Nonlinear dynamics and bifurcation structure of ultrasonically excited lipid coated microbubbles

Influence of the liquid ionic strength on the resonance frequency and shell parameters of lipid-coated microbubbles

The correct measurement of the resonance frequency and shell properties of coated microbubbles (MBs) is essential in understanding and optimizing their response to ultrasound (US) exposure parameters. In diagnostic and therapeutic ultrasound, MBs are typically surrounded by blood; however, the influence of the medium charges on the MB resonance frequency has not been systematically studied using controlled measurements. This study aims to measure the medium charge interactions on MB behavior by measuring the frequency-dependent attenuation of the same size MBs in mediums with different charge densities. In-house lipid-coated MBs with C3F8 gas core were formulated. The MBs were isolated to a mean size of 2.35 μm using differential centrifugation. MBs were diluted to ≈8×105 MBs/mL in distilled water (DW), Phosphate-Buffered Saline solution (PBS1x) and PBS10x. The frequency-dependent attenuation of the MBs solutions was measured using an aligned pair of PVDF transducers with a center frequency of 10MHz and 100% bandwidth in the linear oscillation regime (7 kPa pressure amplitude). The MB shell properties were estimated by fitting the linear equation to experiments. Using a pendant drop tension meter, the surface tension at the equilibrium of ≈6 mm diameter size drops of the same MB shell was measured inside DW, PBS1x and PBS10x. The surface tension at the C3F8/solution interface was estimated by fitting the Young-Laplace equation from the recorded images. The frequency of the peak attenuation at different salinity levels was 13, 7.5 and 6.25 MHz in DW, PBS1x and PBS-10x, respectively. The attenuation peak increased by ≈140% with increasing ion density. MBs' estimated shell elasticity decreased by 64% between DW and PBS-1x and 36% between PBS-1x and PBS-10x. The drop surface tension reduced by 10.5% between DW and PBS-1x and by 5% between PBS-1x and PBS-10x, respectively. Reduction in the shell stiffness is consistent with the drop surface tension measurements. The shell viscosity was reduced by ≈40% between DW and PBS-1x and 42% between PBS-1x and PBS-10x. The reduction in the fitted stiffness and viscosity is possibly due to the formation of a densely charged layer around the shell, further reducing the effective surface tension on the MBs. The changes in the resonance frequency and estimated shell parameters were significant and may potentially help to better understand and explain bubble behavior in applications.

Modulating Nonlinear Acoustic Response of Phospholipid-Coated Microbubbles with pH for Ultrasound Imaging

Activatable microbubble contrast agents for contrast-enhanced ultrasound have a potential role for measuring physiologic and pathologic states in deep tissues, including tumor acidosis. In this study, we describe a novel observation of increased harmonic oscillation of phosphatidylcholine microbubbles (PC-MBs) in response to lower ambient pH using a clinical ultrasound scanner. MB echogenicity and nonlinear echoes were monitored at neutral and acidic pH using B-mode and Cadence contrast pulse sequencing (CPS), a harmonic imaging technique at 7.0 and 1.5 MHz. A 3-fold increase in harmonic signal intensity was observed when the pH of PC-MB suspensions was decreased from 7.4 to 5.5 to mimic normal and pathophysiological levels that can be encountered in vivo. This pH-mediated activation is tunable based on the chemical structure and length of phospholipids composing the MB shell. It is also reliant on the presence of phosphate groups, as the use of lipids without phosphate instead of phospholipids completely abrogated this phenomenon. The increased harmonic signal likely is the result of increased MB oscillation caused by a decrease of the interfacial tension induced at a lower pH, altering the lipid conformation. While relative signal changes are interpreted clinically as mostly related to blood flow, pH effects could be significant contributors, particularly when imaging tumors. While our observation can be used clinically, it requires further research to isolate the effect of pH from other variables. These findings could pave the way toward for the development of new smart ultrasound contrast agents that expand the clinical utility of contrast-enhanced ultrasound.

