Quantitative elasticity imaging: what can and cannot be inferred from strain images

We examine the inverse problem associated with quantitative elastic modulus imaging: given the equilibrium strain field in a 2D incompressible elastic material, determine the elastic stiffness (shear modulus). We show analytically that a direct formulation of the inverse problem has no unique solution unless stiffness information is known a priori on a sufficient portion of the boundary. This implies that relative stiffness images constructed on the assumption of constant boundary stiffness are in error, unless the stiffness is truly constant on the boundary. We show further that using displacement boundary conditions in the forward incompressible elasticity problem leads to a nonunique inverse problem. Indeed, we give examples in which exactly the same strain field results from different elastic modulus distributions under displacement boundary conditions. We also show that knowing the stress on the boundary can, in certain configurations, lead to a well-posed inverse problem for the elastic stiffness. These results indicate what data must be taken if the elastic modulus is to be reconstructed reliably and quantitatively from a strain image.