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Mechanical Significance of the Trabecular Microstructure of the Human Mandibular CondyleDepartment of Functional Anatomy, Academic Center for Dentistry Amsterdam (ACTA), Meibergdreef 15, 1105 AZ Amsterdam, Netherlands; Correspondence: * corresponding author, l.j.vanruijven{at}amc.uva.nl
The human mandibular condyle has a parasagittal plate-like trabecular structure. We tested the hypothesis that this structure reflects the mechanical loading of the condyle. We developed a finite element model of the condyle to analyze the strains occurring during static compressive loading. The principal strains in the trabecular bone were primarily oriented in the sagittal plane. The first component was compressive and oriented supero-inferiorly. The second component was negligibly small and oriented medio-laterally. The third component was tensile, oriented antero-posteriorly, and almost equal to the compressive strain. This tensile strain was caused by antero-posterior bulging of the cortex. This means that the trabecular structure is also subjected to significant tensile forces. The orientation of the parasagittal strains followed the direction of the applied load. It was concluded that the trabecular structure of the mandibular condyle is optimal in resisting the compressive and tensile strains to which it is subjected.
Key Words: human • mandible • finite element analysis • strain
According to the law of bone remodeling (Wolff, 1892), the trabecular structure of bone is optimized to offer maximum resistance to stresses and strains with a minimum amount of bone mass. By the use of finite element models, the correspondence among trabecular structure, bone density, and mechanical loading has been demonstrated in several studies (e.g., Smit et al., 1997; Mullender et al., 1998; Huiskes et al., 2000). Recently, the three-dimensional structure of the trabecular bone of the human mandibular condyle was analyzed by Giesen and Van Eijden (2000). They found that the trabecular bone mainly consists of parallel plates perpendicular to the medio-lateral condylar axis. This means that the trabecular structure can withstand larger stresses in parasagittal planes than in the medio-lateral direction, as indeed has been demonstrated by mechanical tests (Giesen et al., 2001). This suggests that the orientation of the plate-like structure is related to the orientation of the stresses, i.e., that the condyle is optimally adapted to sustain stresses and strains occurring in vivo. Until now, however, a detailed analysis of the stresses and/or strains in the human mandibular condyle has never been performed. The available finite element models (e.g., Korioth et al., 1992; Beek et al., 2000) do not provide enough details for study of the distribution of strains in the bone of the condyle. In the present study, we determined the strains occurring in the mandibular condyle due to static loads to verify that the parasagittal plate-like structure of the trabecular bone is optimized to sustain these loads. For this purpose, we developed a three-dimensional finite element model of the mandibular condyle and calculated, for three different load cases, the strain matrix, the principal strains, and the total principal strain in the condyle. Furthermore, the deformation of the condyle due to loading was analyzed.
Finite Element Model The model was constructed from a dried dentate mandible of a male adult. The use of the human mandible conforms to a written protocol that was reviewed and approved by the Department of Anatomy and Embryology of the Academic Medical Center of the University of Amsterdam. Its right condyle was separated at the neck with an electrical saw. The dimensions of the condyle as well as the bone volume fraction of its trabecular bone were compared with these values for the mandibles used in our previous study (Giesen and van Eijden, 2000). Since the values fell within the range of one standard deviation, our condyle can be considered as a typical specimen. A micro-CT scan (µCT 20, Scanco Medical AG, Bassersdorf, Switzerland) was made of the condyle with a pixel size of 34 x 34 µm2 and an interslice distance of 34 µm. Bone tissue was identified from background noise with a uniform threshold (Giesen and Van Eijden, 2000). Using a bitmap editor, we marked the boundary between the cortical and trabecular bone in the CT scans. Next, all points from the outer surface of the cortex (outer surface), the boundary surface between the cortical and the trabecular bone (inner surface), and the saw plane were extracted. A point belonged to a surface if it contained bone and at least one of the six neighboring points was empty. Through the points of the saw plane, a plane was fitted. To model the inner and outer cortical surfaces, we randomly sslected 30,000 points from both surfaces. We used a method developed by Hoppe (1994) and Schweitzer (1996) to fit subdivision surfaces through these data points.
