Regularized Phase Gradient Analysis for Reverberant Shear Wave Elastography

Reverberant shear wave elastography (R-SWE) evaluates tissue stiffness by creating a reverberant shear wave field in all directions and estimating the shear wave speed (SWS). A previous study established a linear relationship between the field's phase and the local wave number through the phase gradient (PG) method. This study introduces regularization into R-SWE to enhance the quality of the SWS maps across frequencies. Two regularization methods were analyzed: the single-channel Total Variation (TV) and the multi-frequency channel-based Total Nuclear Variation (TNV). The regularization framework involves unwrapping the phase in axial and lateral directions, calculating the ℓ₂-norm of the phase gradient, and proposing a minimizing cost function to address the denoising problem. Metrics were compared to PG with data from simulated and breast phantoms. Results show that regularized methods visually improved the quality of the SWS images while reducing the variability of the estimations. The contrast-to-noise ratio (CNR) was used as an overall metric. For the simulations, the average CNR values were 7.58 (PG-TNV), 2.38 (PG-TV) and 0.88 (PG). For the breast phantom, the CNR values were 4.36 (PG-TNV), 3.01 (PG-TV) and 2.39 (PG). These results suggest that regularization in R-SWE improves SWS imaging, with PG-TNV producing the best SWS maps across all frequency channels, enhancing the trade-off between variability and spatial resolution by incorporating information from multiple frequencies.