McGough, Robert J.

Rapid simulations of three-dimensional (3D) shear wave propagation in viscoelastic media with a Kelvin-Voigt model are enabled by Green’s functions accelerated with graphics processing unit parallelization and high-performance computing resources. Cross-correlation analysis of the wave motion yields an estimate of shear elasticity, where the errors in the estimated values increase with rising shear viscosity. To reduce errors in the shear elasticity estimates, a nonlinear least-squares routine that accounts for the effects of propagation, attenuation, and dispersion is applied to 3D simulated shear wave data in viscoelastic media. The nonlinear least-squares approach also yields an estimate of the shear viscosity when applied to simulated particle velocity waveforms. Post-processing, cross-correlation analysis, and the nonlinear least-squares routine are evaluated on a desktop computer using MATLAB. Estimation of viscoelastic parameters is performed at the focal depth for several combinations of shear elasticity and shear viscosity. The errors in the estimated shear elasticities obtained from the nonlinear least-squares routine and cross-correlation are determined as a function of the input shear elasticity and shear viscosity values. The estimates provided by the nonlinear least-squares routine for shear elasticity and shear viscosity are shown to approach the simulation input values for each combination of viscoelastic parameters.

Characterization of mechanical properties of tissue has become an important area of study over the last two decades. Shear wave elastography (SWE) is being used in many clinical applications to characterize, and multiple mechanical properties are being reported. To more comprehensively characterize soft tissues, we have focused on frequency-domain methods for evaluating viscoelasticity. We measure the shear wave velocity and attenuation and the variation with frequency. We will present results with the Kelvin-Voigt fractional derivative model and general power laws to characterize data from SWE experiments in tissue-mimicking phantoms and exvivo and in vivo tissue. Tissue mimicking phantoms from CIRS, inc., that have varying amounts of viscoelasticity will be investigated. Measurements from ex vivo and in vivo liver and kidney will also be used for this study. We will examine the range of the power-law exponents associated with these samples and the different models to provide understanding of the range of these parameters and how well tissue mimicking phantoms realize the parameters measured in soft tissues.

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.

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.

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.]

To characterize the attenuation and dispersion behavior of shear waves in excised swine liver samples, the complex shear modulus was measured with a Rheospectris C500 + from 10 Hz to 2000 Hz. The shear wave attenuation and shear wave speed were calculated from the complex modulus measurements. A power law fit was evaluated for the shear wave attenuation, and a power law fit with and without a constant offset was evaluated for the shear wave speed. The power law closely matches the measured shear wave attenuation over most of the frequency range evaluated, although some differences are observed in several of the samples below 400 Hz. The power law without the constant offset closely matches the measured shear wave speed above 200 Hz in all measurements, where the addition of the constant offset achieves a much closer fit in all measurements that contain discrepancies below 200 Hz. The results demonstrate that shear wave attenuation in swine liver follows a power law and that a power law with a constant offset is an effective model for the shear wave speed in swine liver. [Work supported in part by NIH Grants DK092255, EB023051, and EB012079.]

We recently developed a fast approach for simulating shear waves generated by an acoustic radiation force. The input for these simulations is generated in FOCUS, the “Fast object-oriented C + + simulator” (www.egr.msu.edu/~fultras-web), which rapidly models the three-dimensional (3D) pressure, intensity, and acoustic radiation force on a desktop computer. The shear waves generated by the acoustic radiation force are then quickly simulated with Green's functions for viscoelastic media in a two-dimensional (2D) plane on a high-performance graphics processing unit (GPU). For these simulations, an L7-4 linear array is electronically focused at a depth of 25 mm to generate an acoustic radiation force “push” for 200 microseconds. The simulation results are compared to shear wave data measured in three viscoelastic shear wave phantoms with low, medium, and high values of the shear viscosity using a Verasonics Vantage ultrasound system. The measured and simulated shear wave data are compared for several different combinations of the simulation parameters, where these comparisons are enabled by the rapid acoustic radiation force and shear wave simulations. The results show that the shear waves simulated in 2D planes achieve good agreement with the measured shear wave data. [Work supported in part by NIH Grants EB023051 and DK092255.]

Shear wave elasticity imaging determines the mechanical parameters of soft tissue by analyzing measured shear waves induced by an acoustic radiation force. Currently, the widely used time-of-flight method calculates the correlation between shear waveforms at adjacent lateral observation points to estimate the shear elasticity value. Although this method provides accurate estimates of the shear elasticity in purely elastic media, our experience suggests that this approach overestimates the shear elasticity values in viscoelastic media because the effects of diffraction, attenuation, and dispersion are not taken into account. To address this problem, we have developed an approach that directly accounts for all of these effects when estimating the shear elasticity. This new approach simulates shear waveforms using a Green’s function-based approach with a Voigt model, while the shear elasticity and viscosity values are estimated using an optimization-based approach by comparing measured shear waveforms with simulated shear waveforms in the time-domain. This operation is then performed on a point-by-point basis to generate images. The results indicate that there is good agreement between the simulated and measured shear velocity waveforms, and that this approach yields improved images of the shear elasticity and shear viscosity. [Work supported, in part, by NIH Grant R01DK092255.]

Shear wave elastography imaging (SWEI) with ultrasound uses an applied acoustic radiation force to obtain quantitative images of liver, breast, prostate, and other soft tissues. In SWEI, shear velocity waveforms are collected along a line, and then, the shear speed is estimated either with a k-space or a correlation-based method. Although these yield effective estimates in elastic media, both methods consistently overestimate the shear speed in viscoelastic soft tissue because neither considers the shear viscosity. To address this problem, we have created a new shear speed estimation approach that also accounts for the effect of shear viscosity. The new approach is evaluated in a computational model that calculates the three-dimensional (3D) intensity field generated by a linear array transducer in FOCUS (http://www.egr.msu.edu/~fultras-web) and then simulates the shear wave velocities in a viscoelastic medium with a 3D finite difference model. In the viscoelastic shear wave simulation model, the shear speed constant is 1.5 m/s, and the shear viscosity is 1 Pa·s. Correlation-based shear speed estimates are compared to the viscosity-compensated estimates, and the results show that the viscosity-compensated approach consistently achieves improved shear speed estimates relative to the correlation-based method. [Work supported in part by NIH Grant Nos. R01EB012079 and R01DK092255.]

Many methods for characterization of the mechanical properties of soft tissues using propagating shear waves have been developed over the past two decades. Most of these methods assume that the shear wave is traveling in an elastic, isotropic tissue. However, many soft tissues are viscoelastic and have material properties that are directionally dependent or anisotropic. We have been developing methods to measure waves propagating in soft tissues to estimate the anisotropic viscoelastic material properties. To refine our measurement methods, we have also developed techniques and models for simulating the wave propagation in these types of materials. We have developed specialized finite element and pseudo-spectral models that can simulate viscoelastic transversely isotropic materials and compared our results with measurements in ex vivo porcine muscle. Additionally, we have developed a finite element model that simulates wave propagation in the myocardium by creating a model of layered transverse isotropic media oriented at different angles. We examined the effect of frequency of the waves for estimating of the angle of propagation. We compared these simulation results with experimental results from an ex vivo porcine left ventricular wall. These simulation methods provide insight for optimizing our measurement methods for in vivo characterization of material properties.