Propagation of elastic waves inside concretions to simulate dynamic failure of kidney stones and gallstones during lithotripsy

Abstract

Reflection and superposition of stress waves is analyzed using finite difference techniques to better understand the effect of stone parameters and geometry on the distribution of strains within kidney stones and gallstones during lithotripsy. Concretions of irregular geometries are subjected to ultrasonic wave sources that simulate lithotripter pulses. The time evolution of strain is calculated inside cylinders of rectangular and circular cross sections, due to an incident radially diverging source in the liquid surrounding the solid. Two schemes are considered to explicitly account for the liquid–solid interface conditions. Both schemes account for varying grid sizes and give identical results for straight interfaces, but the second scheme also handles irregular interfaces. The time sequence obtained numerically for strain at the center of a rectangular cylinder also matches well with the experimental results [S. M. Gracewski etal., J. Acoust. Soc. Am. 94, 652–661 (1993)]. In addition, strain contours are plotted for the propagation of P (longitudinal) and SV (shear vertical) waves inside a circular cylinder. It is shown that the reflection from the concave back surface of the circular cylinder has a focusing effect with the subsequent formation of focal zones (caustics).