it is caused by horizontal plane tension secondary to bilateral rigid fixation surrounding the support. In contrast, the contact simulation with friction allows the beam to slide along the supports and thus, can provide a reasonable stress distribution that show strong compression at the support contacts.
3.
Restorative interface
Tooth–restoration complex stress has been analyzed in numerous studies to predict the failure risk at the interface and within the bonded tooth structures. In a linear static FE analysis, the interface between individual structures of different elastic properties shares the same node, which represents a perfect bond. However, FE results may be erroneously interpreted using this conventional approach [26]. When an external force is directed onto a linear static model, the stress inside the model may immediately rise and transfer continuously without a gap. However, in a discontinuous volume comprising hard tissues and materials that are not perfectly bonded, the interfacial surfaces can move in and out of contact with each other in a microscopic level, and the interface stress transfer may not be continuous; the stress distribution of the collective volume may change dramatically. Subsequent failure can be influenced by the stress calculated according to the changing contact. The application of the contact elements to the tooth–restoration model was essential to directly compare the normal and tangential stresses at the interfacial surface and reported bond strengths. Theoretically, the initial interface debonding occurs when either the normal stress exceeds the tensile strength or the tangential stress overtakes the shear bond strength.
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The stress at the restoration and cavity wall interface was calculated for the composite polymerization contraction and the luting cement [27]. The study found that the traction vector components produced by the inlay were higher than the directly placed bulk restoration, presumably secondary to the visco-elastic flow of uncured resin. Another contact analysis at the Class II restoration and cavity wall interface indicated that the contact condition may increase the fracture potential compared to the bonded interface calculated using linear analysis; as the cavity depth increased, an unfavorable stress gradient became evident [28]. Alternatively, the adhesive layer was modeled in the FE program using spring elements that connecting the natural tooth cavity wall nodes to those of the composite restoration [29]. Although this was not the contact analysis, the normal and shear adhesive layer stiffness was simulated by connecting each node pair to three different springs, one directed normally to the composite–tooth interface and the other two parallel to the interface. As a result, the adhesive layer strain affected the attenuation in the polymerization and occlusal loading stresses. Another study using linear static analysis defined the interface safety factor by dividing the bond strength values by the normal and shear stresses occurring beyond 10% of the interface surface area [30]. Adaptive growth, which is a repetitive auto-modification of cavity geometry with the optimization cycles, was incorporated into 2D premolar FE models [31]. Nonlinear contact conditions were created along the tooth–restoration interface to replicate imperfect bonding, and frictional coefficients defined the interfacial condition; for example, a very low frictional coefficient of 0.01 simulated absent bonding. As a result, the maximum interface stress was significantly reduced using the optimization technique. Another group similarly used 2D FE models of non-carious cervical lesions based on the discrete solver to model automatic transition from continuous to discrete under high stress field in a timedependent failure simulation [32,33]. The model revealed that the strain softened secondary to material micro damage, and the highly stiff restorative materials in current use concentrated stress in a manner that could induce cervical margin failure. In another recent study, the stress-peaks centered at the restorative interface were investigated relative to the cavity wall configuration factor [34]. The contact analysis indicated that an increased factor did not increase the calculated stress-peaks in rectangular Class I cavity walls. The method also simulated an incomplete bond between a feldspathic ceramic veneer and zirconia substructure in a complete crown, with an interface frictional coefficient of 0.3 [35]. The incomplete bond increased the maximum stress within the ceramic veneer compared to the stress under a perfect bond condition. The benefit of contact simulation was demonstrated recently in studies investigating the influence of debonding on the stress distribution within all-ceramic restorations [36,37]. In one study, all-ceramic crown failure attributed to cement degradation was examined using digital image correlation, with visualized corresponding results shown by the FE contact simulation [37]. The stress analysis found that the maximum principal stress at the ceramic interface dramatically increased secondary to lost bond strength.
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Until recently, studies did not perform contact analysis of the root fracture risk of post-and-core materials in endodontically treated teeth. A series of studies by Dejak et al. employed contact analysis at the luting cement and dentin interface in post-and-core tooth models. Cast metal posts [38] and ceramic endocrowns [39] decreased dentin stress within restored teeth more than the FRC posts did. In addition, the tooth with ferrule effect substantially influenced stress reduction [40]. Because contact analysis is a nonlinear and time-dependent simulation, the technique may reveal the stress distribution course from the initial failure to the condition post-loading. To date, however, the gap and contact sites have rarely been demonstrated in stress contour visualization. A comparison between the contacted and perfectly bonded conditions may alternatively characterize the influence of debonding on the post-and-core stress in tooth models. The elastic modulus mismatch between the root dentin and the post-and-core materials is a strong potential factor that may induce interface debonding [41]. Recent studies using contact simulation at the bonded interface indicated that the adhesion of the endodontic restoration to tooth structures was critical in root fracture development [41,42]. Contact analysis has been also employed in removable prosthodontics. The sliding phenomena of the occlusal rest and the rest seat were simulated to demonstrate the timedependent stresses in the abutting tooth periodontium [43]. The 3D model indicated that tooth displacement and periodontal ligament stress, which are caused by forces exerted by removable prostheses, were small and within the tissue physiological limits. In another study, the interfacial stress between denture base resin and the Co–Cr alloy framework created by polymerization shrinkage was calculated using surface-to-surface contact analysis [44]. More recently, the efficacy of the retaining method for bar implant-overdenture was assessed based on a simulated frictional contact coefficient of 0.334 between overdentures and underlying mucosa during sliding and rotational movements [45]. However, the effect of the assumed friction on the resultant stress was not thoroughly explained in the study.
