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Research Papers

Generalized Anisotropic Inverse Mechanics for Soft Tissues

[+] Author and Article Information
Ramesh Raghupathy

Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455

Victor H. Barocas1

Department of Biomedical Engineering, University of Minnesota, Minneapolis, MN 55455baroc001@umn.edu

1

Corresponding author.

J Biomech Eng 132(8), 081006 (Jun 15, 2010) (6 pages) doi:10.1115/1.4001257 History: Received October 13, 2009; Revised February 05, 2010; Posted February 11, 2010; Published June 15, 2010; Online June 15, 2010

Elastography, which is the imaging of soft tissues on the basis of elastic modulus (or, more generally, stiffness) has become increasingly popular in the last decades and holds promise for application in many medical areas. Most of the attention has focused on inhomogeneous materials that are locally isotropic, the intent being to detect a (stiff) tumor within a (compliant) tissue. Many tissues of mechanical interest, however, are anisotropic, so a method capable of determining material anisotropy would be attractive. We present here an approach to determine the mechanical anisotropy of inhomogeneous, anisotropic tissues, by directly solving the finite element representation of the Cauchy stress balance in the tissue. The method divides the sample domain into subdomains assumed to have uniform properties and solves for the material constants in each subdomain. Two-dimensional simulated experiments on linear anisotropic inhomogeneous systems demonstrate the ability of the method, and simulated experiments on a nonlinear model demonstrate the ability of the method to capture anisotropy qualitatively even though only a linear model is used in the inverse problem. As with any inverse problem, ill-posedness is a serious concern, and multiple tests may need to be done on the same sample to determine the properties with confidence.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 6

Different types of partition forming operations: (a) Skinning surface elements along a chain of boundary nodes starting at the red (square) node, which has a resultant force. (b) Current boundary is shown with thick solid lines. Candidate element (shaded pink) can be trimmed by donating to surrounding solved partitions (A, B, C).

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Figure 5

Noise sensitivity: (a) Cijkl values sensitivity to noise; (b) Normalized standard deviation of Cijkl; (c) Principal direction of Cijkl; (c) Correlation between standard deviation and confidence intervals

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Figure 4

Simulated nonlinear test: (a) Variation of eigenvector angle over the domain. (b) Anisotropy plot comparison with averaged input.

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Figure 3

Simulated linear test: (a) Displacement vector; (b) Strain εyy; (c) Partitions produced by GAIM

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Figure 2

Solving the interior sub-domains: (a) The red (square) node has a resultant force and is a starting point for forming an interior partition. The first n layers of elements from it are selected as the partition (shown shaded for n=3). (b) Removal of solved elements gives resultant forces on the updated boundary. Further partitions are formed from such boundary nodes.

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Figure 1

(a) Biaxial stretch setup: Force measured on each arm is the total force on that part of Γu in the direction of pull. (b) Initial boundary partition for bottom arm: All equations of circular (blue) nodes are solved. Sum of equations of square (red) nodes in the direction of pull are solved.

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