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., determine the Lame parameters lambda and mu. We consider several special cases of this problem: (a) for mu known a priori,. is determined by a single deformation field up to a constant. (b) Conversely, for lambda known a priori, mu is determined by a single deformation field up to a constant. This includes as a special case that for which the term lambda del (.) u 0. (c) Finally, if neither. nor mu is known a priori, but Poisson's ratio nu is known, then mu and lambda 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 mu based on the compressible elasticity equations is unstable in the limit lambda -> infinity. Finally, we use the exact solutions as a basis to compute non-trivial modulus distributions in a simulated example.