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

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.