Journal of Biomechanics 33 (2000) 307}316
Three-dimensional "nite element analysis of the human
temporomandibular joint disc
M. Beek*, J.H. Koolstra, L.J. van Ruijven, T.M.G.J. van Eijden
Department of Functional Anatomy, Academic Center for Dentistry Amsterdam (ACTA), Meibergdreef 15, 1105 AZ Amsterdam, The Netherlands
Received 22 February 1999; accepted 5 September 1999
Abstract
A three-dimensional "nite element model of the articular disc of the human temporomandibular joint has been developed. The
geometry of the articular cartilage and articular disc surfaces in the joint was measured using a magnetic tracking device. First,
polynomial functions were "tted through the coordinates of these scattered measurements. Next, the polynomial description was
transformed into a triangulated description to allow application of an automatic mesher. Finally, a "nite element mesh of the articular
disc was created by "lling the geometry with tetrahedral elements. The articulating surfaces of the mandible and skull were modeled
by quadrilateral patches. The "nite element mesh and the patches were combined to create a three-dimensional model in which
unrestricted sliding of the disc between the articulating surfaces was allowed. Simulation of statical joint loading at the closed jaw
position predicted that the stress and strain distributions were located primarily in the intermediate zone of the articular disc with the
highest values in the lateral part. Furthermore, it was predicted that considerable deformations occurred for relatively small joint
loads and that relatively large variations in the direction of joint loading had little in#uence on the distribution of the deformations. ( 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Temporomandibular joint; Finite element method; Articular disc
1. Introduction
The human mandible is connected to the skull by two
temporomandibular joints. The articulating surfaces of
these joints are highly incongruent, which provides the
mandible with a large degree of movability with respect
to the skull (Ostry and Flanagan, 1989; Koolstra and
Van Eijden, 1999). Between the articulating surfaces
a cartilaginous articular disc is situated, which is assumed to decrease the contact pressure by increasing the
contact area between the incongruent joint surfaces, similar to the menisci in the knee joint (Scapino et al., 1996).
Experimental as well as analytic studies have demonstrated, that the human temporomandibular joint is
loaded during masticatory function (Hatcher et al., 1986;
Smith et al., 1986; Faulkner et al., 1987; Koolstra et al.,
1988; Ferrario and Sforza, 1994; Throckmorton and
Dechow, 1994). Development and degeneration of joint
* Corresponding author: Tel.: #31-205665357;
206911856.
E-mail address: [email protected] (M. Beek)
fax:
#31-
tissues and overloading caused by parafunctions like
bruxism, are supposedly in#uenced by these loads
(Mo!et jr. et al., 1964; O'Ryan and Epker, 1984; Nickel et
al., 1988; McCormack and Mansour, 1998; Newberry et
al., 1998). However, detailed data about the distribution
of the loads are still lacking, which means that the main
causes of the processes mentioned cannot be fully understood.
Experimental studies regarding the distribution of the
loads in the temporomandibular joint have been performed in animal models (e.g. Hylander, 1979; Brehnan et
al., 1981; Hohl and Tucek, 1982; Boyd et al., 1990). The
number of experimental studies is limited, because the
joint is di$cult to reach and the application of experimental devices, such as strain gauges, inside the joint will
introduce damage to its tissues, which will in#uence their
mechanical behavior.
Mathematical models of the human masticatory system including the temporomandibular joint have been
demonstrated to be a powerful tool to predict the loads
acting on this joint. Many studies, however, have oversimpli"ed the geometry of the articulating surfaces and
assumed them as being rigid (Koolstra et al., 1988;
0021-9290/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 1 6 8 - 2
308
M. Beek et al. / Journal of Biomechanics 33 (2000) 307}316
Ferrario and Sforza, 1994). Therefore, the tissue deformations and the distribution of loads inside the joint could
not be analyzed.
The "nite element (FE) method has been proven to be
a suitable tool for approximating such mechanical quantities in structures with a complex geometry (Huiskes and
Chao, 1983). Few FE analyses of the temporomandibular
joint including a movable articular disc have been published (Chen and Xu, 1994; DeVocht et al., 1996; Chen et
al., 1998). These analyses, however, were limited to the
two-dimensional sagittal plane and thus unsuitable to
investigate, for example, the in#uence of variations of the
loading direction out of the sagittal plane or the development of peak stresses located medially or laterally.
