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Number Crunching with the Human Masticatory SystemDepartment of Functional Anatomy-ACTA, Meibergdreef 15, 1105 AZ Amsterdam, the Netherlands; j.h.koolstra{at}amc.uva.nl
Key Words: masticatory mechanics • biomechanical modeling • computers
The human masticatory system is a mechanical system par excellence. Dentists know that all too well. They are constantly engaged in repairing the results of or consolidating the possibilities for mechanical functions like the breaking and crunching of food. These functions, incidentally, lead to mechanically damaged dental elements, proof that the forces involved can be very great. The forces generated by the masticatory muscles and temporomandibular joints cannot be measured. Nevertheless, assessment of these forces is crucial for an understanding of masticatory mechanics. The mechanical behavior of the masticatory system can be approximated mathematically with a mechanical model. This provides a means whereby we can estimate the unmeasurable forces. These models enable investigators both to test hypotheses concerning masticatory function and to analyse the influence of variations in morphology or the effects of surgical or orthodontic interventions. Models have been used since the beginning of the 20th century, although the mathematical approach was considered tiresome in presentation and difficult to understand (Gysi, 1921). For example, the demonstration that, during a static bite, the temporomandibular joints are loaded was apparently not convincing, because much later the problem of joint loading was still controversial (Hylander, 1975). Relatively simple problems can be computed by hand, but for a more detailed analysis, the model may grow too complex, and a computer is indispensable. Initially, application of computer technology required specialized knowledge. With the introduction of the personal computer, however, this requirement diminished and made mechanical modeling and its interpretation—for instance, for the analysis of masticatory mechanics—better accessible. Presently, it encourages the development of hypotheses about the functions of all the phenomena we do not yet properly understand.
When I joined the Department of Functional Anatomy of the Academic Center for Dentistry Amsterdam (ACTA), analysis of the mechanical behavior of the human masticatory system was limited to the sagittal plane (Barbenel, 1972; Pruim et al., 1980; Throckmorton and Throckmorton, 1985). While masticatory function is generally non-symmetric, it was hypothesized that its relevant mechanical aspects are also non-symmetric. Therefore, a three-dimensional mechanical model of the human masticatory system had to be developed. The Department had anticipated the need for computing power for this purpose and had invested in a LSI-11 mini-computer system [~ 0.5 Mflop/s (million floating point operations per second—a measure of computing power independent of computer type)]. The masticatory system is mechanically redundant. There are more muscles than are strictly necessary to generate a certain bite force. Therefore, to produce a bite force, different muscle activation patterns can be applied. Since joint loading changes with muscle activation, it is impossible to estimate the joint forces reliably without proper assumptions regarding muscle recruitment. The problem of mechanical modeling of the masticatory system, therefore, is the formulation of adequate assumptions to facilitate the prediction of the required variables without imposing unrealistic functional conditions. Confronted with this problem, there were two ways we could proceed. The first was to develop an adequate theory of how the masticatory muscles are activated to perform relevant tasks. The second was to define trustworthy boundary conditions, sufficiently reducing the muscle activation possibilities. Unfortunately, there was (and still is) no reliable method for muscle recruitment prediction. Despite various proposals (Osborn and Baragar, 1985; Laboissière et al., 1996; Koolstra and van Eijden, 2001), we still do not know how the central nervous system controls the masticatory system. Since muscle recruitment prediction appeared unreliable, we decided to apply (physiological) constraints to reduce the mechanical redundancy. Bite force, for instance, is limited, because muscle force is constrained by its maximum. Consequently, when the mechanically indeterminate masticatory system performs a maximum possible bite force, it becomes mechanically determinate. A model was constructed based upon the assumption that the central nervous system is able to activate the masticatory muscles mutually independently up to their maximum. It predicted maximum possible bite force, the necessary muscle forces, and the concomitant joint forces (Koolstra et al., 1988). Although this was not the first three-dimensional mathematical model of the human masticatory system (Osborn and Baragar, 1985), it was the first one applied in a three-dimensional analysis. If maximum possible bite force was measured simultaneously with masticatory muscle EMG (Møller, 1966; Pruim et al., 1978), the model could be tested. A three-dimensional force transducer was used to record both the bite force magnitude and direction (Graf et al., 1974). Unfortunately, the smallest commercially available transducer required a relatively large (12–14 mm) separation of the elements. Despite this size, it has been applied successfully for several years for bite force measurement (van Eijden et al., 1988, 1990; Blanksma et al., 1992; Farella et al., 2002).
