A Three-Dimensional Finite Element Scheme to Investigate the Apparent Mechanical Properties of Trabecular Bone

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Abstract
A computational technique is described for investigating the apparent mechanical properties of trabecular bone based on tissue geometry obtained from the marching cubes volume rendering scheme. Using this scheme, a 3D representation of the trabecular bone was extracted from two-dimensional cross-sections of the tissue originating from a quantitative serial sectioning procedure. Surface information consists of node coordinates and polygon connectivity in a 3D space. A custom, adaptive mesh generation technique using a normal offset was used to prepare 3D finite element volume meshes (4-node tetrahedral elements) of variable mesh density from the extracted surface geometry. Nine target mesh resolutions (32 µm to 107 µm ) were examined for a (1·5 mm × 1·5 mm × 2 mm) volume of trabecular bone. A mesh density of 50,000 elements/mm3 of bone tissue was found to be adequate for convergence of apparent (bulk) modulus for 1 % uniaxial compression. For this convergent case, the maximum local normal compressive tissue stress was 400 MPa which was six hundred-fold greater than the computed apparent stress. Variation in the apparent modulus was less than 5% when Poisson's ratio values were varied between 0·1 and 0·4. Poisson's ratio values greater than 0·4 had a more marked effect on the apparent modulus. Based upon these results, approximately 1 million, 4-node tetrahedral elements are required to analyze a continuum scale model of trabecular bone (5 mm cube).