Optimal control of the nonspherical oscillation of encapsulated microbubbles for drug delivery

Encapsulated microbubbbles (EMBs) are used in biomedicine for both imaging and diagnostic purposes, including ultrasound imaging and drug delivery. Optimal control theory is used to determine the ultrasound input that causes an EMB to rupture with a minimum amount of acoustic energy in an effort to reduce unwanted side effects enhance patient safety. The optimal control problem is applied to models of small-amplitude nonspherical oscillations for both an encapsulated microbubble and a free gas bubble. These models are solved subject to a cost function that maximizes the incidence of rupture and minimizes the acoustic energy input. Using commercial optimization software, the optimal control problem is solved numerically using pseudospectral collocation methods. Both single-frequency forcing and broadband acoustic forcing schemes are explored and compared. The results show that for a free gas bubble, single-frequency forcing is almost as efficient as broadband forcing for inciting instability in terms of acoustic effort. Also, encapsulated microbubbles require much more effort to incite instability relative to free gas bubbles due to the stabilizing influence of the shell. However, for EMBs, broadband acoustic forcing significantly reduces the acoustic effort required to incite EMB rupture relative to single-frequency schemes. Furthermore, the optimal forcing for single-frequency forcing appears to not be at the natural frequency but rather at a frequency slighly below this value.