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Combined finite-element and rigid-body analysis of human jaw joint dynamics
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Journal of Biomechanics 38 (2005) 2431–2439 www.elsevier.com/locate/jbiomech www.JBiomech.com
Combined finite-element and rigid-body analysis of human jaw joint dynamics J.H. Koolstra, T.M.G.J. van Eijden Department of Functional Anatomy, Academic Centre for Dentistry Amsterdam (ACTA), Universiteit van Amsterdam and Vrije Universiteit, Meibergdreef 15, 1105 AZ Amsterdam, The Netherlands Accepted 7 October 2004
Abstract The jaw joint plays a crucial role in human mastication. It acts as a guidance for jaw movements and as a fulcrum for force generation. The joint is subjected to loading which causes tensions and deformations in its cartilaginous structures. These are assumed to be a major determinant for development, maintenance and also degeneration of the joint. To analyze the distribution of tensions and deformations in the cartilaginous structures of the jaw joint during jaw movement, a dynamical model of the human masticatory system has been constructed. Its movements are controlled by muscle activation. The articular cartilage layers and articular disc were included as finite-element (FE) models. As this combination of rigid-body and FE modeling had not been applied to musculoskeletal systems yet, its benefits and limitations were assessed by simulating both unloaded and loaded jaw movements. It was demonstrated that joint loads increase with muscle activation, irrespective of the external loads. With increasing joint load, the size of the stressed area of the articular surfaces was enlarged, whereas the peak stresses were much less affected. The results suggest that the articular disc enables distribution of local contact stresses over a much wider area of the very incongruent articular surfaces by transforming compressive principal stress into shear stress. r 2004 Elsevier Ltd. All rights reserved. Keywords: Jaw joint; Finite-element modeling; Rigid-body modeling; Dynamics
1. Introduction Biomechanical analysis of musculoskeletal system dynamics has been performed widely by applying rigid-body dynamics (for example, Koolstra and van Eijden, 1995, 1997, 1999; Anderson and Pandy, 1999; Peck et al., 2000; McLean et al., 2003). This method, which basically transforms forces into movements, is very flexible and enables to investigate the influence of muscle activation on body movements. The distribution of forces in irregularly shaped joint structures, however, cannot be analyzed, and the deformations of articular cartilaginous layers cannot be taken into account (Pandy et al., 1997). Therefore, often simplified joints are applied. For investigation of the mechanics of Corresponding author. Tel.: +31 20 5665370; fax: +31 20 6911856.
E-mail address: [email protected] (J.H. Koolstra). 0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2004.10.014
irregularly shaped deformable structures in joints, the finite-element (FE) method is more applicable (Huiskes and Chao, 1983; Li et al., 1999; Beek et al., 2000, 2001b; Donzelli et al., 2004). This method enables the prediction of the internal forces and deformations. These are generated when a priori defined displacements are applied that occur during joint movement. The rigidbody and FE method are supplementary. They cannot replace each other and generally they have a different area of application. The deformations in the cartilaginous structures in joints are caused by the mutual displacements of the articulating body segments. These displacements are the result of muscle forces, external forces, forces of inertia and joint reaction forces. The latter forces are directly dependent on the mechanical behavior of the deformable joint structures. They affect the displacements of the articulating segments, which implies that the
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deformations in articular cartilage are influenced by their own mechanical properties. This influence will be larger when the joint reaction force does not act in line with the muscle forces and the moments of inertia of (at least) one of the articulating segments are relatively small with respect to the corresponding joint torques. These circumstances are present in, for instance, the human masticatory system. Recently, it has become possible to connect FE method routines to rigid-body models in commercially available simulation software. This enables the analysis of the dynamics of the bony structures in a musculoskeletal system simultaneously with the local distribution of joint forces. Moreover, it permits the evaluation of the mutual influence of muscle activation patterns, rigidbody dynamics and the effects of deformations of articular cartilage. To our knowledge, this combination has not yet been applied to musculoskeletal systems. The purpose of the present study was to test the applicability of this new development for biomechanical analysis in a relatively complex musculoskeletal system as the human masticatory system. In particular, it was studied whether it can enlighten the role of the articular disc present in the temporomandibular joint during jaw movement, as this is still ill-understood.
2. Materials and methods 2.1. The model A three-dimensional biomechanical model of the human masticatory system (Fig. 1) was constructed using MADYMO (TNO Automotive, the Netherlands), a simulation program which combines the capabilities of multi-body motion and FE modeling. It contained two rigid bodies, the skull and the mandible, which
articulated at two six degree-of-freedom temporomandibular joints. Mutually impermeable dentures were connected to both of them. Twelve pairs of muscle portions were able to move the mandible with respect to the skull (Koolstra and van Eijden, 1995, 1997, 1999). They were: superficial, deep anterior and deep posterior masseter, anterior and posterior temporalis, medial pterygoid, superior and inferior lateral pterygoid, digastric, geniohyoid, and anterior and posterior mylohyoid. The muscle models were of the Hill-type consisting of a contractile element, a parallel elastic element, and a series elastic element. The architectural parameters (attachments, maximum force, fiber length, sarcomere length) had been obtained from eight human cadavers (van Eijden et al., 1995, 1996, 1997). Optimum length of the contractile element was defined by [(optimum sarcomere length fiber length)/sarcomere length]. The series elastic element was modeled as an inextensible wire (muscle length–fiber length) (Table 1). The characteristics of the contractile and parallel elastic elements were shaped according to van Ruijven and Weijs (1990). The two temporomandibular joints consisted of two deformable articular cartilage layers of 0.5 mm (Hansson et al., 1977) which were connected to the (rigid) temporal bone above and the mandibular condyle below, respectively. Between the two cartilaginous layers, a freely movable deformable cartilaginous articular disc was situated. It was connected medially and laterally to the adjacent mandibular condyle with pairs of inextensible wires representing the lower part of the articular capsule. The geometry of the deformable joint structures had been obtained from the right temporomandibular joint of one cadaver (Beek et al., 2000, 2001b). The left side joint was constructed as a mirror image of the right one. The volumes of the deformable structures were divided into tetrahedral
Fig. 1. The model. (A) anterolateral view: red lines: muscle contractile element; black lines: muscle serial elastic element; Ta: anterior temporalis; Tp: posterior temporalis; Ms: superficial masseter; Mpa: anterior deep masseter; Mpp: posterior deep masseter; Pm: medial pterygoid; Pls: superior lateral pterygoid; Pli: inferior lateral pterygoid; Dig: digastric; GH: geniohyoid; MHa: anterior mylohyoid; MHp: posterior mylohyoid; thin black lines: part of articular capsule; Dig, GH, and Mhp are connected to the hyoid bone (not shown), MHa to the mylohyoid raphe (black line). (B) Cartilaginous structures of the jaw joint; blue: temporal cartilage layer; orange: articular disc; red: condylar cartilage layer. (C) Sagittal cross-section of the jaw joint.
