Mathematical models for nonlinear ultrasound contrast imaging with microbubbles

TL;DR

This paper develops and analyzes mathematical models for nonlinear ultrasound contrast imaging involving microbubbles, including existence proofs and numerical experiments on bubble dynamics and ultrasound interactions.

Contribution

It introduces a hierarchy of coupled nonlinear models and proves local existence of solutions for the combined acoustic and bubble dynamics equations.

Findings

Existence of solutions for coupled nonlinear models established.

Numerical simulations demonstrate bubble behavior under ultrasound.

Models capture nonlinear interactions relevant for contrast imaging.

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.