Data-driven acoustic control of an encapsulated microbubble using a Koopman linear quadratic regulator*

Encapsulated microbubbles (EMBs) are used in biomedicine for both diagnostic and therapeutic purposes that include ultrasound imaging and drug delivery. A data-driven method to control the oscillations of EMBs using the applied acoustic field is presented based on Koopman operator theory, which is a method for transforming nonlinear dynamical systems into linear systems on an infinite-dimensional function space. This linearization allows classical linear control methods to be applied to the underlying nonlinear dynamical system. Here, we apply a Koopman linear quadratic regulator (KLQR) to control the oscillations of a spherical EMB based on the Marmottant model through the applied ultrasound. Compared to similar methods employed by the authors to control unencapsulated microbubbles, the control of EMBs presents novel challenges due to the presence of a slow manifold that arises in the phase plane spanned by the bubble radius and radial velocity. This slow manifold confounds control in its vicinity and requires special care in formulating the KLQR controller through the judicious selection of Koopman eigenfunctions to capture the relevant dynamics. Results are presented that demonstrate the effectiveness of the modified KLQR controller in driving an EMB to follow arbitrarily-prescribed radial oscillations and stabilize at nonequilibrium radii.