Presented at NACOB 98: North American Congress on Biomechanics Canadian Society for Biomechanics - American Society of Biomechanics University of Waterloo Waterloo, Ontario, Canada August 14-18, 1998

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THE IMPACT OF BOUNDARY CONDITIONS AND MESH SIZE ON THE ACCURACY
OF CANCELLOUS BONE TISSUE MODULUS DETERMINATION USING
LARGE SCALE FINITE ELEMENT MODELING

C. Jacobs ¹ , B. Davis ¹ , C. Rieger ¹ , J. Francis ¹ , M. Saad ² , D. Fyhrie ^2
1 Musculoskeletal Research Laboratory, Penn State University, Hershey, PA 17033
² Breech Research Laboratory, Henry Ford Hospital, Detroit, MI

INTRODUCTION

The combination of experimentally measured apparent stiffness and large-scale finite element modeling has been proposed as a method for determining the average hard tissue Young's Modulus of cancellous bone. In this study we evaluate the potential accuracy of this approach and how it is influenced by boundary conditions and mesh size.

REVIEW AND THEORY

The bulk or apparent properties of a porous material such as cancellous bone are a result of the combined effects of both the properties of the hard tissue and its microstructural organization. This fact makes it possible to indirectly quantify hard tissue properties given an experimental measurement of apparent properties and a mathematical model of microstructural organization. Previously, it was suggested that the latter could be accomplished by combining recent large-scale finite element solution techniques with voxel-based automatic meshing generation (Van Reitbergen et al., 1995). Voxel-based large-scale finite element solutions have been successfully applied to cancellous bone with increasing frequency and model size has steady increased (Fyhrie et al., 1997, Van Reitbergen et al., 1997). However, the accuracy of this method has not been directly characterized using a material with known material properties. Furthermore, it has been shown boundary conditions can have a significant impact on the accuracy obtained in the experimental determination of cancellous bone apparent properties (Keaveny et al, 1997). In a previous study we addressed this issue by determining the trabecular tissue modulus of plastic analogues with material properties which were know independently (Jacobs et al, 1998). In this project we have extended our previous study by assessing the impact on accuracy of an increased mesh density and the assumed boundary conditions.

PROCEDURES

A porous structure was obtained in the manner previously described (Jacobs et al, 1998). Briefly, this consists of introducing controlled voids into cubic specimens of a commercial bone cement. Rigid endcaps of glass-mica ceramic (McMaster-Carr, New Brunswick, NJ) were affixed to five specimens with cyanoacrylate glue. Five specimens were left without endcaps. The porous cubes were microCT scanned with a 25µm pixel resolution. A high density mesh was obtained with a 25µm element size by converting each voxel with a microCT density above the bone cement threshold into an element. A 50µm element size mesh was obtained for each specimen by averaging groups of eight adjacent elements together prior to thresholding (Fig. 1). Parallel plate compression was applied to each computer model by assuming either clamped (endcaps) or frictionless (no endcaps) boundary conditions. Poisson's ratio was assumed to be 0.3. Solutions were obtained with custom written software implementing an iterative element-by-element scheme with a conjugate gradient preconditioner. The apparent stiffness was measured experimentally in the standard fashion. Finally, the trabecular hard tissue Young's Modulus input to each of the finite element models was adjusted until the apparent stiffness predicted by the computer model exactly matched that measured in the experimental test, resulting in the average trabecular hard tissue Young's Modulus for each specimen.

Figure 1: A typical FE model with 50 µ m element size.

RESULTS

The Young's Modulus of the bone cement was determined to be 2.15GPa (SEM = 0.08GPa) from the void free specimens. The modulus determined from the indirect method was compared with that determined directly to compute the error in the method (Fig 2.). The relative increase in computational effort associated with the finer mesh is reflected in an increase in the average number of nodes per model from 186,000 to 1,385,000.

Figure 2: The mean error (±SEM) for the four experimental groups

DISCUSSION

These results demonstrate that this approach is an effective method for determining the average trabecular tissue properties of cancellous bone with a high degree of accuracy. In the best case (with end caps and a fine mesh) the error was less than one percent. Accuracy was substantially improved by the use of endcaps at each mesh density indicating that boundary conditions have a significant impact on the accuracy of this approach. In fact, the accuracy improvement due to the use of endcaps was larger than that resulting from an increase in problem size by a factor of six.

REFERENCES

Jacobs C. et al., Trans. ORS, 23, in press, 1998.

Fyhrie D. et al., Trans. ORS, 22, 815, 1997.

Keaveny T. et al., JOR, 15, 101-110, 1997.

Van Reitbergen B. et al. J. Biomech., 28, 69 -81, 1995.

Van Reitbergen B. et al. Trans. ORS, 22, 62, 1997.

ACKNOWLEDGMENTS

Supported by NIH grant RR11769. Cement provided at no cost by Zimmer Inc.