An analytical solution to the dispersion-by-inversion problem in magnetic resonance elastography

Purpose: Magnetic resonance elastography (MRE) measures stiffness of soft tissues by analyzing their spatial harmonic response to externally induced shear vibrations. Many MRE methods use inversion-based reconstruction approaches, which invoke first- or second-order derivatives by finite difference operators (first- and second-FDOs) and thus give rise to a biased frequency dispersion of stiffness estimates. Methods: We here demonstrate analytically, numerically, and experimentally that FDO-based stiffness estimates are affected by (1) noise-related underestimation of values in the range of high spatial wave support, that is, at lower vibration frequencies, and (2) overestimation of values due to wave discretization at low spatial support, that is, at higher vibration frequencies. Results: Our results further demonstrate that second-FDOs are more susceptible to noise than first-FDOs and that FDO dispersion depends both on signal-to-noise ratio (SNR) and on a lumped parameter A, which is defined as wavelength over pixel size and over a number of pixels per stencil of the FDO. Analytical FDO dispersion functions are derived for optimizing A parameters at a given SNR. As a simple rule of thumb, we show that FDO artifacts are minimized when A/2 is in the range of the square root of 2SNR for the first-FDO or cubic root of 5SNR for the second-FDO. Conclusions: Taken together, the results of our study provide an analytical solution to a long-standing, well-recognized, yet unsolved problem in MRE postprocessing and might thus contribute to the ongoing quest for minimizing inversion artifacts in MRE.