Complex-valued stiffness reconstruction for magnetic resonance elastography by algebraic inversion of the differential equation

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Abstract
Noninvasive quantitation of the mechanical properties of tissue could improve early detection of pathology. Previously a method for detecting displacement from propagating shear waves using a phase‐contrast MRI technique was developed. In this work it is demonstrated how a collection of data representing the full vector displacement field could be used to potentially estimate the full complex stiffness tensor. An algebraic inversion approach useful for piece‐wise homogeneous materials is described in detail for the general isotropic case, which is then specialized to incompressible materials as a model for tissue. Results of the inversion approach are presented for simulated and experimental phantom data that show the technique can be used to obtain shear wave‐speed and attenuation in regions where there is sufficient signal‐to‐noise ratio in the displacement and its second spatial derivatives. The sensitivity to noise is higher in the attenuation estimates than the shear wave‐speed estimates. Magn Reson Med 45:299–310, 2001.