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

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 Lame parameters λ and μ. We consider several special cases of this problem: (a) for μ known a priori, λ is determined by a single deformation field up to a constant. (b) Conversely, for λ known a priori, 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 a priori, 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 non-trivial modulus distributions in a simulated example