| Transient Elastography is a promising new technique for characterizing the elasticity of soft tissues, Using this method, an "ultrafast imaging" system (up to 10,000 frames/s) follows in real time the propagation of a low frequency shear wave. The displacement of the propagating shear wave is measured as a function of time and space. Assuming a wave equation model, we establish that Lipschitz continuous arrival times of the shear wave satisfy the Eikonal equation, which is only a first order partial differential equation. We first develop techniques to find the arrival time surface given the displacement data from a transient elastography experiment, We then propose a family of methods to solve the inverse Eikonal equation: Given the arrival times of a propagating wave, find the wave speed. Combining the techniques for finding the arrival times and the methods for solving the inverse Eikonal equation results in a complete algorithm for shear wave speed recovery. We use this algorithm to generate wave speed recoveries on synthetic data, and give a reconstruction example using a phantom experiment accomplished by Mathias Fink's group (the Laboratoire Ondes et Acoustique, ESPCI, Universite Paris VII). |