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Table 1 Model parameters Moments of inertia Lower jaw
Mass (kg) 0.44
Muscles
Muscle length (mm)
Superficial masseter Deep anterior masseter Deep posterior masseter Anterior temporalis Posterior temporalis Medial pterygoid Superior lateral pterygoid Inferior lateral pterygoid Anterior digastric Geniohyoid Anterior mylohyoid Posterior mylohyoid
48.0 29.5 30.9 57.4 62.9 43.3 29.1 27.2 51.9 48.5 21.8 44.8
Ixx (kg m2)
Iyy (kg m2)
0.00086
0.00029
Max. force (N)
a
CE optimum length (mm)
272.8 73.8 65.8 308.0 222.0 240.0 38.0 112.8 46.4 38.8 63.6 21.2
22.6 21.8 15.0 30.7 31.3 14.1 21.5 22.3 42.6 35.3 24.0 39.7
Izz (kg m2) 0.00061 a
SE length (mm) 25.8 17.1 13.3 24.2 28.8 27.6 9.4 9.0 3.0 5.4 0.0 0.0
Number of finite elements
Temporal cartilage
Articular disc
Condylar cartilage
Right joint Left joint
2190 2320
2167 2157
1853 1960
a
CE—contractile element, and SE—series elastic element.
finite elements with edges of about 0.5 mm (HyperMesh 6.0, Altair Engineering GmbH, Bo¨blingen, Germany) as specified in Table 1. MADYMO does not apply a material model for cartilage. As both articular cartilage and the cartilage of the articular disc are subject to large deformations, their material properties were approximated according to the Mooney–Rivlin material model (Chen et al., 1998). To determine the constants C1 and C2 that describe the behavior of this rubber-like material model, we performed simulations of indentation experiments on human temporomandibular joint discs (Beek et al., 2003) with this material. With C1 ¼ 9 105 and C2 ¼ 9 102 Pa the reaction forces upon indentation approximated the experimental observations (Beek et al., 2001a) most closely. For the articular cartilage layers we applied C1 ¼ 4.5 105 and C2 ¼ 4.5 102 Pa (vide infra). 2.2. Simulations The jaw was closed at the start of all simulations. From this position, symmetrical jaw-open movements were simulated. These were performed by a 10%, 50% or 100% activation of the digastric, geniohyoid, mylohyoid and lateral pterygoid muscles simultaneously. Jaw-closing movements without food resistance were simulated by a 1% or 10% simultaneous activation of the masseter, medial pterygoid and temporalis muscles after a maximum gape had been obtained in a preceding (100%) jaw-open movement. The influence of food resistance was investigated by adding a restraint, which generated a constant force during jaw closure. A
force of 50 N was applied symmetrically between the central incisors or 80 N unilaterally between the right second molars of upper and lower jaw. Loaded jawclosing movements were simulated by a bilateral 25% activation of all jaw-closing muscles, as this appeared to be sufficient to overcome this restraint. In all simulations, activation of each muscle was individually defined as a function of time. These functions included activation and deactivation ramps of 45 and 75 ms, respectively, to incorporate activation dynamics (Winters and Stark, 1987). 2.3. Analysis The joint forces during the movements in response to the applied muscle loads were characterized by the resultant reaction force generated by the contacting elements in the joint. The stress in the deformable structures was characterized by the Von Mises stress criterion and the maximum principal stress. Strain of a cartilaginous structure was quantified by computing the number of elements that were strained by more than 1% and the mean maximum principal strain of these elements. The results were analyzed and visualized using HyperWorks 6.0 (Altair Engineering GmbH, Bo¨blingen, Germany).
3. Results The movements of the jaw predicted by the model as a consequence of muscle activation were relatively fast, but similar to natural movements. The mandibular
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load in each joint increased to about 145 N when a 50 N load was applied at the central incisors (Fig. 2D). When a 80 N load was applied at the right molar region, it increased to about 145 and 110 N for the contralateral and ipsilateral joint, respectively (Fig. 2E). As the jaw was at its open position, the largest stresses were generated around the thin intermediate zone of the disc (Fig. 3, 1st rows). Medio-laterally, this area was extended from the middle of the joint to its lateral side. When the jaw closed this area shifted in medial direction. The largest stresses in the cartilage layers were associated with the articular eminence. In the condylar cartilage this area shifted with condylar rotation. In the temporal cartilage the mandibular fossa region became also stressed as the jaw closed. In the disc, the Von Mises stresses were much larger than in the articular layers. In the latter the (compressive) principal stresses were larger. The area of large principal stress was proportional to the amount of muscle activation. In contrast to the articular cartilage layers where the maximum principal stress was almost exclusively compressive, the disc showed both tensile and compressive stresses. Furthermore, the disc areas with large principal stress were not necessarily the same as the areas with large Von Mises stress. At the end of the unilaterally loaded closing movement, the Von
condyles moved forward as the jaw-opening muscles opened the jaw in about 50 ms. The articular disc moved together with the mandibular condyle along the articular eminence. After a 100%, 50% and 10% activation of the jaw-opening muscles, the maximum (inter-incisal) jaw opening was 3.0, 2.6 and 1.8 cm and the joints were loaded up to 85, 45 and 15 N, respectively (Fig. 2A). Jaw closing took, dependent on muscle activation and mandibular load, between 60 and 125 ms. In the first stage, the mandibular condyle moved backwards over the articular eminence. Thereafter it remained in the mandibular fossa as the jaw closed further. If only the masseter, temporalis and medial pterygoid muscles were activated, the mandibular condyle moved backwards out of the mandibular fossa as the jaw was almost closed. This could be prevented by simultaneous activation of the lateral pterygoid muscle. A 100% activation of this muscle was necessary to prevent dislocation when the jaw-closing muscles had been activated to 10% or more. During unloaded jaw closing, the load in each joint was proportional to the muscle activation. It became about 10 or 90 N when the jaw closers were activated by 1% or 10%, respectively (Fig. 2B and C). When the jaw had to close against a resistance at the dental elements the joints became more heavily loaded. The resultant 200 (A)
opening
100
100% 50% 10%
0
Joint force (N)
200 (B)
unloaded, 1%
(C) unloaded, 10%
(D) 50 N, central incisor
(E) 80 N, right molar
100
0 200
contralateral
100 ipsilateral
0 0
5
10
15
20
25
0
5
10
15
20
25
Jaw gape (degrees) Fig. 2. Joint forces as a function of jaw gape. (A) jaw-opening movement: dotted line: 10% jaw-opener activation; dashed line: 50% jaw-opener activation; continuous line: 100% jaw-opener activation. (B) Unloaded jaw-closing movement, 1% jaw-closer activation. (C) Unloaded jaw-closing movement, 10% jaw-closer activation. (D) Jaw-closing movement with a 50 N symmetrical load. (E) Jaw-closing movement with a 80 N unilateral load; dotted line: contralateral joint, continuous line: ipsilateral joint; thin dashed lines: jaw-opening movement (100% jaw-opener activity).