Numerical investigation of the effect of bubble properties on the linear resonance frequency shift due to inter-bubble interactions in ultrasonically excited lipid coated microbubbles

Ultrasonically excited microbubbles (MBs) have numerous applications in various fields, such as drug delivery, and imaging. Ultrasonically excited MBs are known to be nonlinear oscillators that generate secondary acoustic emissions in the media when excited by a primary ultrasound wave. The propagation of acoustic waves in the liquid is limited to the speed of sound, resulting in each MB receiving the primary and secondary waves at different times depending on their distance from the ultrasound source and the distance between MBs. These delays are referred to as primary and secondary delays, respectively. A previous study demonstrated that the inclusion of secondary delays in a model describing the interactions between MBs exposed to ultrasound results in an increase in the linear resonance frequency of MBs as they approach each other. This work investigates the impact of various MB properties on the change in linear resonance frequency resulting from changes in inter-bubble distances. The effects of shell properties, including the initial surface tension, surface dilatational viscosity of the shell monolayer, elastic compression modulus of the shell, and the initial radius of the MBs, are examined. MB size is a significant factor influencing the rate of linear resonance frequency increase with increasing concentration. Moreover, it is found that the shell properties of MBs play a negligible role in the rate of change in linear resonance frequency of MBs as the inter-bubble distances change.The findings of this study have implications for various applications of MBs in the biomedical field. By understanding the impact of inter-bubble distances and shell properties on the linear resonance frequency of MBs, the utilization of MBs in applications reliant on their resonant behavior can be optimized.

Investigation of interaction effects on dual-frequency driven cavitation dynamics in a two-bubble system

The cavitation dynamics of a two-bubble system in viscoelastic media excited by dual-frequency ultrasound is studied numerically with a focus on the effects of inter-bubble interactions. Compared to the isolated bubble cases, the enhancement or suppression effects can be exerted on the amplitude and nonlinearity of the bubble oscillations to different degrees. Moreover, the interaction effects are found to be highly sensitive to multiple paramount parameters related to the two-bubble system, the dual-frequency ultrasound and the medium viscoelasticity. Specifically, the larger bubble of a two-bubble system shows a stronger effect on the smaller one, and this effect becomes more pronounced when the larger bubble undergoes harmonic and/or subharmonic resonances as well as the two bubbles get closer (e.g., d0 < 100 μm). For the influences of the dual-frequency excitation, the results show that the bubbles can achieve enhanced harmonic and/or subharmonic oscillations as the frequency combinations with small frequency differences (e.g., Δf < 0.2 MHz) close to the corresponding resonance frequencies of bubbles, and the interaction effects are consequently intensified. Similarly, the bubble oscillations and the interaction effects can also be enhanced as the acoustic pressure amplitude of each frequency component is equal and the pressure amplitude pA increases. Above a pressure threshold (pA = 215 kPa), a larger bubble undergoes period 2 (P2) oscillations, which can force a smaller bubble to change its oscillation pattern from period 1 (P1) into P2 oscillations. In addition, it is found that the medium viscosity dampens the bubble oscillations while the medium elasticity affects the bubble resonances, accordingly exhibiting stronger interaction effects at smaller viscosities (e.g., μ < 4 mPa·s) or certain elasticities (approximately G = 70-120 kPa, G = 160-200 kPa and G = 640-780 kPa) at which the bubble resonances occur. The study can contribute to a better understanding of the complex dynamic behaviors of interacting cavitation bubbles in viscoelastic tissues for high efficient cavitation-mediated biomedical applications using dual-frequency ultrasound.

Bubble pulsation characteristics in multi-bubble systems affected by bubble size polydispersity and spatial structure

This study seeks to explore the bubble pulsation characteristics in multi-bubble environment with a special focus on the influences of the size polydispersity and the two-dimensional structure of bubbles. Three representative configurations of three interacting bubbles are formed by setting the initial radii of cavitation bubbles and inter-bubble distances appropriately, then the pulsation characteristics of a small bubble are investigated and compared by the bifurcation analysis. The results illustrate that the bubble size polydispersity and two-dimensional structure would greatly affect the bubble pulsations (i.e., the amplitude and nonlinearity of pulsations). Furthermore, the effects of two-dimensional structure are strong at a small inter-bubble distance of the large and small bubbles while the bubble size polydispersity always significantly affects the bubble pulsations for all cases. Moreover, the influences of both bubble size polydispersity and two-dimensional structure can be enhanced as the acoustic pressure increases, which can also become stronger when the large bubble is located at the same side as the small bubble and the initial radius of large bubble increases. Additionally, the effects would also be increased when the tissue viscoelasticity varies within a certain range. The present findings shed new light on the dynamics of multiple polydisperse microbubbles in viscoelastic tissues, potentially contributing to an optimization of their applications with ultrasound excitation.

