In contrast-enhanced echography, the simulation of nonlinear propagation of ultrasound through a population of oscillating microbubbles imposes a computational challenge. Also, the numerical complexity increases because each scatterer has individual properties. To address these problems, the Iterative Nonlinear Contrast Source (INCS) method has been extended to include a large population of nonlinearly responding microbubbles. The original INCS method solves the Westervelt equation in a four-dimensional spatiotemporal domain by generating increasingly accurate field corrections to iteratively update the acoustic pressure. The field corrections are computed by the convolution of a nonlinear contrast source with the Green's function of the linear background medium. Because the convolution integral allows a coarse discretization, INCS can efficiently deal with large-scale problems. To include a population of microbubbles, these are considered as individual contrast point sources with their own nonlinear response. The field corrections are computed as before, but now, in each iteration, the temporal signature of each contrast point source is computed by solving the bubble's Marmottant equation. Physically, each iteration adds an order of multiple scattering. Here, the performance of the extended INCS method and the significance of multiple scattering is demonstrated through various results from different configurations.

An in vivo range verification technology for proton beam cancer therapy, preferably in real-time and with submillimeter resolution, is desired to reduce the present uncertainty in dose localization. Acoustical imaging technologies exploiting possible local interactions between protons and microbubbles or nanodroplets might be an interesting option. Unfortunately, a theoretical model capable of characterising the acoustical field generated by an individual proton on nanometer and micrometer scales is still missing. In this work, such a model is presented. The proton acoustic field is generated by the adiabatic expansion of a region that is locally heated by a passing proton. To model the proton heat deposition, secondary electron production due to protons has been quantified using a semi-empirical model based on Rutherford's scattering theory, which reproduces experimentally obtained electronic stopping power values for protons in water within 10% over the full energy range. The electrons transfer energy into heat via electron-phonon coupling to atoms along the proton track. The resulting temperature increase is calculated using an inelastic thermal spike model. Heat deposition can be regarded as instantaneous, thus, stress confinement is ensured and acoustical initial conditions are set. The resulting thermoacoustic field in the nanometer and micrometer range from the single proton track is computed by solving the thermoacoustic wave equation using k-space Green's functions, yielding the characteristic amplitudes and frequencies present in the acoustic signal generated by a single proton in an aqueous medium. Wavefield expansion and asymptotic approximations are used to extend the spatial and temporal ranges of the proton acoustic field.

For several decades, microbubbles have been the primary choice for ultrasound contrast agents both for efficiency and safety reasons. To optimize their performance in various applications e.g. proton therapy, it is important to understand the dynamics of populations of nonlinearly oscillating microbubbles. Especially for high concentration clouds, multiple scattering can significantly influence the propagation of medical ultrasound. To numerically study the higher-order interaction between nonlinear microbubbles, we present a modified version of the Iterative Nonlinear Contrast Source (INCS) method. Based on a Neumann iterative scheme, the acoustic pressure is obtained by treating the nonlinearly scattering bubbles as individual contrast sources in a “linearized” background medium. In each iteration, the full nonlinear pressure field is updated by applying the 4D spatiotemporal convolution between the background Green’s function and the contrast sources obtained from the previous iteration. In this study, all microbubbles may exhibit an individual behavior. To accommodate this, in each iteration the solution of the Marmottant equation for each microbubble is obtained and used as the temporal signature of the respective contrast source. Using this scheme, each iteration adds an order of multiple scattering. The accuracy and the efficiency of INCS results will be demonstrated for monodisperse and polydisperse concentrations.

There is an ongoing development of ultrasound contrast agents (UCAs) for specific medical diagnostic and therapeutic applications. Recent researches involve all kinds of targeted and loaded microbubbles, monodisperse microbubbles, and phase-change nanodroplets. To design and improve applications that use UCAs, it is necessary to have a thorough understanding of the mutual interaction of ultrasound fields and UCAs. The individual response of microbubbles and nanodroplets is extensively studied and, in the case of bubbles, multiple variants of the Rayleigh-Plesset equation are available to describe their behavior. However, novel applications, such as therapeutic proton beam localization, may strongly depend on the mutual interaction in a population of UCAs. In this presentation, a numerical approach for the analysis of ultrasound-bubble interaction in a population with many (order one million) microbubbles will be discussed. This approach is based on the earlier developed Iterative Nonlinear Contrast Source (INCS) method for simulating nonlinear ultrasound waves. In the current case, the population of microbubbles will be represented by a set of pressure-dependent contrast point sources that are iteratively updated by the INCS method, where each iteration adds an order of multiple scattering. Results will be presented for the linear scattering in populations with different concentrations of microbubbles.

Multiple scattering of sound by a population of particles attracts scientific interest for many decades. In contrast-enhanced echography, the simulation of ultrasound propagation through a dense cloud of nonlinearly oscillating microbubbles imposes a numerical challenge. This is particularly the case for polydisperse concentrations in which each scatterer has individual and independent properties. To address this problem, the Iterative Nonlinear Contrast Source (INCS) method has been adapted. Originally, this solved the Westervelt equation by letting its nonlinear term represent a contrast source in a “linearized” medium, and iteratively updating the pressure by computing the 4D spatiotemporal convolution between this source and the Green’s function. Because convolution allows a coarse discretization, INCS is suitable to deal with large-scale problems. In this talk, microbubbles are regarded as nonlinearly responding point sources that act as contrast sources. The total scattered pressure is computed iteratively, as in the original method, but now in each iteration the temporal signature of the contrast sources is calculated by solving every bubble’s own Marmottant equation. Physically, each iteration accounts for an order of multiple scattering. Numerical results will also be presented, demonstrating that INCS can accurately and efficiently simulate ultrasound propagation through a 3D population of polydisperse nonlinear microbubbles.