bone level and loss of osseointegration, occur precisely at the osseointegrated interface. Fig. 2 illustrates the maximum principal and equivalent stress distributions of the tensile and compressive sides of the bone–implant crestal interface under an oblique bite force. The linear elastic model generated a perfectly integrated bone–implant complex (left), while the contact element was applied to the interface of the nonlinear model (right). The resultant stress distributions were completely different; a stress concentration appeared in the cervical bone on the tensile side for the perfectly bonded model, while a micro-gap was observed in the contact analysis. The relatively high stress concentration on the compression side of the contact model suggested a hard contact of the implant neck onto the cervical cortical bone. The linear static model with a perfect bonded interface generated an unrealistic condition, particularly for an unhealed bone. Contact analyses have also evaluated the influence of bone quality on micro-motion and bone strain. For this purpose, two extreme scenarios have been modeled for the bone– implant interface [46]. The first is a complete bond with an infinite tension strength and shear at the osseointegration interface or in bioactive material; the second is a free contact simulation of relative displacement without tensile force resistance to describe healing or an immediately loaded condition. The latter condition was simulated using contact analysis. Implant and bone served as deformable bodies, and non-penetration constraint was automatically imposed by the program; boundaries from one body were unable to cross the surface of another body. These studies reported increased local strain or micro-motion in the contact model compared to the completely bonded model. Duyck et al. assigned a 1 MPa interface tensile strength, which was surpassed when the tensile stress normal to the interface exceeded the strength [47]. Implant displacement relative to the alveolus with a resultant gap space was clearly demonstrated in the strain contour graphic of this study. Contact analysis has been used in a report investigating the trabecular bone architecture generated from the segmented mCT scans that was accounted for its contact interactions with the implant [48].
4.2.
4.
Bone–implant integration
4.1.
Contact option at the osseointegration
Cortical and cancellous bone with one or more installed implants is one of the most generated FE models currently, likely because numerous unsolved biomechanical questions remain within implant dentistry. Conventionally, linear elastic models simulated the bone–implant complex as a completely bonded structure. Due to its higher Young’s modulus compared to cortical bone, a titanium implant immediately adjacent to the interface theoretically absorbs mechanical energy and stress and may lower the bone surface strain and displacement. However, this simulation is questionable because this inhibitory effect can occur only in a bone–implant complex that is perfectly bonded. In reality, most short- and long-term implant failures, such as decreased
Coefficient of friction
Contact analyses using different frictional coefficients have simulated different integration qualities during the osseointegration process [49]. The bonded model simulates perfect osseointegration between the implant and surrounding cortical bone preventing sliding and separation at the implant–cortical bone interface. The contact model by contrast, replicates an imperfect osseointegration permitting contact interface separation with frictionless sliding. In this scenario, the interface stress and micro-motion were estimated in the immediately loaded implants. Different frictional coefficients represented the varying roughness of the implant surface (a lower coefficient for a smooth metal surface and higher for a porous or excessively rough surface), in addition to simulating the bonded interface [50]. Different friction coefficients also simulated the initial stability of immediately loaded implants and implants placed within the freshly extracted alveolus. Under this scenario,
journal of prosthodontic research 58 (2014) 92–101
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Fig. 2 – Stress distributions by the linear static and nonlinear contact analyses of the bone–implant complex. Simulated bite force was directed onto the abutment in an oblique direction from left to right (red arrow in upper figure). The implant was rigidly fixed to the cortical bone in the linear static model, while the contact elements were applied to the bone–implant interface in the nonlinear model. The stress contour graphics are presented by the cross-sectioned views. The linear analysis indicated a relatively mild stress concentration on the both tensile and compressive sides of the bone surface (middle and lower figures on left). The contact analysis revealed a micro-gap at the bone–implant interface (middle on right) with very little stress on the tensile side and a remarkable stress concentration on the compression side (lower right).
contact analysis assessed the biomechanical effects of implant diameter and length [51,52], abutment angulation [53], tapering angles [54], specific design factors of the orthodontic mini-screw [55], implant topology optimization [56], implant material [57,58], thread design [58,59], and the anterior implant position and orientation [60]. The studies found that the peak bone stress surrounding the immediately loaded implant (contact bone–implant interfaces) was higher than that surrounding the osseointegrated implant (bonded interfaces). In another study, however, the bone strain surrounding the immediately loaded implant in contact with the bone surface was comparable to that surrounding the completely bonded implant–bone complex, when were used as abutments to bar-retained maxillary overdentures [61]. Notably, one study compared the bone strain in the human cadavers measured using strain gauges to that calculated
using FE contact analysis [62]. The two different methods did not suggest qualitative inconsistency in the detection of strain, and despite using a single friction coefficient in the FE analysis, the agreement between the bone strains was high. However, there is no study evaluating the relationship between the friction coefficient, bone quality, and the implant surface roughness. The validity and reliability of the mathematical models using adapted friction coefficients to simulate bone quality and osseointegration magnitude should further be tested before this technique is made a standard biomechanical assessment.
4.3.