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Fig. 3. Stress in the jaw joint during jaw closing as a function of jaw gape. (A) unloaded jaw-closing movement, 10% jaw-closer activation. Columns 1–2: Von Mises stress; Columns 3–6: maximum principal stress; Columns 1, 5: articular disc, top view; Columns 2, 3: sagittal section, right view; Column 4: temporal cartilage layer, top view; Column 6: condylar layer, top view. a: anterior; p: posterior; m: medial; l: lateral. Rows: 1—maximum open, 2—201 open, 3—151 open, 4—101 open, 5—51 open, 6—closed. (B) Jaw-closing movement with a 50 N symmetrical load. Rows and columns as (A). (C) Jaw-closing movement with a 80 N unilateral load. Rows as A. Columns 1–6: right joint as (A). Columns 7–12: left joint as Columns 1–6 in reverse order. Legends: Stress in Pa.
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(A)
0.2
unloaded, 1%
unloaded, 10%
50 N, central incisor
80 N, right molar
0.15 temporal cartilage 0.1 0.05
Mean element strain
0 0.2 0.15 disc 0.1 0.05 0 0.2 0.15 condylar cartilage 0.1 0.05 0 0
5
10 15 20 25
0
5
10 15 20 25
0
5
10 15 20 25
0
5
10 15 20 25
Jaw gape (degrees) (B) 2500
unloaded, 1%
unloaded, 10%
50 N, central incisor
80 N, right molar
2000 1500 1000
temporal cartilage
500 0 Strained elements
2500 2000 1500 1000
disc
500 0 2500 2000 1500 1000 condylar cartilage
500 0 0
5
10 15 20 25
0
5
10 15 20 25
0
5
10 15 20 25
0
5
10 15 20 25
Jaw gape (degrees) Fig. 4. Strain in the jaw joint structures as a function of jaw gape. (A) mean strain in temporal cartilage layer (row 1), articular disc (row 2) and condylar cartilage layer (row 3) during unloaded close with 1% jaw-closer activity (column 1), unloaded close with 10% jaw-closer activity (column 2), jaw-closing movement with 50 N symmetrical load (column 3) and jaw-closing movement with 80 N unilateral load (column 4). Continuous lines: ipsilateral joint; Dotted lines: contralateral joint; Thin dashed lines: jaw-opening movement. Elements strained less than 1% were excluded. (B) Number of elements that were strained by more than 1% as a representation of the strained area. Organization as in (A).
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Mises stress was less and the maximum principal stress larger than in the symmetrically loaded one (Fig. 3B and C, bottom row). During all simulated movements, the mean principal (compressive and tensile) strain in the articular disc was proportional to the joint load and almost all of the articular disc was deformed with more than 1% (Fig. 4A and B, 2nd row). The largest strains were observed in the disc during loaded jaw-closing movements. Here the mean disc strain increased gradually and reached 20% when the teeth bumped against each other. The mean strain in the articular layer of the condyle varied much less and in the mandibular fossa it was almost constant (Fig. 4A, 1st and 3rd row). In contrast, the size of the strained area of the fossa layer changed with joint load more directly than that of the condyle (Fig. 4B, 1st and 3rd row).
4. Discussion Until recently, analysis of the interaction of muscle, joint and bone mechanics, all contributing to the mechanical behavior of the same musculoskeletal system, has been laborious. With the availability of simulation software such as MADYMO, such analysis can be performed with one integral model. The present model, which is the first three-dimensional dynamical model of the human masticatory system that includes naturally shaped deformable structures in the joints, enables the performance of such analysis during habitual movements. 4.1. Muscle recruitment and joint force In a jaw-opening movement, the jaw-opening muscles have to overcome the increasing passive tensions of the jaw-closing muscles (Langenbach and Hannam, 1999; Koolstra and van Eijden, 2004). Therefore, the final jaw gape is proportional to the activation of the jaw-opening muscles. In contrast, the jaw-closing movement requires very little muscle activation, because the passive tensions of the jaw-opening muscles are negligible. The amount of necessary muscle force is reflected in the joint force and herewith in the amount of stress in the joint. When the jaw is maximally open, both articulating surfaces show their largest convexity in the sagittal plane. This causes the occurring stresses to be concentrated more than when the condyle is situated in the mandibular fossa or against the less-curved posterior surface of the articular eminence. The model did not contain the structures that normally are present behind the mandibular fossa, or the capsule of the jaw joint. These may contribute to prevent dislocation of the joint in posterior direction which occurred, for instance, in simulations where the
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temporalis muscle was activated beyond 25% while the teeth experienced a vertical (but not horizontal) resistance. This indicates that even during forceful biting, a subtle regulation of muscle force remains necessary. This is reflected by observations of large interindividual differences in masticatory muscle activity for the same task. It is not possible to define one unique pattern as generally applicable. Therefore, the applied muscle activation patterns were relatively simple. Proprioceptive modulation and timing of muscle activation (Hof, 2003) were not incorporated as the recent model does not allow to include instantaneous interactive muscle control (Koolstra and van Eijden, 2001). This can be considered as a considerable limitation. 4.2. Deformations and tensions The model predicted that when the jaw joint is loaded, its articular disc deforms much more than the (softer) cartilaginous layers of the temporal bone and mandibular condyle. This is most likely caused by the fact that the articular cartilage layers are bonded to the bone on one side but the articular disc is not. The compressive forces applied to the articular cartilage layers can be transmitted to the bone more directly and do not propagate through the structure. In contrast, in the articular disc they can propagate easily parallel to the articular surfaces, thereby causing large deformations accompanied by large local tensions. This finding suggests that the difference in mechanical properties between disc and cartilaginous layers is advantageous for the joint. While the freely moving disc is more susceptible to large deformations when it is compressed between the articular surfaces of temporal bone and condyle, it needs to be stiffer than the