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.

翻译：超声造影成像是一种专门的成像技术，它将微泡造影剂应用于传统医学超声检查中，实现对血流和血管的实时可视化。充气微泡被注入体内，在超声波作用下经历压缩与稀疏，并与超声波发生非线性相互作用。因此，声波在含泡液体中的传播是一个强非线性问题，可通过非线性声波方程对压力波的传播进行建模，该方程通过源项与描述气泡动力学的Rayleigh-Plesset型非线性常微分方程耦合。在本研究中，我们首先基于本构关系推导了此类耦合模型的层次结构。随后，我们重点研究了Westervelt声学方程与Rayleigh-Plesset型方程的耦合问题，并在初始压力-微泡数据及终止时间满足适当条件的前提下，严格证明了局部时间解的存在性。最后，我们设计并讨论了单气泡动力学以及微泡与超声波相互作用的数值实验。