Shocking stones with the Khokhlov–Zabolotskaya–Kuznetsov equation.

Abstract

In shock wave lithotripsy (SWL) high-amplitude acoustic waves generated outside the body are focused onto kidney stones in order to fragment them into pieces that are small enough to pass naturally. In order to investigate mechanisms of fragmentation and collateral damage to the soft tissue it is necessary to understand the acoustic field delivered to the tissue in and around the kidney. This is not easily accomplished experimentally and motivates the development of numerical models. The application of the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation to the SWL problem is discussed as there are two underlying assumptions that are challenged: the peak pressures are on the order of 50 MPa resulting in an appreciable acoustic Mach number of 0.02, and large focusing gains are employed which violate the paraxial approximation. Predictions using the KZK equation, with a layered model of the tissue path to the kidney, demonstrate that the waveform shape, in particular, the risetime, is strongly affected by the tissue path. However, once the waveform enters the urine in the collecting space of the kidney, the waveform heals and a sharp shock results within about 5 mm of propagation. [Work supported in part by NIH DK-43881.]