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Research Papers

# Comparison of Mixed and Kinematic Uniform Boundary Conditions in Homogenized Elasticity of Femoral Trabecular Bone Using Microfinite Element Analyses

[+] Author and Article Information
Jarunan Panyasantisuk

Institute for Surgical Technology
and Biomechanics,
University of Bern,
Stauffacherstr. 78,
Bern CH-3014, Switzerland
e-mail: jarunan.panyasantisuk@istb.unibe.ch

Dieter H. Pahr

Institute for Lightweight Design
and Structural Biomechanics,
Vienna University of Technology,
Gusshausstr. 27-29/317,
Vienna A-1040, Austria
e-mail: pahr@ilsb.tuwien.ac.at

Thomas Gross

Institute for Lightweight Design
and Structural Biomechanics,
Vienna University of Technology,
Gusshausstr. 27-29/317,
Vienna A-1040, Austria
e-mail: tgross@ilsb.tuwien.ac.at

Philippe K. Zysset

Institute for Surgical Technology
and Biomechanics,
University of Bern, Stauffacherstr. 78,
Bern CH-3014, Switzerland
e-mail: philippe.zysset@istb.unibe.ch

Manuscript received May 30, 2014; final manuscript received October 23, 2014; accepted manuscript posted November 3, 2014; published online December 10, 2014. Assoc. Editor: Blaine Christiansen.

J Biomech Eng 137(1), 011002 (Jan 01, 2015) Paper No: BIO-14-1235; doi: 10.1115/1.4028968 History: Received May 30, 2014; Revised October 23, 2014; Accepted November 03, 2014; Online December 10, 2014

## Abstract

Mechanical properties of human trabecular bone play an important role in age-related bone fragility and implant stability. Microfinite element (μFE) analysis allows computing the apparent elastic properties of trabecular bone for use in homogenized FE (hFE) analysis, but the results depend unfortunately on the type of applied boundary conditions (BCs). In this study, 167 human femoral trabecular cubic regions with a side length of 5.3 mm were extracted from three proximal femora and analyzed using μFE analysis to compare systematically their stiffness with kinematic uniform BCs (KUBCs) and periodicity-compatible mixed uniform BCs (PMUBCs). The obtained elastic constants were then used in the volume fraction and fabric-based orthotropic Zysset–Curnier model to identify their respective model parameters. As expected, PMUBCs lead to more compliant apparent elastic properties than KUBCs, especially in shear. The differences in stiffness decreased with bone volume fraction and mean intercept length (MIL). Unlike KUBCs, PMUBCs were sensitive to heterogeneity of the biopsies. The Zysset–Curnier model fitted the apparent elastic constants successfully in both cases with adjusted coefficients of determination ($radj2$) of 0.986 for KUBCs and 0.975 for PMUBCs. The proper use of these BCs for hFE analysis of whole bones will need to be investigated in future work.

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## Figures

Fig. 1

Top: a μFE mesh converted directly from a μCT image with a resolution of 36 μm; middle: the uni-axial loading case in 1-direction applying PMUBCs; bottom: the shear loading case in 1- and 3-direction applying PMUBCs; ui,ti, and ei are displacements, tractions, and unit directions, respectively. North–South plane is normal to West–East and Bottom–Top planes.

Fig. 2

Examples of highly heterogeneous bone biopsies which were not included in the filtered data set: BV/TV = 11.21% (left) and 24.07% (right)

Fig. 3

(a) The relationship between BV/TV and degree of anisotropy of the femoral trabecular bone cubes from Gross et al. [25] and the current study. The complete data set of the current study includes 167 femoral bone biopsies. The filtered data set excludes highly heterogeneous bone samples, and includes the 126 remaining bone cubes. (b) The linear regression of the relationship between model stiffness ln(SmodelKUBC) and μFE stiffness ln(SFEorthoKUBC) based on KUBCs in log space. The data sets of Gross et al. and of the current study are in line despite the distinct μFE model resolution and the rotation of the μCT image necessary to align the biopsies with the fabric tensor in the current study.

Fig. 4

A comparison of μFE stiffness tensor components based on KUBCs and PMUBCs in log space. As expected, the stiffness from PMUBCs are more compliant than from KUBCs. Yet, the μFE stiffness tensor components of the two BCs highly correlate.

Fig. 5

Δ denotes the difference of μFE stiffness tensor components between KUBCs and PMUBCs; (a) the relationship of Δ and BV/TV: Δ decreases with increasing BV/TV. The shear components have the highest Δ, while the diagonal terms of the normal components have the lowest Δ; (b) the relationship of Δ and fabric: Δ in the major fabric direction is lower than that in the minor directions.

Fig. 6

The linear regressions of the relationship between the model and μFE stiffness based on PMUBCs. The filtered data set based on a heterogeneity criterion excludes successfully the outliers in the Zysset–Curnier model.

Fig. 7

The filtered data set: the linear regression of the relationship between the model stiffness and μFE stiffness based on PMUBCs. Using the split linear systems improved the model fitting.

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