A stabilized forward elasticity finite element formulation yields a stable and convergent inverse elasticity solution.

Abstract

“Elastography” refers to the procedure of using ultrasound to image tissue deformation. Images showing tissue deformation are sensitive to distributions of tissue mechanical properties. The mechanical properties themselves may be quantified by solving an inverse elasticity problem. An iterative approach to solving the inverse problem can be formulated by repeated solutions of the forward problem. That is, the shear modulus distribution sought is that which predicts a displacement field most consistent with the measured displacement field. Here we show that given plane displacement measurements for plane stress elasticity, the continuous elastic equilibrium equation uniquely determines the modulus distribution. On the other hand, we shall also demonstrate that the discrete elasticity equations from standard FEM discretization does not, even at the limit of infinite mesh refinement. We diagnose the problem as an underenforcement of the elasticity equations. With this knowledge, we have been able to design new forward elasticity FEM formulations that provide provably convergent forward and inverse elasticity solutions.