Effective stress and strain thresholds
Including the contact concept into FE analysis undoubtedly improves the accuracy of the local stress and strain within the
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Fig. 3 – Superimposed image of cell activity and strain distribution. A histological image of a loaded mouse maxilla (left), the corresponding regional strain distribution obtained using a CT-based finite element model (center), and the two images superimposed (right).
bone–implant complex. Therefore, stress and strain is expected to predict the loss of osseointegration at the bone– implant interface. If the calculated maximum tensile or compressive stresses surpass the osseointegrated interface strength, then one may hypothesize that interface failure will occur. It is generally accepted that stimulations based on peak strain and strain energy density are consistent with in vivo data [63]. However, no in vivo data evaluating the effective stress as a critical stimulus of bone degradation and formation currently exists. Notably, the majority of FE oral tissue models were interpreted based on an unsolved presumption, that a higher maximum stress, illustrated by red coloration in stress contour graphics, indicates tissue failure or damage. Although this presumption may be true in solid materials such as ceramics, polymers, and alloys, it may not be true in hard and soft living tissues. There is no conclusive evidence that the bone resorption is caused by highly concentrated stress, including compressive stress. On the contrary, it has been well recognized that the bone architecture atrophies or hypertrophies depending on the habitual mechanical stimulation magnitude and frequency. Prolonged stress below the normal range would decrease bone mass and increase bone stress [64]. This concept, termed adaptive modeling, was later formulated as a trigger criterion or threshold for bone resorption and apposition [65,66]. The approach lends itself to numerical modeling approaches including the FE method, which improves theories and the development of new models [67–71]. The most famous model describing bone growth and loss may be the Mechanostat theory promoted by Frost [72]. This concept asserts that bone growth and loss are stimulated by local mechanical elastic deformation, defined as disuse, adaption, overload, and fracture, and controlled by the strain level. Although the theoretical formulas were mostly advocated within the orthopedics field, there is no effective stress magnitude directly predicting bone adaptability. In dentistry, models have been formulated to assess the internal bone strains associated with dental implants [47,73] and removable prosthodontics [74]. Investigation of the bone response in the rabbit tibiae under static and dynamic loads,
histomorphometrical quantifications, and contact analysis of the bone–implant interface all evidence the continuing process of bone resorption [47]. To assess the effect of mechanical stimulation on the denture-bearing residual ridge, a combination of bone histomorphometry and the FE method simulating the external loads on the mouse maxilla were performed [74]. The study revealed that osteoclastic resorption was location-dependent and sensitive to the local strain intensity. Despite increasingly numerous studies, the mechanism of bone resorption associated with implant and prosthodontic interventions and the effective stress and strain levels have not been established. The peri-implant bone level before and after occlusal loading serves as the most common and reliable outcome for the dental implants. While excessive bite force or overloading have been proposed as a cause of late implant failures, the association between overload and peri-implant bone level has not been proven [75,76]. The current consensus is that the biomechanical scales (stress, strain) are not determinant factors of the implant failure or success; they are less important than the microbial accumulation around the implant. However, the significance of the stress and strain generated in the human hard and soft tissues is presently being explored in numerous studies. Increasingly, studies employ numerically modeled histological scales of animal subjects and cultured cells to determine the mechanical influences on biological activity. This approach may strengthen proposed rationale explaining the biomechanical phenomena observed in in vivo. These dental studies may raise potential concerns in bone formation and deposition in loaded animals. Fig. 3 illustrates a histological view of a loaded mouse maxilla (left), the strain distribution of the corresponding region obtained using CT-based finite element modeling (center), and the two images super-imposed (right). The combined image may facilitate assessment of the biomechanical factor effects on cell activity and the morphometric outcome. These studies will include animal histomorphometric and patient-based approaches to discern the association between bone morphometry and stress in the long-term basis.
journal of prosthodontic research 58 (2014) 92–101
Conflict of interest [14]
The authors have no conflict of interest with respect to the manuscript content or funding.
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Acknowledgements
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The authors acknowledge Drs. Kengo Fujiki and Yusuke Toyoshima (Removable Partial Prosthodontics, Tokyo Medical and Dental University) for performing mathematical and biological analyses for figures. This work was partially supported by grants (No. 25893068 to N.M. and No. 24592902 to N.W.) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.
