Keywords: soft tissue elastography, inverse problem, FEM, reconstruction.

Soft tissue elastography requires a solution of an inverse problem, based on the displacements (or strains) measured at observation points after a load application to the soft tissue, where the objective is to reconstruct elastic properties of the soft tissue, and thereby identify regions of diseased tissue (e.g. tumours) in a non-invasive way [1].

One of the current approaches starts from a guess of an initial trial solution of the distribution of material stiffness, which is then used in a series of Finite Element Method (FEM) solutions to solve the forward problem [1]. The computed displacements are then compared with the observed target displacements, initial guess is modified and some optimisation method (usually a modified Levenberg-Marquardt method) is employed to iteratively arrive at the distribution of elastic properties of the soft tissue. Such an approach requires considerable computing time. An alternative approach is to compute elastic properties of the material at every point directly from the equilibrium equations [2], however this approach is associated with considerable theoretical difficulties if the noise in the measured displacements is considered.

This paper presents a novel approach, similar to FEM, which employs a weak form of equilibrium equations, with elasticity parameters of each element acting as the unknown variables. The solution is obtained by minimising an objective function, defined as the sum of the residual norms at all nodes, where the nodal residual is represented as a linear function of elasticity parameters of the associated elements. As a result, the soft tissue elastography can be obtained directly by solving the resulting set of linear equations and no iterations are required. When the measured displacement is exact according to the forward problem, this approach can obtain the exact elasticity distribution of the soft tissue.

1

L. Han, J.A. Noble, M. Burcher, "The elastic reconstruction of soft tissue", IEEE International Symposium on Biomedical Imaging, 2002.

2

P.E. Barbone, N.H. Gokhale, "Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverse problem in two dimensions", Inverse Problems, 20(1), 283-296, 2004. doi:10.1088/0266-5611/20/1/017

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