Probing the pressure dependence of sound speed and attenuation in bubbly media: Experimental observations, a theoretical model and numerical calculations

The problem of attenuation and sound speed of bubbly media has remained partially unsolved. Comprehensive data regarding pressure-dependent changes of the attenuation and sound speed of a bubbly medium are not available. Our theoretical understanding of the problem is limited to linear or semi-linear theoretical models, which are not accurate in the regime of large amplitude bubble oscillations. Here, by controlling the size of the lipid coated bubbles (mean diameter of ≈5.4μm), we report the first time observation and characterization of the simultaneous pressure dependence of sound speed and attenuation in bubbly water below, at and above microbubbles resonance (frequency range between 1-3 MHz). With increasing acoustic pressure (between 12.5-100 kPa), the frequency of the peak attenuation and sound speed decreases while maximum and minimum amplitudes of the sound speed increase. We propose a nonlinear model for the estimation of the pressure dependent sound speed and attenuation with good agreement with the experiments. The model calculations are validated by comparing with the linear and semi-linear models predictions. One of the major challenges of the previously developed models is the significant overestimation of the attenuation at the bubble resonance at higher void fractions (e.g. 0.005). We addressed this problem by incorporating bubble-bubble interactions and comparing the results to experiments. Influence of the bubble-bubble interactions increases with increasing pressure. Within the examined exposure parameters, we numerically show that, even for low void fractions (e.g. 5.1×10-6) with increasing pressure the sound speed may become 4 times higher than the sound speed in the non-bubbly medium.

Theoretical prediction of the scattering of spherical bubble clusters under ultrasonic excitation

Due to the nonlinear vibration of ultrasound contrast agent bubbles, a nonlinear scattered sound field will be generated when bubbles are driven by ultrasound. A bubble cluster consists of numerous bubbles gathering in a spherical space. It has been noted that the forward scattering of a bubble cluster is larger than its backscattering, and some studies have experimentally found the angular dependence of a bubble cluster's scattering signal. In this paper, a theory is proposed to explain the difference of acoustic scattering at different directions of a bubble cluster when it is driven by ultrasound, and predicts the angular distribution of scattered acoustic pressure under different parameters. The theory is proved to be correct under circumstances of small clusters and weak interactions by comparing theoretical results with numerical simulations. This theory not only sheds light on the physics of bubble cluster scattering, but also may contribute to the improvement of ultrasound imaging technology, including ultrasonic harmonic imaging and contrast-enhanced ultrasonography.

Resonance behaviors of encapsulated microbubbles oscillating nonlinearly with ultrasonic excitation

The resonance behaviors of a few lipid-coated microbubbles acoustically activated in viscoelastic media were comprehensively examined via radius response analysis. The size polydispersity and random spatial distribution of the interacting microbubbles, the rheological properties of the lipid shell and the viscoelasticity of the surrounding medium were considered simultaneously. The obtained radius response curves present a successive occurrence of linear resonances, nonlinear harmonic and sub-harmonic resonances with the acoustic pressure increasing. The microbubble resonance is radius-, pressure- and frequency-dependent. Specifically, the maximum bubble expansion ratio at the main resonance peak increases but the resonant radius decreases as the ultrasound pressure increases, while both of them decrease with the ultrasound frequency increasing. Moreover, compared to an isolated microbubble case, it is found that large microbubbles in close proximity prominently suppress the resonant oscillations while slightly increase the resonant radii for both harmonic and subharmonic resonances, even leading to the disappearance of the subharmonic resonance with the influences increasing to a certain degree. In addition, the results also suggest that both the encapsulating shell and surrounding medium can substantially dampen the harmonic and subharmonic resonances while increase the resonant radii, which seem to be affected by the medium viscoelasticity to a greater degree rather than the shell properties. This work offers valuable insights into the resonance behaviors of microbubbles oscillating in viscoelastic biological media, greatly contributing to further optimizing their biomedical applications.

Concentration-Dependent Viscoelasticity of Poloxamer-Shelled Microbubbles

The oscillation of shelled microbubbles during exposure to ultrasound is influenced by the mechanical properties of the shell components. The oscillation behavior of bubbles coated with various phospholipids and other amphiphiles has been studied. However, there have been few investigations of how the adsorption conditions of the shell molecules relate to the viscoelastic properties of the shell and influence the oscillation behavior of the bubbles. In the present study, we investigated the oscillation characteristics of microbubbles coated with a poloxamer surfactant, that is, Pluronic F-68, at several concentrations after the adsorption kinetics of the surfactant at the gas-water interface had reached equilibrium. The dilatational viscoelasticity of the shell during exposure to ultrasound was analyzed in the frequency domain from the attenuation characteristics of the acoustic pulses propagated in the bubble suspension. At Pluronic F-68 concentrations lower than 2.0 × 10^-2 mol L^-1, the attenuation characteristics typically exhibited a sharp peak. At concentrations higher than 2.0 × 10^-2 mol L^-1, the peak flattened. The dilatational elasticity and viscosity of the shell were estimated by fitting the theoretical model to the experimental values, which revealed that both the elasticity and viscosity increased markedly at approximately 2.0 × 10^-2 mol L^-1. This suggests that the adsorption properties of Pluronic F-68 strongly affect the oscillation characteristics of microbubbles of a size suitable for medical ultrasound diagnostics.

