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Biomaterials & Bioengineering

Prediction of Mechanical Properties of the Cancellous Bone of the Mandibular Condyle

L.J. van Ruijven*, E.B.W. Giesen1, M. Farella2 and T.M.G.J. van Eijden

Department of Functional Anatomy, Academic Center for Dentistry Amsterdam (ACTA), Meibergdreef 15, 1105 AZ Amsterdam, the Netherlands;
1 Department of Orthodontics and Oral Biology, University Medical Centre Nijmegen, Nijmegen, the Netherlands;
2 Department of Orthodontics, University of Naples "Federico II", Naples, Italy;

Correspondence: *corresponding author, l.j.vanruijven{at}amc.uva.n


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS & METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
The mechanical properties of cancellous bone depend on the bone structure. The present study examined the extent to which the apparent stiffness of the cancellous bone of the human mandibular condyle can be predicted from its structure. Two models were compared. The first, a structure model, used structural parameters such as bone volume fraction and anisotropy to estimate the apparent stiffness. The second was a finite element model (FEM) of the cancellous bone. The bone structure was characterized by micro-computed tomography. The calculated stiffnesses of 24 bone samples were compared with measured stiffnesses. Both models could predict 89% of the variation in the measured stiffnesses. From the stiffness approximated by FEM in combination with the measured stiffness, the stiffness of the bone tissue was estimated to be 11.1 ± 3.2 GPa. It was concluded that both models could predict the stiffness of cancellous bone with adequate accuracy.

Key Words: cancellous bone • mandible • finite element model • apparent Young’s modulus


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS & METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
The cancellous bone of the mandibular condyle consists of plate-like trabeculae, which are primarily oriented vertically and antero-posteriorly (Giesen and Van Eijden, 2000). This plate-like trabecular structure is optimal to resist compressive and tensile deformations during loading of the temporomandibular joint (Van Ruijven et al., 2002). The load-bearing capacities of cancellous bone are determined by its mechanical properties, such as apparent stiffness (i.e., the stiffness of cancellous bone measured in a standard compression test) and strength. These properties depend on the density, microstructure, and stiffness of the bone tissue. Due to the combined variations in density and structure, the stiffness of cancellous bone can vary over a large range (Kabel et al., 1999a). This stiffness can be measured on excised bone specimens in vitro. In addition, since the introduction of modern imaging techniques for the characterization of three-dimensional cancellous bone structure (for example, micro-computed tomography [micro-CT], Rüegsegger et al., 1996), non-destructive methods have been developed to predict the mechanical properties of cancellous bone from its structure.

One method of linking the bone structure to its mechanical properties uses one or more structural parameters, extracted from the cancellous microstructure. Examples of parameters that have been used are bone volume fraction, fabric tensor, and anisotropy (Cowin, 1985; Hodgskinson and Currey, 1990; Turner et al., 1990; Goulet et al., 1994; Van Rietbergen et al., 1998; Kabel et al., 1999a; Uchiyama et al., 1999; Ulrich et al., 1999; Borah et al., 2000). Correlation of measured stiffnesses with these structural parameters has yielded highly significant coefficients (r2: 0.70-0.90). Until now, however, these relations have not been generally applicable (Van Rietbergen et al., 1998; Kabel et al., 1999a), i.e., a relation developed for one bone is not automatically valid for another bone.

An alternative method is based on the finite element method. High-resolution images (~ 20-µm resolution) obtained from micro-CT scanning of cancellous bone samples can be digitized and converted to micro-finite element models (FEM) on which compression tests are simulated numerically (Hollister et al., 1994; Van Rietbergen et al., 1995, 1996). This method also requires that the bone tissue stiffness be known. The FEM-predicted stiffnesses have been shown to be highly correlated (r2 = 0.90) with experimentally determined stiffnesses (Hou et al., 1998; Kabel et al., 1999b). Inversely, when the apparent stiffness of a specimen has been measured, the finite element method can also be used to determine the tissue stiffness, because of the linear relationship between the apparent and the tissue stiffnesses (Ladd et al., 1998). Thus far, no information has been reported on the tissue stiffness of the cancellous bone of the mandible.

The aim of the present study was to examine the extent to which the mechanical properties of cancellous bone of the mandibular condyle can be predicted from its structure. For this purpose, predicted stiffnesses were compared with measured stiffnesses. Furthermore, the tissue stiffness of the cancellous bone was determined.


