A stacked frequency approach for inhomogeneous time-dependent MRE: an inverse problem for the elastic shear modulus

Abstract

Abstract We derive and analyse a new way to calculate the shear modulus of an inhomogeneous elastic material from time-dependent magnetic resonance elastography (MRE) measurements of its interior displacement. Even with such a rich data source, this is a challenging inverse problem because the coefficient of the shear modulus in the governing equations can be small (or potentially zero). Our approach overcomes this by combining different data sets into an overdetermined matrix–vector equation. It uses finite differences to approximate space derivatives and a Fourier interpolant in time, and we do not need to assume that the inhomogeneous material is ‘locally homogeneous’. Crucially, our construction ensures that the computed value of the (real) shear modulus is real: approximation methods based on the frequency domain version of the problem often give a complex shear modulus for the elastic case and this can be hard to interpret, especially if its imaginary part dominates. We carry out careful numerical tests on a one (space) dimensional analogue of the problem and on experimental MRE data for an inhomogeneous gel phantom.