Figure 1. Basic 2-D Finite Element Model of the TMJ
(Click
inside the figure to see full view.)
The temporal bone, mandibular condyle, and articular disc were all modeled as deformable bodies using four-noded brick elements. The material properties that were used in the 2-D model are shown in the following table:
Attachments to the disc were modeled using spring elements. The nodes of the cut edges of the temporal bone were fixed in place. Motion was induced into the model by specifying the position of the lower left corner of the modeled part of the mandible and applying a constant vertical force to the lower right corner. The disc moved only passively due to its geometric relationship with the mandibular condyle and the temporal bone. During normal opening of the mouth, the mandibular condyle initially rotates some 15 degrees and then translates along the temporal bone until it is just anterior to the articular eminence. This motion was simulated in the model by defining 20 different steps and doing a static analysis for each step.
ABAQUS/Standard does not have the ability to model arbitrary contact between two deformable 3-D structures - one such structure must be a rigid surface. Consequently, the temporal bone and mandible were modeled as rigid surfaces with the articular disc in between being modeled as a 3-D dimensional solid, deformable object using 8-noded brick elements.
The geometry for the basic 3-D model was taken from the Visible Human Project data. The transverse, high resolution, color photographs containing any part of the right temporomandibular joint and every second or third photograph of lower sections of the right half of the mandible were downloaded from the Internet. These images were uncompressed and loaded into Adobe Photoshop and converted into PICT file format. All of the images were reviewed to determine the size of the smallest rectangle that could be uniformly positioned on each image and still contain all of the TMJ or mandible. This was done to provide a means of comparison from image to image since there was no reference point in each image. Using this approach, the boundaries of the image itself served as the reference points.
The selected and cropped images were then converted into grayscale and imported into an image analysis program named VIDA, (Volumetric Image Display and Analysis, Hoffman et al. 1992). On each image, the boundaries of the mandible and relevant part of the temporal bone were roughly outlined by use of the mouse. An edge detecting routine was then utilized to more precisely fit the rough outline to the actual structure. The routine was based on starting on the outside going in along a normal to each point along the initial outline. The algorithm looked for the first point that was half the maximum grayscale value of the average of the ones before it. Since the entire boundary of each structure was not extremely distinct, this automated approach did not work well in some regions. These regions were then manually edited. Smoothing algorithms for the boundary as a whole were also found to be helpful. Finished boundaries were then plotted on top of the adjacent images to help serve as a guide when defining the boundaries on the next image. This ensured that the changes in geometry from image to consecutive image were all reasonable.
The boundaries for the temporal bone and mandible were then printed out in hard copy. The individual nodes for the finite element model were hand marked along each contour of each structure. These hard copy plots were then placed on a CalComp III Drawing Board digitizing tablet. The coordinates for each node were determined using custom developed software written in Microsoft QuickBASIC for the PC that was driving the tablet. Rigid surfaces were then defined for the mandible and temporal bone using 3 noded triangular elements that could be defined in three dimensions but that had no thickness, as shown in Figures 2 and 3.
Figure 2. Rigid Surface Defining the Mandible
(Click inside
the figure to see full view.)
Figure 3. Finite Element Model of the TMJ
(Click inside the
figure to see full view.)
The modeled geometry of the mandibular condyle as shown in Figures 2 and 3 has manually altered to remove an abnormal bone formation that appeared to be an arthritic spur. Additionaly, post mortem changes had caused the mandible to slip out of the glenoid fossa. The nodes were collectively moved to place the mandibular condyle back in normal position. The disc was then manually created to fill the space between the mandibular condyle and the temporal bone as shown in Figure 3. The mandible, temporal bone, and disc consist of 750, 360, and 912 elements respectively.
A more refined mesh had been used previously to generate these rigid surface but the resulting model would not converge to a solution. It was eventually determined that although the model appeared to be quite smooth with reasonable contacting surfaces, the displayed view was actually deceiving. When shown on the computer screen, the model had straight lines connecting the individual nodes, as do the views shown in Figures 2 and 3. However, when the model was being executed, the actual contacting surface was a Bezier surface containing all of the nodes that were used to define the surface. Since Bezier surfaces are defined in such a way as to force them to be smooth with continuous curvatures, the closely positioned nodes required many peaks and valleys to be placed in the contacting surfaces making a very difficult contact problem. Consequently, the courser mesh shown in Figures 2 and 3 was used.
Although the 3-D model could be made to converge to a solution with the courser mesh, it could only be done by defining all degrees of freedom of the mandible. Therefore the exact position, including the orientation of the mandible had to be defined for every step. This was more artificial than was the approach used in the 2-D model and greatly restricted the range of studies that could be done with the existing 3-D model. As with the 2-D model, the motion of normal opening of the mouth was simulating by making a number (19) of small steps and obtaining a static solution for each.
Figure 4. Von Mises Stress Distribution in Step 7 of Normal
Opening
(Click inside the figure to see full view.)
The von Mises stress is a mathematical combination of all components of both axial and shear stresses. It is commonly used to represent the total stress a given region is experiencing and therefore is a reasonable indicator of where failure will occur. A similar plot was made for each of the 20 steps representing normal opening of the mouth. Shown in rapid sequence, these results from each of 20 static solutions simulate the motion of normal opening, as well as showing the changing stress distribution patterns. This effect is illustrated in the animation that can be displayed by clicking on the following icon.
