Complex-valued quantitative stiffness estimation using dynamic displacement measurements and local inversion of conservation of momentum

It has been shown that elasticity can be estimated from direct displacement measurements measured using either ultrasonic or magnetic resonance (MRE) elastography methods. Most of the inversion techniques presented in the ultrasound literature are based on inverting a displacement dataset that satisfies a static equation of shear motion resulting in only relative estimates of shear modulus. We present a method for estimating absolute shear-wave speed and attenuation from dynamic displacement data based on local inversion of the linearized differential equation satisfied by small-amplitude displacements in a source-free region. The inversion requires knowledge of the spatial derivatives of the displacement which we calculate using local, least-square polynomial fits to the data. We apply this direct inversion technique to a two dimensional simulated and a three-dimensional measured MRE dataset. The MRE dataset was collected from a phantom composed of a marine-algae gel mixture (1.5% by weight) with an embedded cylinder and sphere composed of the same gel with a higher concentration (3%). In the simulated experiment we obtain good agreement with the input values of shear-wave speed while in the phantom dataset we obtain an average value of 4.2 ± 0.3 m/s wave-speed in the background gel and 9.7 ± 1.0 m/s in the cylinder and 8.8 ± 1.0 m/s in the sphere. Finally, we report attenuation values for the 1.5% background gel as 17 ± 12 m^-1). Averaging regions inside the 3%-gel cylinder and 3%-gel sphere gives values of 21 ± 10 m^-1 and 25 ± 22 m^-1 respectively.