Yang, Yiqun

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.]

Shear wave elastography with ultrasound applies an acoustic radiation force to generate shear waves in viscoelastic soft tissue. To address the need for more effective simulation tools that model shear waves generated by an applied acoustic radiation force, three-dimensional (3D) finite difference programs that simulate propagation of shear waves in an isotropic, homogeneous medium have been created. These programs simulate shear wave propagation in elastic and viscoelastic soft tissue models. The 3D finite difference implementation combines an explicit time-stepping approach and a staggered spatial grid with an absorbing boundary condition that reduces boundary reflections. The acoustic radiation force in these simulations is quickly and accurately simulated in FOCUS (http://www.egr.msu.edu/~fultras-web) for a focused linear ultrasound array with an f/# of 2. The compressional wave speed is 1500 m/s and the shear wave speed is 1.5m/s in the elastic and viscoelastic tissue models, and the shear viscosity is 1 Pa·s in the viscoelastic model. The simulation completes 90,000 time steps of a 10 ms simulation in an 80 mm × 80 mm × 80 mm region with spatial sampling equal to the wavelength of compressional component in three days on a 3.4 MHz Intel i7 processor. [Supported in part by NIH Grants R01EB012079 and R01DK092255.]

Green’s functions effectively simulate shear waves produced by an applied acoustic radiation force in elastic and viscoelastic soft tissue. In an effort to determine the optimal parameters for these simulations, the convergence of Green's function-based calculations is evaluated for realistic spatial distributions of the initial radiation force “push.” The input to these calculations is generated by FOCUS, the “Fast Object-oriented C + + Ultrasound Simulator,” which computes the approximate intensity fields generated by a Phillips L7-4 ultrasound transducer array for both focused and unfocused beams. The radiation force in the simulation model, which is proportional to the simulated intensity, is applied for 200 μs, and the resulting displacements are calculated with the Green's function model. Simulation results indicate that, for elastic media, convergence is achieved when the intensity field is sampled at roughly one-tenth of the wavelength of the compressional component that delivers the radiation force “push.” Aliasing and oscillation artifacts are observed in the model for an elastic medium at lower sampling rates. For viscoelastic media, spatial sampling rates as low as two samples per compressional wavelength are sufficient due to the low-pass filtering effects of the viscoelastic medium. [Supported in part by NIH Grants R01EB012079 and R01DK092255.]

The Green's function approach describes propagating shear waves generated by an acoustic radiation force in elastic and viscoelastic soft tissue. Calculations with the Green's function approach are evaluated in elastic and viscoelastic soft tissue models for a line source and for a simulated focused beam. The results for the line source input are evaluated at 200 time samples in a 101 by 101 point grid that is perpendicular to the line source. For a shear wave speed of 1.4832 m/s and a compressional wave speed of 1500 m/s, shear wave simulations for a line source input in elastic and viscoelastic soft tissue models are completed in 431 and 2487 s with Matlab scripts, respectively, where the shear viscosity is 0.1 Pa.s in the viscoelastic model. Simulations are evaluated at a single point for an acoustic radiation force generated by a 128 element linear array operating at 4.09 MHz, and these simulations require 228 s and 1327 s for elastic and viscoelastic soft tissue models, respectively. The results show that these are effective models for simulating shear wave propagation in soft tissue, and plans to accelerate these simulations will also be discussed. [Supported in part by NIH Grants R01 EB012079 and R01 DK092255.]

Improved approximate frequency domain expressions are derived for cylindrical shear wave propagation in viscoelastic media. These expressions extend prior results that describe cylindrical wave propagation in lossless media. Previously, an analytical expression for a cylindrical wave was obtained in terms of a Hankel function and a large argument approximation was applied to the result. A leading frequency-dependent term was then treated as a constant with respect to frequency. In the improved expression, the frequency-dependence of the leading term is retained. For lossless media, the leading term is a fractional integrator, and for viscoelastic media, the leading term is either a fractional integrator or an integer-order integrator, depending on the frequency range. The lossless and the viscoelastic models are evaluated in the frequency domain for simplified source geometries and compared to numerical results. The comparison shows that the agreement between the analytical and the numerical models is excellent. Implications for time-domain calculations in viscous media will also be discussed. [This work was supported in part by NIH Grant Nos. R01 EB012079 and R01 DK092255.]