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Multiharmonic algorithms for contrast-enhanced ultrasound

Vanja Nikolić, Teresa Rauscher

TL;DR

This work rigorously establishes the existence of time-periodic solutions for this Westervelt-ODE system, derives a multiharmonic representation of the system under time-periodic excitation and develops iterative algorithms that rely on the successive computation of higher harmonics under the assumption of real-valued or complex solution fields.

Abstract

Harmonic generation plays a crucial role in contrast-enhanced ultrasound, both for imaging and therapeutic applications. However, accurately capturing these nonlinear effects is computationally very demanding when using traditional time-domain approaches. To address this issue, in this work, we develop algorithms based on a time discretization that uses a multiharmonic Ansatz applied to a model that couples the Westervelt equation for acoustic pressure with a volume-based approximation of the Rayleigh--Plesset equation for the dynamics of microbubble contrast agents. We first rigorously establish the existence of time-periodic solutions for this Westervelt-ODE system. We then derive a multiharmonic representation of the system under time-periodic excitation and develop iterative algorithms that rely on the successive computation of higher harmonics under the assumption of real-valued or complex solution fields. In the real-valued setting, we characterize the approximation error in terms of the number of harmonics and a contribution owing to the fixed-point iteration. Finally, we investigate these algorithms numerically and illustrate how the number of harmonics and presence of microbubbles influence the propagation of acoustic waves.

Multiharmonic algorithms for contrast-enhanced ultrasound

TL;DR

This work rigorously establishes the existence of time-periodic solutions for this Westervelt-ODE system, derives a multiharmonic representation of the system under time-periodic excitation and develops iterative algorithms that rely on the successive computation of higher harmonics under the assumption of real-valued or complex solution fields.

Abstract

Harmonic generation plays a crucial role in contrast-enhanced ultrasound, both for imaging and therapeutic applications. However, accurately capturing these nonlinear effects is computationally very demanding when using traditional time-domain approaches. To address this issue, in this work, we develop algorithms based on a time discretization that uses a multiharmonic Ansatz applied to a model that couples the Westervelt equation for acoustic pressure with a volume-based approximation of the Rayleigh--Plesset equation for the dynamics of microbubble contrast agents. We first rigorously establish the existence of time-periodic solutions for this Westervelt-ODE system. We then derive a multiharmonic representation of the system under time-periodic excitation and develop iterative algorithms that rely on the successive computation of higher harmonics under the assumption of real-valued or complex solution fields. In the real-valued setting, we characterize the approximation error in terms of the number of harmonics and a contribution owing to the fixed-point iteration. Finally, we investigate these algorithms numerically and illustrate how the number of harmonics and presence of microbubbles influence the propagation of acoustic waves.

Paper Structure

This paper contains 27 sections, 9 theorems, 151 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Existence of time-periodic solutions
  3. Notation
  4. Auxiliary existence results for linear time-periodic problems
  5. Analysis of the Westervelt-ODE system
  6. Time discretization via a multiharmonic Ansatz for real fields
  7. A multiharmonic cut-off algorithm
  8. A linearized multiharmonic cut-off algorithm
  9. Convergence of the linearized multiharmonic algorithm for real fields
  10. Time discretization via a multiharmonic Ansatz for complex fields
  11. A multiharmonic cut-off algorithm
  12. A linearized multiharmonic cut-off algorithm
  13. A two-harmonic approximation scheme
  14. Numerical results
  15. Numerical framework
  16. ...and 12 more sections

Key Result

Lemma 2.1

Let $T>0$ and $f \in L^2(0,T; L^\infty(\Omega))$. Furthermore, let $\delta$, $\omega_0>0$. Then, the periodic ODE problem has a unique solution that satisfies

Figures (11)

  • Figure 1: Overview of multiharmonic algorithms in this work.
  • Figure 2: Dependence of the norm $\left\| \Re(p)\right\|_{L^{\infty}(L^2(\Omega))}$ on the mesh size $h_{\textup{FEM}}$ for different numbers of harmonics $N \in \left\lbrace 1,2, 5, 10\right\rbrace$; the pressure is computed using the scheme in \ref{['multiharm_complex']} with $\omega = \omega_0$, $r_{\delta} = 0.004$, and $A=1500$.
  • Figure 3: Five-harmonic expansion of $\mathfrak{R}(p(x,y_0,t_0))$ resulting from complex fields in \ref{['system_v_0_0']} vs. real setting in \ref{['eq: multi_sim1']}.
  • Figure 4: Real part of the pressure $\mathfrak{R}(p(x,y_0,t_0))$ obtained from \ref{['system_v_0_0']} using complex fields for the two-, three- and ten-harmonic expansions.
  • Figure 5: Real part of the pressure from the five-harmonic expansion computed using \ref{['system_v_0_0']} without bubbles ($n_0=0$, dashed green line) and with bubbles (blue line) over time at three spatial points.
  • ...and 6 more figures

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Proposition 2.1: see Rainer2024nonlinear
  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • ...and 10 more