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

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 elastic material, determine the shear modulus. We show that for the standard case, N = 1, the general solution contains two arbitrary functions which must be prescribed to make the solution unique. In practice, the data required to evaluate the necessary functions are impossible to obtain. For N = 2, on the other hand, the general solution contains at most four arbitrary constants, and so very few data are required to find the unique solution. For N = 4, the general solution contains only one arbitrary constant. Our results apply to both quasistatic and dynamic deformations.