2-D Normal Opening with Springs
Due to the extremely complex nature of the in vivo structures, the attachments to the disc are undoubtedly the most unrealistic aspect of the model. Several different variations were made but in every case the disc was observed to follow along with the mandible as it does in this sequence depicting normal opening. This was true even in the case when there were no attachments to the disc at all, as shown in the animation that can be displayed by clicking on the following icon.
2-D Normal Opening with no Attachments
to the Disc
These results support the contentions of Osborn (1985) that muscle contraction is not required to pull the disc forward during normal opening of the mouth and that the elastin fibers in the posterior attachments to the disc are not required to pull the disc back during normal closing of the mouth.
A series of parametric studies were done by varying one of the parameters used in the model while holding all the others constant. This general approach is one of the main strengths of the finite element method - to demonstrate the effect of a varying a single parameter while everything else remains exactly the same. In the course of this work many parametric studies were done including the modulus of elasticity of the disc, Poisson's ratio of the disc, the stiffness of the springs attached to the disc, the amount of vertical force applied to the mandibular condyle, and the amount of anterior force applied to the anterior edge of the disc. Additional work was done to experimentally determine the modulus of elasticity of the disc and to provide experimental validation of the results of the model. Parametric studies with anterior force on the disc were done to simulate the contraction of the superior head of the lateral pterygoid muscle. That muscle apparently at least sometimes has fibers that attach directly to the anterior edge of the disc. A simulation that was done with an anterior force on the disc. Step 7 of that simulation is compared to step 7 from an identical simulation but with no anterior force in Figure 5.
FIGURE 5: 2-D Normal Opening with Anterior Force Applied to the
Disc
(Click inside the figure to see full view.)
It can be seen that the disc in the simulation with the anterior force is positioned more anteriorly than when no anterior force is used. It would seem reasonable to assume that if sufficient anterior force were used that the disc could be observed to displace anteriorly relative to the mandibular condyle - the most commonly seen in vivo TMJ derangement. Unfortunately, the model could not be made to converge to a solution when using higher anterior forces, even when the springs representing medial and lateral attachments to the disc were removed. However, it can be observed by comparing the simulation involving the anterior force (ICON 3) to the others (ICONS 1 and 2) that the overall stress in the TMJ is higher. This higher stress with the anterior force applied suggests that hypertonicity of the superior head of the lateral pterygoid muscle could eventually lead to abnormally high stresses and therefore result in excessive wear and erosion of components of the TMJ as described by Oberg et al.(1971).
As was previously discussed, the existing 3-D model is much more limited in its versatility than the 2-D model. This is because the 3-D model will not converge to a solution unless every degree of freedom of the mandible is defined. This restriction makes it impractical to do parametric studies as done with the 2-D model be because the course of the mandible will be, by definition, exactly the same no matter what changes are made to any of the parameters. However, even with this limitation, it is still possible to demonstrate what effect having or removing the spring attachments to the disc has on the movement of the disc. Clicking on the following icon will display an animation depicting a lateral view of the 3-D model going through normal opening with no attachments to the disc.
3-D Normal Opening with no Attachments to
the Disc (Lateral View)
Only the condylar portion of the mandible is shown in the animation in an effort to make the animation less complicated and easier to interpret. Note that the disc does again follow along with the mandible even though there are no conditions placed on any part of the disc other than the geometric interaction with the mandible and temporal bone. It is more difficult to visualize the entire structure of a 3-D model with a single view so a second animation was made of the same analysis which can be seen by clicking on the following icon.
3-D normal Opening with no Attachments to
the Disc (Superior View)
In this animation you are looking down on the TMJ from directly above. Since the temporal bone and the mandible were modeled as rigid structures, there can be no stress information calculated for them. However, they are displayed as wire meshes so as to portray their position and still allow you to see through them so you can see the stress patterns in the disc. In this view, the disc is below the temporal bone and above the mandibular condyle. These results also support Osborn's contentions that contraction of the superior head of the lateral pterygoid muscle is not required to move the disc forward during normal opening of the mouth and that the elastin fibers in the posterior attachments to the disc are not required to pull the disc back during closing of the mouth.
DeVocht JW, Goel VK, Zeitler DL, Lew D (1996): A Study of the Control of Disc Movement Within the Temporomandibular Joint Using the Finite Element Technique. Scheduled to be published in the December, 1996 issue of the Journal of Oral and Maxillofacial Surgery .
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Hoffman EA, Gnanaprakasam D, Gupta KB, Hoford JD, Kugelmass SD, Kulawiec RS (1992): VIDA: An environment for multidimensional image display and analysis. SPIE Proc. Biomed. Image Processing and 3-D Microscopy, vol. 1660:1-18.
Katzberg RW, Westesson P-L (1993): Diagnosis of the temporomandibular joint. Philadelphia:WB Saunders, p 5, 21.
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Osborn J (1985): The disc of the human temporomandibular joint: design, function and failure. J Oral Rehab 12:279-293.