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Mathematical models for nonlinear ultrasound contrast imaging with microbubbles

Vanja Nikolić, Teresa Rauscher

TL;DR

This work develops a rigorous mathematical framework for nonlinear ultrasound contrast imaging in bubbly media by deriving and analyzing a coupled PDE–ODE system that links the Westervelt nonlinear acoustic equation for pressure $p(x,t)$ to Rayleigh–Plesset–type bubble dynamics for radius $R(t)$ with $v=\frac{4}{3}\pi R^3$. It presents a hierarchy of bubbly acoustic models, derives explicit inhomogeneous Westervelt and Kuznetsov equations as bubbly extensions, and proves local well-posedness of the Westervelt–Rayleigh–Plesset system via a Banach fixed-point approach under small-data assumptions. The paper also provides a comprehensive numerical study comparing coated and non-coated microbubbles under sinusoidal and Westervelt-driven pressures, highlighting how driving amplitude, frequency, and shell properties shape bubble dynamics and harmonic content. The results yield a rigorous foundation and practical insights for predicting nonlinear ultrasound–bubble interactions, with implications for imaging quality, safety, and cavitation management in clinical ultrasound applications.

Abstract

Ultrasound contrast imaging is a specialized imaging technique that applies microbubble contrast agents to traditional medical sonography, providing real-time visualization of blood flow and vessels. Gas-filled microbubbles are injected into the body, where they undergo compression and rarefaction and interact nonlinearly with the ultrasound waves. Therefore, the propagation of sound through a bubbly liquid is a strongly nonlinear problem that can be modeled by a nonlinear acoustic wave equation for the propagation of the pressure waves coupled via the source terms to a nonlinear ordinary differential equation of Rayleigh-Plesset type for the bubble dynamics. In this work, we first derive a hierarchy of such coupled models based on constitutive laws. We then focus on the coupling of Westervelt's acoustic equation to Rayleigh-Plesset type equations, where we rigorously show the existence of solutions locally in time under suitable conditions on the initial pressure-microbubble data and final time. Thirdly, we devise and discuss numerical experiments on both single-bubble dynamics and the interaction of microbubbles with ultrasound waves.

Mathematical models for nonlinear ultrasound contrast imaging with microbubbles

TL;DR

This work develops a rigorous mathematical framework for nonlinear ultrasound contrast imaging in bubbly media by deriving and analyzing a coupled PDE–ODE system that links the Westervelt nonlinear acoustic equation for pressure to Rayleigh–Plesset–type bubble dynamics for radius with . It presents a hierarchy of bubbly acoustic models, derives explicit inhomogeneous Westervelt and Kuznetsov equations as bubbly extensions, and proves local well-posedness of the Westervelt–Rayleigh–Plesset system via a Banach fixed-point approach under small-data assumptions. The paper also provides a comprehensive numerical study comparing coated and non-coated microbubbles under sinusoidal and Westervelt-driven pressures, highlighting how driving amplitude, frequency, and shell properties shape bubble dynamics and harmonic content. The results yield a rigorous foundation and practical insights for predicting nonlinear ultrasound–bubble interactions, with implications for imaging quality, safety, and cavitation management in clinical ultrasound applications.

Abstract

Ultrasound contrast imaging is a specialized imaging technique that applies microbubble contrast agents to traditional medical sonography, providing real-time visualization of blood flow and vessels. Gas-filled microbubbles are injected into the body, where they undergo compression and rarefaction and interact nonlinearly with the ultrasound waves. Therefore, the propagation of sound through a bubbly liquid is a strongly nonlinear problem that can be modeled by a nonlinear acoustic wave equation for the propagation of the pressure waves coupled via the source terms to a nonlinear ordinary differential equation of Rayleigh-Plesset type for the bubble dynamics. In this work, we first derive a hierarchy of such coupled models based on constitutive laws. We then focus on the coupling of Westervelt's acoustic equation to Rayleigh-Plesset type equations, where we rigorously show the existence of solutions locally in time under suitable conditions on the initial pressure-microbubble data and final time. Thirdly, we devise and discuss numerical experiments on both single-bubble dynamics and the interaction of microbubbles with ultrasound waves.
Paper Structure (20 sections, 6 theorems, 109 equations, 9 figures, 1 table)

This paper contains 20 sections, 6 theorems, 109 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Derivation of nonlinear acoustic models in bubbly media
  3. Approach in the derivation
  4. Derivation of acoustic equations in bubbly liquids
  5. Simplifying the conservation of momentum equation
  6. Simplifying the conservation of mass equation
  7. Combining the simplified equations
  8. Nonlinear acoustic equations for bubbly fluids
  9. Coupling to microbubble dynamics
  10. Analysis of the Westervelt--Rayleigh--Plesset system
  11. Numerical study
  12. Numerical framework for solving Rayleigh--Plesset equations
  13. Single-microbubble dynamics: Coated vs. non-coated
  14. The influence of ultrasound on microbubbles
  15. Outlook
  16. ...and 5 more sections

Key Result

Proposition 3.1

\newlabelProp:WellP_West0 Assume that $\Omega \subset \mathbb{R}^d$, $d \in \{1, 2,3\}$ is a bounded and $C^{1,1}$-regular domain. Let $c$, $b>0$ and $k \in L^\infty(\Omega)$. Furthermore, let $(p_0, p_1) \in {H_\diamondsuit^2(\Omega)} \times H_0^1(\Omega)$ and $f \in L^2(0,T; L^2(\Omega))$. Ther then there is a unique solution of in ${X}_p = L^\infty(0,T; {H_\diamondsuit^2(\Omega)}) \cap W^{1

Figures (9)

  • Figure 1: Single-microbubble dynamics; adapted from lauterborn2023acoustic
  • Figure 1: Sensitivity of a coated microbubble to driving amplitude and frequency.
  • Figure 2: Behavior of a coated microbubble for high frequencies: Radius-time $R(t)$ curves and the corresponding FFT-spectra.
  • Figure 3: Comparison of the dynamics of a coated and non-coated microbubble.
  • Figure 4: Behavior of a non-coated microbubble: Radius-time $R(t)$ curves and the corresponding FFT-spectra.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Proposition 3.1
  • Proof 1
  • Theorem 3.2
  • Proof 2
  • Remark 3.3: On nonlocal acoustic attenuation
  • Lemma 1.1
  • Proof 3
  • Lemma 1.2
  • Proof 4
  • Proposition 1.3: see Theorem 3.1 in kaltenbacher2022inverse
  • ...and 2 more