articular cartilage. In the articular disc the largest Von Mises stresses occurred in the intermediate zone. In the adjacent cartilage layers the Von Mises stress was much less, but here the principal (compressive) stress was larger than in the disc. This indicates that the disc is primarily stressed with shear (Von Mises stress is also indicated as octahedral shear stress). This can be regarded as a mechanism to spread the compressive stress of the cartilage layers over a larger area. The medio-lateral shift of the largest discal stresses with jaw movement was similar to the observations of Beek et al. (2001b). However, generalization of this finding should be performed with care as the same joint geometry was applied. It is suggested that the mandibular condyles bend inwards during contraction of the jaw closers (van Eijden, 2000). Such bending, which was not included in the present model, most probably influences the medio-lateral position of the largest stresses. During a unilaterally loaded closing movement, the contralateral joint is more heavily loaded than the
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ipsilateral one. Despite this difference, the average strain in the contralateral temporal cartilage is not larger than in the ipsilateral one, but the strained area is larger. This illustrates the articular disc’s capacity to prevent overloading of articular surfaces by a load distribution which becomes more effective when the amount of applied load increases. This effect is more dominant for the temporal cartilage than for the condyle. The concavity of the mandibular fossa is less than the convexity of the mandibular condyle. Furthermore, the bone of the mandibular condyle is suitable to distribute loads internally (Giesen and van Eijden, 2000). Although this may also be the case for the articular eminence, the roof of the mandibular fossa is hardly able to withstand large forces because it is very thin. Apparently, the articular disc is also able to compensate for these differences. 4.3. Consequences of model inaccuracies Cartilage exhibits complex material behavior. The articular disc has been demonstrated to possess both viscoelastic and hyperelastic properties and undergoes large deformations (Beek et al., 2001a). Unfortunately, MADYMO does not incorporate a material model that can model these properties accurately. Therefore, the hyperelastic Mooney–Rivlin material model has been applied as it behaves most reliably under large deformations. Its hyperelasticity, however, is mainly directed to stretch. Compressive hyperelasticity, as exhibited by cartilage, remains underrated. As the material constants were optimized for a 30% compression, this implies that compressive strains smaller than 30% may be underestimated while larger ones are overestimated. The material behavior of the articular cartilage layers in the temporomandibular joint is not known. Although they have been obtained from condylar cartilage of rabbits by nanoindentation (Hu et al., 2001), these are not reflecting the behavior under large deformations (Hasler et al., 1999). Therefore, we have also applied a Mooney–Rivlin material model for these structures. The relevant constants were chosen to be half of those of the articular disc, as articular cartilage is considered softer than that of the disc (Chen et al., 1998; Beek et al., 2001b). This introduces a quantitative uncertainty in the predicted stresses and strains. However, it has been demonstrated that the location where these stresses and strains occur is less dependent on the applied material constants (Beek et al., 2001b). The FE models of the deformable cartilaginous structures contained elements with edges of about 0.5 mm length. This is of the same order of magnitude as the thickness of the cartilage layers and the intermediate zone part of the articular disc. Consequently, the resulting meshes were relatively coarse. This
may reduce the quantitative accuracy of the predicted stresses and strains. As such, accuracy is dominantly affected by the applied material models, and a lawful mesh architecture could be maintained despite the relatively large deformations, this was considered acceptable. 4.4. Benefits and limitations To our knowledge, the present model for the first time enables the analysis of instantaneous deformations and tensions in the joints of a musculoskeletal system where the movements are not defined a priori, but controlled by muscle tensions. Furthermore, the reaction forces in the joints, generated by the deformed cartilaginous structures, are also a determinant for the resulting movement. Presently, it enabled the simulation of tasks that incorporated muscle forces up to about 25% of their capacity causing local deformations in the joints of more than 20%. Compared to the majority of models that incorporated FE modeling, this is a considerable extension of force generation and deformation capacity. The limitations of the present method include the inability to control muscle activation in feedback to the position and velocity of the moving jaw. Furthermore, the applicability (especially in a quantitative sense) is limited by the absence of a more realistic material model for articular and discal cartilage. Inclusion of compressive hyperelasticity, for instance, could extend the range of applicable muscle forces because the materials would become stiffer and less susceptible to deformations as loading increases.
Acknowledgments The authors gratefully thank Dr. G.E.J. Langenbach for his constructive comments on the manuscript. This research was institutionally supported by the Interuniversity Research School of Dentistry, through the Academic Centre for Dentistry Amsterdam (ACTA). References Anderson, F.C., Pandy, M.G., 1999. A dynamic optimization solution for vertical jumping in three dimensions. Computer Methods in Biomechanics and Biomedical Engineering 2, 201–231. Beek, M., Koolstra, J.H., van Ruijven, L.J., van Eijden, T.M.G.J., 2000. Three-dimensional finite element analysis of the human temporomandibular joint disc. Journal of Biomechanics 33, 307–316. Beek, M., Aarnts, M.P., Koolstra, J.H., Feilzer, A.J., van Eijden, T.M.G.J., 2001a. Dynamical properties of the human temporomandibular joint disc. Journal of Dental Research 80, 876–880. Beek, M., Koolstra, J.H., van Ruijven, L.J., van Eijden, T.M.G.J., 2001b. Three-dimensional finite element analysis of the cartilaginous structures in the human temporomandibular joint. Journal of Dental Research 80, 1913–1918.
