Influence of Poisson's Ratio on Elastographic Direct and Inverse Problems

We consider the displacement of an elastic material under an external compression (axial or almost axial stress). We assume that only one component of the displacement is observed, in the direction of compression (axial displacement), or alternatively, that two components are observed in a plane. These hypotheses are in accordance with an imaging modality, namely ultrasonic elastography. In the case of a homogeneous medium we show that any value of Poisson's ratio allows us to predict the observed value of the axial displacement. When two components of the displacement are measured in a plane, the Poisson's ratio of the plane strain model that predicts the observed displacement is not the same as the tri-dimensional material. These facts are illustrated by numerical experiments in the case of an inhomogeneous medium. We also present results on experimental phantom data, where the inverse problem of reconstructing the Young's modulus is solved assuming different values for Poisson's ratio.