Numerical investigation of the translational motion of bubbles: The comparison of capabilities of the time-resolved and the time-averaged methods

In the present study, the accuracies of two different numerical approaches used to model the translational motion of acoustic cavitational bubble in a standing acoustic field are compared. The less accurate but less computational demanding approach is to decouple the equation of translational motion from the radial oscillation, and solve it by calculating the time-averaged forces exerted on the bubble for one acoustic cycle. The second approach is to solve the coupled ordinary differential equations directly, which provides more accurate results with higher computational effort. The investigations are carried out in the parameter space of the driving frequency, pressure amplitude and equilibrium radius. Results showed that both models are capable to reveal stable equilibrium positions; however, in the case of oscillatory solutions, the difference in terms of translational frequency may be more than three fold, and the amplitude of translational motion obtained by the time-averaged method is roughly 1.5 times higher compared to the time-resolved solution at particular sets of parameters. This observation implies that where the transient behaviour is important, the time-resolved approach is the proper choice for reliable results.

Influence of rigid wall on the nonlinear pulsation of nearby bubble

This paper mainly focuses on the nonlinear pulsation of a bubble near the rigid wall. Dynamics of near-wall bubble and free bubble are discussed and compared in details. Investigation reveals as the driving acoustic pressure amplitude increases, nonlinear pulsation of bubble becomes intense gradually. Besides, decreasing the viscosity of host liquid is advantageous for the nonlinear pulsation of bubble. Bifurcation diagrams of bubble radius show acoustic reflection of the rigid wall makes the initial bifurcation appear at low driving acoustic amplitude and on bubble with small ambient radius, and makes the bifurcation still exist for bubble in high-viscosity liquids. That indicates the rigid wall will produce enhancement on the nonlinearity of nearby bubble. As the bubble approaches the wall, the enhancement becomes strong. Moreover, research on the influence of driving frequency shows the rigid wall makes the frequency band corresponding to chaos around the resonant frequency of free bubble shift downward.

Enhancing cavitation dynamics and its mechanical effects with dual-frequency ultrasound

Objective. Acoustic cavitation and its mechanical effects (e.g. stress and strain) play a primary role in ultrasound applications. Introducing encapsulated microbubbles as cavitation nuclei and utilizing dual-frequency ultrasound excitation are highly effective approaches to reduce cavitation thresholds and enhance cavitation effects. However, the cavitation dynamics of encapsulated microbubbles and the resultant stress/strain in viscoelastic tissues under dual-frequency excitation are poorly understood, especially for the enhancement effects caused by a dual-frequency approach. The goal of this study was to numerically investigate the dynamics of a lipid-coated microbubble and the spatiotemporal distributions of the stress and strain under dual-frequency excitation. Approach. The Gilmore–Zener bubble model was coupled with a shell model for the nonlinear changes of both shell elasticity and viscosity to accurately simulate the cavitation dynamics of lipid-coated microbubbles in viscoelastic tissues. Then, the spatiotemporal evolutions of the cavitation-induced stress and strain in the surrounding tissues were characterized quantitatively. Finally, the influences of some paramount parameters were examined to optimize the outcomes. Main results. We demonstrated that the cavitation dynamics and associated stress/strain were prominently enhanced by a dual-frequency excitation, highlighting positive correlations between the maximum bubble expansion and the maximum stress/strain. Moreover, the results showed that the dual-frequency ultrasound with smaller differences in its frequencies and pressure amplitudes could enhance the bubble oscillations and stress/strain more efficiently, whereas the phase difference manifested small influences under these conditions. Additionally, the dual-frequency approach seemed to show a stronger enhancement effect with the shell/tissue viscoelasticity increasing to a certain extent. Significance. This study might contribute to optimizing the dual-frequency operation in terms of cavitation dynamics and its mechanical effects for high-efficient ultrasound applications.