Presentation of three-dimensional results proved to be another problem. Since the maximum bite force depends on its direction, the complete collection of maximum bite force vectors could be depicted as a series of longer and shorter pins jabbed into a cushion (Fig. 1A
The mechanical model described was limited to the analysis of static bites. However, many of the unresolved mechanical aspects of masticatory function, such as muscle control of the loaded and unloaded jaw, are related to jaw movement, and for this, a three-dimensional dynamic model of the human masticatory system needed to be constructed. Basically, it was simply a matter of applying Newton’s laws to the masticatory system (Koolstra, 2002). The most difficult part was modeling a joint without movement restrictions. Since it was not the purpose at that time to analyze the force distribution within the joint, a geometrically relatively simple joint model was applied (Fig. 2A  ). A sensitivity analysis revealed that this simplification did not qualitatively affect the results (Koolstra and van Eijden, 1995).
The forces and torques applied to the mandible cause accelerations. They must be computed with respect to some origin. For static analyses, this origin was usually located near the temporomandibular joints. For dynamic analysis, this is inappropriate when the moments of inertia of the moving structures become relevant (Stern, 1974). The center of gravity of the lower jaw is the only appropriate origin location. By the addition of the dimension time, a dynamic analysis requires much more computational labor than a static one. The first relatively simple dynamic model (one pair of muscles) required hours of computing time on a state-of-the-art personal computer (~ 0.01 Mflop/s) to simulate less than one second of jaw motion. Full masticatory function analysis required the inclusion of all masticatory muscles with their relevant dynamic properties. This was possible after the fine architecture of these muscles had been measured (van Eijden et al., 1995, 1996, 1997). The resulting model behaved in a manner consistent with experimental measurements, and we learned, among other things, how the muscles interact with the joints to move the jaw in a certain direction (Koolstra and van Eijden, 1997, 1999). It became far too complex to be computed with the available desktop computers. Even an IBM SP1 computer system needed several hours’ computing time on eight parallel processors (~ 200 Mflop/s) for one second of simulation. Presently, the need for high-performance computing facilities for this purpose has been countered by the availability of ever-faster desktop computers.
Thus far, the distribution of the internal joint forces could not be analyzed. This particularly concerns the articular disc, which is considered of major importance. Therefore, a more elaborate model of the jaw joint was necessary. Its articular layers and disc have complex shapes (van Ruijven et al., 1999, 2000). Furthermore, these structures are very "deformable". For internal deformations and forces to be computed in such complex geometric structures, the so-called Finite Element method can be applied. With this method, arbitrarily shaped structures are divided into a large number of regularly shaped elements with known mechanical properties and interactions. The distribution of tensions and deformations in the structures results from the individual components, with a reliability that is strongly related to the accuracy of the model geometry and the mathematical description of the mechanical properties of its materials. The Finite Element method was used earlier for analysis of two-dimensional temporomandibular joint mechanics (Chen and Xu, 1994; DeVocht et al., 1996; Chen et al., 1998). Because the mechanics of the temporomandibular joint are not restricted to a plane, a three-dimensional Finite Element model of the temporomandibular joint was developed (Fig. 2B  ). This model made possible, for the first time, the examination of the force distribution and deformations inside the joint during static bites. It predicted that, locally, these deformations exceeded the 40% and had a clear mediolateral distribution. Similarly, large deformations had been measured experimentally (Beek et al., 2001a), and the largest deformations coincided with the locations where the disc is most often affected by wear (Beek et al., 2000, 2001b). The Finite Element method requires large numbers of computations, which increase rapidly with the model complexity. In the temporomandibular joint, the complexity is relatively large, because it possesses variously curved structures, and its articular disc can move freely between adjacent articular surfaces. The computations, therefore, were executed on the (Dutch) National Supercomputer, a Cray C916 (~ 16,000 Mflop/s), the largest computer system to which we could get access, thanks to the support of the National Computing Facilities Foundation.
