Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverseproblem in two dimensions

Abstract

We examine the uniqueness of an N-field generalization of a 2D inverse problem associated with elastic modulus imaging:given N linearly independent displacement fields in an incompressible elasticmaterial, determine the shear modulus. We show that for the standard case,N=1, the general solution contains two arbitrary functions whichmust be prescribed to make the solution unique. In practice, the datarequired to evaluate the necessary functions are impossible to obtain. ForN=2, on the other hand, the general solution contains at most four arbitraryconstants, and so very few data are required to find the unique solution. ForN=4, the general solution contains only one arbitrary constant. Our results apply to bothquasistatic and dynamic deformations.