Session T12: Biomedical, Cavitation and Acoustics I

Generalized Rayleigh-Plesset equation for bubbles in viscoelastic liquids

The spherical dynamics of microbubbles in liquids is well understood. Yet, the problem becomes markedly more complex if the bubble is embedded in viscoelastic materials. This is the case in many biomedical applications including ultrasound imaging, drug delivery, and ablation therapies. Several studies have proposed models based on extended forms of the Rayleigh-Plesset equation that account for viscoelastic stresses. However, the validity of these models remain restricted to a particular choice of constitutive model and/or to small deformations. Here, we derive a generalized equation for bubbles in viscoelastic materials by borrowing concepts from finite-strain theory and stress relaxation functions. The proposed theory is applicable to viscoelastic media with arbitrary complex modulus and remains valid for large bubble deformations. We demonstrate the effectiveness of our approach by comparing it to previously published models of viscoelastic solids and liquids with specific constitutive equations.   

Investigating the hydrodynamic parameters and thermodynamic properties of a collapsing bubble at nanoscales using Molecular Dynamics Simulations

Abstract

There has been a generalized discrepancy of radius parameters and thermodynamic properties at the point of final collapse and rebound of the bubble during cavitation below the micro-scale. Here we will investigate these parameters and properties by doing a parametric study using molecular dynamics (MD) simulations. MD simulations allow us to resolve the bubble down to nanoscales. The results from the simulations will be compared with the Rayleigh-Plesset equation and other analytical models, that describe the evolution of a collapsing and growing bubble. This study will help determine the validity of continuum models at nano-scales.   

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.   

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.   

Ambient pressure dependent subharmonic response of contrast microbubbles for blood pressure estimation using ultrasound imaging

Subharmonic Aided Pressure Estimation (SHAPE) is a non-invasive method to measure blood pressure inside critical organs using ultrasound imaging. This method utilizes the strong sensitivity of subharmonic response from ultrasound contrast microbubbles to the change in the ambient pressure i.e. blood pressure. The subharmonic generation is known to be a threshold phenomenon with the threshold depending on the acoustic parameters as well as the physical properties of bubbles such as size, resonance frequency, and shell initial condition. In this study, we characterized the subharmonic generation and its sensitivity to the ambient pressure for different microbubbles over 25 – 700 kPa acoustic pressures, 3 MHz frequency, and hydrostatic pressure range of 0-25 kPa. The subharmonic showed different trends of increase and decrease with hydrostatic pressure depending on the maximum overpressure magnitude, number of pressure cycles, and initial subharmonic amplitude (before and after threshold). These findings could play important role in accurate determination of subharmonic correlation with ambient pressure resulting in improved pressure estimation in SHAPE.   

Effects of Filling Gas Composition on Microbubble Material Properties

Lipid-Shelled microbubbles are highly echogenic gas-filled particles ranging from 1 to 10 microns. The shell typically contains varying amounts of PEGylated lipids to improve biocompatibility and reduce coalescence. The filling gas type and PEG concentration immensely affect fundamental characteristics such as size distribution, material properties, attenuation, and stability. Here we investigated the extent of these changes by studying four different PFB-air-filled microbubble groups produced by mechanical agitation with varying PFB-air concentration and PEG ratios. We measured the attenuation of microbubbles under low ultrasonic excitation with a 5 MHz flat transducer. The measured attenuation was used to determine microbubble material properties using an interfacial rheological model. The interfacial elasticity was found to be significantly higher for lower PFB-air ratios, while the effects of PEG ratios were little. Additionally, the yield of microbubbles was higher for lower PFB-air ratios, although the mean size remains similar across the various groups studied. The findings offer new insights into both the stabilization of freshly formed microbubbles and their in vitro stability.   

Shape stability of a microbubble under the effect of an acoustic field

Therapeutic applications of microbubbles and ultrasound can be used in the field of cancer treatment to improve the treatment efficacy while minimizing side effects. Ultrasound-mediated microbubbles can temporarily increase the permeability of cell membranes to enhance the delivery of drugs and genes to specific target tissues. The non-spherical stable shape oscillation of a microbubble, known as shape modes, is one of the core mechanisms of medical applications. Microbubbles are encapsulated in a lipid or protein shell to increase their lifetime in blood. This study aims to investigate the shape stability of an encapsulated bubble under the effect of an acoustic field. Power law and Kelvin-Voigt constitutive laws are employed for blood and bubble shell, respectively. An external acoustic field is applied to obtain volume oscillations. The spherical interface of the microbubble is perturbed with a harmonic non-spherical perturbation. After the spherical oscillations reach a steady periodic state, Floquet theory is applied to find the eigenvalues of the system. Phase diagrams are obtained for different pressure amplitudes of the acoustic field and initial bubble radius to investigate stable and unstable regions. It is observed that for larger pressure amplitudes of the acoustic field and initial radius values, the bubble becomes unstable, although the detailed structure of the phase diagram is quite complicated.