The variance of quantitative estimates in shear wave imaging: Theory and experiments

In this paper, we investigate the relationship between the estimated shear modulus produced in shear wave imaging and the acquisition parameters. Using the framework of estimation theory and the Cramer-Rao lower bound applied both to the estimation of the velocity field variance and to the estimation of the shear wave travel time, we can derive the analytical formulation of the shear modulus variance σ_μ² using relevant physical parameters such as the shear wave frequency, bandwidth, and ultrasonic parameters. This variance corresponds to the reproducibility of shear modulus reconstruction for a deterministic, quasi-homogeneous, and purely elastic medium. We thus consider the shear wave propagation as a deterministic process which is then corrupted during its observation by electronic noise and speckle decorrelation caused by shearing. A good correlation was found between analytical, numerical, and experimental results, which indicates that this formulation is well suited to understand the parameters´ influence in those cases. The analytical formula stresses the importance of high-frequency and wideband shear waves for good estimation. Stiffer media are more difficult to assess reliably with identical acquisition signal-to-noise ratios, and a tradeoff between the reconstruction resolution of the shear modulus maps and the shear modulus variance is demonstrated. We then propose to use this formulation as a physical ground for a pixel-based quality measure that could be helpful for improving the reconstruction of real-time shear modulus maps for clinical applications.