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[52] Lee JS, Cho IH, Kim YS, Heo SJ, Kwon HB, Lim YJ. Bone– implant interface with simulated insertion stress around an immediately loaded dental implant in the anterior maxilla: a three-dimensional finite element analysis. Int J Oral Maxillofac Implants 2012;27:295–302. [53] Kao HC, Gung YW, Chung TF, Hsu ML. The influence of abutment angulation on micromotion level for immediately loaded dental implants: a 3-D finite element analysis. Int J Oral Maxillofac Implants 2008;23:623–30. [54] Atieh MA, Shahmiri RA. Evaluation of optimal taper of immediately loaded wide-diameter implants: a finite element analysis. J Oral Implantol 2013;39:123–32. [55] Lin CL, Yu JH, Liu HL, Lin CH, Lin YS. Evaluation of contributions of orthodontic mini-screw design factors based on FE analysis and the Taguchi method. J Biomech 2010;43:2174–81. [56] Chang CL, Chen CS, Huang CH, Hsu ML. Finite element analysis of the dental implant using a topology optimization method. Med Eng Phys 2012;34:999–1008. [57] Chang CL, Chen CS, Yeung TC, Hsu ML. Biomechanical effect of a zirconia dental implant-crown system: a threedimensional finite element analysis. Int J Oral Maxillofac Implants 2012;27:e49–57. [58] Fuh LJ, Hsu JT, Huang HL, Chen MY, Shen YW. Biomechanical investigation of thread designs and interface conditions of zirconia and titanium dental implants with bone: three-dimensional numeric analysis. Int J Oral Maxillofac Implants 2013;28:e64–71. [59] Chang PK, Chen YC, Huang CC, Lu WH, Chen YC, Tsai HH. Distribution of micromotion in implants and alveolar bone with different thread profiles in immediate loading: a finite element study. Int J Oral Maxillofac Implants 2012;27: e96–101. [60] Hussein MO, Rabie ME. 3-Dimensional non-linear contact FEA analysis of mandibular all-on-4 design with different anterior implant positions and angulations. J Oral Implantol 2013. PMID: 24032471. [61] Akca K, Eser A, Eckert S, Cavusoglu Y, Cehreli MC. Immediate versus conventional loading of implantsupported maxillary overdentures: a finite element stress analysis. Int J Oral Maxillofac Implants 2013;28:e57–63. [62] Eser A, Akca K, Eckert S, Cehreli MC. Nonlinear finite element analysis versus ex vivo strain gauge measurements on immediately loaded implants. Int J Oral Maxillofac Implants 2009;24:439–46. [63] Mellal A, Wiskott HW, Botsis J, Scherrer SS, Belser UC. Stimulating effect of implant loading on surrounding bone. Comparison of three numerical models and validation by in vivo data. Clin Oral Implants Res 2004;15:239–48. [64] Rubin CT, Lanyon LE. Regulation of bone mass by mechanical strain magnitude. Calcif Tissue Int 1985;37: 411–7. [65] Frost HM. Bone mass and the mechanostat: a proposal. Anat Rec 1987;219:1–9. [66] Frost HM. Wolff law and bones structural adaptations to mechanical usage – an overview for clinician. Angle Orthod 1994;64:175–88. [67] Carter DR, Rapperport DJ, Fyhrie DP, Schurman DJ. Relation of coxarthrosis to stresses and morphogenesis. A finite element analysis. Acta Orthop Scand 1987;58:611–9. [68] Cowin SC. Bone stress adaptation models. J Biomech Eng 1993;115:528–33. [69] Mosley JR, Lanyon LE. Strain rate as a controlling influence on adaptive modeling in response to dynamic loading of the ulna in growing male rats. Bone 1998;23:313–8. [70] Brunski JB. In vivo bone response to biomechanical loading at the bone/dental–implant interface. Adv Dent Res 1999;13:99–119.
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Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/jpor
Review
Finite element contact analysis as a critical technique in dental biomechanics: A review Natsuko Murakami DDS, PhD, Noriyuki Wakabayashi DDS, PhD* Removable Partial Prosthodontics, Oral Health Sciences, Graduate School of Medical and Dental Sciences, Tokyo Medical and Dental University, Japan
article info
abstract
Article history:
Purpose: Nonlinear finite element contact analysis is used to mathematically estimate stress
Received 27 February 2014
and strain in a time- and status-dependent mechanical model. However, the benefits and
Accepted 10 March 2014
limitations of this method have not been thoroughly examined.
Available online 4 April 2014
Study selection: The current review summarizes the utility of contact analysis in character-
Keywords:
implant integration.
Finite element analysis
Results: Opposing tooth contact, friction, and sliding phenomena were simulated to esti-
izing individual stressors: (1) tooth-to-tooth contact, (2) restorative interface, and (3) bone–
Stress
mate stress distribution and assess the failure risk for tooth enamel, composite, and ceramic
Dental
restorations. Mechanical tests such as the flexural tests were simulated with the contact
Implant
analysis to determine the rationale underlying experimental findings. The tooth–restoration complex was modeled with interface contact elements that simulate imperfect bonding,
Nonlinear
and the normal and tangential stresses were calculated to predict failure progression. Previous studies have used a friction coefficient to represent osseointegration adjacent to dental implants, but the relationship between interface friction and the bone quality is unknown. An understanding of the local stress and strain may better predict loss of osseointegration, however, the effective stress as a critical contributor to bone degradation and formation has not been established. Conclusions: Contact analysis provides numerous benefits for dental and oral health sciences, however, a fundamental understanding and improved methodology are necessary. # 2014 Published by Elsevier Ireland on behalf of Japan Prosthodontic Society.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . Tooth-to-tooth contact . . Restorative interface . . . . Bone–implant integration
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* Corresponding author at: Removable Partial Prosthodontics, Oral Health Sciences, Graduate School of Medical and Dental Sciences, Tokyo Medical and Dental University, 1-5-45 Yushima, Bunkyo, Tokyo 113-8549, Japan. Tel.: +81 358034935. E-mail address: [email protected] (N. Wakabayashi). http://dx.doi.org/10.1016/j.jpor.2014.03.001 1883-1958/# 2014 Published by Elsevier Ireland on behalf of Japan Prosthodontic Society.