The purpose of the present study was to develop
a three-dimensional "nite element model of the human
temporomandibular joint in which unrestricted sliding of
a deformable articular disc between the articulating surfaces was allowed. This model was used to investigate the
three-dimensional load distribution in the articular disc
during statical loading tasks. The results might contribute to a better understanding of the normal and abnormal functioning of the disc.
2. Materials and methods
Fig. 1. The polynomial reconstructions of the articulating surfaces of
the condyle and of the skull (mandibular fossa and articular eminence)
in the human temporomandibular joint. Both panels display a frontal
view of the reconstructions (sup"superior, inf"inferior, med"medial, lat"lateral).
2.1. Geometry
The geometry of the model was obtained from the
right temporomandibular joint of an embalmed male
cadaver (age: 77 yr), showing no abnormalities, using
a magnetic tracking device (Polhemus 3SPACE
Digitizer). To avoid disturbing the geometry of the soft
tissues by applying force with this device during the
measurements, tight "tting plaster casts of the articular
surfaces were made. To ensure that the separate measurements of the casts could be combined into one joint
model afterwards, four reference points were measured
on both the mandible and the skull before separation.
Thereafter, the articular capsule was removed and the
mandible was separated from the skull. This way the
articular surface of the mandibular condyle and the lower
surface of the articular disc, which remained in contact
with the articulating surface of the skull, were revealed
and casts could be made. After removing the disc from
the preparation the articular surface on the skull (fossa
plus eminence) was revealed and a third cast could be
made. The casts were sampled in a random sequence with
about 10 000 points with a frequency of 30 Hz and an
accuracy of about 0.1 mm (Luo et al., 1996; Van Ruijven
et al., 1999b).
The articulating surfaces were reconstructed from the
unstructured measurements according to Van Ruijven et
al. (1999a). Brie#y, this method presumes that the spatial
coordinates of the surfaces could be approximated by
eighth-order polynomial functions of their surface coordinates. The coe$cients of these functions were determined simultaneously using a nonlinear least-squares
optimization method, by minimizing the di!erence
between the measured spatial coordinates and their
equivalent in the polynomial approximation. This reconstruction procedure "ltered the measurement noise by
a factor ten due to the large amount of measurements.
The resulting reconstructions of the articulating surfaces
on the condyle and the skull are displayed in Fig. 1.
2.2. Finite element mesh
Removal of the disc from the articular surface of the
skull deformed its geometry in such a way that direct
measurement of its top surface was unreliable. This limited direct measurement of the articular disc geometry to
its bottom surface. In a closed jaw position, the top
surface of the disc is situated against the articulating
surface of the skull. Therefore, the geometric data of the
latter surface were used to represent this top surface.
Due to the complex geometry of the articular disc, an
automatic mesher (Mentat 3.2, MARC Analysis
Research Corporation, Palo Alto, USA) was needed to
create the FE mesh. To be able to apply this mesher, the
polynomial descriptions of the surfaces were transformed
to triangular patches. This was performed by applying
M. Beek et al. / Journal of Biomechanics 33 (2000) 307}316
309
Fig. 2. A: typical example of possible deterioration of the surface
representation after transformation from two-dimensional surface coordinates to three-dimensional spatial coordinates. The dotted line
identi"es the mathematical shape of the surface. B: surface representation after correction by the smoothing procedures mentioned in the
text.
a two-dimensional Delaunay algorithm (Lee and Schachter, 1980; Cavendish et al., 1985; Chew, 1989) upon the
two-dimensional surface coordinates of the reconstructed
surfaces. After transformation of the patches to the
three-dimensional space, occasionally occurring surface
distortions (see Fig. 2) were corrected by applying the
following procedures. Pairs of adjoining triangles were
redivided according to the shortest diagonal. Edges larger than 1.7 times the mean edge length were subdivided.
The new nodes were placed on the surface using the
polynomial description. The resulting triangulated reconstructions of the surfaces are displayed in Fig. 3. The
borders of the top and bottom surfaces of the disc were
then connected to each other by triangular patches of the
same size as those on the surfaces themselves. After
removal of a small region on the lateral side of the disc
that appeared to be too thin to allow the creation of
a valid mesh, the geometry of the disc was totally surrounded by 3028 two-dimensional triangular patches
and was subsequently meshed with three-dimensional
linear hexahedral elements. The mesh consisted of about
16 000 elements. The joint surfaces on the condyle and
the skull were modeled by quadrilateral patches.
