# Multiharmonic Algorithms for Contrast-Enhanced Ultrasound

## TL;DR

This paper introduces new algorithms to efficiently model nonlinear effects in contrast-enhanced ultrasound using multiharmonic methods.

## Contribution

The paper proposes multiharmonic algorithms that reduce computational costs in modeling microbubble dynamics in ultrasound.

## Key findings

- The existence of time-periodic solutions for the Westervelt-ODE system is rigorously established.
- Multiharmonic algorithms are derived for the system under time-periodic excitation.
- Numerical investigations show how harmonics and microbubbles influence acoustic wave propagation.

## 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 demanding when using traditional time-domain approaches. To address this issue, 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 assuming either real-valued or complex-valued solution fields. In the real-valued setting, we characterize the approximation error in terms of the number of harmonics and a contribution arising from the fixed-point iteration. Finally, we investigate these algorithms numerically and illustrate how the number of harmonics and the presence of microbubbles influence the propagation of acoustic waves.