Exact solutions of the compressible inverse elasticity problem

Abstract

In elastic modulus imaging or quantitative elastography, tissue stiffness is inferred from an (ultrasonically) measured displacement field. Doing so requires the solution of an elastic inverse problem. We present several exact solutions of the compressible inverse elasticity problem: Given the (possibly transient) dispalcement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density ρ, determine the Lamé parameters λ(x) and μ(x). The cases we treat are: (a) for μ(x) known a priori, find λ; (b) for λ(x) known a priori, find μ; (c) neither λ(x) nor μ is known a priori, but Poisson’s ratio ν is known; (d) neither λ(x) nor μ nor ν is known a priori. We present several example applications including 2D and 3D problems, quasistatic and time dependent displacement fields. Many of the results here are drawn from P.E. Barbone and A.A. Oberai [‘‘Elastic Modulus Imaging: Some exact solutions of the compressible elastography inverse problem,’’ Phys. Med. Biol. (in press)].