LAUSR

Ultrasound elastography uses propagating P-waves to measure the deformation of soft elastic solids, including soft tissues. In some cases, the deformation of interest is quasistatic while in others it is a propagating S-wave. From the measured deformation field, it is of interest to infer the distribution of tissue rigidity and/or shear wave speed distribution within the imaged region. We review several variational formulations that have been proposed to solve this inverse problem, and their respective mathematical properties. The primary focus will be on variational formulations that lead to direct solution of inverse problem. These include the virtual fields method (and the related method of Romano, Shirron, and Bucaro), the direct error in constitutive equation formulation, and the adjoint-weighted variational formulation. We will consider consistency with the original boundary value problem, stability of the variational formulation, and required continuity of both data functions and the unknown modulu...