McGough, Robert J.

Shear wave elastography generates shear waves with an acoustic radiation force to obtain estimates of the shear wave speed. Shear wave elastography often utilizes time-of-flight calculations that apply cross-correlations or time-to-peak methods to obtain estimates of the shear wave speed in the time-domain. However, these time-domain methods fail to consider the effect of dispersion caused by the shear viscosity, which can introduce substantial errors into estimates of the shear wave speed. To solve this problem, a new viscoelastic method for estimating shear wave parameters in the time-domain is introduced. The performance of the new method, which is applied to shear wave particle velocity waveforms, is established by evaluating a simulated three-dimensional viscoelastic Green's function model for shear waves in the time-domain. Results show that the new viscoelastic time-domain method is applicable to locations outside of the push beam region at the focal depth and throughout a two-dimensional plane. The results also indicate that estimates of the shear wave speed are either comparable to or significantly improved relative to values obtained from cross-correlations while simultaneously providing estimates of the shear viscosity in the time-domain.

Shear wave elasticity imaging is a non-invasive imaging modality that characterizes the mechanical properties of soft tissue, such as the shear wave speed or shear modulus. Present time-domain methods for shear wave parameter estimation divide the propagation distance by the propagation time to obtain an estimate of the shear wave speed. However, soft tissues are viscoelastic, and the shear viscosity frequently produces large errors in the estimated shear wave speed. To address this problem, a simplified time-domain model for shear wave parameter estimation in viscoelastic media is applied to simulated three-dimensional (3D) shear waves in a viscoelastic medium, where these 3D simulations require high performance computing to evaluate an approximate time-domain Green’s function solution of Navier’s equation that employs a Kelvin-Voigt rheological model. Parameters for the time-domain model are determined from an optimization procedure, then the shear wave speed is estimated from estimated time delays that account for the shear viscosity. Estimates of the shear wave speed are shown at the focal depth and in a two-dimensional region for various parameter combinations. The simplified time-domain viscoelastic model consistently yields smaller errors for the shear wave speed than cross-correlations while simultaneously providing estimates of the shear viscosity.

In soft tissue, the attenuation of shear waves satisfies a power law. However, the values of power law exponents for shear waves in various soft tissues remain unknown due to windowing artifacts that limit the effectiveness of present methods that are based on two-dimensional fast Fourier transforms. To solve this problem, a time-domain signal processing approach is proposed. Robust shear wave parameter estimates are obtained by minimizing the least-squares error between measured time-domain shear wave data and stable probability density functions that describe the solution to a closely related fractional-order diffusion equation. Assessments are performed with measured shear wave data collected from ex vivo pig liver using a Verasonics system with an L7-4 linear array transducer. Power law exponents are obtained from measured shear wave data at the focal depth between 1.2 and 13.9 mm from the center of the push beam. The mean estimated power law exponent for these locations is 0.74, and the standard deviation is 0.05. The results suggest that fractional calculus models that support power law exponents less than one are needed for shear waves in pig liver.

When applied to compressional waves, the Caputo time-fractional wave equation describes power law attenuation in soft tissue. However, when applied to shear waves, the Caputo wave equation produces some unexpected behavior. To demonstrate examples of the time-domain waveforms generated by this fractional calculus model, the Caputo wave equation is numerically evaluated for multiple orders of the time-fractional derivative between 0 and 1, for an extended range of values for the relaxation time, and for a fixed value of the shear wave speed extracted from ex vivo pig liver. The computed waveforms are then compared to the measured shear wave particle velocities obtained from ex vivo pig liver. These comparisons reveal that the full-width at half maximum (FWHM) of the positive component of the main shear wave particle velocity waveform measured in pig liver is at least an order of magnitude greater than the FWHM obtained from the Caputo wave equation for all parameter combinations evaluated. These preliminary results indicate that the Caputo wave equation produces time-domain waveforms that are inconsistent with measured shear wave data in pig liver and that further efforts are required to establish more effective fractional calculus models for shear waves in soft tissue.

Time-domain estimates of the shear wave speed are obtained by performing cross-correlations or by evaluating the time-to-peak for two laterally separated shear wave particle velocity waveforms, where the distance divided by the propagation time yields an approximate value for the shear wave speed. Challenges associated with these time-domain estimation approaches include increasing errors as the shear viscosity increases and the absence of an estimated value for the shear viscosity parameter which is responsible for the rapid attenuation of shear waves in soft tissue. To obtain a time-domain estimate of the shear viscosity that also provides an estimate for the shear wave speed, a nonlinear least squares estimation approach is applied to three-dimensional (3D) shear wave particle velocities computed for a shear elasticity of 1.5 kPa, shear viscosities of 1, 2, 3, and 4 Pa.s, and a realistic simulated 3D acoustic radiation force excitation. The results show cross-correlations tend to over-estimate the value of the shear wave speed, the nonlinear least squares approach tends to under-estimate the value of the shear wave speed, and the nonlinear least square approach produces more accurate estimates as the shear viscosity increases. Two-dimensional maps of each estimated parameter are also shown.