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Research Papers

Analytical Approach to Recovering Bone Porosity From Effective Complex Shear Modulus

[+] Author and Article Information
Carlos Bonifasi-Lista

Department of Bioengineering, University of Utah, 72 South Central Campus Drive, Room 2646, Salt Lake City, UT 84112cb28@utah.edu

Elena Cherkaev1

Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT 84112elena@math.utah.edu

Yener N. Yeni

Bone and Joint Center, Henry Ford Hospital, 2799 West Grand Boulevard, Detroit, MI 48202yeni@bjc.hfh.edu

1

Corresponding author.

J Biomech Eng 131(12), 121003 (Oct 29, 2009) (8 pages) doi:10.1115/1.4000082 History: Received January 20, 2009; Revised May 14, 2009; Posted September 01, 2009; Published October 29, 2009

This work deals with the study of the analytical relations between porosity of cancellous bone and its mechanical properties. The Stieltjes representation of the effective shear complex modulus of cancellous bone is exploited to recover porosity. The microstructural information is contained in the spectral measure in this analytical representation. The spectral function can be recovered from the effective measurements over a range of frequencies. The problem of reconstruction of the spectral measure is very ill-posed. Regularized algorithm is derived to ensure stability of the results. The proposed method does not use any specific assumptions about the microgeometry of bone. The approach does not rely on correlation analysis, it uses analytical relationships. For validation purposes, complex shear modulus over a range of frequencies was calculated by the finite element method using micro-computed tomography (micro-CT) images of human cancellous bone. The calculated values were used in numerical algorithm to recover bone porosity. At the microlevel, bone was modeled as a heterogeneous medium composed of trabeculae tissue and bone marrow treated as transversely isotropic elastic and isotropic viscoelastic materials, respectively. Recovered porosity values are in excellent agreement with true porosity found from the corresponding micro-CT images.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Example of a micro-CT scan of a T12 vertebra used in the numerical simulations. The subdomain Ω1 is filled with bone marrow, and Ω2 is filled with trabecular tissue.

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Figure 2

Loss modulus of the effective complex shear corresponding to hexagonal structure composed of cylinders filled with trabecular tissue and matrix representing bone marrow. Results calculated by finite element method of solution of Eq. 5 are given by circles (●), and the solid line (−) represents the analytical solution derived in Ref. 11.

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Figure 3

Standard four-parameter Maxwell model

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Figure 8

Eigenvectors ui for i=1,4. Filled dots ● show eigenvectors calculated without noise, and open circles ○ show eigenvectors calculated with noise added. Noise levels: noiseμ1=27.35±11.32, noiseμ2=30.82±15.91, and noiseμ∗=26.49±14.77. (a) Eigenvector corresponding to the first eigenvalue. (b) Eigenvector corresponding to the fourth eigenvalue.

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Figure 4

Steady flow viscosity: filled circles ● show values derived from flow curve of human marrow (40 years old man) at 37°C(26); continuous solid line shows approximation obtained by nonlinear regression using Eq. 12. Correlation factor r2=0.99. Parameters for our model are listed in Table 1.

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Figure 7

Curvature κ versus regularization parameter α. Two local maxima are shown together with the corresponding recovered porosities. True porosity=0.8987.

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Figure 6

Effective complex shear modulus of cancellous bone (bone marrow+trabecular tissue) determined computationally using the finite element method. Values at only four frequencies were computed and used in further calculations. Filled dots ● show storage modulus, and squares ◻ indicate loss modulus.

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Figure 5

Complex shear modulus for bone marrow: filled dots ● show storage modulus, and squares ◻ indicate loss modulus

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