Data-driven acoustic control of a spherical bubble using a Koopman linear quadratic regulator.

Koopman operator theory has gained interest as a framework for transforming nonlinear dynamics on the state space into linear dynamics on abstract function spaces, which preserves the underlying nonlinear dynamics of the system. These spaces can be approximated through data-driven methodologies, which enables the application of classical linear control strategies to nonlinear systems. Here, a Koopman linear quadratic regulator (KLQR) was used to acoustically control the nonlinear dynamics of a single spherical bubble, as described by the well-known Rayleigh-Plesset equation, with several objectives: (1) simple harmonic oscillation at amplitudes large enough to incite nonlinearities, (2) stabilization of the bubble at a nonequilibrium radius, and (3) periodic and quasiperiodic oscillation with multiple frequency components of arbitrary amplitude. The results demonstrate that the KLQR controller can effectively drive a spherical bubble to radially oscillate according to prescribed trajectories using both broadband and single-frequency acoustic driving. This approach has several advantages over previous efforts to acoustically control bubbles, including the ability to track arbitrary trajectories, robustness, and the use of linear control methods, which do not depend on initial guesses.

Shape oscillation and stability of an encapsulated microbubble translating in an acoustic wave.

Encapsulated microbubbles (EMBs) are associated with a wide variety of important medical applications, including sonography, drug delivery, and sonoporation. The nonspherical oscillations, or shape modes, of EMBs strongly affect their stability and acoustic signature, and thus are an important factor to consider in the design and utilization of EMBs. Under acoustic forcing, EMBs often translate with significant velocity, which can excite shape modes, yet few studies have addressed the effect of translation on the shape stability of EMBs. In this work, the shape stability of an EMB subject to translation is investigated through development of an axisymmetric model for the case of small deformations. The potential flow in the bulk volume of the external flow is modeled using an asymptotic analysis. Viscous effects within the thin boundary layer at the interface are included, owing to the no-slip boundary condition, using Prosperetti's theory [Q. Appl. Math. 34, 339 (1977)]. In-plane stress and bending moment due to the encapsulation are incorporated into the model through the dynamic boundary condition at the interface. The evolution equations for radial oscillation, translation, and shape oscillation of an EMB are derived, which can be reduced to model an uncoated gas bubble by neglecting the encapsulation properties. These equations are solved numerically to analyze the shape mode stability of an EMB and a gas bubble subject to an acoustic, traveling plane wave. The findings demonstrate the counterintuitive result that translation has a more destabilizing effect on an EMB than on a gas bubble. The no-slip condition at the encapsulating membrane is the main factor responsible for mediating this interfacial instability due to translation.

High Efficiency Molecular Delivery with Sequential Low-Energy Sonoporation Bursts.

Microbubbles interact with ultrasound to induce transient microscopic pores in the cellular plasma membrane in a highly localized thermo-mechanical process called sonoporation. Theranostic applications of in vitro sonoporation include molecular delivery (e.g., transfection, drug loading and cell labeling), as well as molecular extraction for measuring intracellular biomarkers, such as proteins and mRNA. Prior research focusing mainly on the effects of acoustic forcing with polydisperse microbubbles has identified a "soft limit" of sonoporation efficiency at 50% when including dead and lysed cells. We show here that this limit can be exceeded with the judicious use of monodisperse microbubbles driven by a physiotherapy device (1.0 MHz, 2.0 W/cm(2), 10% duty cycle). We first examined the effects of microbubble size and found that small-diameter microbubbles (2 µm) deliver more instantaneous power than larger microbubbles (4 & 6 µm). However, owing to rapid fragmentation and a short half-life (0.7 s for 2 µm; 13.3 s for 6 µm), they also deliver less energy over the sonoporation time. This translates to a higher ratio of FITC-dextran (70 kDa) uptake to cell death/lysis (4:1 for 2 µm; 1:2 for 6 µm) in suspended HeLa cells after a single sonoporation. Sequential sonoporations (up to four) were consequently employed to increase molecular delivery. Peak uptake was found to be 66.1 ± 1.2% (n=3) after two sonoporations when properly accounting for cell lysis (7.0 ± 5.6%) and death (17.9 ± 2.0%), thus overcoming the previously reported soft limit. Substitution of TRITC-dextran (70 kDa) on the second sonoporation confirmed the effects were multiplicative. Overall, this study demonstrates the possibility of utilizing monodisperse small-diameter microbubbles as a means to achieve multiple low-energy sonoporation bursts for efficient in vitro cellular uptake and sequential molecular delivery.