   MATERIALS & METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS & METHODS
RESULTS
 DISCUSSION
 REFERENCES
 
Specimen Preparation
Twenty-four cancellous bone specimens were obtained from the mandibular condyles of 24 embalmed human cadavers with no known history of temporomandibular joint disorders (19 female, five male, 74.8 ± 11.7 yrs [means ± SD]). The use of the condyles 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. Only cadavers with a substantial number of dental elements were selected; the mean numbers of teeth were 8.5 in the upper jaw and 10.7 in the lower jaw. Cylindrical specimens were excised with a custom-made hollow drill (Fig. 1Go); the orientations of the excisions were evenly distributed in space. The specimens had a diameter of 3.57 ± 0.08 mm and a length of 4.88 ± 0.04 mm. The specimens were stored in embalming fluid.


Figure 1
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  		Figure 1. A schematic overview of the different steps in the analysis. A condyle is separated from a human mandible. Cylindrical specimens were taken from the condyle at different orientations and locations. The specimens were scanned in a micro-CT system. The scans were used to construct finite element (FE) models and to calculate the Mean Intercept Lengths {lambda}1, {lambda}2, and {lambda}3 to describe the cancellous bone structure.

 
Simulations
Micro-CT
To obtain the three-dimensional cancellous bone structure, we scanned the intact specimens in a micro-CT system (µCT20, Scanco Medical AG, Zürich, Switzerland) and determined their bone volume fraction experimentally using Archimedes’ method. For an extensive description of the method, we refer to Giesen et al. (2003). Briefly, the specimens were placed in fluid to avoid dehydration and scanned at a resolution of 18 µm. Bone and marrow were discriminated from each other by a fixed threshold, which we obtained experimentally by matching the bone volume fraction from the CT scans with that measured according to Archimedes’ method.

Structure model
The fabric tensor, anisotropy (II), and the bone volume fraction (VF) of the samples were calculated from the CT scans (Software Revision 3.1, Scanco Medical AG, Zürich, Switzerland). The bone volume fraction is the number of voxels containing bone, divided by the total number of voxels. The fabric tensor is a mathematical representation of an arbitrarily oriented ellipsoid. It is used to approximate the angular dependence of a structural parameter. The fabric tensor was calculated with the mean intercept length (MIL) method, i.e., a collection of vectors, their lengths representing the number of marrow-bone intersections in their direction, was constructed. Thereafter, an ellipsoid was fitted through the end-points (Harrigan and Mann, 1984). From the ellipsoid (Fig. 1Go), the lengths {lambda}1, {lambda}2, and {lambda}3 of the main axes and their normalized directions e->1, e->2, and e->3 were calculated. The axes’ lengths were normalized so that {lambda}1 + {lambda}2 + {lambda}3 = 1. These lengths are called the Mean Intercept Lengths. The anisotropy was defined as


Formula 1(1)

The anisotrophy reaches its minimum of 0.0 when {lambda}1 = 1 and {lambda}2 = {lambda}3 = 0. Its maximum of 1/3 is reached when {lambda}1 = {lambda}2 = {lambda}3 = 1/3. The three apparently normal stiffnesses of the trabecular bone were calculated as (Kabel et al., 1999a)


Formula 2(2)

with the stiffness of the bone tissue Et = 11.1 GPa and the model constants km(m = 1,2,3,4), depending on the bone volume fractions (VF) according to


Formula 3(3)

The tissue density {rho} = 2.0 g/cm3 (Giesen et al., 2001). Let a-> = a1e->1 + a2e->2 + a3e->3 be a normalized vector defining the orientation of the cylindrical specimens relative to the fabric tensor. The apparent stiffness Kstruct of every cylindrical specimen was then calculated as


Formula 4(4)

In this equation, the shear stresses are neglected, because they were not measured. An iterative least-squares fit was used to adjust the kma and kmb until the sum of the squared differences between the measured and calculated stiffnesses was minimal.

Finite element model
Using the CT scans, we constructed a mesh of brick elements of 36 x 36 x 36 µm3 (Fig. 1Go). Every element corresponded to a block of 2 x 2 x 2 voxels, of which more than 50% contained bone. The number of elements in the models ranged from 120,000 to 260,000. Parallel-plate compression of the cylinders along their main axes was simulated with frictionless (lubricated) endcaps at the top and bottom surfaces. All element material properties were assumed to be isotropic and linear-elastic. The Young’s modulus was 1 GPa, and the Poisson ratio was 0.3 (Carter and Hayes, 1977; Kabel et al., 1999b). The apparent stiffness KFEM of the cancellous bone was calculated with use of the external force and shortening from the simulations as well as the length and cross-section of the specimens.

