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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.

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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.

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Correspondence to Ghislain Maquer.

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Appendices

Appendix 1

See Table 3.

Appendix 2

See Table 4.

Appendix 3

An extra analysis was performed on the complete data set after mesh refinement of the 2.63, 5.29, 7.96 and 10.58 mm CVEs. Each hexahedral finite element from the original mesh (37 \(\upmu \hbox {m}\) size) was divided into eight smaller elements (18.5 \(\upmu \hbox {m}\) size). The Zysset–Curnier model was then fitted to the KUBCs, PMUBCs and in situ stiffness tensors to evaluate the impact of the refinement (Tables 5, 6). To verify the impact of the mesh on the Hill condition, the average of the product of the stress and strain tensors for all elements (internal strain energy \(U_{\mathrm{micro}})\) and the product of the stress and strain averages (macro-level strain energy \(U_{{ macro}})\) were computed. The relative difference between \(U_{\mathrm{micro}}\) and \(U_{{ macro}}\) for 5.29 mm IVEs as a function of BV/TV is presented in Fig. 6.

Table 5 Zysset–Curnier model parameters calculated for 5.29 mm CVEs after mesh refinement
Table 6 Relative changes in the Zysset–Curnier model parameters after mesh refinement computed for constant \(k=2.3116\) and \(l=1.313\)
Fig. 6
figure 6

Relative difference between internal strain energy \(U_{\mathrm{micro}}\) and macro-level strain energy \(U_{{ macro}}\) before and after mesh refinement as a function of BV/TV for a KUBCs, b in situ, c PMUBCs. The Hill condition is only approximately fulfilled. The difference between strain energy at macro- and micro-level decreases after mesh refinement and with increasing BV/TV

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Daszkiewicz, K., Maquer, G. & Zysset, P.K. The effective elastic properties of human trabecular bone may be approximated using micro-finite element analyses of embedded volume elements. Biomech Model Mechanobiol 16, 731–742 (2017). https://doi.org/10.1007/s10237-016-0849-3

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