Analytical Minimization-Based Regularized Subpixel Shear-Wave Tracking for Ultrasound Elastography

Ultrasound elastography is a convenient and affordable method for imaging mechanical properties of tissue, which are often correlated with pathologies. An emerging novel elastography technique applies an external acoustic radiation force to generate a shear wave in the tissue and uses ultrasound imaging to track the shear wave. Accurate tracking of the small tissue motion is a critical step in shear-wave elastography (SWE), but it is challenging due to various sources of noise in the ultrasound data. We formulate tissue displacement estimation as an optimization problem and propose two computationally efficient approaches to estimate the displacement field. The first algorithm is referred to as dynamic programming analytic minimization (DPAM), which utilizes first-order Taylor series expansion of the highly nonlinear cost function to allow for its efficient optimization, and was previously proposed for quasistatic elastography. The second algorithm is a novel technique that utilizes second-order derivatives of the nonlinear cost function. We call the new algorithm second-order analytic minimization elastography (SESAME). We compare DPAM and SESAME to the standard normalized cross correlation (NCC) approach in the context of displacement and speed estimation of wave propagation in SWE. The results of micrometer-order displacement estimation in a uniform simulation phantom illustrate that SESAME outperforms DPAM, which in turn outperforms NCC in terms of signal-to-noise ratio (SNR) and jitter. In addition, the relative difference between true and reconstructed shear modulus (averaged over excitations at different focal depths and several scatterer realizations at each depth) is approximately 3.41%, 1.12%, and 1.01%, respectively, for NCC, DPAM, and SESAME. The performance of the proposed methods is also assessed with real data acquired using a tissue-mimicking phantom, wherein, in comparison to NCC, DPAM and SESAME improve the SNR of displacement estimates by 7.6 and 9.5 dB, respectively. Experimental results on a tissue-mimicking phantom also show that shear modulus reconstruction substantially improved with the proposed DPAM technique over NCC and with some further improvement achieved by utilizing the second-order Taylor series approximation in SESAME instead of the first-order DPAM.