For each sample, the tissue stiffness was calculated as the ratio between the measured stiffness and the calculated KFEM times 1 GPa. Subsequently, KFEM was scaled with the average of these tissue stiffnesses.

Mechanical Testing
Destructive mechanical compression tests were carried out on the specimens with a material testing machine (858 Mini Bionix, MTS Systems Corporation, Minneapolis, MN, USA) equipped with a 1-kN load cell. For an extensive description of the test, we refer to Giesen et al. (2001). Briefly, each specimen was placed between 2 steel loading rods lubricated with low-viscosity mineral oil so that low friction would be obtained. An extensometer (model 632.11F-20, MTS) was attached to the loading rods close to the specimen so that we could monitor its deformation while the load cell registered the applied force. After being pre-conditioned, the specimen was compressed at a constant strain rate of 0.002 s–1, until a shortening of 3% was reached. From the stress-strain curve, the apparent stiffness (Kexp) and the ultimate stress were calculated. The ultimate stress was defined as the maximal stress during the test. The part of the stress-strain curve below the ultimate stress was fitted to a fifth-order polynomial. The apparent stiffness was calculated from the maximum of the slope of this stress-strain curve.

Data Analysis
The structure model was implemented in an Excel spreadsheet. The FEMs were solved on a SGI3800 supercomputer equipped with MARC2001 (MSC Software, Los Angeles, CA, USA). SPSS 10.0.7 (SPSS Inc, Chicago, IL, USA) was used for calculation of the descriptive statistics, adjusted correlations, and statistical significances.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS & METHODS
 RESULTS
DISCUSSION
 REFERENCES
 
Both structural and mechanical parameters showed a large variation (Table). The variability of the mechanical parameters was larger than that of the structural parameters. The variability of the anisotropy was very small. The mean tissue modulus of the bone was Et = 11.1 ± 3.2 GPa. The mean apparent stiffness (Kexp) of the specimens was 374 MPa. The Kexp, KFEM, and Kstruct displayed almost the same values for the mean, standard deviation, and range.

The stiffnesses predicted by the structure model (r = 0.95, p < 0.001) and the FEM (r = 0.94, p < 0.001) correlated strongly with the measured stiffnesses (Fig. 2Go). The ultimate stress correlated strongly with the measured stiffness (r = 0.94, p < 0.001), the FEM stiffness (r = 0.97, p < 0.001), and the structure model stiffness (r = 0.92, p < 0.001). The lowest correlation (r = 0.90, p < 0.001) was found between the apparent stiffnesses predicted by the FEM and the structure model. Both the structure model and the finite element model predicted 89% (r2 = 0.89) of the variation in the measured stiffness.


Figure 2
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  		Figure 2. Scatter plots of the apparent stiffness as predicted by the structure of the cancellous bone (top) and a finite element analysis (bottom) vs. the experimentally determined stiffnesses. The lines depict all points where the two stiffnesses are equal.

 
The dependence on the bone volume fraction of the apparent stiffness of the cancellous bone as predicted by the structure model and the bone volume fraction was most pronounced along the first main axis (Fig. 3Go). Along the other main axes, the stiffness depended much less on the bone volume fraction.


Figure 3
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  		Figure 3. The dependence of the ratios between the apparent stiffnesses (ci) and the tissue stiffness (Et) for the 3 main orientations of the trabecular structure on the bone volume. The curves were constructed based on Eq. (2)Go and the values given in the Table for the mean intercept lengths {lambda}1, {lambda}2, and {lambda}3. Note that the curves in Fig. 3Go show ratios.

 

   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS & METHODS
 RESULTS
 DISCUSSION
REFERENCES
 
Until now, only one study has attempted to predict the apparent stiffness of cancellous bone of the human mandibular condyle (Giesen et al., 2003). After a principal component analysis to determine the main structural features and a linear regression analysis, 72% of the variance in the measured stiffnesses could be explained. In the present study, both the structure model as well as the FEM yielded significant improvements, since they could explain 89% of the variance in the measured stiffnesses. This improvement is most likely due to the more complex nature of the models used in this study.

