Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

Abstract

Abstract. We consider several inverse problems motivated by elastography. Given the (possibly transient) displacement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density ρ, determine the Lamé parameters λ and µ. We consider several special cases of this problem: (a) For µ known apriori, λ is determined by a single deformation field up to a constant. (b) Conversely, for λ known apriori, µ is determined by a single deformation field up to a constant. This includes as a special case that for which the term λ∇ · u ≡ 0. (c) Finally, if neither λ nor µ is known apriori, but Poisson's ratio ν is known, then µ and λ are determined by a single deformation field up to a constant. This includes as a special case plane stress deformations of an incompressible material. Exact analytical solutions valid for 2D, 3D, and transient deformations are given for all cases in terms of quadratures. These are used to show that the inverse problem for µ based on the compressible elasticity equations is unstable in the limit λ → ∞. Finally, we use the exact solutions as a basis to compute nontrivial modulus distributions in a simulated example.

Adjoint-weighted variational formulation for the direct solution of plane stress inverse elasticity problems

“…As such, the prescribed data, that is the tensor A(x), must satisfy a solvability condition. That condition is derived in [11], along with the exact analytical solution of this inverse problem.…”

A Fourier‐series‐based virtual fields method for the identification of 2‐D stiffness distributions

“…One approach for the linear elastic case is to solve for the modulus analytically while exploiting the information from elastography [1]. Exact distributions of the shear modulus valid for two-dimensions (2-D) and three-dimensions (3-D) are achieved up to a constant, by solving the momentum equation written for compressible elastic materials, given static (or time-dependent) full-field displacements u(x, y), the density ρ(x, y) and the Lamé parameter λ(x, y).…”

Divergence of finite element formulations for inverse problems treated as optimization problems

“…In this paper, we limit our attention to plane stress incompressible elasticity. In this case, an exact analytical solution is available in nearly closed form (up to a quadrature) [1,5]. Thus it is known that the solutions are unique and that the problems are well-posed, provided the data is sufficiently smooth.…”

“…When the given displacement field u m (x) satisfies certain solvability conditions, an exact (strong) solution for µ(x) exists, which is given by [1]: In (3), µ(x o ) is the single specified constant required to make the solution unique, and is the (2D) measured strain tensor. It is thus clear from the form of equation (3) that the solution of the inverse problem is unique and well-defined for any reasonably smooth measured displacement field that satisfies the solvability conditions described in [1].…”

“…It is thus clear from the form of equation (3) that the solution of the inverse problem is unique and well-defined for any reasonably smooth measured displacement field that satisfies the solvability conditions described in [1].…”