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4.1. Contact option at the osseointegration . 4.2. Coefficient of friction. . . . . . . . . . . . . . . 4.3. Effective stress and strain thresholds . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
The finite element (FE) method features a series of computational procedures that calculate the stress and strain within a structural model caused by external force, pressure, thermal change, magnetic field power, and other factors. The method is extremely useful in estimating the biomechanical characteristics of dental prostheses and supporting oral tissues that are difficult to measure in vivo. The stress and strain estimated through model structures can be analyzed using visualization software within the FE environment to evaluate a variety of physical parameters. Until recently, linear static models have been employed primarily in dental biomechanics. A constant elastic modulus representing the linear stress–strain relationship of each material or oral tissue may be entered into a FE program. Linear analyses are valid in model structures with a linear stress–strain relationship up to a stress level known as the proportional limit and a within a bonded single unit volume. However, most oral physical phenomena are not adequately simulated by linear static structures; oral tissues and biomaterials exhibit time-dependent and status-dependent characteristics under mechanical stress. A more realistic simulation generates nonlinearities especially in principal categories such as changing status. This structural behavior is commonly observed in intra-oral contacts such as between occluding antagonistic teeth, adjacent teeth, and frictional contact between the denture and supporting tissues. Stress and strain may change dramatically with the changing status of each contact. Therefore, accurate calculation of the mechanical output from model simulations is essential. Finite element contact analysis has recently appeared in numerous dental and prosthodontic studies. The method has become an increasingly powerful predictor of the realistic structural stress and strain that cannot be estimated in a linear static model. However, the benefits and limitations have not been thoroughly examined, particularly for data interpretation. The key elements required for the design and appropriate utilization of this method should be fully discussed. The present review describes the recent developments in the application of contact analysis to prosthodontics research of tooth-to-tooth contact, restorative interface, and bone–implant integration.
2.
Tooth-to-tooth contact
The fracture risk of the enamel and ceramic restorations is determined by calculating the stress and strain distributions associated with tooth-to-tooth contact under occlusal loading. To estimate occlusal surface stresses using a linear elastic model, force is applied onto a node or an element, or the pressure upon an area of the occlusal surface is estimated to
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simulate occlusal loading. The resultant stress may be erroneous because this assumed loading condition is likely to excessively concentrate stress peripherally around the loading site, which is far from the reality. These problems can be partially resolved by using contact analysis, which estimates sliding and friction phenomena during mastication using contact elements to simulate the contacting maxillary and mandibular teeth. However, the method is highly nonlinear and difficult to solve due to several limitations. First, the contact regions are unknown until a sequence of the problem has been solved. Depending upon the load, material, and environmental factors, the surfaces can move in and out of contact with each other unpredictably. Secondly, friction should be considered in most contact analyses. The frictional responses can be chaotic and make solution convergence difficult. In addition, contact simulations often require significant computer resources to perform. Contact analyses enhance the estimated accuracy of the Hertzian damage experienced by the enamel and dental ceramics. Fig. 1 demonstrates the stress distributions created by a simple point loading on the occlusal surface of a linear static model (left) and a nonlinear contact simulation using the opposing tooth cusp (right). The linear analysis predicted an unrealistically high stress concentration and a tensile area surrounding the loaded point that does not reflect a real condition. The contact analysis by contrast predicted a compressive area from the contact cusp. Contact analyses of tooth surfaces appeared within a twodimensional model devised by Magne and Belser [1]. They demonstrated the stress distribution in maxillary and mandibular molar teeth during working, non-working, and vertical closure of the jaw. The same group later comprised computerized tomography (CT)-based FE models using standard triangle language (STL) and Boolean operations (volume addition, intersection, or subtraction) [2–4]. The method recently estimated the fracture risk of ultra-thin composite resin occlusal veneers [5]. The forces acting upon the molar teeth during mastication constantly change direction, magnitude, and location, depending on the specific contact between opposing tooth surfaces. Dejak et al. conducted a nonlinear contact simulation of the interface between molar occlusal surfaces and morsels, then analyzed the induced stresses on a mandibular molar during clenching and chewing of morsels of various elastic moduli [6]. The same group later assessed the mechanism underlying cervical lesion formation [7]. Damaged elements were removed from the computer tooth model based on the Tsai–Wu criterion. In addition, the strength of mandibular molars restored with composite resin inlays was compared to those restored with ceramic inlays, according to the Mohr–Coulomb failure criterion [8]. Recently, the strength of thin-walled molar crowns comprising various materials was analyzed under simulated mastication [9].
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Fig. 1 – Linear static and nonlinear contact simulations of occlusal tooth contact. A static load (red arrow) was directed onto a surface spot of a linear static model (upper left) and a nonlinear analysis, conducted by contact simulation of opposing tooth cusps with a friction coefficient of 0.3 (upper right). The maximum first principal stress distributions of the functional cusp are illustrated using stress contour graphics, with the surface area of the highest stress level (tensile) colored red and that of the lowest level (compressive) colored blue (color bar by stress level). The linear static model shows an unrealistically sharp concentration of tension surrounding the loading spot (lower left). In contrast, the nonlinear model realistically represents an area of compressive contact (lower right).
Patient-specific or patient-based models have been used to assess the occlusal surface stress under mechanical function. The surface contacts between antagonistic maxillary and mandibular teeth during the power stroke has been kinetically analyzed using Occlusal Fingerprint Analyzer (OFA) software, which identified the occlusal contact areas in human or cadaver-based 3D FE models [10–12]. The combination of the FE method and OFA revealed the changing force distributions during occlusion created by interproximal wear facets. The interproximal contacts between natural teeth and implants were simulated to enable precise calculation of bone strain at the implant–bone interface [13]. The strain distributions based on FE model were compared to those determined experimentally using a full-field surface strain measurement technique (digital image correlation). A nonlinear frictionless condition was assumed, allowing free sliding and gap
formation between their relative displacements. The authors concluded that surface strain measurement effectively validated the FE models of dental and implant restorations. Contact analysis was also employed in a model of tooth enamel fracture with graded elastic modulus and toughness [14,15]. Another model analyzed variable wear using contact analysis by focusing on the relationship between tooth wear and food properties [16]. The model simulated a hemispherical shell (enamel) on a compliant interior (dentin) and replicated occlusal loading with contact to a flat or curved and hard or soft indenter. The effect of enamel microstructure prism orientation on surface stresses was also explored in yet another study [17]. Prism orientation is location-dependent, and the friction coefficient and wear resistance change accordingly. The study found greater wear resistance at the intercuspal region but less at the lateral enamel [17].
