Short communicationStatistical osteoporosis models using composite finite elements: A parameter study
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 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 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 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 rods with circular cross section with diameter (Hildebrand et al., 1999) and length 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.