Forward and Inverse Problems in Nonlinear Acoustics

TL;DR

This paper discusses modeling nonlinear ultrasound propagation in tissue, addressing fractional damping effects, and explores inverse problems for nonlinearity parameter imaging using quasilinear wave equations.

Contribution

It introduces models for nonlinear and fractional damping effects in ultrasound and analyzes inverse problems for imaging nonlinearity parameters.

Findings

Modeling of nonlinear ultrasound with fractional damping.

Analysis of PDE challenges and parameter asymptotics.

Formulation of inverse problems for coefficient identification.

Abstract

The importance of ultrasound is well established in the imaging of human tissue. In order to enhance image quality by exploiting nonlinear effects, recently techniques such as harmonic imaging and nonlinearity parameter tomography have been put forward. As soon as the pressure amplitude exceeds a certain bound, the classical linear wave equation loses its validity and more general nonlinear versions have to be used. Another characteristic property of ultrasound propagation in human tissue is frequency power law attenuation, leading to fractional derivative damping models in time domain. In this contribution we will first of all dwell on modeling nonlinearity on the one hand and fractional damping on the other hand. Moreover we will give an idea on the challenges in the analysis of the resulting PDEs and discuss some parameter asymptotics. Finally, we address a relevant inverse problems in this context, the above mentioned task of nonlinearity parameter imaging, which leads to a coefficient identification problem for a quasilinear wave equation.