journal of prosthodontic research 58 (2014) 92–101
Contact analyses have been used extensively to evaluate dental ceramic failure mechanisms. Fracture strengths and failure modes obtained from indentation tests of dental ceramics have been interpreted using tailored FE models [18–20]. The stress distributions from the nonlinear calculations were frequently implicated in the failure mechanics. Contact-induced plastic deformation and cracking in yttriastabilized zirconia samples were demonstrated using both an experimental and FE simulated Hertzian contact model; the findings suggested that cyclic loading caused significant longterm strength degradation [21]. Contact analysis findings have been compared to those of SEM and EDX analyzing the veneered zirconia-based crowns fracture resistance of with either luted or fused veneer assessed by fractography [22]. Contact analysis has been used to simulate basic mechanical tests such as three-point and four-point flexural tests, which evaluate the rectangular beam specimen and loading pins contact surfaces [23], passive gripping of micro tensile devices and the gripping jig [24], and the contact area between the piston and dental ceramic disk in the piston-on-three-ball and piston-on-ring tests [25]. The contact analysis is essential to explain material failure and justify and verify in vitro experiments. If a three-point bending test is simulated by linear static model, the specimen beam is fixed by two columnar supports. This condition may create a high tensile stress on the lower beam not only centrally, but also near the supports. Yet, the tensile stress at the supports is unrealistic; it is caused by horizontal plane tension secondary to bilateral rigid fixation surrounding the support. In contrast, the contact simulation with friction allows the beam to slide along the supports and thus, can provide a reasonable stress distribution that show strong compression at the support contacts.
3.
Restorative interface
Tooth–restoration complex stress has been analyzed in numerous studies to predict the failure risk at the interface and within the bonded tooth structures. In a linear static FE analysis, the interface between individual structures of different elastic properties shares the same node, which represents a perfect bond. However, FE results may be erroneously interpreted using this conventional approach [26]. When an external force is directed onto a linear static model, the stress inside the model may immediately rise and transfer continuously without a gap. However, in a discontinuous volume comprising hard tissues and materials that are not perfectly bonded, the interfacial surfaces can move in and out of contact with each other in a microscopic level, and the interface stress transfer may not be continuous; the stress distribution of the collective volume may change dramatically. Subsequent failure can be influenced by the stress calculated according to the changing contact. The application of the contact elements to the tooth–restoration model was essential to directly compare the normal and tangential stresses at the interfacial surface and reported bond strengths. Theoretically, the initial interface debonding occurs when either the normal stress exceeds the tensile strength or the tangential stress overtakes the shear bond strength.
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The stress at the restoration and cavity wall interface was calculated for the composite polymerization contraction and the luting cement [27]. The study found that the traction vector components produced by the inlay were higher than the directly placed bulk restoration, presumably secondary to the visco-elastic flow of uncured resin. Another contact analysis at the Class II restoration and cavity wall interface indicated that the contact condition may increase the fracture potential compared to the bonded interface calculated using linear analysis; as the cavity depth increased, an unfavorable stress gradient became evident [28]. Alternatively, the adhesive layer was modeled in the FE program using spring elements that connecting the natural tooth cavity wall nodes to those of the composite restoration [29]. Although this was not the contact analysis, the normal and shear adhesive layer stiffness was simulated by connecting each node pair to three different springs, one directed normally to the composite–tooth interface and the other two parallel to the interface. As a result, the adhesive layer strain affected the attenuation in the polymerization and occlusal loading stresses. Another study using linear static analysis defined the interface safety factor by dividing the bond strength values by the normal and shear stresses occurring beyond 10% of the interface surface area [30]. Adaptive growth, which is a repetitive auto-modification of cavity geometry with the optimization cycles, was incorporated into 2D premolar FE models [31]. Nonlinear contact conditions were created along the tooth–restoration interface to replicate imperfect bonding, and frictional coefficients defined the interfacial condition; for example, a very low frictional coefficient of 0.01 simulated absent bonding. As a result, the maximum interface stress was significantly reduced using the optimization technique. Another group similarly used 2D FE models of non-carious cervical lesions based on the discrete solver to model automatic transition from continuous to discrete under high stress field in a timedependent failure simulation [32,33]. The model revealed that the strain softened secondary to material micro damage, and the highly stiff restorative materials in current use concentrated stress in a manner that could induce cervical margin failure. In another recent study, the stress-peaks centered at the restorative interface were investigated relative to the cavity wall configuration factor [34]. The contact analysis indicated that an increased factor did not increase the calculated stress-peaks in rectangular Class I cavity walls. The method also simulated an incomplete bond between a feldspathic ceramic veneer and zirconia substructure in a complete crown, with an interface frictional coefficient of 0.3 [35]. The incomplete bond increased the maximum stress within the ceramic veneer compared to the stress under a perfect bond condition. The benefit of contact simulation was demonstrated recently in studies investigating the influence of debonding on the stress distribution within all-ceramic restorations [36,37]. In one study, all-ceramic crown failure attributed to cement degradation was examined using digital image correlation, with visualized corresponding results shown by the FE contact simulation [37]. The stress analysis found that the maximum principal stress at the ceramic interface dramatically increased secondary to lost bond strength.
