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Research Papers

Elastic Anisotropy of Human Cortical Bone Secondary Osteons Measured by Nanoindentation

[+] Author and Article Information
Giampaolo Franzoso

Laboratory of Biological Structure Mechanics (LaBS), Structural Engineering Department, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italygiampaolo.franzoso@polimi.it

Philippe K. Zysset

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

J Biomech Eng 131(2), 021001 (Dec 09, 2008) (11 pages) doi:10.1115/1.3005162 History: Received December 14, 2007; Revised September 20, 2008; Published December 09, 2008

The identification of anisotropic elastic properties of lamellar bone based on nanoindentation data is an open problem. Therefore, the purpose of this study was to develop a method to estimate the orthotropic elastic constants of human cortical bone secondary osteons using nanoindentation in two orthogonal directions. Since the indentation modulus depends on all elastic constants and, for anisotropic materials, also on the indentation direction, a theoretical model quantifying the indentation modulus from the stiffness tensor of a given material was implemented numerically (Swadener and Pharr, 2001, “Indentation of Elastically Anisotropic Half-Spaces by Cones and Parabolae of Revolution,” Philos. Mag. A, 81(2), pp. 447–466). Nanoindentation was performed on 22 osteons of the distal femoral shaft: A new holding system was designed in order to indent the same osteon in two orthogonal directions. To interpret the experimental results and identify orthotropic elastic constants, an inverse procedure was developed by using a fabric-based elastic model for lamellar bone. The experimental indentation moduli were found to vary with the indentation direction and showed a marked anisotropy. The estimated elastic constants showed different degrees of anisotropy among secondary osteons of the same bone and these degrees of anisotropy were also found to be different than the one of cortical bone at the macroscopic level. Using the log-Euclidean norm, the relative distance between the compliance tensors of the estimated mean osteon and of cortical bone at the macroscopic level was 9.69%: Secondary osteons appeared stiffer in their axial and circumferential material directions, and with a greater bulk modulus than cortical bone, which is attributed to the absence of vascular porosity in osteonal properties. The proposed method is suitable for identification of elastic constants from nanoindentation experiments and could be adapted to other (bio)materials, for which it is possible to describe elastic properties using a fabric-based model.

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Figures

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Figure 1

Conical indentation in an anisotropic material: The contact perimeter around the indenter has different depths. (e1,e2,e3) is the MCS and (a1,a2,a3) is the corrected ICS.

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Figure 2

Definition of the angle α for the EPAC orientation in the first guess ICS (e1′,e2′,e3′); after an α rotation around a3≡e3′ the corrected ICS must coincide with (a1,a2,a3).

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Figure 3

Indentation modulus EcI(S,n) (left) and EPACs (right) for cortical bone (at the macroscopic level) computed by applying the Swadener and Pharr theory (9) to the orthotropic stiffness tensor found in literature (1). On the right side, the red lines represent the shorter semi-axes of the EPACs: Because the computed eccentricities were always close to the unity in order to give a representation of cortical bone anisotropy that could be appreciated, the semi-axis ratio was elevated to power 10; therefore, to draw visible ellipses on a sphere with unitary radius, for each ellipse a2=0.015 was chosen and a1=0.015(a1∕a2)10 was used.

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Figure 4

Sensitivity analysis: IM of cortical bone computed for variations of the orthotropic elastic constants. Indentation direction is represented by circle, box, or triangle. On the x-axis, the literature values of the elastic constants are reported (1).

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Figure 5

Fabric-based orthotropy in a cylindrical coordinate system. (a) The sketch of a sliced osteon shows the lamellar organization; at each material point, the MCS (e1,e2,e3) and the the fabric principal directions (m1,m2,m3) are assumed coincident. (b) Three particular cases for the fiber organization at a material point of lamellar bone: Below each case the relation among fabric eigenvalues is reported as well as the resultant elastic symmetry (orthotropic or transverse isotropic).

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Figure 6

Theoretical curves describing the IM in the circumferential (m2) and in the axial (m3) direction of lamellar bone tissue as function of the second eigenvalue m2 of the fabric tensor M computed using the normalized stiffness tensor S¯(M)

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Figure 7

Left: N-face and P-face of a bone cube after polishing: The surfaces are orthogonal and have a common edge. Individual osteons with the sliced Haversian canals are easy to identify. The arrow indicates the upper direction of the femoral shaft. Right: A secondary osteon after nanoindentation in its axial (on the N-face) and circumferential (on the P-face) directions.

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Figure 8

Home-designed holder with two cubic specimens glued into its groove. A sketch of the N-face and the P-face is superimposed over the left specimen for a better comprehension of the mounting direction.

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Figure 9

Three-dimensional representation of the elongation modulus ε(n) as function of a direction vector n with the bulk modulus κ(n) superimposed: The shape of the surfaces represents ε, while the color represents κ. (a) Mean osteon in cortical bone tissue measured by means of nanoindentation. (b) Cortical bone at the macroscopic level (1). Units are in gigapascals.

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