Elsevier

Journal of Biomechanics

Volume 42, Issue 13, 18 September 2009, Pages 2205-2209
Journal of Biomechanics

Short communication
Statistical osteoporosis models using composite finite elements: A parameter study

https://doi.org/10.1016/j.jbiomech.2009.06.017Get rights and content

Abstract

Osteoporosis is a widely spread disease with severe consequences for patients and high costs for health care systems. The disease is characterised by a loss of bone mass which induces a loss of mechanical performance and structural integrity. It was found that transverse trabeculae are thinned and perforated while vertical trabeculae stay intact. For understanding these phenomena and the mechanisms leading to fractures of trabecular bone due to osteoporosis, numerous researchers employ micro-finite element models. To avoid disadvantages in setting up classical finite element models, composite finite elements (CFE) can be used.

The aim of the study is to test the potential of CFE. For that, a parameter study on numerical lattice samples with statistically simulated, simplified osteoporosis is performed. These samples are subjected to compression and shear loading.

Results show that the biggest drop of compressive stiffness is reached for transverse isotropic structures losing 32% of the trabeculae (minus 89.8% stiffness). The biggest drop in shear stiffness is found for an isotropic structure also losing 32% of the trabeculae (minus 67.3% stiffness).

The study indicates that losing trabeculae leads to a worse drop of macroscopic stiffness than thinning of trabeculae. The results further demonstrate the advantages of CFEs for simulating micro-structured samples.

Introduction

Osteoporosis is a widely spread disease (Randell et al., 1995) characterised by a loss of bone mass which induces a loss of stiffness and structural integrity (Karlsson et al., 2005, Marcus and Majumdar, 2001).

Numerous researchers investigated the mechanical properties of trabecular bone using voxel-based micro-finite element models (μFE) models (Müller et al., 1994, Guo and Kim, 2002, van Rietbergen et al., 2003, Morgan et al., 2005, Thurner et al., 2006, Woo et al., 2007, Chevalier et al., 2007). Direct conversion of voxels gained from computed tomography into a hexahedral FE mesh is a robust method but results in a rough mesh with nonsmooth surfaces. Subsequent smoothing (Boyd and Müller, 2006) can lead to distorted elements and thus possibly to a corruption of the results. Generating “good” tetrahedral meshes is a nontrivial problem (Bern and Eppstein, 1992, Teng and Wong, 2000, Shewchuk, 2002). Moreover, they are inherently unstructured. This prohibits the application of geometric multigrid methods (Brandt, 1977, Xu, 1989, Brandt and Ron, 2002) for efficient numerical computation.

To overcome these disadvantages in classical μFE models, composite finite elements (CFE) introduced in Hackbusch and Sauter, 1997b, Hackbusch and Sauter, 1997a, Hackbusch and Sauter, 1998 can be used. A 3D implementation in case of an image based domain description is presented in Liehr et al. (2009) and Preusser et al. (2007).

In contrast to μFE models of trabecular structures, Yeh and Keaveny (1999), Guo and Kim (2002) and Diamant et al., 2005, Diamant et al., 2007 proposed lattice models to simulate osteoporotic and nonosteoporotic trabecular bone by varying trabecular thickness, spacing or random material removal. Compared to lattice models, the volume-based CFE approach permits a much better resolution of the elastic behaviour at trabecular crossings.

This study aims to test the potential of CFE on parameter studies on artificial lattice samples with statistically simulated, simplified osteoporosis. These samples are meant to investigate the influence of structural changes, such as degradation or thinning of trabeculae on the macroscopic stiffness, being one influence factor (among others) on biomechanical stability.

Section snippets

Geometries of samples

We consider artificial micro-structured, elastic specimens consisting of equidistant 10×10×10 rods with circular cross section with diameter d=0.134mm (Hildebrand et al., 1999) and length l=0.335mm for the starting configuration. These specimens represent one cell of a periodic micro-structure big enough for determining macroscopic material parameters (Harrigan et al., 1988).

Displacement boundary conditions are applied to all free trabecular ends on two opposite faces of the bounding box: The

Results

Maximum loss of compressive stiffness of 89.8% is obtained for specimens with transverse isotropic diameters subjected to isotropic degradation. For shear, maximum loss of initial stiffness of 67.3% is reached for an isotropic structure subjected to isotropic degradation. Changing degradation from isotropic, to mainly transverse, to only transverse reduces the maximum loss of compressive stiffness from 76.6% to 50.3% to 6.3% for structures with trabecular diameter-to-length ratio (cf. Section

Discussion

The main advantage of CFE over classical FE is the representation of the geometric complexity of the specimen considered. By using uniform hexahedral grids (and treating the complicated shape in the basis functions), efficient data storage and cache-optimal data retrieval are achieved and all involved matrices have a uniform sparsity structure. Most importantly, geometric multigrid solvers can be used for efficient computations. Given the level set representation of the specimen, grid

Conflict of interest statement

The authors declare that none of them has any potential conflict of interest regarding that article.

Acknowledgments

The authors would like to thank Martin Lenz for fruitful discussions and Stefan Sauter for his advice on CFE. This work was supported by the DFG Projects RU-567/8-2, WI-1352/9-1 and WI-1352/13-1.

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    Both authors contributed equally to the manuscript and share first authorship.

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