Authors’ abstract: The imaging problem of elastography is an inverse problem. The nature of an inverse problem is that it is ill-conditioned. In this investigation, the authors 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 implicitly in this problem of imaging in elastography. In this treatment, they consider the tissue to be linearly elastic, isotropic, and spatially heterogeneous. In this investigation, they determine the conditioning of this problem of function reconstruction, in particular for the stiffness function. They further examine the conditioning when determining both stiffness and density. In addition, they examine the Fréchet derivative of the nonlinear mapping, which enables them to describe the properties of how the field affects the individual maps to the stiffness and density functions. Moreover, they illustrate how use of the implicit function theorem can considerably simplify the analysis of Fréchet differentiability and regularity properties of this underlying operator. Finally, they 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 their work here is analytical. The validity of the Newton-Kantorovich and optimization methods for the computational solution of this inverse problem is directly linked to the Fréchet differentiability of the appropriate nonlinear operator, which they justify.