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

Abstract

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.

MR elastography: Principles, guidelines, and terminology

“…If there are not enough pixels sampled per wavelength, discretization effects when estimating derivatives can bias the results. Depending on the inversion and the derivative kernel used, this can cause about a 4% error when there are 9‐10 pixels per wavelength, but the error rapidly increases with fewer pixels per wavelength 79,83,84 . Conversely, when the wavelength becomes much larger than the pixel size, the discrete derivative operations required for some inversion methods are highly affected by low SNR.…”

“…79,83,84 Conversely, when the wavelength becomes much larger than the pixel size, the discrete derivative operations required for some inversion methods are highly affected by low SNR. It has been reported that the accuracy and precision of MRE measurements with direct inversion are optimal when six to nine voxels per wavelength are obtained and the SNR is sufficiently high 84 ; more recent work studies the effects of kernel width and the order of the derivative operators in the inversion. 84 Discussion of results should address this question, and whether the wavelength/pixel ratio is in an appropriate range.…”

Multiple motion encoding in phase-contrast MRI: A general theory and application to elastography imaging

“…Wrap-free phase images from unwrapping algorithms and from dual and multiple encoding methods were used for reconstruction of shear wave speed (SWS) maps based on phase-gradient wavenumber recovery to avoid noise amplification by the Laplacian operator which is inevitable in direct inversion techniques [40,41]. SWS is related to tissue stiffness and will be termed as such in the following.…”

Inversion‐recovery MR elastography of the human brain for improved stiffness quantification near fluid–solid boundaries

“…As demonstrated in prior work, high shear strain at slip boundaries leads to reconstruction of very low elasticity in the vicinity of the interface 16 . However, most inversion approaches analyze shear strain in a spatially extended neighborhood, causing blurring of localized properties (partial volume effects) due to inversion 28 . Applied to surface regions in cerebral MRE maps, this partial volume–related averaging of very soft (slip interface) properties of CSF with solid cortex properties results in overall softening of surface voxels in conventional MRE of the brain.…”