Numerical Simulation of Wave Focusing and Scattering in Shock Wave Lithotripsy

Abstract

In this work we simulate shock wave focusing and scattering that occurs during shock wave lithotripsy, a noninvasive medical treatment for kidney stone disease. Shock waves are generated outside the body of the patient and are focused at the kidney stone with the intention of pulverizing the stone while it remains inside the patient. The patient can then ostensibly pass the debris naturally. We use a multidimensional second-order method of the Godunov type with slope limiters and shock capturing capability to solve the inviscid Euler equations. Because we begin with the fundamental statements of conservation of mass, momentum, and energy, we simulate all the relevant acoustics occurring during a typical treatment. Lithotripters, the machines that generate and focus shock waves, can be classified according to the mechanism of shock generation. In this work, we simulate three different types of lithotripters: electrohydraulic, piezoelectric, and electromagnetic. We choose one representative of each lithotripter type: the Dornier HM3, a research piezoelectric lithotripter array, and the XX-Es, respectively. We first study a model of the in vitro setting for each lithotripter, where shock waves are generated and focus in a bath of pure water. Next, we introduce different heterogeneous materials near the focus of the lithotripter to model the effect of the body of an actual patient, i.e., the in vivo condition. We use two approaches in this modeling effort. One approach is to use simple geometrical models for the body cavity and kidney that we created ourselves. The other approach is to import real anatomical data made available from the VOXEL-MAN Group. In studying the focal region acoustics, we specifically examine the maximum calculated pressures. These pressures represent the forces that will ultimately cause the kidney stone to break. We also study the pulse intensity integral, i.e., the energy density carried by the focusing shock wave. In addition to these pressures and energy densities, we are interested in investigating how soft tissue in the focal region may potentially be damaged by the resulting wavefields. We isolate two mechanisms that are thought to be important in soft tissue injury: shearing and cavitation. We calculate estimates for the maximum principal normal and shear strains in the focal region in addition to the corresponding strain rates and use these as metrics for the potential for damage via shearing. We study the calculated negative pressure fields in this region as a surrogate for potential damage caused by cavitation. We find that our simple geometrical anatomical models cause little deviation from the acoustics observed in a water bath. However, when the real anatomical data of the VOXEL-MAN Group is used, the fields of the various relevant flow quantities become more highly oscillatory and produce secondary extrema that could produce damage not predicted from the water bath case. In addition to the conclusions from our own work, we discuss how our results motivate future studies that will hopefully help elucidate specific mechanisms by which kidney stones break and soft tissue becomes damaged.