For a volume mesh to be made, a closed surface mesh had to be made first. Triangular meshes were generated from the subdivision surfaces of the inner and the outer surfaces of the cortex. The boundary vertices of these meshes were projected rectangularly onto the saw plane. The meshes and the projections of the vertices were imported in an automated mesher (Mentat 3.2, MSC Software, Los Angeles, CA, USA). First, the spaces between the projected boundaries and the surface boundaries were closed with a triangular mesh. Next, the area within the projection of the inner boundary as well as the area between the projected boundaries was closed with planar triangular meshes. Finally, tetrahedron meshes were created for the trabecular (44,000 tetrahedrons) and the cortical (13,000 tetrahedrons) volumes (Fig. 1
The trabecular and cortical bone volumes were modeled as isotropic homogeneous material. For cortical bone, a Young’s modulus of 19.0 GPa (Rho et al., 1993) was used. With a mean volume fraction of 0.17 for the trabecular bone of the mandibular condyle (Giesen and Van Eijden, 2000), its estimated Young’s modulus was 1.11 GPa (Rho et al., 1993). Poisson ratios of 0.32 and 0.20 (Carter and Hayes, 1977; Kabel et al., 1999) were used for the cortical and the trabecular bone, respectively.
Simulations The articular surface was divided into three non-overlapping parts with equal surface areas (anterior, apical, and posterior) that were loaded separately with a constant pressure. Since the parts had a surface area of 7 x 10-5 m2, and the estimated reaction force in the condyle during static clenching is approximately 300 N (Koolstra et al., 1988), a pressure of 4.28 MPa was used. The nodes belonging to the saw plane were fixed. The finite element problem was solved with MARC (MSC Software) on a Beowulf Linux-PC cluster. Principal strains as well as the total principal strain were calculated for every trabecular and cortical bone element. The total principal strain of an element is equal to the change in volume of that element. Three-dimensional plots were made with The Visualization Toolkit v3.1 (Kitware Inc., New York, NY, USA) on a Windows/ Intel computer.
The three load cases resulted in three different deformations (Fig. 2  ). In all simulations, the condyle was compressed along the mean loading direction and extended perpendicular to this direction. The amounts of compression and extension were estimated to be approximately 0.2%. Frontal cross-sections did not reveal a deformation in the medio-lateral direction. For all load cases together, 95% of the elements of the cortical bone had a strain less than 1600 µstrain, and 95% of the elements of the trabecular bone had a strain less than 4600 µstrain.
The principal strains in the trabecular bone elements were sorted by magnitude (Table ). The first principal strain was always compressive. The third principal strain was always tensile and only 10-20% smaller in absolute value than the first. The second principal strain alternated between compressive and tensile. Its mean magnitude as well as its standard deviation, however, was negligibly small compared with those of the other principal strains.
The total principal strain was dependent on the load case (Fig. 3 ). It was minimal at the apical load case, and maximal at the posterior load case. Furthermore, the total principal strain changed across the condyle. In every cross-section, the largest total principal strains were found cranially in the region just below the loaded area. Medio-lateral differences were relatively small.
Within the trabecular bone, the directions of the first and third principal strains coincided with those of the sagittal planes. Compression occurred in the supero-inferior direction (direction of the load); extension occurred in the antero-posterior direction (rectangular to the direction of the load). The orientation and magnitudes of the principal strains in the sagittal plane depended on the load case. The angle between the orientations at the anterior and the posterior load was 80°. The apical load case resulted in the smallest principal strains. Within one load case, the orientation was approximately constant throughout the condyle. The magnitudes of the principal strains varied throughout the condyle. They were maximal in the cranial region just below the area where the load was applied. In the medio-lateral direction, the differences were small.