The local incongruencies due to the #at patches used in
the triangular description, caused point contacts generating large stress gradients and thus resulting in poor
convergence. Therefore, an initial adaptation step was
performed by moving the articular surface of the condyle
towards the articular surface of the skull over a distance
according to the incongruencies. During this operation,
the elastic modulus of the articular disc was temporarily
decreased considerably. This way, the articular disc regained its congruency with the articulating surfaces.
Thereafter, the strains and stresses were neutralized and
the articular disc was remeshed. This way, an initial state
with conditions necessary for a proper use of a contact
algorithm was obtained. Fig. 4 displays the FE model in
its initial state.
Fig. 3. Triangulated reconstructions of the bottom surface and the top
surface of the articular disc in the human temporomandibular joint in
a frontal view. Both reconstructions consist of triangles of which the
majority has a nearly equilateral shape (sup"superior, inf"inferior,
med"medial, lat"lateral).
2.3. Simulations
In order to investigate load distributions and deformations in the articular disc during joint loading, simulations were performed using the commercially available
FE software MARC K6.2 (MARC Analysis Research
Corporation, Palo Alto, USA). Because large deformations were expected to occur, the Cauchy stresses and the
logarithmic strains were calculated using the "nite deformation theory (MARC, 1996). The contact phenomenon
occurring at the condyle}disc and the disc}fossa interfaces, was solved using the Direct Constraint Method, in
which direct constraints are placed on the motion of
a body in contact using boundary conditions * kinematic constraints as well as nodal forces (MARC, 1996).
The material behavior of the articular disc was assumed to be linearly elastic. The values of the elastic
modulus and the Poisson's ratio applied were 6 MPa and
0.40, respectively. Due to lack of consensus in the literature and to allow comparison with other studies, the
value of the elastic modulus was chosen between the
values used by other investigators (Chen and Xu, 1994;
DeVocht et al., 1996; Chen et al., 1998). A relatively low
value was chosen for the Poisson's ratio, because the #uid
inside the disc was supposed to play a minor role in
determining its mechanical behavior in (quasi-)statical
situations. The joint surfaces on the condyle and the skull
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M. Beek et al. / Journal of Biomechanics 33 (2000) 307}316
Fig. 4. Three-dimensional lateral view of the FE model of the right human temporomandibular joint. The articulating surfaces of the condyle and the
skull are represented by quadrilateral patches and the geometry of the articular disc is "lled with approximately 16 000 tetrahedral elements.
were assumed to be rigid and the joint capsule was not
included. Furthermore, no friction between the articular
disc and the articulating surfaces was applied (Mow et al.,
1993; Nickel and McLachlan, 1994).
In a con"guration concomitant with the jaw in
a closed position, the condyle was displaced in a direction
according to the estimated direction of the joint reaction
force, i.e. a line originating from the center of the condyle
and perpendicular to the joint surface and to the axis
through the poles of the condyle (Koolstra and Van
Eijden, 1992). During the simulations, the articular surface of the skull was "xed.
Four additional simulations were performed in which
the direction of the movement was changed with respect
to the reference simulation over 303 in anterior, medial,
lateral and posterior direction, respectively. The results
of these variations will yield information about the sensitivity of loading direction on the stress and strain distributions in the disc. The in#uence of the material
properties of the articular disc on the load distribution in
the joint was investigated by increasing the value of the
elastic modulus by a factor 10, and by applying values of
0.30 and 0.47 for the Poisson's ratio. Finally, to investigate the convergence of the simulations, the elements in
the region that was mostly deformed in the reference
simulation were transformed from linear (containing four
nodes) to quadratic (ten nodes) tetrahedral ones.
3. Results
3.1. Reference simulation
When the joint was loaded in a direction according to
the estimated joint reaction force, a displacement of
0.20 mm was simulated. This distance corresponded with
almost half the thickness of the so-called intermediate
zone of the articular disc (thickness: 0.46 mm). In this
reference simulation, a force of 35.5 N was acting on the
condyle. This force was calculated by summation of the
nodal forces that were generated by the contact
algorithm. In Fig. 5 the resulting stress distribution
M. Beek et al. / Journal of Biomechanics 33 (2000) 307}316
311
Fig. 5. Distribution of the Von Mises stress (p ) after nine increments of joint loading in a frontal and inferior view of the disc. The stress is
VM
concentrated in the lateral part of the thin ("intermediate) zone of the articular disc (sup"superior, inf"inferior, med"medial, lat"lateral).