The model constants (Eq. 2Go) found in this study all differ from the constants found for a model developed for several bones (Kabel et al., 1999a). This underlines that a model developed for one bone is not always applicable to another bone. We found, however, that when Eq. (2)Go was simplified to ci = Et[k4{lambda}i2], the correlation coefficient decreased only slightly, to r = 0.93. Apparently, Eq. (2)Go is redundant, i.e., it contains more variables than necessary. This does not imply that the solution is incorrect, but rather that different solutions may predict the same stiffnesses. As a consequence, the model could be invalid for bone specimens whose structural parameters fall outside the range of values used in this study. This redundancy can also explain why a model developed for one bone cannot be used automatically for another bone.

The amount of unexplained variance may have different origins. First, it may be ascribed to noise in the stiffness measurements. This hypothesis is supported by the observation that the ultimate stress also had a squared correlation coefficient of 0.89 with the measured stiffnesses. Second, the structure model used a simplified algorithm to calculate the apparent stiffness of the specimens. In this algorithm, only normal stiffnesses were used. We chose to neglect the other components in the stiffness matrix, because they were not measured or otherwise known. Inclusion of these components in the fitting algorithm would increase the number of variables to be fitted to 18 and may add to the total explained variance. Third, our FEM neglects intraspecimen as well as interspecimen variations in the bone tissue stiffness. The tissue stiffness has been shown to correlate with the apparent density (r = –0.48, Nicholson et al., 1997), and large variations have been shown within one sample (Rho et al., 1997; Zysset et al., 1999). The r2 can decrease by several percent when interspecimen variations are not accounted for (Kabel et al., 1999b). Variations of the tissue stiffness within a specimen can influence the apparent stiffness considerably (Van der Linden et al., 2001; Jaasma et al., 2002); however, the magnitude of its effect on r2 is not known.

By combining the FEM apparent stiffnesses with the measured apparent stiffnesses, we could determine the actual tissue stiffness of the cancellous bone of the mandibular condyle. This value has not been published before. The use of FEM on other bones resulted in tissue stiffnesses ranging from 1.17 GPa for pig lumbar vertebrae to 18.7 GPa for bovine tibia (Hou et al., 1998; Ladd et al., 1998; Kabel et al., 1999a; Borah et al., 2000; Niebur et al., 2000). Stiffnesses determined by nanoindentation ranged from 8 GPa for the human fifth lumbar vertebra to 14 GPa for the human distal radius (Rho et al., 1997; Hoffler et al., 2000). Our value of 11.1 GPa for dentate mandibles is within the range of values found in other studies, indicating that the tissue stiffness of the cancellous bone of the mandibular condyle has the same order of magnitude as that of other bones. It should be noted that the embalming procedure could have slightly increased the tissue stiffness (Linde, 1994).

The apparent stiffness of the mandibular bone ranged from 59 to 809 GPa. This variation can be ascribed to variations in bone volume fraction in the axis along which the stiffness was measured, and to variations in the fabric tensor. Notably, the variation in bone volume fraction primarily affected the stiffness along the main orientation of the cancellous bone (Fig. 3Go). This orientation coincides with a vertical direction, i.e., the direction in which the condyle is primarily loaded during function (Giesen and Van Eijden, 2000).

The methods presented in this study can be used to predict the mechanical properties of cancellous bone with good accuracy without destroying the bone. The structure model has the advantage that it is easy to use, but for a reliable result, the application of the model should be restricted to values that fall within the ranges listed in the Table, since it was optimized for values only within these ranges. The finite element method is much more robust, but it requires a powerful computer. Only the tissue stiffness is required to get an accurate result for any specimen. The finite element method can also be used inversely, i.e., when the apparent stiffness of a specimen is known, the average stiffness of the bone tissue can be estimated. This way, the method can be used to analyze, for example, the influence of age or dentition on the quality of the bone tissue.


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  		Table. Descriptive Statistics of Bone Parameters Determined by Micro-CT, Mechanical Tests, FEM, and Structure Model
 

   ACKNOWLEDGMENTS
 
This work was sponsored by the National Computing Facilities Foundation (NCF) for the use of supercomputing facilities. This research was institutionally supported by the Inter-University Research School of Dentistry, through the Academic Centre for Dentistry Amsterdam. We thank the Academic Computer Services Amsterdam for the use of their technical support. We are grateful to Geerling Langenbach and Jan Harm Koolstra for their comments on the manuscript.

Received for publication October 9, 2002. Revision received May 30, 2003. Accepted for publication July 25, 2003.


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 MATERIALS & METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 

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Journal of Dental Research, Vol. 82, No. 10, 819-823 (2003)
DOI: 10.1177/154405910308201011
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