Alternating direction method of multipliers for displacement estimation in ultrasound strain elastography

Ultrasound strain imaging, which delineates mechanical properties to detect tissue abnormalities, involves estimating the time delay between two radio-frequency (RF) frames collected before and after tissue deformation. The existing regularized optimization-based time-delay estimation (TDE) techniques suffer from at least one of the following drawbacks: (1) The regularizer is not aligned with the tissue deformation physics due to taking only the first-order displacement derivative into account; (2) The $$ -norm of the displacement derivatives, which oversmooths the estimated time-delay, is utilized as the regularizer; (3) The modulus function defined mathematically should be approximated by a smooth function to facilitate the optimization of $$ -norm.

Our purpose is to develop a novel TDE technique that resolves the aforementioned shortcomings of the existing algorithms.

Herein, we propose employing the alternating direction method of multipliers (ADMM) for optimizing a novel cost function consisting of $$ -norm data fidelity term and $$ -norm first- and second-order spatial continuity terms. ADMM empowers the proposed algorithm to use different techniques for optimizing different parts of the cost function and obtain high-contrast strain images with smooth backgrounds and sharp boundaries. We name our technique ADMM for totaL variaTion RegUlarIzation in ultrasound STrain imaging (ALTRUIST). ALTRUIST's efficacy is quantified using absolute error (AE), Structural SIMilarity (SSIM), signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), and strain ratio (SR) with respect to GLUE, OVERWIND, and $$ -SOUL, three recently published energy-based techniques, and UMEN-Net, a state-of-the-art deep learning-based algorithm. Analysis of variance (ANOVA)-led multiple comparison tests and paired $$ -tests at $$ overall significance level were conducted to assess the statistical significance of our findings. The Bonferroni correction was taken into account in all statistical tests. Two simulated layer phantoms, three simulated resolution phantoms, one hard-inclusion simulated phantom, one multi-inclusion simulated phantom, one experimental breast phantom, and three in vivo liver cancer datasets have been used for validation experiments. We have published the ALTRUIST code at http://code.sonography.ai.

ALTRUIST substantially outperforms the four state-of-the-art benchmarks in all validation experiments, both qualitatively and quantitatively. ALTRUIST yields up to $$ , $$ , and $$ SNR improvements and $$ , $$ , and $$ CNR improvements over $$ -SOUL, its closest competitor, for simulated, phantom, and in vivo liver cancer datasets, respectively, where the asterisk (*) indicates statistical significance. In addition, ANOVA-led multiple comparison tests and paired $$ -tests indicate that ALTRUIST generally achieves statistically significant improvements over GLUE, UMEN-Net, OVERWIND, and $$ -SOUL in terms of AE, SSIM map, SNR, and CNR.

A novel ultrasonic displacement tracking algorithm named ALTRUIST has been developed. The principal novelty of ALTRUIST is incorporating ADMM for optimizing an $$ -norm regularization-based cost function. ALTRUIST exhibits promising performance in simulation, phantom, and in vivo experiments.