Application of nonlinear sliding mode control to ultrasound contrast agent microbubbles.

A sliding mode control system is developed and applied to a spherical model of a contrast agent microbubble that simulates its radial response to ultrasound. The model uses a compressible form of the Rayleigh-Plesset equation combined with a thin-shell model. A nonlinear control law for the second-order model is derived and used to design and simulate the controller. The effect of the controller on the contrast agent response is investigated for various control scenarios. This work demonstrates the feasibility of using a nonlinear control system to modulate the dynamic response of ultrasound contrast agents, but highlights the need for improved feedback mechanisms and control input methods. Possible applications of the nonlinear control system to contrast agents illustrated in this work include radius stabilization in the presence of an acoustic wave, radial growth and subsequent collapse, and generation of periodic radial oscillations while a contrast agent is within an acoustic forcing regime known to cause a chaotic response.

Dynamical analysis of the nonlinear response of ultrasound contrast agent microbubbles.

The nonlinear response of spherical ultrasound contrast agent microbubbles is investigated to understand the effects of common shells on the dynamics. A compressible form of the Rayleigh-Plesset equation is combined with a thin-shell model developed by Lars Hoff to simulate the radial response of contrast agents subject to ultrasound. The responses of Albunex, Sonazoid, and polymer shells are analyzed through the application of techniques from dynamical systems theory such as Poincaré sections, phase portraits, and bifurcation diagrams to illustrate the qualitative dynamics and transition to chaos that occurs under certain changes in system parameters. Corresponding calculations of Lyapunov exponents provide quantitative data on the system dynamics. The results indicate that Albunex and polymer shells sufficiently stabilize the response to prevent transition to the chaotic regime throughout typical clinical ranges of ultrasound pressure and frequency. By contrast, Sonazoid shells delay the onset of chaos relative to an unshelled bubble but do not prevent it. A contour plot identifying regions of periodic and chaotic behavior over clinical ranges of ultrasound pressure and frequency is provided for Sonazoid. This work characterizes the nonlinear response of various ultrasound contrast agents, and shows that shell properties have a profound influence on the dynamics.

Interaction of lithotripter shockwaves with single inertial cavitation bubbles.

The dynamic interaction of a shockwave (modelled as a pressure pulse) with an initially spherically oscillating bubble is investigated. Upon the shockwave impact, the bubble deforms non-spherically and the flow field surrounding the bubble is determined with potential flow theory using the boundary-element method (BEM). The primary advantage of this method is its computational efficiency. The simulation process is repeated until the two opposite sides of the bubble surface collide with each other (i.e. the formation of a jet along the shockwave propagation direction). The collapse time of the bubble, its shape and the velocity of the jet are calculated. Moreover, the impact pressure is estimated based on water-hammer pressure theory. The Kelvin impulse, kinetic energy and bubble displacement (all at the moment of jet impact) are also determined. Overall, the simulated results compare favourably with experimental observations of lithotripter shockwave interaction with single bubbles (using laser-induced bubbles at various oscillation stages). The simulations confirm the experimental observation that the most intense collapse, with the highest jet velocity and impact pressure, occurs for bubbles with intermediate size during the contraction phase when the collapse time of the bubble is approximately equal to the compressive pulse duration of the shock wave. Under this condition, the maximum amount of energy of the incident shockwave is transferred to the collapsing bubble. Further, the effect of the bubble contents (ideal gas with different initial pressures) and the initial conditions of the bubble (initially oscillating vs. non-oscillating) on the dynamics of the shockwave-bubble interaction are discussed.

Theoretical study of BOLD response to sinusoidal input.

This is a theoretical study of a compelling model of blood oxygen level-dependent (BOLD) response dynamics, measured in functional magnetic resonance imaging (fMRI). The novelty of this study involves the way the model is driven sinusoidally, in order to avoid onset and offset transients that pose difficulties in data analysis and interpretation. The driving frequency ranges over the natural time scales of the hemodynamic response (0.01-1 Hz), which also corresponds to the period in typical boxcar stimulus designs. At low stimulus amplitude, the predicted BOLD response is quasi-linear. The amplitude exhibits a mild peak near the modulation frequency 0.1 Hz, and falls rapidly for higher frequencies. The phase lag relative to the stimulus is a monotonically increasing function of the modulation frequency. These findings illustrate the dynamical nature of the BOLD response, and could be used to optimize experimental designs that admit sinusoidal modulation. Higher stimulus amplitude elicits nonlinear behavior characterized by a double peak during the positive deflection of the BOLD response. This finding is particularly interesting, because similar double peaks are seen frequently in BOLD data.