ARTICLE IN PRESS J.H. Koolstra, T.M.G.J. van Eijden / Journal of Biomechanics 38 (2005) 2431–2439 Beek, M., Koolstra, J.H., van Eijden, T.M.G.J., 2003. Human temporomandibular joint disc cartilage as a poroelastic material. Clinical Biomechanics 18, 69–76. Chen, J., Akyuz, U., Xu, L., Pidaparti, R.M.V., 1998. Stress analysis of the human temporomandibular joint. Medical Engineering and Physics 20, 565–572. Donzelli, P.S., Gallo, L.M., Spilker, R.L., Palla, S., 2004. Biphasic finite element simulation of the TMJ disc from in vivo kinematic and geometric measurements. Journal of Biomechanics 37, 1787–1791. Giesen, E.B.W., van Eijden, T.M.G.J., 2000. The three-dimensional cancellous bone architecture of the human mandibular condyle. Journal of Dental Research 79, 957–963. Hansson, T., Oberg, T., Carlsson, G.E., Kopp, S., 1977. Thickness of the soft tissue layers and the articular disc in the temporomandibular joint. Acta Odontologica Scandinavia 35, 77–83. Hasler, E.M., Herzog, W., Wu, J.Z., Mu¨ller, W., Wyss, U., 1999. Articular cartilage biomechanics: theoretical models, material properties, and biosynthetic response. Critical Reviews in Biomedical Engineering 27, 415–488. Hof, A.L., 2003. Muscle mechanics and neuromuscular control. Journal of Biomechanics 36, 1031–1038. Hu, K., Radhakrishnan, P., Patel, R.V., Mao, J.J., 2001. Regional structural and viscoelastic properties of fibrocartilage upon dynamic nanoindentation of the articular condyle. Journal of Structural Biology 136, 46–52. Huiskes, R., Chao, E.Y.S., 1983. A survey of finite element analysis in orthopedic research: the first decade. Journal of Biomechanics 16, 385–409. Koolstra, J.H., van Eijden, T.M.G.J., 1995. Biomechanical analysis of jaw closing movements. Journal of Dental Research 74, 1564–1570. Koolstra, J.H., van Eijden, T.M.G.J., 1997. Dynamics of the human masticatory muscles during a jaw open-close movement. Journal of Biomechanics 30, 883–889. Koolstra, J.H., van Eijden, T.M.G.J., 1999. Three dimensional dynamical capabilities of the human masticatory system. Journal of Biomechanics 32, 145–152. Koolstra, J.H., van Eijden, T.M.G.J., 2001. A method to predict muscle control in the kinematically and mechanically indeterminate
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human masticatory system. Journal of Biomechanics 34, 1179–1188. Koolstra, J.H., van Eijden, T.M.G.J., 2004. Functional significance of the coupling between head and jaw movements. Journal of Biomechanics 37, 1387–1392. Langenbach, G.E.J., Hannam, A.G., 1999. The role of passive muscle tensions in a three-dimensional dynamic model of the human jaw. Archives of Oral Biology 44, 557–573. Li, G., Gil, J., Kanamori, A., Woo, S.L.-Y., 1999. A validated threedimensional computational model of a human knee joint. Journal of Biomechanical Engineering 121, 657–662. McLean, S.G., Su, S., van den Bogert, A.J., 2003. Development and validation of a 3-D model to predict knee joint loading during dynamic movement. Journal of Biomechanical Engineering 125, 864–874. Pandy, M.G., Sasaki, K., Kim, S., 1997. A three-dimensional musculoskeletal model of the human knee joint. Part 1: theoretical construction. Computer Methods in Biomechanics and Biomedical Engineering 1, 87–108. Peck, C.C., Langenbach, G.E.J., Hannam, A.G., 2000. Dynamic simulation of muscle and articular properties during human wide jaw opening. Archives of Oral Biology 45, 963–982. van Eijden, T.M.G.J., 2000. Biomechanics of the mandible. Critical Reviews in Oral Biology and Medicine 11, 123–136. van Eijden, T.M.G.J., Koolstra, J.H., Brugman, P., 1995. Architecture of the human pterygoid muscles. Journal of Dental Research 74, 1489–1495. van Eijden, T.M.G.J., Koolstra, J.H., Brugman, P., 1996. Threedimensional structure of the human temporalis muscle. Anatomical Record 246, 565–572. van Eijden, T.M.G.J., Korfage, J.A.M., Brugman, P., 1997. Architecture of the human jaw-closing and jaw-opening muscles. Anatomical Record 248, 464–474. van Ruijven, L.J., Weijs, W.A., 1990. A new model for calculating muscle forces from electromyograms. European Journal of Applied Physiology 61, 479–485. Winters, J.M., Stark, L., 1987. Muscle models, what is gained and what is lost by varying model complexity. Biological Cybernetics 55, 403–420.
Journal of Biomechanics 38 (2005) 2431–2439 www.elsevier.com/locate/jbiomech www.JBiomech.com
Combined finite-element and rigid-body analysis of human jaw joint dynamics J.H. Koolstra, T.M.G.J. van Eijden Department of Functional Anatomy, Academic Centre for Dentistry Amsterdam (ACTA), Universiteit van Amsterdam and Vrije Universiteit, Meibergdreef 15, 1105 AZ Amsterdam, The Netherlands Accepted 7 October 2004
Abstract The jaw joint plays a crucial role in human mastication. It acts as a guidance for jaw movements and as a fulcrum for force generation. The joint is subjected to loading which causes tensions and deformations in its cartilaginous structures. These are assumed to be a major determinant for development, maintenance and also degeneration of the joint. To analyze the distribution of tensions and deformations in the cartilaginous structures of the jaw joint during jaw movement, a dynamical model of the human masticatory system has been constructed. Its movements are controlled by muscle activation. The articular cartilage layers and articular disc were included as finite-element (FE) models. As this combination of rigid-body and FE modeling had not been applied to musculoskeletal systems yet, its benefits and limitations were assessed by simulating both unloaded and loaded jaw movements. It was demonstrated that joint loads increase with muscle activation, irrespective of the external loads. With increasing joint load, the size of the stressed area of the articular surfaces was enlarged, whereas the peak stresses were much less affected. The results suggest that the articular disc enables distribution of local contact stresses over a much wider area of the very incongruent articular surfaces by transforming compressive principal stress into shear stress. r 2004 Elsevier Ltd. All rights reserved. Keywords: Jaw joint; Finite-element modeling; Rigid-body modeling; Dynamics
1. Introduction Biomechanical analysis of musculoskeletal system dynamics has been performed widely by applying rigid-body dynamics (for example, Koolstra and van Eijden, 1995, 1997, 1999; Anderson and Pandy, 1999; Peck et al., 2000; McLean et al., 2003). This method, which basically transforms forces into movements, is very flexible and enables to investigate the influence of muscle activation on body movements. The distribution of forces in irregularly shaped joint structures, however, cannot be analyzed, and the deformations of articular cartilaginous layers cannot be taken into account (Pandy et al., 1997). Therefore, often simplified joints are applied. For investigation of the mechanics of Corresponding author. Tel.: +31 20 5665370; fax: +31 20 6911856.
E-mail address: [email protected] (J.H. Koolstra). 0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2004.10.014
irregularly shaped deformable structures in joints, the finite-element (FE) method is more applicable (Huiskes and Chao, 1983; Li et al., 1999; Beek et al., 2000, 2001b; Donzelli et al., 2004). This method enables the prediction of the internal forces and deformations. These are generated when a priori defined displacements are applied that occur during joint movement. The rigidbody and FE method are supplementary. They cannot replace each other and generally they have a different area of application. The deformations in the cartilaginous structures in joints are caused by the mutual displacements of the articulating body segments. These displacements are the result of muscle forces, external forces, forces of inertia and joint reaction forces. The latter forces are directly dependent on the mechanical behavior of the deformable joint structures. They affect the displacements of the articulating segments, which implies that the
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deformations in articular cartilage are influenced by their own mechanical properties. This influence will be larger when the joint reaction force does not act in line with the muscle forces and the moments of inertia of (at least) one of the articulating segments are relatively small with respect to the corresponding joint torques. These circumstances are present in, for instance, the human masticatory system. Recently, it has become possible to connect FE method routines to rigid-body models in commercially available simulation software. This enables the analysis of the dynamics of the bony structures in a musculoskeletal system simultaneously with the local distribution of joint forces. Moreover, it permits the evaluation of the mutual influence of muscle activation patterns, rigidbody dynamics and the effects of deformations of articular cartilage. To our knowledge, this combination has not yet been applied to musculoskeletal systems. The purpose of the present study was to test the applicability of this new development for biomechanical analysis in a relatively complex musculoskeletal system as the human masticatory system. In particular, it was studied whether it can enlighten the role of the articular disc present in the temporomandibular joint during jaw movement, as this is still ill-understood.
