On an inverse problem from magnetic resonance elastic imaging

The imaging problem of elastography is an inverse problem. The nature of an inverse problem is that it is illconditioned. We consider properties of the mathematical map which describes how the elastic properties of the tissue being reconstructed vary with the field measured by Magnetic Resonance Imaging (MRI). This map is a nonlinear mapping and our interest is in proving certain conditioning and regularity results for this operator which occurs naturally in this problem of imaging in elastography. In this treatment we consider the tissue to be linearly elastic, isotropic and spatially heterogeneous. We determine the conditioning of this problem of function reconstruction; in particular for the density and stiffness functions. We examine the Fn!chet derivative of the nonlinear mapping, which enables us to describe the properties of how the field affects the individual maps to the density and stiffness functions. We illustrate how use of the implicit function theorem can considerably simplify the analysis of Frechet differentiability and regularity properties of this underlying operator. We present new results which show that the stiffness map is mildly ill-posed, whereas the density map suffers from medium ill-conditioning. Computational work has been done previously to study the sensitivity of these maps but our work here is analytical. The validity of the Newton-Kantorovich methods for the computational solution of this inverse problem is directly linked to the Frechet differentiability of the appropriate nonlinear operator, which we justify