Local well-posedness of a coupled Westervelt-Pennes model of nonlinear ultrasonic heating

TL;DR

This paper proves the local well-posedness of a coupled nonlinear ultrasonic heating model combining the Westervelt wave equation and Pennes bioheat equation, capturing thermal lensing effects in HIFU treatments.

Contribution

It introduces and analyzes a new coupled mathematical model for ultrasonic heating, establishing its well-posedness under certain conditions.

Findings

The model is well-posed locally in time.

The model remains non-degenerate under small pressure data.

It captures the thermal lensing effect in ultrasonic heating.

Abstract

High-Intensity Focused Ultrasound (HIFU) waves are known to induce localized heat to a targeted area during medical treatments. In turn, the rise in temperature influences their speed of propagation. This coupling affects the position of the focal region as well as the achieved pressure and temperature values. In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on the Westervelt wave equation coupled to the Pennes bioheat equation that captures this so-called thermal lensing effect. We prove that this quasi-linear model is well-posed locally in time and does not degenerate under a smallness assumption on the pressure data.