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Shear wave speed recovery in transient elastography and supersonic imaging using propagating fronts

Transient elastography and supersonic imaging are promising new techniques for characterizing the elasticity of soft tissues. Using this method, an ‘ultrafast imaging’ system (up to 10 000 frames s−1) 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. The objective of this paper is to develop and test algorithms whose ultimate product is images of the shear wave speed of tissue mimicking phantoms. The data used in the algorithms are the front of the propagating shear wave. Here, we first develop techniques to find the arrival time surface given the displacement data from a transient elastography experiment. The arrival time surface satisfies the Eikonal equation. We then propose a family of methods, called distance methods, to solve the inverse Eikonal equation: given the arrival times of a propagating wave, find the wave speed. Lastly, we explain why simple inversion schemes for the inverse Eikonal equation lead to large outliers in the wave speed and numerically demonstrate that the new scheme presented here does not have any large outliers. We exhibit two recoveries using these methods: one is with synthetic data; the other is with laboratory data obtained by Mathias Fink's group (the Laboratoire Ondes et Acoustique, ESPCI, Université Paris VII).