Koopman linear quadratic regulator control of an acoustically driven encapsulated microbubble

In biomedical applications, microbubbles are often encapsulated with lipid or protein compounds to increase their longevity within the bloodstream. These encapsulated microbubbles (EMBs) are used in ultrasound imaging and drug delivery. Here, a data-driven method is presented for controlling EMBs with an acoustic field based on Koopman operator theory, which is a mathematical framework for transforming nonlinear dynamical systems into linear systems on an infinite-dimensional function space. This linearization, in turn, allows classical linear control methods to be directly applied to strongly nonlinear systems. In this work, we use a Koopman linear quadratic regulator (KLQR) to design acoustic control signals for a spherical EMB based on the Marmottant model. It is shown that KLQR is able to effectively drive EMBs to specific target behaviors, such as amplifying a subharmonic resonance or exhibiting a quasiperiodic oscillation. These results are compared to previous work by the authors on unencapsulated microbubbles, and it is discovered that EMBs present unique difficulties caused by the presence of a slow manifold in their dynamics. This slow manifold disrupts the controller when a target trajectory nears it; as a result, the controller must be built using Koopman eigenfunctions that are carefully constructed so as to capture the relevant dynamics.