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Until recently, studies did not perform contact analysis of the root fracture risk of post-and-core materials in endodontically treated teeth. A series of studies by Dejak et al. employed contact analysis at the luting cement and dentin interface in post-and-core tooth models. Cast metal posts [38] and ceramic endocrowns [39] decreased dentin stress within restored teeth more than the FRC posts did. In addition, the tooth with ferrule effect substantially influenced stress reduction [40]. Because contact analysis is a nonlinear and time-dependent simulation, the technique may reveal the stress distribution course from the initial failure to the condition post-loading. To date, however, the gap and contact sites have rarely been demonstrated in stress contour visualization. A comparison between the contacted and perfectly bonded conditions may alternatively characterize the influence of debonding on the post-and-core stress in tooth models. The elastic modulus mismatch between the root dentin and the post-and-core materials is a strong potential factor that may induce interface debonding [41]. Recent studies using contact simulation at the bonded interface indicated that the adhesion of the endodontic restoration to tooth structures was critical in root fracture development [41,42]. Contact analysis has been also employed in removable prosthodontics. The sliding phenomena of the occlusal rest and the rest seat were simulated to demonstrate the timedependent stresses in the abutting tooth periodontium [43]. The 3D model indicated that tooth displacement and periodontal ligament stress, which are caused by forces exerted by removable prostheses, were small and within the tissue physiological limits. In another study, the interfacial stress between denture base resin and the Co–Cr alloy framework created by polymerization shrinkage was calculated using surface-to-surface contact analysis [44]. More recently, the efficacy of the retaining method for bar implant-overdenture was assessed based on a simulated frictional contact coefficient of 0.334 between overdentures and underlying mucosa during sliding and rotational movements [45]. However, the effect of the assumed friction on the resultant stress was not thoroughly explained in the study.
bone level and loss of osseointegration, occur precisely at the osseointegrated interface. Fig. 2 illustrates the maximum principal and equivalent stress distributions of the tensile and compressive sides of the bone–implant crestal interface under an oblique bite force. The linear elastic model generated a perfectly integrated bone–implant complex (left), while the contact element was applied to the interface of the nonlinear model (right). The resultant stress distributions were completely different; a stress concentration appeared in the cervical bone on the tensile side for the perfectly bonded model, while a micro-gap was observed in the contact analysis. The relatively high stress concentration on the compression side of the contact model suggested a hard contact of the implant neck onto the cervical cortical bone. The linear static model with a perfect bonded interface generated an unrealistic condition, particularly for an unhealed bone. Contact analyses have also evaluated the influence of bone quality on micro-motion and bone strain. For this purpose, two extreme scenarios have been modeled for the bone– implant interface [46]. The first is a complete bond with an infinite tension strength and shear at the osseointegration interface or in bioactive material; the second is a free contact simulation of relative displacement without tensile force resistance to describe healing or an immediately loaded condition. The latter condition was simulated using contact analysis. Implant and bone served as deformable bodies, and non-penetration constraint was automatically imposed by the program; boundaries from one body were unable to cross the surface of another body. These studies reported increased local strain or micro-motion in the contact model compared to the completely bonded model. Duyck et al. assigned a 1 MPa interface tensile strength, which was surpassed when the tensile stress normal to the interface exceeded the strength [47]. Implant displacement relative to the alveolus with a resultant gap space was clearly demonstrated in the strain contour graphic of this study. Contact analysis has been used in a report investigating the trabecular bone architecture generated from the segmented mCT scans that was accounted for its contact interactions with the implant [48].
4.2.
4.
Bone–implant integration
4.1.
Contact option at the osseointegration
Cortical and cancellous bone with one or more installed implants is one of the most generated FE models currently, likely because numerous unsolved biomechanical questions remain within implant dentistry. Conventionally, linear elastic models simulated the bone–implant complex as a completely bonded structure. Due to its higher Young’s modulus compared to cortical bone, a titanium implant immediately adjacent to the interface theoretically absorbs mechanical energy and stress and may lower the bone surface strain and displacement. However, this simulation is questionable because this inhibitory effect can occur only in a bone–implant complex that is perfectly bonded. In reality, most short- and long-term implant failures, such as decreased
Coefficient of friction
Contact analyses using different frictional coefficients have simulated different integration qualities during the osseointegration process [49]. The bonded model simulates perfect osseointegration between the implant and surrounding cortical bone preventing sliding and separation at the implant–cortical bone interface. The contact model by contrast, replicates an imperfect osseointegration permitting contact interface separation with frictionless sliding. In this scenario, the interface stress and micro-motion were estimated in the immediately loaded implants. Different frictional coefficients represented the varying roughness of the implant surface (a lower coefficient for a smooth metal surface and higher for a porous or excessively rough surface), in addition to simulating the bonded interface [50]. Different friction coefficients also simulated the initial stability of immediately loaded implants and implants placed within the freshly extracted alveolus. Under this scenario,
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Fig. 2 – Stress distributions by the linear static and nonlinear contact analyses of the bone–implant complex. Simulated bite force was directed onto the abutment in an oblique direction from left to right (red arrow in upper figure). The implant was rigidly fixed to the cortical bone in the linear static model, while the contact elements were applied to the bone–implant interface in the nonlinear model. The stress contour graphics are presented by the cross-sectioned views. The linear analysis indicated a relatively mild stress concentration on the both tensile and compressive sides of the bone surface (middle and lower figures on left). The contact analysis revealed a micro-gap at the bone–implant interface (middle on right) with very little stress on the tensile side and a remarkable stress concentration on the compression side (lower right).
