Travelling Wave Expansion: A Model Fitting Approach to the Inverse Problem of Elasticity Reconstruction

In this paper, a novel approach to the problem of elasticity reconstruction is introduced. In this approach, the solution of the wave equation is expanded as a sum of waves travelling in different directions sharing a common wave number. In particular, the solutions for the scalar and vector potentials which are related to the dilatational and shear components of the displacement respectively are expanded as sums of travelling waves. This solution is then used as a model and fitted to the measured displacements. The value of the shear wave number which yields the best fit is then used to find the elasticity at each spatial point. The main advantage of this method over direct inversion methods is that, instead of taking the derivatives of noisy measurement data, the derivatives are taken on the analytical model. This improves the results of the inversion. The dilatational and shear components of the displacement can also be computed as a byproduct of the method, without taking any derivatives. Experimental results show the effectiveness of this technique in magnetic resonance elastography. Comparisons are made with other state-of-the-art techniques.