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Research Papers

# Transversely Isotropic Elasticity Imaging of Cancellous Bone

[+] Author and Article Information
Spencer W. Shore

Department of Mechanical Engineering, Boston University, 110 Cummington Street, Boston, MA 02215 e-mail: swshore@bu.edu

Paul E. Barbone

Department of Mechanical Engineering, Boston University, 730 Commonwealth Avenue, Boston, MA 02446 e-mail: barbone@bu.edu

Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, 4013 CII, 110 8th Street, Troy, NY 12180 e-mail: oberaa@rpi.edu

Elise F. Morgan

Department of Mechanical Engineering, Boston University, 110 Cummington Street, Boston, MA 02215 e-mail: efmorgan@bu.edu

J Biomech Eng 133(6), 061002 (Jun 16, 2011) (11 pages) doi:10.1115/1.4004231 History: Received January 25, 2011; Revised May 12, 2011; Published June 16, 2011; Online June 16, 2011

## Abstract

To measure spatial variations in mechanical properties of biological materials, prior studies have typically performed mechanical tests on excised specimens of tissue. Less invasive measurements, however, are preferable in many applications, such as patient-specific modeling, disease diagnosis, and tracking of age- or damage-related degradation of mechanical properties. Elasticity imaging (elastography) is a nondestructive imaging method in which the distribution of elastic properties throughout a specimen can be reconstructed from measured strain or displacement fields. To date, most work in elasticity imaging has concerned incompressible, isotropic materials. This study presents an extension of elasticity imaging to three-dimensional, compressible, transversely isotropic materials. The formulation and solution of an inverse problem for an anisotropic tissue subjected to a combination of quasi-static loads is described, and an optimization and regularization strategy that indirectly obtains the solution to the inverse problem is presented. Several applications of transversely isotropic elasticity imaging to cancellous bone from the human vertebra are then considered. The feasibility of using isotropic elasticity imaging to obtain meaningful reconstructions of the distribution of material properties for vertebral cancellous bone from experiment is established. However, using simulation, it is shown that an isotropic reconstruction is not appropriate for anisotropic materials. It is further shown that the transversely isotropic method identifies a solution that predicts the measured displacements, reveals regions of low stiffness, and recovers all five elastic parameters with approximately 10% error. The recovery of a given elastic parameter is found to require the presence of its corresponding strain (e.g., a deformation that generates $ɛ12$ is necessary to reconstruct C1212 ), and the application of regularization is shown to improve accuracy. Finally, the effects of noise on reconstruction quality is demonstrated and a signal-to-noise ratio (SNR) of $40dB$ is identified as a reasonable threshold for obtaining accurate reconstructions from experimental data. This study demonstrates that given an appropriate set of displacement fields, level of regularization, and signal strength, the transversely isotropic method can recover the relative magnitudes of all five elastic parameters without an independent measurement of stress. The quality of the reconstructions improves with increasing contrast, magnitude of deformation, and asymmetry in the distributions of material properties, indicating that elasticity imaging of cancellous bone could be a useful tool in laboratory studies to monitor the progression of damage and disease in this tissue.

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Copyright © 2011 by American Society of Mechanical Engineers
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## Figures

Figure 1

Three dimensional μCT renderings at (a) 0% and (b) -2% strain

Figure 2

Renderings of a 2-mm-thick, longitudinal (x-z) section of the specimen in Fig. 1. Images taken at (a) 0% and (b) 2% compressive strain were used to generate longitudinal (c) and shear (d) strain images using DVC. The reconstructed distribution of secant modulus (e) shows a region of low modulus corresponding to the regions of high strain and of deformations large enough to be apparent to the eye (highlighted by a red circle). Secant modulus values are relative, rather than absolute, because no force data are used.

Figure 3

Positions of the reference images for the results presented in Experiments 2-4 (a) and schematics of the applied loadings for compression in the 2-direction (b) and shear in the 1-2 plane (c). The 1-axis is the principal material direction and the 2 and 3 axes define the transverse, isotropic plane.

Figure 4

Isotropic reconstruction of synthetic data corresponding to an isotropic distribution of material properties with 1% applied strain in the 2-direction (see Fig. 3). C11 and E1 are calculated from the independent λ and μ distributions.

Figure 5

Isotropic reconstruction of synthetic data corresponding to a transversely isotropic distribution of elastic parameters with 1% applied strain in the 2-direction (see Fig. 3)

Figure 6

Isotropic reconstruction corresponding to a transversely isotropic distribution of elastic parameters using three separate synthetic data sets with 1% applied compressive strain applied in each principal material direction

Figure 7

Target distributions for the five independent stiffness coefficients of a transversely isotropic specimen with a compliant spherical inclusion

Figure 8

Transversely isotropic reconstruction of a transversely isotropic distribution of elastic parameters with a uniaxial compressive strain applied in each (1-, 2-, 3-) principal material direction

Figure 9

Transversely isotropic reconstruction of a transversely isotropic distribution of elastic parameters with a uniaxial compressive strain applied in each (1-, 2-, 3-) principal material direction and the addition of a 1-2 shear strain (see Fig. 3)

Figure 10

Transversely isotropic reconstruction of a transversely isotropic distribution of elastic parameters with a uniaxial compressive strain applied in each (1-, 2-, 3-) principal material direction and the addition of a 1-2 shear strain. In this case, regularization (α = 1e-9, β = 5e-4) is also applied.

Figure 11

Target distributions for the five independent stiffness coefficients, Cn , of a transversely isotropic specimen with a compliant band

Figure 12

Transversely isotropic reconstruction of a transversely isotropic distribution of elastic parameters with a uniaxial compressive strain applied in each [1-3] principal material direction. Regularization: α = 1e-8, β = 1e-6 for all Cn except C55 (α3 = 1e-9, β3 = 1e-5).

Figure 13

Shear strain (ɛ31) for (a) a specimen with a compliant spherical inclusion and (b) a specimen with a compliant band with an applied uniaxial compressive load in the 1 direction. Shear strains are not induced in the spherical inclusion whereas they are for the compliant band.

Figure 14

Change in reconstruction accuracy with increasing signal-to-noise ratio (SNR) at two levels of applied strain with and without regularization (α = 1e-8, β = 1e-5) for default loading conditions. The dotted line indicates the level of error for a reconstruction corresponding to zero-noise displacements using tailored regularization (α = 1e-8, β = 1e-6; C55 : α = 1e-9, β = 1e-5).

Figure 15

Improvement in the qualitative fidelity of transversely isotropic reconstruction with increasing signal-to-noise ratio (SNR) for default loading conditions, 1.0% applied strain and regularization (α = 1e-8, β = 1e-5). Marked improvement is achieved at 40dB.

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