Nonlinear Acoustic Analysis Could Help Destroy Kidney Stones

Mathematical analysis and simulation of models governing the propagation of sound in fluids could enhance lithotripsy, according to a new study.

Researchers at Alpen-Adria-Universität (Klagenfurt am Wörthersee, Austria) are developing an application-oriented comprehensive mathematical analysis of the underlying model equations the propagation of sound in fluids, since n-depth numerical simulation and optimization could result in a better understanding and control of the physical effects and complication risks during medical applications. As such, the mathematical analysis of the underlying partial differential equation (PDE) models of high intensity focused ultrasound (HIFU) is crucial for proper design of devices, such as in lithotripsy.

For example, optimization methods can be used to design the three dimensional (3-D) geometrical shape an acoustic lens so that the acoustic pressure generated is focused directly onto a kidney stone, while the surrounding tissue remains intact. Another issue the researchers plan to investigate is the coupling of nonlinear acoustics to other physical fields, such as excitation mechanisms, focusing devices, heat generation, and interaction with the kidney stones. The project is supported by the Austrian Science Fund (FWF; Vienna, Austria).

“These models are based on partial differential equations. The better our grasp of these equations, the more successfully one can avoid complications during the application of ultrasound technology,” said lead author Rainer Brunnhuber, MSc, of the department of mathematics. “To give an example, mathematical optimization methods can be used to enhance the shape of an acoustic lens in such a way that the acoustic pressure is focused precisely on the location of the kidney stone, and the surrounding tissue retains as little damage as possible.”

Nonlinear acoustics deal with large amplitude sound waves that require using comprehensive governing equations of fluid dynamics (for sound waves in liquids and gases) and elasticity (for sound waves in solids). The effects of nonlinearity cause sound waves to become distorted as they travel through a material, but geometric spreading and absorption effects usually overcome the self-distortion, so linear behavior usually prevails. Nonlinear acoustic propagation therefore occurs only for very large amplitudes and only near the source.

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