Numerical simulation of ultrasonic shock wave propagation in lossy liquids obeying a frequency power law

Abstract

Ultrasonic shock waves are applied for several noninvasive therapies like extracorporal lithotripsy. In order to design and to optimize therapeutical devices it is essential to predict accurately the shockwave propagation through the arrangement transducer/coupling medium/human tissue. An explicit high-order FDTD-discretization of a set of nonlinear acoustic equations has been shown to model the lithotripter pulse propagation in water in a quality that beats all other known methods. To consider attenuation caused by tissue structures, realistic broadband models of soft tissue attenuation are provided by frequency power laws that include strong dispersion. In this contribution a new method to include these frequency power laws into the explicit FDTD model is presented. It is based on a causal continuous time domain formulation of the attenuation law which contains a Riemann–Liouville derivative operator of fractional order. This operator is discretized by using a quadrature formula. The resulting FDTD algorithm is verified. It turns out to be an efficient tool to compute the dispersive and attenuating effects of soft tissue on the propagation of high-intensity ultrasonic pulses and shock waves. Exemplary nonlinear dispersive propagation of a lithotripter pulse through tissue is simulated.