A Laplacian-based SNR measure: Shear stiffness estimation in MR elastography

Magnetic resonance elastography (MRE) is a phase-contrast MRI based technique that allows quantitative, noninvasive assessment of the mechanical properties of tissues by the introduction of shear waves into the body and measurement of the resulting displacements. In MRE, the calculated stiffness values are affected by noise, which is amplified by the inversion process. It would be useful to know that beyond some SNR threshold, the stiffness values are accurate within some confidence limit. The most common methods to calculate SNR values in MRE are variations of displacement SNR, which estimate the noise in the measured displacement. However, the accuracy of stiffness determination depends not only on the displacement SNR, but also on the wavelength of the shear wave, in turn dependent on the stiffness of the underlying material. More recently, the SNR of the octahedral shear strain (OSS) has been proposed as a more appropriate measure, since shear deformation is the signal in MRE. We also propose here another measure based on the SNR of the Laplacian of the data, since this is the most noise sensitive quantity calculated when performing direct inversion of the Helmholtz equation. The three SNR measures were compared on simulated data for materials of different stiffness with varying amounts of noise using three inversion algorithms commonly used in MRE (phase gradient, local frequency estimation, and direct inversion). We demonstrate that the proper SNR measure for MRE depends on the inversion algorithm used, and, more precisely, on the order of derivatives used in the inversion process.