# The effective elastic properties of human trabecular bone may be approximated using micro-finite element analyses of embedded volume elements

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## Abstract

Boundary conditions (BCs) and sample size affect the measured elastic properties of cancellous bone. Samples too small to be representative appear stiffer under kinematic uniform BCs (KUBCs) than under periodicity-compatible mixed uniform BCs (PMUBCs). To avoid those effects, we propose to determine the effective properties of trabecular bone using an embedded configuration. Cubic samples of various sizes (2.63, 5.29, 7.96, 10.58 and 15.87 mm) were cropped from \(\mu \hbox {CT}\) scans of femoral heads and vertebral bodies. They were converted into \(\mu \hbox {FE}\) models and their stiffness tensor was established via six uniaxial and shear load cases. PMUBCs- and KUBCs-based tensors were determined for each sample. “In situ” stiffness tensors were also evaluated for the embedded configuration, i.e. when the loads were transmitted to the samples via a layer of trabecular bone. The Zysset–Curnier model accounting for bone volume fraction and fabric anisotropy was fitted to those stiffness tensors, and model parameters \(\nu _{0}\) (Poisson’s ratio) \(E_{0}\) and \(\mu _{0}\) (elastic and shear moduli) were compared between sizes. BCs and sample size had little impact on \(\nu _{0}\). However, KUBCs- and PMUBCs-based \(E_{0}\) and \(\mu _{0}\), respectively, decreased and increased with growing size, though convergence was not reached even for our largest samples. Both BCs produced upper and lower bounds for the in situ values that were almost constant across samples dimensions, thus appearing as an approximation of the effective properties. PMUBCs seem also appropriate for mimicking the trabecular core, but they still underestimate its elastic properties (especially in shear) even for nearly orthotropic samples.

## Keywords

Trabecular bone Elastic properties Boundary conditions Micro-finite elements In situ Embedded configuration## Notes

### Acknowledgments

The script manager Medtool (www.dr-pahr.at) was used to generate the \(\mu \hbox {FE}\) models, start the computations and analyse the results. The authors would like to thank Prof. Bert van Rietbergen for sharing the \(\mu \hbox {CT}\) data. Karol Daszkiewicz is supported by grants from the Faculty of Civil and Environmental Engineering, Gdańsk University of Technology. The Swiss National Foundation (Grant No. 143769) and the Gebert Rüf Foundation (GRS-079/14) are also gratefully acknowledged.

### Compliance with ethical standards

### Conflict of interest

The authors have no conflict of interest to report.

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