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Forward and Inverse Problems in Nonlinear Acoustics

Barbara Kaltenbacher

TL;DR

This work develops and analyzes forward and inverse problems in nonlinear acoustics, unifying classical and advanced nonlinear models (e.g., Kuznetsov, Westervelt, JMGT, Blackstock–Crighton) with fractional attenuation mechanisms to describe ultrasound in tissue. It systematically addresses parameter limits (e.g., vanishing relaxation, vanishing diffusivity, fractional limits) and establishes well-posedness and convergence results, highlighting transitions between hyperbolic and parabolic regimes. In the frequency domain, it employs multiharmonic expansions to capture higher-harmonic generation and to structure the inverse problem, culminating in linearized and local nonlinear uniqueness results and practical reconstruction experiments from two harmonics for nonlinearity-parameter imaging (ANT). The analysis underscores the memory effects and nonlocality inherent in fractional models and demonstrates their potential to enhance tissue characterization and imaging via higher-harmonic information. Overall, the paper provides a rigorous mathematical framework for modeling, analysis, and inverse imaging in nonlinear, fractional-attenuation acoustics with implications for medical ultrasound.

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.

Forward and Inverse Problems in Nonlinear Acoustics

TL;DR

This work develops and analyzes forward and inverse problems in nonlinear acoustics, unifying classical and advanced nonlinear models (e.g., Kuznetsov, Westervelt, JMGT, Blackstock–Crighton) with fractional attenuation mechanisms to describe ultrasound in tissue. It systematically addresses parameter limits (e.g., vanishing relaxation, vanishing diffusivity, fractional limits) and establishes well-posedness and convergence results, highlighting transitions between hyperbolic and parabolic regimes. In the frequency domain, it employs multiharmonic expansions to capture higher-harmonic generation and to structure the inverse problem, culminating in linearized and local nonlinear uniqueness results and practical reconstruction experiments from two harmonics for nonlinearity-parameter imaging (ANT). The analysis underscores the memory effects and nonlocality inherent in fractional models and demonstrates their potential to enhance tissue characterization and imaging via higher-harmonic information. Overall, the paper provides a rigorous mathematical framework for modeling, analysis, and inverse imaging in nonlinear, fractional-attenuation acoustics with implications for medical ultrasound.

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.

Paper Structure

This paper contains 17 sections, 6 theorems, 45 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Models of nonlinear acoustics
  3. Physical Principles
  4. Classical Models of Nonlinear Acoustics
  5. Advanced Models of Nonlinear Acoustics
  6. Parameter asymptotics
  7. Vanishing relaxation time in the Jordan-Moore-Gibson-Thompson equation
  8. Some further parameter limits
  9. Fractional attenuation models in ultrasonics
  10. Fractional models of (linear) viscoelasticity
  11. Fractional models of (linear) acoustics via pressure -- density relation
  12. Fractional models of nonlinear acoustics via temperature -- heat flux relation
  13. Fractional derivatives
  14. Describing nonlinear wave propagation in frequency domain by multiharmonic expansions
  15. Inverse problems
  16. ...and 2 more sections

Key Result

Theorem 3.1

Let $c^2$, $b$, $\tau>0$, and let $T>0$. Let the assumptions non-degeneracy_assumption, eq:alphagammaf_reg_Wes, reg_init hold. Then there exists a unique solution The solution fullfils the estimate If additionally $\|\nabla \alpha\|_{L^\infty L^3}<\frac{\underline{\alpha}}{C^\Omega_{H^1,L^6} }$ holds, then $C(\alpha, T, {\tau})$ is independent of ${\tau}$.

Figures (9)

  • Figure 1: Linear versus nonlinear wave propagation in space (top) and time (bottom) domain.
  • Figure 2: Higher harmonics in frequency domain.
  • Figure 3: Snapshots of pressure $p=\varrho \psi_t$ for fixed relaxation time $\tau=0.1 \, \mu$s
  • Figure 4: Pressure wave for different relaxation parameters $\tau$ at final time $t=45\mu$s.
  • Figure 5: Relative errors as $\tau\to0$
  • ...and 4 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 5.1
  • Theorem 6.1
  • Theorem 6.2