contact analysis assessed the biomechanical effects of implant diameter and length [51,52], abutment angulation [53], tapering angles [54], specific design factors of the orthodontic mini-screw [55], implant topology optimization [56], implant material [57,58], thread design [58,59], and the anterior implant position and orientation [60]. The studies found that the peak bone stress surrounding the immediately loaded implant (contact bone–implant interfaces) was higher than that surrounding the osseointegrated implant (bonded interfaces). In another study, however, the bone strain surrounding the immediately loaded implant in contact with the bone surface was comparable to that surrounding the completely bonded implant–bone complex, when were used as abutments to bar-retained maxillary overdentures [61]. Notably, one study compared the bone strain in the human cadavers measured using strain gauges to that calculated
using FE contact analysis [62]. The two different methods did not suggest qualitative inconsistency in the detection of strain, and despite using a single friction coefficient in the FE analysis, the agreement between the bone strains was high. However, there is no study evaluating the relationship between the friction coefficient, bone quality, and the implant surface roughness. The validity and reliability of the mathematical models using adapted friction coefficients to simulate bone quality and osseointegration magnitude should further be tested before this technique is made a standard biomechanical assessment.
4.3.
Effective stress and strain thresholds
Including the contact concept into FE analysis undoubtedly improves the accuracy of the local stress and strain within the
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Fig. 3 – Superimposed image of cell activity and strain distribution. A histological image of a loaded mouse maxilla (left), the corresponding regional strain distribution obtained using a CT-based finite element model (center), and the two images superimposed (right).
bone–implant complex. Therefore, stress and strain is expected to predict the loss of osseointegration at the bone– implant interface. If the calculated maximum tensile or compressive stresses surpass the osseointegrated interface strength, then one may hypothesize that interface failure will occur. It is generally accepted that stimulations based on peak strain and strain energy density are consistent with in vivo data [63]. However, no in vivo data evaluating the effective stress as a critical stimulus of bone degradation and formation currently exists. Notably, the majority of FE oral tissue models were interpreted based on an unsolved presumption, that a higher maximum stress, illustrated by red coloration in stress contour graphics, indicates tissue failure or damage. Although this presumption may be true in solid materials such as ceramics, polymers, and alloys, it may not be true in hard and soft living tissues. There is no conclusive evidence that the bone resorption is caused by highly concentrated stress, including compressive stress. On the contrary, it has been well recognized that the bone architecture atrophies or hypertrophies depending on the habitual mechanical stimulation magnitude and frequency. Prolonged stress below the normal range would decrease bone mass and increase bone stress [64]. This concept, termed adaptive modeling, was later formulated as a trigger criterion or threshold for bone resorption and apposition [65,66]. The approach lends itself to numerical modeling approaches including the FE method, which improves theories and the development of new models [67–71]. The most famous model describing bone growth and loss may be the Mechanostat theory promoted by Frost [72]. This concept asserts that bone growth and loss are stimulated by local mechanical elastic deformation, defined as disuse, adaption, overload, and fracture, and controlled by the strain level. Although the theoretical formulas were mostly advocated within the orthopedics field, there is no effective stress magnitude directly predicting bone adaptability. In dentistry, models have been formulated to assess the internal bone strains associated with dental implants [47,73] and removable prosthodontics [74]. Investigation of the bone response in the rabbit tibiae under static and dynamic loads,
histomorphometrical quantifications, and contact analysis of the bone–implant interface all evidence the continuing process of bone resorption [47]. To assess the effect of mechanical stimulation on the denture-bearing residual ridge, a combination of bone histomorphometry and the FE method simulating the external loads on the mouse maxilla were performed [74]. The study revealed that osteoclastic resorption was location-dependent and sensitive to the local strain intensity. Despite increasingly numerous studies, the mechanism of bone resorption associated with implant and prosthodontic interventions and the effective stress and strain levels have not been established. The peri-implant bone level before and after occlusal loading serves as the most common and reliable outcome for the dental implants. While excessive bite force or overloading have been proposed as a cause of late implant failures, the association between overload and peri-implant bone level has not been proven [75,76]. The current consensus is that the biomechanical scales (stress, strain) are not determinant factors of the implant failure or success; they are less important than the microbial accumulation around the implant. However, the significance of the stress and strain generated in the human hard and soft tissues is presently being explored in numerous studies. Increasingly, studies employ numerically modeled histological scales of animal subjects and cultured cells to determine the mechanical influences on biological activity. This approach may strengthen proposed rationale explaining the biomechanical phenomena observed in in vivo. These dental studies may raise potential concerns in bone formation and deposition in loaded animals. Fig. 3 illustrates a histological view of a loaded mouse maxilla (left), the strain distribution of the corresponding region obtained using CT-based finite element modeling (center), and the two images super-imposed (right). The combined image may facilitate assessment of the biomechanical factor effects on cell activity and the morphometric outcome. These studies will include animal histomorphometric and patient-based approaches to discern the association between bone morphometry and stress in the long-term basis.
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Conflict of interest [14]
The authors have no conflict of interest with respect to the manuscript content or funding.
[15]
Acknowledgements
[16]
The authors acknowledge Drs. Kengo Fujiki and Yusuke Toyoshima (Removable Partial Prosthodontics, Tokyo Medical and Dental University) for performing mathematical and biological analyses for figures. This work was partially supported by grants (No. 25893068 to N.M. and No. 24592902 to N.W.) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.
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