Existence of solutions to k-Wave models of nonlinear ultrasound propagation in biological tissue

TL;DR

This paper proves the local existence of solutions for nonlinear ultrasound propagation models in biological tissue, incorporating fractional absorption and viscosity, with implications for imaging inverse problems.

Contribution

It introduces a Galerkin approximation for these models and establishes solution existence, including the vanishing viscosity limit, under variable tissue parameters.

Findings

Proved local existence of solutions for nonlinear ultrasound models.

Established the vanishing viscosity limit for the models.

Supported variable tissue parameters in the theoretical framework.

Abstract

We investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k-Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. We first consider a system that incorporates additional viscosity in the equation for momentum conservation. By constructing a Galerkin approximation procedure, we prove the local existence of its solutions. In view of inverse problems arising from imaging tasks, the theory allows for the variable background mass density, speed of sound, and the nonlinearity parameter in the systems. Secondly, under stronger conditions on the data, we take the vanishing viscosity limit of the problem, thereby rigorously establishing the existence of solutions for the limiting system as well.