Nonlinear ultrasound simulations using a time-explicit discontinuous Galerkin (DG) method

Abstract

Histotripsy with ultrasound is an emerging noninvasive therapeutic modality that uses cavitation to precisely destroy diseased soft tissue. Accurate simulations of histotripsy are needed for treatment planning and device design. These simulations must model transient pressure fields, span hundreds of wavelengths, and must handle strong shocks and discontinuities between materials, such as the brain and the skull. The discontinuous Galerkin (DG) method is an outstanding candidate for such simulations. DG methods possess the following qualities: 1) high order accuracy, 2) geometric flexibility, 3) excellent dissipation properties, and 4) excellent scalability on massively parallel machines. The objective of this work is to introduce our efforts to develop a model nonlinear ultrasound simulations that are ultimately intended for histotripsy simulations in the brain. We have developed a 3D nonlinear wave equation solver using a time-explicit DG method [1]. The governing equations are expressed in first-order flux form, which models the effects of diffraction, attenuation, and nonlinearity. A Rusanov numerical flux is formulated and the eigenvalues of the flux Jacobian are calculated. A third-order, strong stability preserving Runge-Kutta (RK) time-integrator is used for time-discretization. Frequencysquared attenuation is modeled via a second-order diffusion term, which is evaluated using the local DG method. To stabilize the method and guarantee a non-oscillatory solution near shocks, a parameter-free stabilization scheme is implemented [2]. Full-wave 3D simulations are simulated for both linear and nonlinear problems. Numerical results for a planar waveguide and a pulsed rectangular piston are presented and compared to existing analytical solutions and to the FOCUS package [3]-[7]. Our nonlinear DG captures strong shocks and resolves diffraction, absorption, and nonlinear for all problems considered. An approach for coupling DG with FOCUS is proposed. These results suggest DG is a competitive method for transient biomedical acoustics simulations.