McGough, Robert J.

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.]

We implemented a three-dimensional solver for simulating the ultrasound radiation force induced shear wave motion in viscoelastic media using a mapped Chebyshev pseudospectral method. The solver used a body force field that is simulated by the Field II program as the excitation. Then, a velocity-stress formulation was used to calculate the shear wave motion. We used a Voigt model to describe tissue viscoelasticity, and the viscosity is introduced by memory variables and auxiliary differential equations. The top boundary of the domain had a fixed boundary condition to mimic the interaction between an ultrasound probe and the tissue surface. All other sides of the domain were surrounded by perfectly matched layers to minimize the reflections. Time stepping was implemented with the LSODE integrator with variable step sizes. Results show that the waveform of the simulated shear wave is close to experimental observations and equivalent FEM simulations. We used a Fourier-based method to estimate the phase velocities of the shear waves between 100–1000 Hz and a nonlinear regression to estimate the Voigt model parameters. The reconstructions matched within 10% error compared with the true values.

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.]

Rapid calculations of the intensity are required for models of the radiation force generated by linear ultrasound phased arrays. To determine the computation times and numerical errors produced by two medical ultrasound simulation packages, namely, FIELD II and FOCUS, comparisons of the calculated pressures and intensities are performed using a simulated Vermon L5 phased array probe with 128 active elements. In these simulations, the array generates a fixed focus 25 mm from the array for a 5-MHz excitation, and the pressure field is computed on a 133 × 133 × 261 rectilinear grid that is centered at the focus. Half wavelength sampling is employed in all three directions. In FIELD II, this calculation takes approximately 4 h on a 64-bit laptop running Linux, and in FOCUS, the same calculation takes less than 4 min on the same computer. The numerical errors are also much smaller with FOCUS. Details of the simulation approaches employed for this transducer model will be presented, including selection of FIELD II and FOCUS simulation parameters and calculations of the reference pressure field for error evaluations. The impact of numerical errors on shear wave calculations will also be discussed.