2. Materials and methods 2.1. The model A three-dimensional biomechanical model of the human masticatory system (Fig. 1) was constructed using MADYMO (TNO Automotive, the Netherlands), a simulation program which combines the capabilities of multi-body motion and FE modeling. It contained two rigid bodies, the skull and the mandible, which
articulated at two six degree-of-freedom temporomandibular joints. Mutually impermeable dentures were connected to both of them. Twelve pairs of muscle portions were able to move the mandible with respect to the skull (Koolstra and van Eijden, 1995, 1997, 1999). They were: superficial, deep anterior and deep posterior masseter, anterior and posterior temporalis, medial pterygoid, superior and inferior lateral pterygoid, digastric, geniohyoid, and anterior and posterior mylohyoid. The muscle models were of the Hill-type consisting of a contractile element, a parallel elastic element, and a series elastic element. The architectural parameters (attachments, maximum force, fiber length, sarcomere length) had been obtained from eight human cadavers (van Eijden et al., 1995, 1996, 1997). Optimum length of the contractile element was defined by [(optimum sarcomere length fiber length)/sarcomere length]. The series elastic element was modeled as an inextensible wire (muscle length–fiber length) (Table 1). The characteristics of the contractile and parallel elastic elements were shaped according to van Ruijven and Weijs (1990). The two temporomandibular joints consisted of two deformable articular cartilage layers of 0.5 mm (Hansson et al., 1977) which were connected to the (rigid) temporal bone above and the mandibular condyle below, respectively. Between the two cartilaginous layers, a freely movable deformable cartilaginous articular disc was situated. It was connected medially and laterally to the adjacent mandibular condyle with pairs of inextensible wires representing the lower part of the articular capsule. The geometry of the deformable joint structures had been obtained from the right temporomandibular joint of one cadaver (Beek et al., 2000, 2001b). The left side joint was constructed as a mirror image of the right one. The volumes of the deformable structures were divided into tetrahedral
Fig. 1. The model. (A) anterolateral view: red lines: muscle contractile element; black lines: muscle serial elastic element; Ta: anterior temporalis; Tp: posterior temporalis; Ms: superficial masseter; Mpa: anterior deep masseter; Mpp: posterior deep masseter; Pm: medial pterygoid; Pls: superior lateral pterygoid; Pli: inferior lateral pterygoid; Dig: digastric; GH: geniohyoid; MHa: anterior mylohyoid; MHp: posterior mylohyoid; thin black lines: part of articular capsule; Dig, GH, and Mhp are connected to the hyoid bone (not shown), MHa to the mylohyoid raphe (black line). (B) Cartilaginous structures of the jaw joint; blue: temporal cartilage layer; orange: articular disc; red: condylar cartilage layer. (C) Sagittal cross-section of the jaw joint.
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Table 1 Model parameters Moments of inertia Lower jaw
Mass (kg) 0.44
Muscles
Muscle length (mm)
Superficial masseter Deep anterior masseter Deep posterior masseter Anterior temporalis Posterior temporalis Medial pterygoid Superior lateral pterygoid Inferior lateral pterygoid Anterior digastric Geniohyoid Anterior mylohyoid Posterior mylohyoid
48.0 29.5 30.9 57.4 62.9 43.3 29.1 27.2 51.9 48.5 21.8 44.8
Ixx (kg m2)
Iyy (kg m2)
0.00086
0.00029
Max. force (N)
a
CE optimum length (mm)
272.8 73.8 65.8 308.0 222.0 240.0 38.0 112.8 46.4 38.8 63.6 21.2
22.6 21.8 15.0 30.7 31.3 14.1 21.5 22.3 42.6 35.3 24.0 39.7
Izz (kg m2) 0.00061 a
SE length (mm) 25.8 17.1 13.3 24.2 28.8 27.6 9.4 9.0 3.0 5.4 0.0 0.0
Number of finite elements
Temporal cartilage
Articular disc
Condylar cartilage
Right joint Left joint
2190 2320
2167 2157
1853 1960
a
CE—contractile element, and SE—series elastic element.
finite elements with edges of about 0.5 mm (HyperMesh 6.0, Altair Engineering GmbH, Bo¨blingen, Germany) as specified in Table 1. MADYMO does not apply a material model for cartilage. As both articular cartilage and the cartilage of the articular disc are subject to large deformations, their material properties were approximated according to the Mooney–Rivlin material model (Chen et al., 1998). To determine the constants C1 and C2 that describe the behavior of this rubber-like material model, we performed simulations of indentation experiments on human temporomandibular joint discs (Beek et al., 2003) with this material. With C1 ¼ 9 105 and C2 ¼ 9 102 Pa the reaction forces upon indentation approximated the experimental observations (Beek et al., 2001a) most closely. For the articular cartilage layers we applied C1 ¼ 4.5 105 and C2 ¼ 4.5 102 Pa (vide infra). 2.2. Simulations The jaw was closed at the start of all simulations. From this position, symmetrical jaw-open movements were simulated. These were performed by a 10%, 50% or 100% activation of the digastric, geniohyoid, mylohyoid and lateral pterygoid muscles simultaneously. Jaw-closing movements without food resistance were simulated by a 1% or 10% simultaneous activation of the masseter, medial pterygoid and temporalis muscles after a maximum gape had been obtained in a preceding (100%) jaw-open movement. The influence of food resistance was investigated by adding a restraint, which generated a constant force during jaw closure. A
force of 50 N was applied symmetrically between the central incisors or 80 N unilaterally between the right second molars of upper and lower jaw. Loaded jawclosing movements were simulated by a bilateral 25% activation of all jaw-closing muscles, as this appeared to be sufficient to overcome this restraint. In all simulations, activation of each muscle was individually defined as a function of time. These functions included activation and deactivation ramps of 45 and 75 ms, respectively, to incorporate activation dynamics (Winters and Stark, 1987). 2.3. Analysis The joint forces during the movements in response to the applied muscle loads were characterized by the resultant reaction force generated by the contacting elements in the joint. The stress in the deformable structures was characterized by the Von Mises stress criterion and the maximum principal stress. Strain of a cartilaginous structure was quantified by computing the number of elements that were strained by more than 1% and the mean maximum principal strain of these elements. The results were analyzed and visualized using HyperWorks 6.0 (Altair Engineering GmbH, Bo¨blingen, Germany).