312
M. Beek et al. / Journal of Biomechanics 33 (2000) 307}316
(Von Mises) on the surface of the disc is displayed. The
articular disc appeared to be loaded predominantly in
the intermediate zone. The largest values of the stresses
were located in the lateral part of this zone. The strains
were not as concentrated as the stresses, but were also
mainly located in the intermediate zone. The maximum
value of the equivalent elastic strain was 0.44.
3.2. Variation of the direction
A change in the direction of joint loading over 303 had
little in#uence on the distribution of strains and stresses
in the articular disc (Fig. 6). Only in the cases with
a laterally or medially directed load, a small shift of the
strain pattern in the concomitant direction could be
noticed visually. However, the direction in#uenced the
amount of displacement that was needed to obtain a similar amount of deformation. This displacement was
smallest (0.18 mm) when the direction of the movement
was changed over 303 in anterior direction and largest
(0.34 mm) when the direction was changed in posterior
direction.
3.3. Variation of material properties
A force of 327 N was acting on the condyle after
a displacement of 0.2 mm, when the elastic modulus had
a value of 60 MPa. The predicted strains did not di!er
substantially from the strains predicted for the reference
disc elastic modulus. The predicted stresses increased
proportionally with the elastic modulus.
By comparing the results of the simulations obtained
by application of the three di!erent Poisson's ratio
values, we observed that the predicted force acting on the
condyle were proportional to the applied Poisson's ratio,
when the same amount of displacement was simulated.
3.4. Mesh accuracy
In Fig. 7 the Von Mises stress at node 1512 is displayed
as a function of the increment number. This node was
located on the top surface of the disc in the middle of the
region which bore the highest loads. In this area elements
were transformed from linear to quadratic in order to
check the convergence. As can be seen, the Von Mises
stress at this node predicted in the model containing
quadratic elements, increased similarly with the joint
load as in the original model. The amount of stress at the
"nal increment di!ered by only 0.36% from the reference
simulation.
4. Discussion
The present model, as far as we know, is the "rst
three-dimensional FE model of the temporomandibular
joint disc. The disc was shaped according to its anatomical geometry which was sampled with high resolution.
The transition zone between the "brous attachment of
the superior head of the lateral pterygoid muscle and the
articular disc was also included in the present model.
This probably explains why the disc of the present model
and that reported by other researchers (e.g., Chen and
Xu, 1994) had a thicker appearance anteriorly. The disc
was allowed to move unrestrictedly between the articulating surfaces of the condyle and the skull, only kept in
place by the contact occurring between the disc and the
surfaces.
Although movement of the disc was not impeded by
ligaments or friction, the disc remained in the fossa during loading in this position. Variation of the loading
direction over 603 led to only small changes in the stress
and strain distributions. Therefore, it can be concluded
that during clenching, the highest deformations are always located in the same region. The sketches in Fig. 8
visualize the possible reason why a large variation of
loading direction has little in#uence on the deformation
of the disc. Whatever the loading direction, the deformation of the intermediate zone remains large compared to
the other regions in the disc. Fig. 6 might suggest that
during various, both symmetric as well as asymmetric,
loading conditions in a closed jaw position, large stress
concentrations occur predominantly in the lateral side of
the intermediate zone of the disc. This is supported by
a dissection study by Werner et al. (1991), who reported
that wear leading to perforations was mainly located in
this region.
Thus far, no experimental measurements of stress and
strain distributions in the loaded articular disc have
become available. Therefore, it is yet impossible to compare our predictions with such data. It is also still unknown whether the articular disc is preloaded when the
jaw is in the closed position. Therefore, we assumed the
initial loads to be zero and our results should be interpreted as relative values with respect to the initial con"guration.
Comparison of the results with previous "nite element
analyses (Chen and Xu, 1994; DeVocht et al., 1996; Chen
et al., 1998) is di$cult, because these simulations di!er
substantially from those described in the present study.
