Virtual fields based-method for reconstructing the elastic modulus in quasi-static ultrasound elastography

Reconstructing tissue elastic properties from dis-placements measured in quasi-static ultrasound elastography is a challenging task. Indeed, it requires to solve an ill-posed inverse problem, with generally no available boundary information and solely 2D estimated displacements, whereas the problem is inherently three-dimensional. In this paper, a method based on the virtual work principle is investigated to reconstruct Young's modulus maps from the knowledge of internal displacements and the force applied. The media examined are assumed to be linear elastic and isotropic. Moreover, for these first developments, the plane stress problem is investigated to overcome the lack of 3D data. The developed method is assessed with plane-stress and 3D simulations, as well as with experimental data. For all the media examined, regions of different stiffnesses are clearly revealed in the reconstructed Young's modulus maps. The stiffness contrast between regions is accurately estimated for the plane stress simulations but underestimated for the 3D simulations, which could be expected as plane stress conditions are no longer satisfied in this last case. Finally, similar comments can be made for the phantom results, with an inclusion-to-background Young's modulus ratio of 2.4 lower than the reference ratio of around 3, provided by the manufacturer.