3. Results The movements of the jaw predicted by the model as a consequence of muscle activation were relatively fast, but similar to natural movements. The mandibular
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load in each joint increased to about 145 N when a 50 N load was applied at the central incisors (Fig. 2D). When a 80 N load was applied at the right molar region, it increased to about 145 and 110 N for the contralateral and ipsilateral joint, respectively (Fig. 2E). As the jaw was at its open position, the largest stresses were generated around the thin intermediate zone of the disc (Fig. 3, 1st rows). Medio-laterally, this area was extended from the middle of the joint to its lateral side. When the jaw closed this area shifted in medial direction. The largest stresses in the cartilage layers were associated with the articular eminence. In the condylar cartilage this area shifted with condylar rotation. In the temporal cartilage the mandibular fossa region became also stressed as the jaw closed. In the disc, the Von Mises stresses were much larger than in the articular layers. In the latter the (compressive) principal stresses were larger. The area of large principal stress was proportional to the amount of muscle activation. In contrast to the articular cartilage layers where the maximum principal stress was almost exclusively compressive, the disc showed both tensile and compressive stresses. Furthermore, the disc areas with large principal stress were not necessarily the same as the areas with large Von Mises stress. At the end of the unilaterally loaded closing movement, the Von
condyles moved forward as the jaw-opening muscles opened the jaw in about 50 ms. The articular disc moved together with the mandibular condyle along the articular eminence. After a 100%, 50% and 10% activation of the jaw-opening muscles, the maximum (inter-incisal) jaw opening was 3.0, 2.6 and 1.8 cm and the joints were loaded up to 85, 45 and 15 N, respectively (Fig. 2A). Jaw closing took, dependent on muscle activation and mandibular load, between 60 and 125 ms. In the first stage, the mandibular condyle moved backwards over the articular eminence. Thereafter it remained in the mandibular fossa as the jaw closed further. If only the masseter, temporalis and medial pterygoid muscles were activated, the mandibular condyle moved backwards out of the mandibular fossa as the jaw was almost closed. This could be prevented by simultaneous activation of the lateral pterygoid muscle. A 100% activation of this muscle was necessary to prevent dislocation when the jaw-closing muscles had been activated to 10% or more. During unloaded jaw closing, the load in each joint was proportional to the muscle activation. It became about 10 or 90 N when the jaw closers were activated by 1% or 10%, respectively (Fig. 2B and C). When the jaw had to close against a resistance at the dental elements the joints became more heavily loaded. The resultant 200 (A)
opening
100
100% 50% 10%
0
Joint force (N)
200 (B)
unloaded, 1%
(C) unloaded, 10%
(D) 50 N, central incisor
(E) 80 N, right molar
100
0 200
contralateral
100 ipsilateral
0 0
5
10
15
20
25
0
5
10
15
20
25
Jaw gape (degrees) Fig. 2. Joint forces as a function of jaw gape. (A) jaw-opening movement: dotted line: 10% jaw-opener activation; dashed line: 50% jaw-opener activation; continuous line: 100% jaw-opener activation. (B) Unloaded jaw-closing movement, 1% jaw-closer activation. (C) Unloaded jaw-closing movement, 10% jaw-closer activation. (D) Jaw-closing movement with a 50 N symmetrical load. (E) Jaw-closing movement with a 80 N unilateral load; dotted line: contralateral joint, continuous line: ipsilateral joint; thin dashed lines: jaw-opening movement (100% jaw-opener activity).
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Fig. 3. Stress in the jaw joint during jaw closing as a function of jaw gape. (A) unloaded jaw-closing movement, 10% jaw-closer activation. Columns 1–2: Von Mises stress; Columns 3–6: maximum principal stress; Columns 1, 5: articular disc, top view; Columns 2, 3: sagittal section, right view; Column 4: temporal cartilage layer, top view; Column 6: condylar layer, top view. a: anterior; p: posterior; m: medial; l: lateral. Rows: 1—maximum open, 2—201 open, 3—151 open, 4—101 open, 5—51 open, 6—closed. (B) Jaw-closing movement with a 50 N symmetrical load. Rows and columns as (A). (C) Jaw-closing movement with a 80 N unilateral load. Rows as A. Columns 1–6: right joint as (A). Columns 7–12: left joint as Columns 1–6 in reverse order. Legends: Stress in Pa.
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(A)
0.2
unloaded, 1%
unloaded, 10%
50 N, central incisor
80 N, right molar
0.15 temporal cartilage 0.1 0.05
Mean element strain
0 0.2 0.15 disc 0.1 0.05 0 0.2 0.15 condylar cartilage 0.1 0.05 0 0
5
10 15 20 25
0
5
10 15 20 25
0
5
10 15 20 25
0
5
10 15 20 25
Jaw gape (degrees) (B) 2500
unloaded, 1%
unloaded, 10%
50 N, central incisor
80 N, right molar
2000 1500 1000
temporal cartilage
500 0 Strained elements
2500 2000 1500 1000
disc
500 0 2500 2000 1500 1000 condylar cartilage
500 0 0
5
10 15 20 25
0
5
10 15 20 25
0
5
10 15 20 25
0
5
10 15 20 25
Jaw gape (degrees) Fig. 4. Strain in the jaw joint structures as a function of jaw gape. (A) mean strain in temporal cartilage layer (row 1), articular disc (row 2) and condylar cartilage layer (row 3) during unloaded close with 1% jaw-closer activity (column 1), unloaded close with 10% jaw-closer activity (column 2), jaw-closing movement with 50 N symmetrical load (column 3) and jaw-closing movement with 80 N unilateral load (column 4). Continuous lines: ipsilateral joint; Dotted lines: contralateral joint; Thin dashed lines: jaw-opening movement. Elements strained less than 1% were excluded. (B) Number of elements that were strained by more than 1% as a representation of the strained area. Organization as in (A).
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Mises stress was less and the maximum principal stress larger than in the symmetrically loaded one (Fig. 3B and C, bottom row). During all simulated movements, the mean principal (compressive and tensile) strain in the articular disc was proportional to the joint load and almost all of the articular disc was deformed with more than 1% (Fig. 4A and B, 2nd row). The largest strains were observed in the disc during loaded jaw-closing movements. Here the mean disc strain increased gradually and reached 20% when the teeth bumped against each other. The mean strain in the articular layer of the condyle varied much less and in the mandibular fossa it was almost constant (Fig. 4A, 1st and 3rd row). In contrast, the size of the strained area of the fossa layer changed with joint load more directly than that of the condyle (Fig. 4B, 1st and 3rd row).
