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Research Papers

# The Role of Fabric in the Large Strain Compressive Behavior of Human Trabecular Bone

[+] Author and Article Information
Mathieu Charlebois1

Department of Mechanics, Faculty of Civil Engineering, Czech Technical University, Prague Thakurova 7, Prague 166 29, Czech Republicmathieu.charlebois@fsv.cvut.cz

Michael Pretterklieber

Department of Applied Anatomy, Center of Anatomy and Cell Biology, Medical University of Vienna, Währingerstrasse 13, Vienna A-1090, Austriamichael.pretterklieber@meduniwien.ac.at

Philippe K. Zysset

Institute of Lightweight Design and Structural Biomechanics, Vienna University of Technology, Gusshausstrasse 27-29, Vienna A-1040, Austriaphilippe.zysset@ilsb.tuwien.ac.at

1

Corresponding author.

J Biomech Eng 132(12), 121006 (Nov 08, 2010) (10 pages) doi:10.1115/1.4001361 History: Received June 04, 2009; Revised February 19, 2010; Published November 08, 2010; Online November 08, 2010

## Abstract

Osteoporosis-related vertebral body fractures involve large compressive strains of trabecular bone. The small strain mechanical properties of the trabecular bone such as the elastic modulus or ultimate strength can be estimated using the volume fraction and a second order fabric tensor, but it remains unclear if similar estimations may be extended to large strain properties. Accordingly, the aim of this work is to identify the role of volume fraction and especially fabric in the large strain compressive behavior of human trabecular bone from various anatomical locations. Trabecular bone biopsies were extracted from human T12 vertebrae $(n=31)$, distal radii $(n=43)$, femoral head $(n=44)$, and calcanei $(n=30)$, scanned using microcomputed tomography to quantify bone volume fraction $(BV/TV)$ and the fabric tensor $(M)$, and tested either in unconfined or confined compression up to very large strains $(∼70%)$. The mechanical parameters of the resulting stress-strain curves were analyzed using regression models to examine the respective influence of $BV/TV$ and fabric eigenvalues. The compressive stress-strain curves demonstrated linear elasticity, yielding with hardening up to an ultimate stress, softening toward a minimum stress, and a steady rehardening followed by a rapid densification. For the pooled experiments, the average minimum stress was $1.89±1.77 MPa$, while the corresponding mean strain was $7.15±1.84%$. The minimum stress showed a weaker dependence with fabric as the elastic modulus or ultimate strength. For the confined experiments, the stress at a logarithmic strain of 1.2 was $8.08±7.91 MPa$, and the dissipated energy density was $5.67±4.42 MPa$. The latter variable was strongly related to the volume fraction $(R2=0.83)$ but the correlation improved only marginally with the inclusion of fabric $(R2=0.84)$. The influence of fabric on the mechanical properties of human trabecular bone decreases with increasing strain, while the role of volume fraction remains important. In particular, the ratio of the minimum versus the maximum stress, i.e., the relative amount of softening, decreases strongly with fabric, while the dissipated energy density is dominated by the volume fraction. The collected results will prove to be useful for modeling the softening and densification of the trabecular bone using the finite element method.

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## Figures

Figure 4

3D reconstructed images depicting the difference between specimens with low and high axial fabrics; left: calcaneus sample with ρ=0.22 and M33=0.805, right: radius sample with ρ=0.17 and M33=1.60

Figure 5

Graphics of the experimental against the predicted values for the minimum stress. Distinct symbols have been used for the different anatomical sites (see legend in the figure).

Figure 6

Stress-strain curve illustrating the softening difference between samples with high and low axial fabrics

Figure 7

Graphics of the experimental against the predicted values using bone volume fraction and fabric; left: dissipated energy, right: stress at a logarithmic strain of 1.2

Figure 1

Scheme of the unconfined (left) and confined (right) setups. In both cases, samples were glued between metal rods, which were fixed in the testing device using hydraulic grips. The load cell is attached under the fixed lower grip, while the actuator moved the upper grip. The confinement tube was placed by sliding it down over the upper rod, which had a slightly reduced diameter.

Figure 2

Prototype of a stress-strain curve with the defined mechanical parameters; top: low to moderate strains parameters expressed with Green–Lagrange strain and its work conjugated second Piola–Kirchhoff stress, bottom: large strain parameters expressed in logarithmic strain and Cauchy stress

Figure 3

Graphics showing the relation between the volume fraction and the degree of anisotropy (left) and axial fabric (right); CA: calcaneus, FH: femoral head, RA: distal radius, and T12: 12th thoracic vertebra

Figure 8

Photograph depicting the typical difference observed at high strains (∼50%) between sample with low (left) and high (right) axial fabrics

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