Vanja Nikolić, Mathematics, Radboud University

Abstract:

Nonlinear acoustic wave models are increasingly important for medical ultrasound and industrial applications where sound propagates at high intensities or frequencies. Harmonic generation, in particular, plays a key role in contrast-enhanced ultrasound, both for imaging and therapeutic applications. In this talk, we will discuss models for contrast-enhanced ultrasound, where the acoustic field is governed by a nonlinear Westervelt-type wave equation coupled to a Rayleigh-Plesset-type ODE describing the dynamics of microbubble contrast agents. We will go over the analytical and computational questions arising from this coupling, such as the existence of time-periodic solutions and the handling of nonlinearity and differing time scales. Since time-domain approaches are computationally demanding for capturing nonlinear effects, we will also discuss a discretization strategy based on a multiharmonic Ansatz applied to the wave–bubble system. A priori error estimates will be presented that characterize the approximation error in terms of the number of retained harmonics and a contribution arising from the fixed-point iteration. Additionally, numerical experiments will illustrate how the number of retained harmonics and the presence of microbubbles influence acoustic wave propagation. The talk is based on joint work with Teresa Rauscher (University of Graz, Austria).