The Chen-Holm and Treeby-Cox wave equations are space-fractional partial differential equations that describe power law attenuation of the form α(ω)≈α0|ω|y. Both of these space-fractional wave equations are causal, but the phase velocities differ, which impacts the shapes of the time-domain Green's functions. Exact and approximate closed-form time-domain Green's functions are derived for these space-fractional wave equations, and the resulting expressions contain symmetric and maximally skewed stable probability distribution functions. Numerical results are evaluated with ultrasound parameters for breast and liver at different times as a function of space and at different distances as a function of time, where the reference calculations are computed with the Pantis method. The results show that the exact and approximate time-domain Green's functions contain both outbound and inbound propagating terms and that the inbound component is negligible a short distance from the origin. Exact and approximate analytical time-domain Green's functions are also evaluated for the Chen-Holm wave equation with power law exponent y = 1. These comparisons demonstrate that single term analytical expressions containing stable probability densities provide excellent approximations to the time-domain Green's functions for the Chen-Holm and Treeby-Cox wave equations.
Shear wave elasticity imaging applies an acoustic radiation force to generate shear waves, where the shear wave speed is often estimated with time-of-flight calculations. To characterize the errors in time-of-flight estimates of the shear wave speed, three-dimensional simulated shear wave particle velocities and shear wave particle displacements are computed with time-domain Green's functions for a Kelvin-Voigt model. Estimated shear wave speeds are obtained from cross correlations and time-to-peak calculations, and the results demonstrate the effects of time-domain dispersion and noise on estimates of the shear wave speed. Time-domain dispersion of the propagating shear wave produces large errors in estimates performed with shear wave particle velocities and/or cross-correlations close to the push beam, and in the absence of noise, the errors in the shear wave speed estimated at the focal depth diminish with distance from the push beam. However, shear wave attenuation also increases sensitivity to noise as the distance from the push beam increases. Time-to-peak and cross-correlation methods consistently achieve much smaller errors with shear wave particle displacements than with shear wave particle velocities. Furthermore, cross-correlations are much more robust with respect to noise, and larger values of the shear viscosity increase the sensitivity of each method to noise.
Three dimensional (3D) simulations of shear wave elasticity imaging (SWEI) are performed by computing a 3D push beam in FOCUS, the “Fast Object-oriented C++ Ultrasound Simulator” (https://www.egr.msu.edu/~fultras-web) followed by the superposition of Green's function solutions to Navier's equation in viscoelastic media. Results show that these Green's function calculations are particularly amenable to parallel implementations that combine MPI and CUDA.
The time-of-flight approach estimates the shear elasticity in tissue mimicking elastography phantoms and in soft tissue. The time-of-flight approach is effective in elastic phantoms, but the time-of-flight approach tends to overestimate the shear elasticity in viscoelastic phantoms and in viscoelastic soft tissues. To characterize errors in estimated parameters for different values of the shear elasticity and the shear viscosity, three-dimensional (3D) shear wave simulations are evaluated for twelve different parameter combinations. The 3D acoustic radiation force is calculated for an L7-4 transducer using the fast nearfield method and the angular spectrum approach, and then, 3D shear wave propagation in a viscoelastic medium is simulated with Green's functions for a Kelvin-Voigt model. The time-of-flight method is then evaluated within a two-dimensional plane. The results show that the accuracy of the time-of-flight method depends on the values of the shear elasticity and the shear viscosity. In particular, the error in the estimated shear elasticity increases as the shear viscosity increases, where the largest errors are observed when larger values of the shear viscosity are combined with smaller values of the shear elasticity. [Work supported in part by NIH Grants DK092255, EB023051, and EB012079.]
Shear wave elasticity imaging (SWEI) uses an acoustic radiation force to generate shear waves, and then soft tissue mechanical properties are obtained by analyzing the shear wave data. In SWEI, the shear wave speed is often estimated with time-of-flight (TOF) calculations. To characterize the errors produced by TOF calculations, three-dimensional (3D) simulated shear waves are described by time-domain Green's functions for a Kelvin-Voigt model evaluated for multiple combinations of the shear elasticity and the shear viscosity. Estimated shear wave speeds are obtained from cross correlations and time-to-peak (TTP) calculations applied to shear wave particle velocities and shear wave particle displacements. The results obtained from these 3D shear wave simulations indicate that TTP calculations applied to shear wave particle displacements yield effective estimates of the shear wave speed if noise is absent, but cross correlations applied to shear wave particle displacements are more robust when the effects of noise and shear viscosity are included. The results also show that shear wave speeds estimated with TTP methods and cross correlations using shear wave particle velocities are more sensitive to increases in shear viscosity and noise, which suggests that superior estimates of the shear wave speed are obtained from noiseless or noisy shear wave particle displacements.
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