Validation of composite finite elements efficiently simulating elasticity of trabecular bone

Abstract

Patient-specific analyses of the mechanical properties of bones become increasingly important for the management of patients with osteoporosis. The potential of composite finite elements (CFEs), a novel FE technique, to assess the apparent stiffness of vertebral trabecular bone is investigated in this study. Segmented volumes of cylindrical specimens of trabecular bone are compared to measured volumes. Elasticity under uniaxial loading conditions is simulated; apparent stiffnesses are compared to experimentally determined values. Computational efficiency is assessed and recommendations for simulation parameters are given. Validating apparent uniaxial stiffnesses results in concordance correlation coefficients 0.69 ≤ r_𝒸 ≤ 0.92 for resolutions finer than 168 μm, and an average error of 5.8% between experimental and numerical results at 24 μm resolution. As an application, the code was used to compute local, macroscopic stiffness tensors for the trabecular structure of a lumbar vertebra. The presented technique allows for computing stiffness using smooth FE meshes at resolutions that are well achievable in peripheral high resolution quantitative CT. Therefore, CFEs could be a valuable tool for the patient-specific assessment of bone stiffness.

Keywords::

• apparent stiffness
• composite finite elements
• linearised elasticity
• nanoindentation
• trabecular bone
• validation

Acknowledgements

The authors would like to thank the DFG for financial support RU567/8-2 and WI1352/13-1, the Hausdorff Centre for Mathematics at University of Bonn and the ILSB at Vienna University of Technology for providing computational resources, Bettina Willi for suggesting gas micropycnometry, Jenny Bienias for performing the gas micropycnometry measurements and Martin Rumpf for valuable advice on CFE and numerical homogenisation.

Notes

^1. Present address for Lars Ole Schwen: Fraunhofer MEVIS, Bremen, Germany.

^2. Present address for Uwe Wolfram: Institute for Surgical Technology and Biomechanics, University of Bern, Bern, Switzerland.