  We introduce a model-based iterative method to obtain shear modulus images of
tissue using magnetic resonance elastography. The method jointly finds the
displacement field that best fits multifrequency tissue displacement data and
the corresponding shear modulus. The displacement satisfies a viscoelastic wave
equation constraint, discretized using the finite element method. Sparsifying
regularization terms in both shear modulus and the displacement are used in the
cost function minimized for the best fit. The formulated problem is bi-convex.
Its solution can be obtained iteratively by using the alternating direction
method of multipliers. Sparsifying regularizations and the wave equation
constraint filter out sensor noise and compressional waves. Our method does not
require bandpass filtering as a preprocessing step and converges fast
irrespective of the initialization. We evaluate our new method in multiple in
silico and phantom experiments, with comparisons with existing methods, and we
show improvements in contrast to noise and signal to noise ratios. Results from
an in vivo liver imaging study show elastograms with mean elasticity comparable
to other values reported in the literature.