4. Discussion Until recently, analysis of the interaction of muscle, joint and bone mechanics, all contributing to the mechanical behavior of the same musculoskeletal system, has been laborious. With the availability of simulation software such as MADYMO, such analysis can be performed with one integral model. The present model, which is the first three-dimensional dynamical model of the human masticatory system that includes naturally shaped deformable structures in the joints, enables the performance of such analysis during habitual movements. 4.1. Muscle recruitment and joint force In a jaw-opening movement, the jaw-opening muscles have to overcome the increasing passive tensions of the jaw-closing muscles (Langenbach and Hannam, 1999; Koolstra and van Eijden, 2004). Therefore, the final jaw gape is proportional to the activation of the jaw-opening muscles. In contrast, the jaw-closing movement requires very little muscle activation, because the passive tensions of the jaw-opening muscles are negligible. The amount of necessary muscle force is reflected in the joint force and herewith in the amount of stress in the joint. When the jaw is maximally open, both articulating surfaces show their largest convexity in the sagittal plane. This causes the occurring stresses to be concentrated more than when the condyle is situated in the mandibular fossa or against the less-curved posterior surface of the articular eminence. The model did not contain the structures that normally are present behind the mandibular fossa, or the capsule of the jaw joint. These may contribute to prevent dislocation of the joint in posterior direction which occurred, for instance, in simulations where the
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temporalis muscle was activated beyond 25% while the teeth experienced a vertical (but not horizontal) resistance. This indicates that even during forceful biting, a subtle regulation of muscle force remains necessary. This is reflected by observations of large interindividual differences in masticatory muscle activity for the same task. It is not possible to define one unique pattern as generally applicable. Therefore, the applied muscle activation patterns were relatively simple. Proprioceptive modulation and timing of muscle activation (Hof, 2003) were not incorporated as the recent model does not allow to include instantaneous interactive muscle control (Koolstra and van Eijden, 2001). This can be considered as a considerable limitation. 4.2. Deformations and tensions The model predicted that when the jaw joint is loaded, its articular disc deforms much more than the (softer) cartilaginous layers of the temporal bone and mandibular condyle. This is most likely caused by the fact that the articular cartilage layers are bonded to the bone on one side but the articular disc is not. The compressive forces applied to the articular cartilage layers can be transmitted to the bone more directly and do not propagate through the structure. In contrast, in the articular disc they can propagate easily parallel to the articular surfaces, thereby causing large deformations accompanied by large local tensions. This finding suggests that the difference in mechanical properties between disc and cartilaginous layers is advantageous for the joint. While the freely moving disc is more susceptible to large deformations when it is compressed between the articular surfaces of temporal bone and condyle, it needs to be stiffer than the articular cartilage. In the articular disc the largest Von Mises stresses occurred in the intermediate zone. In the adjacent cartilage layers the Von Mises stress was much less, but here the principal (compressive) stress was larger than in the disc. This indicates that the disc is primarily stressed with shear (Von Mises stress is also indicated as octahedral shear stress). This can be regarded as a mechanism to spread the compressive stress of the cartilage layers over a larger area. The medio-lateral shift of the largest discal stresses with jaw movement was similar to the observations of Beek et al. (2001b). However, generalization of this finding should be performed with care as the same joint geometry was applied. It is suggested that the mandibular condyles bend inwards during contraction of the jaw closers (van Eijden, 2000). Such bending, which was not included in the present model, most probably influences the medio-lateral position of the largest stresses. During a unilaterally loaded closing movement, the contralateral joint is more heavily loaded than the
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ipsilateral one. Despite this difference, the average strain in the contralateral temporal cartilage is not larger than in the ipsilateral one, but the strained area is larger. This illustrates the articular disc’s capacity to prevent overloading of articular surfaces by a load distribution which becomes more effective when the amount of applied load increases. This effect is more dominant for the temporal cartilage than for the condyle. The concavity of the mandibular fossa is less than the convexity of the mandibular condyle. Furthermore, the bone of the mandibular condyle is suitable to distribute loads internally (Giesen and van Eijden, 2000). Although this may also be the case for the articular eminence, the roof of the mandibular fossa is hardly able to withstand large forces because it is very thin. Apparently, the articular disc is also able to compensate for these differences. 4.3. Consequences of model inaccuracies Cartilage exhibits complex material behavior. The articular disc has been demonstrated to possess both viscoelastic and hyperelastic properties and undergoes large deformations (Beek et al., 2001a). Unfortunately, MADYMO does not incorporate a material model that can model these properties accurately. Therefore, the hyperelastic Mooney–Rivlin material model has been applied as it behaves most reliably under large deformations. Its hyperelasticity, however, is mainly directed to stretch. Compressive hyperelasticity, as exhibited by cartilage, remains underrated. As the material constants were optimized for a 30% compression, this implies that compressive strains smaller than 30% may be underestimated while larger ones are overestimated. The material behavior of the articular cartilage layers in the temporomandibular joint is not known. Although they have been obtained from condylar cartilage of rabbits by nanoindentation (Hu et al., 2001), these are not reflecting the behavior under large deformations (Hasler et al., 1999). Therefore, we have also applied a Mooney–Rivlin material model for these structures. The relevant constants were chosen to be half of those of the articular disc, as articular cartilage is considered softer than that of the disc (Chen et al., 1998; Beek et al., 2001b). This introduces a quantitative uncertainty in the predicted stresses and strains. However, it has been demonstrated that the location where these stresses and strains occur is less dependent on the applied material constants (Beek et al., 2001b). The FE models of the deformable cartilaginous structures contained elements with edges of about 0.5 mm length. This is of the same order of magnitude as the thickness of the cartilage layers and the intermediate zone part of the articular disc. Consequently, the resulting meshes were relatively coarse. This
may reduce the quantitative accuracy of the predicted stresses and strains. As such, accuracy is dominantly affected by the applied material models, and a lawful mesh architecture could be maintained despite the relatively large deformations, this was considered acceptable. 4.4. Benefits and limitations To our knowledge, the present model for the first time enables the analysis of instantaneous deformations and tensions in the joints of a musculoskeletal system where the movements are not defined a priori, but controlled by muscle tensions. Furthermore, the reaction forces in the joints, generated by the deformed cartilaginous structures, are also a determinant for the resulting movement. Presently, it enabled the simulation of tasks that incorporated muscle forces up to about 25% of their capacity causing local deformations in the joints of more than 20%. Compared to the majority of models that incorporated FE modeling, this is a considerable extension of force generation and deformation capacity. The limitations of the present method include the inability to control muscle activation in feedback to the position and velocity of the moving jaw. Furthermore, the applicability (especially in a quantitative sense) is limited by the absence of a more realistic material model for articular and discal cartilage. Inclusion of compressive hyperelasticity, for instance, could extend the range of applicable muscle forces because the materials would become stiffer and less susceptible to deformations as loading increases.
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