Optimal control of the nonspherical oscillations of encapsulated microbubbles

Abstract

Encapsulated microbubbles (EMBs) were originally developed as contrast agents for ultrasound imaging but are more recently emerging as vehicles for intravenous drug and gene delivery. Ultrasound can excite nonspherical oscillations, or shape modes, that can enhance the acoustic signature of an EMB and also incite rupture, which promotes drug and gene delivery at targeted sites. Therefore, the ability to control shape modes can improve the efficacy of both the diagnosis and treatment mediated by EMBs. This work uses optimal control theory to determine the ultrasound input that maximizes a desired nonspherical EMB response (e.g., to enhance scattering or rupture), while minimizing the total acoustic input in order to enhance patient safety and reduce unwanted side effects. The optimal control problem is applied to a model of an EMB that accounts for small amplitude shape deformations. This model is solved subject to a cost function that maximizes the incidence of rupture or acoustic echo while minimizing the acoustic energy input. The optimal control problem is solved numerically through pseudospectral collocation methods using commercial optimization software. Single frequency and broadband acoustic forcing schemes are explored and compared. The results show that broadband forcing significantly reduces the acoustic effort required to incite EMB rupture relative to single frequency schemes. Furthermore, the acoustic effort required depends strongly on the shape mode that is forced to become unstable.