An improved time-base transformation scheme for computing waveform deformation during nonlinear propagation of ultrasound

Abstract

Nonlinear propagation plays an important role in various applications of medical ultrasound, like higher harmonic imaging and high intensity focused ultrasound (HIFU) treatment. Simulation of nonlinear ultrasound fields can greatly assist in explaining experimental observations and in predicting the performance of novel procedures and devices. Many numerical simulations are based on the generic split-step approach, which takes the ultrasound field at the transducer plane and propagates this forward over successive parallel planes. Usually, the spatial steps between the planes are small and the diffraction, attenuation, and nonlinear deformation may be treated as separate substeps. For the majority of methods, e.g., for all KZK-type methods, the nonlinear substep relies on the implicit solution of the one-dimensional Burgers equation, which is implemented using a time-base transformation. This generally works fine, but when the shock wave regime is approached, reduced spatial steps are required to avoid time points to cross over, and the method can become notoriously slow. This paper analyses the fundamental difficulty with the common time base transformation, and provides an alternative that does not suffer from the mentioned slowdown. Numerical results will be shown to demonstrate that this alternative will allow much larger spatial steps without compromising the numerical accuracy.