The principal task of the temporomandibular joint is to enable the jaws to move. The tensions and deformations it experiences during these movements are likely to play a crucial role in the balance between function and dysfunction. The Finite Element model allows us to analyze joint loading during prescribed displacements of the condyle along the articular surface of the skull and perpendicular to it. These displacements are caused not only by the masticatory muscles and chewing loads, but also by the joint reaction force, which is the direct result of the local tensions in its cartilaginous structures. When the joint is loaded, the condyle will move more closely to the articular eminence than when it is not (Huddleston Slater et al., 1999). This dynamic balance prevents condylar displacements from being correctly described. Therefore, the present static Finite Element model is inadequate for the analysis of temporomandibular joint dynamics.
For a complete mechanical analysis of the joint dynamics, the Finite Element joint models must be integrated with the dynamic musculo-skeletal model. A computer program, developed for automotive crash safety analysis (Madymo), appeared to include this combination. Although it has several limitations and consumes a massive amount of computational power, it becomes possible for the first time to predict the mechanical behavior of both the deformable and undeformable structures of the human masticatory system during jaw movements (Fig. 2C From these developments, I have learned once more that it pays to look beyond the generally accepted borders to reach the apparently unreachable. Novel methods which could help our branch of science forward are often developed for a completely distant, commercially attractive purpose. Now that analysis of temporomandibular joint dynamics has become possible, it is important to know how its cartilaginous structures react mechanically to dynamic loads. Since cartilage is a viscoelastic material, its mechanical behavior changes with time and loading history. The disc, for instance, absorbs energy during repetitive loading (Beek et al., 2001a). This requires the development of a proper qualitative description of the mechanical behavior which incorporates all properties adequately (Beek et al., 2002).
Now that the mechanical analysis of the human masticatory system incorporates more anatomical detail, one might get the impression that the basic mechanical problems are solved. Unfortunately, this is not true. One of the major remaining problems concerns muscle recruitment: In what way do we activate our masticatory muscles and why? A solution to this problem will presumably include the relation between the fiber-type composition of the relevant muscle portions (Korfage et al., 2000) and the speed of the task to be accomplished. The articular layers and disc of the temporomandibular joint combine almost all aspects in which biomechanics deviates from classic mechanics. The cartilaginous materials have complex, time-variant mechanical properties and undergo large deformations. The use of simplified material properties in a mechanical model severely limits its reliability in studies concerning biological structures. Until adequate material properties and material models become available, model predictions will have to be treated with extreme care. As more details of the mechanics of masticatory function are uncovered, new undiscovered areas come into view. For instance, now that we are able to observe the fine architecture of the jaw bone in relation to applied forces, the possibility of analyzing adaptation and its normal and abnormal development becomes apparent. This mandates, on the one hand, that the occurring forces be analyzed over longer periods of time (Langenbach et al., 2002), and, on the other hand, that the mechanics of the masticatory system be analyzed during development and maturation. In conclusion, the availability of computing power is a necessary, but insufficient, condition for progress in the analysis of masticatory function mechanics. I feel fortunate that the necessary creativity and skill in many related disciplines are present in the Department of Functional Anatomy of ACTA, thus enabling us to integrate the various aspects of the system into a whole. This has undoubtedly contributed to the success we have achieved in unraveling many aspects of masticatory function mechanics. Although not all fundamental problems have yet been solved, the application of mathematical models will remain essential to progress in the near future. We have not finished number crunching!
Received for publication February 21, 2003. Revision received April 24, 2003. Accepted for publication June 16, 2003.
Journal of Dental Research, Vol. 82, No. 9,
672-676 (2003) This article has been cited by other articles:
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