The studies were two-dimensional and did not describe
the strain distributions. Furthermore, they mutually differed in the direction and the amount of the displacement
applied to the condyle, and in the mechanical properties
of the articular disc. Although the Von Mises stress is not
a reliable representative of the mechanical stress in
a structure when the deformations are very large, it was
nevertheless calculated in the present study to allow
comparison with the studies mentioned. When we applied an elastic modulus according to Chen and Xu
(1994), we obtained almost similar results. DeVocht et al.
(1996) prescribed a displacement that was much larger
M. Beek et al. / Journal of Biomechanics 33 (2000) 307}316
313
Fig. 6. In#uence of the direction of the displacement of the condyle on the distribution of the equivalent elastic strain (e ) in the articular disc. The
%displacement was directed 303 anteriorly (A), posteriorly (B), medially (C), or laterally (D) with respect to the reference simulation.
than the ones applied in the studies by Chen and Xu
(1994) and Chen et al. (1998) and the present one. While
this displacement was almost parallel to the articulating
surface of the skull only small deformations of the disc
occurred.
4.1. Model assumptions
The present "nite element model was based on the
geometry of the right temporomandibular joint of only
one male cadaver. This particular joint did not deviate
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M. Beek et al. / Journal of Biomechanics 33 (2000) 307}316
Fig. 7. The Von Mises stress (p ) at the location of node 1512 as a function of the increment number. The results obtained using quadratic elements
VM
(striped line) do not di!er very much from those using linear elements (solid line).
Fig. 8. Sagittal sketches of the human temporomandibular joint. Left
panel: anterior variation of the direction of the displacement over 303
compared to the reference direction. Right panel: posterior variation of
the direction over 303 compared to the same reference direction. It is
shown that the deformation of the intermediate zone remains large
compared to other regions of the disc.
from normal morphology and thus the model can be
expected to yield generally applicable results. Due to the
complexity of the joint, several additional assumptions
had to be made to ensure convergence of the model.
The method used to "t a polynomial function through
the scattered measurements has been tested with known
mathematical shapes. It showed an accuracy of about
0.01 mm on average (Van Ruijven et al., 1999a). The
question remains whether the applied eighth-order poly-
nomials were adequate to reconstruct the articulating
surfaces of the temporomandibular joint accurately. Visual inspection ensured us that the reconstructions did not
show detectable aberrations.
The convergence of the simulations was investigated
by application of higher-order (quadratic) tetrahedral
elements in the region that was deformed mostly in the
reference simulation. While the desired mechanical
quantities are related to the displacements of the nodes,
such higher-order elements lead to more accurate results.
The results obtained using quadratic elements di!ered by
only 0.36% from those obtained using linear elements.
Therefore, we concluded that mesh re"nement was not
necessary. On the other hand, the results showed relatively large deformations of the disc upon joint loading.
Loading of the joint beyond the values presented here
caused divergence of the calculations due to severe deterioration of the elements. Therefore, re"nement of the
mesh in this particular region would obviously lead to
more accurate results.
It has been demonstrated that the mechanical behavior of cartilage is time dependent, which can be modeled
as viscoelastic (Kovach, 1996) or biphasic (Mow et al.,
1984). Regarding the articular disc, details about these
characteristics are still lacking. Therefore, it was considered inappropriate to simulate dynamic situations like
mouth opening where the condyle moves together with
the articular disc in anterior direction along the articular
eminence. The present study was principally aimed at the
(quasi-)statical behavior of the disc.
M. Beek et al. / Journal of Biomechanics 33 (2000) 307}316
The model did not include deformable cartilage layers
on the articulating surfaces of mandible and skull, while
we were principally aiming for the mechanical behavior
of the articular disc. It has been suggested that the cartilage layers in joints may play an important role in
the load transmission during joint loading (Schreppers
et al., 1990). According to the two-dimensional study
by Chen et al. (1998) the implementation of cartilage
layers in the model had little in#uence on the stress
distribution in the disc. The joint capsule was also neglected in our model. The study by DeVocht et al. (1996)
showed that even during jaw opening, the discal attachments had little in#uence on the mechanical behavior of
the disc.
Acknowledgements
This work was sponsored by the National Computing
Facilities Foundation (NCF) for the use of supercomputing facilities, with "nancial support from Netherlands
Organization for Scienti"c Research (NWO). This research was institutionally supported by the Interuniversity Research School of Dentistry, through the Academic
Center for Dentistry, Amsterdam.
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