Often the principal orientations and densities of the trabecular bone are compared with stress-related parameters (stress trajectories — Wolff, 1892; principal stresses — Carter et al., 1989; strain energy density — Huiskes et al., 1987; Weinans et al., 1992). In recent studies, strains or strain-derived parameters have become more popular (energy equivalent strain — Miki c’ and Carter, 1995; strain gradients — Turner et al., 1997; strain rate — Mosley and Lanyon, 1998; equivalent strain — Smit and Burger, 2000). This is more in agreement with studies showing that bone cells are sensitive for (pulsating) fluid streams (Klein-Nulend et al., 1995) in the lacunae and canaliculi. These are caused by strain gradients and strain rates (Judex et al., 1997). In the present study, strains were analyzed because strain gradients as well as strain rates are proportional to the strain. Several assumptions had to be made for the simulations. Since the bone volume fraction was known, we assumed that the volume fraction was equal to the bone density and used the Young’s moduli found by Rho et al. (1993) for cortical and trabecular bone. The trabecular bone was modeled isotropically, because only then could a possible correspondence between the orientations of the trabeculae and the principal strains be expected. Experimentally determined Young’s moduli on specimens from the trabecular bone of the condyle (Giesen et al., 2001) were half the value used in this study. For a reliable calculation of the strain distributions, the ratio of the two moduli is more important than the absolute value, and Rho et al. (1993) used the same protocol for the cortical and trabecular bone. To check the influence of the Young’s modulus, we also simulated apical loads with moduli of 2.0 GPa and 4.0 Gpa (data not shown). Increasing the Young’s modulus resulted in an obvious reduction of the strains. At a Young’s modulus of 4.0 GPa, the parasagittal strains had decreased to –599 µstrain (compression) and 401 µstrain (extension). The medio-lateral strain remained negligible. The magnitude of the deformation also decreased, but its shape remained similar to the original deformation. For the Poisson ratios, literature values of other cortical and cancellous bones have been used. To test the influence of the Poisson ratio, we also simulated the apical load with Poisson ratios of 0.3 and 0.4 for the trabecular bone. The increase had a small effect (< 10%) on the magnitudes of the strains, and the shape of the deformation did not change visibly. Finally, there are no experimental data available to describe the contact area and the distribution of the contact forces during static clenching. In the present study, the strain distributions in the mandibular discus calculated by Beek et al. (2000) were used to model the shape of the loaded areas. To investigate the role of cartilaginous structures (disc, articular surfaces) on the occurring strains in the condyle, we are currently developing a three-dimensional dynamic finite element model of the temporomandibular joint, including muscles, bones, and cartilaginous structures. Despite these limitations, the magnitudes of the principal strains found in the present study are in agreement with experimentally determined strains. During in vivo measurements, strains of up to 2000 µstrain have been found in cortical bone (Hylander and Johnson, 1992, 1997). In vivo measurements of trabecular strains are not available, but in vitro experiments show that trabecular bone can be subjected to strains up to 5000 µstrain without being damaged (Wachtel and Keaveny, 1997). In the trabecular bone of the condyle, the tensile strain in the antero-posterior direction was almost as large as the compressive strain in the supero-inferior direction. The Poisson ratio used in the simulations is too small to explain the tensile strain as a result of the compression. This means that, besides the compressive supero-inferiorly oriented compressive stress, there was also an antero-posteriorly oriented tensile stress acting on the trabecular bone. Most likely, the antero-posterior bulging of the cortex caused the tensile stress. This also implies that the bulging was caused not by the trabecular bone volume, but by other factors (e.g., shape of the cortex, shape of the loaded area, etc.). Apparently, the trabecular bone serves not only to resist compression in the supero-inferior direction, but also to resist tension in the antero-posterior direction. In the medio-lateral direction, the strains were negligible. This is probably related to the fact that the compressive and tensile strains in the sagittal plane were almost equal. Compressive strains in the sagittal plane cause extension in the medio-lateral direction, while tensile strains will result in compression in the medio-lateral direction. Since the compressive and tensile strains were almost equal in the simulations, it could be expected that the medio-lateral strain would be negligible. The mean principal strains were smallest for the apical load case. The percentile variation of the mean principal strain was also smallest for the apical load case. This implies that the apical load is distributed more evenly throughout the condyle than the other loads. Probably not only the mean strain but also the ratio maximal strain/mean strain is lower for apical loads. Therefore, the present results suggest that the condyle is more optimized to sustain apical loads than anterior and posterior loads. The strains in the trabecular bone were concentrated in sagittal planes, both tensile and compressive. The orientation of these strains rotated over an angle of 80° in this plane by variations in the load case. To withstand these strains, a plate-like structure with the plates parallel to the sagittal plane is the optimal structure. Note that such a plate-like structure withstands not only compression but also tension in parasagittal planes. This is exactly the structure Giesen and Van Eijden (2000) found in the human mandibular condyle. They also found that the volume fraction and the trabecular thickness are higher in the superior region of the condyle. This also corresponds with the regions where the highest total principal strains were found. Therefore, the results of the present study suggest that the trabecular structure of the mandibular condyle is in agreement with the law of bone remodeling.
This research was institutionally supported by the Interuniversity Research School of Dentistry, through the Academic Centre for Dentistry Amsterdam. We thank the Academic Computing Services Amsterdam (SARA) for the use of their computing facilities, Peter Brugman for his technical assistance, and Jan Harm Koolstra and Geerling Langenbach for their comments on the manuscript. Received for publication March 4, 2002. Revision received July 15, 2002. Accepted for publication July 18, 2002.
Journal of Dental Research, Vol. 81, No. 10,
706-710 (2002)
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