Nonlinear dynamics and acoustic emissions of interacting cavitation bubbles in viscoelastic tissues

The cavitation-mediated bioeffects are primarily associated with the dynamic behaviors of bubbles in viscoelastic tissues, which involves complex interactions of cavitation bubbles with surrounding bubbles and tissues. The radial and translational motions, as well as the resultant acoustic emissions of two interacting cavitation bubbles in viscoelastic tissues were numerically investigated. Due to the bubble-bubble interactions, a remarkable suppression effect on the small bubble, whereas a slight enhancement effect on the large one were observed within the acoustic exposure parameters and the initial radii of the bubbles examined in this paper. Moreover, as the initial distance between bubbles increases, the strong suppression effect is reduced gradually and it could effectively enhance the nonlinear dynamics of bubbles, exactly as the bifurcation diagrams exhibit a similar mode of successive period doubling to chaos. Correspondingly, the resultant acoustic emissions present a progressive evolution of harmonics, subharmonics, ultraharmonics and broadband components in the frequency spectra. In addition, with the elasticity and/or viscosity of the surrounding medium increasing, both the nonlinear dynamics and translational motions of bubbles were reduced prominently. This study provides a comprehensive insight into the nonlinear behaviors and acoustic emissions of two interacting cavitation bubbles in viscoelastic media, it may contribute to optimizing and monitoring the cavitation-mediated biomedical applications.

Cavitation Dynamics and Inertial Cavitation Threshold of Lipid Coated Microbubbles in Viscoelastic Media with Bubble-Bubble Interactions

Encapsulated microbubbles combined with ultrasound have been widely utilized in various biomedical applications; however, the bubble dynamics in viscoelastic medium have not been completely understood. It involves complex interactions of coated microbubbles with ultrasound, nearby microbubbles and surrounding medium. Here, a comprehensive model capable of simulating the complex bubble dynamics was developed via taking the nonlinear viscoelastic behaviors of the shells, the bubble–bubble interactions and the viscoelasticity of the surrounding medium into account simultaneously. For two interacting lipid-coated bubbles with different initial radii in viscoelastic media, it exemplified that the encapsulating shell, the inter-bubble interactions and the medium viscoelasticity would noticeably suppress bubble oscillations. The inter-bubble interactions exerted a much stronger suppressing effect on the small bubble within the parameters examined in this paper, which might result from a larger radiated pressure acting on the small bubble due to the inter-bubble interactions. The lipid shells make the microbubbles exhibit two typical asymmetric dynamic behaviors (i.e., compression or expansion dominated oscillations), which are determined by the initial surface tension of the bubbles. Accordingly, the inertial cavitation threshold decreases as the initial surface tension increases, but increases as the shell elasticity and viscosity increases. Moreover, with the distance between bubbles decreasing and/or the initial radius of the large bubble increasing, the oscillations of the small bubble decrease and the inertial cavitation threshold increases gradually due to the stronger suppression effects caused by the enhanced bubble–bubble interactions. Additionally, increasing the elasticity and/or viscosity of the surrounding medium would also dampen bubble oscillations and result in a significant increase in the inertial cavitation threshold. This study may contribute to both encapsulated microbubble-associated ultrasound diagnostic and emerging therapeutic applications.

GPU accelerated numerical investigation of the spherical stability of an acoustic cavitation bubble excited by dual-frequency

The spherical stability of an acoustic cavitation bubble under dual-frequency excitation is investigated numerically. The radial dynamics is described by the Keller-Miksis equation, which is a second-order ordinary differential equation. The surface dynamics is modelled by a set of linear ordinary differential equation according to Hao and Prosperetti (1999), which takes into account the effect of vorticity by boundary layer approximation. Due to the large amount of investigated parameter combinations, the numerical computations were carried out on graphics processing units. The results showed that for bubble size between R_E=2μm and 4μm, the combination of a low and a high frequency, and the combination of two close but not equal frequencies are important to prevent the bubble losing its shape stability, while reaching the chemical threshold (R_max/R_E=3) (Kalmár et al., 2020). The phase shift between harmonic components of dual-frequency excitation has no effect on the shape stability.

Bursting Microbubbles: How Nanobubble Contrast Agents Can Enable the Future of Medical Ultrasound Molecular Imaging and Image-Guided Therapy

The field of medical ultrasound has undergone a significant evolution since the development of microbubbles as contrast agents. However, due to their size, microbubbles remain in the vasculature, and therefore have limited clinical applications. Building a better - and smaller - bubble can expand the applications of contrast-enhanced ultrasound by allowing bubbles to extravasate from blood vessels - creating new opportunities. In this review, we summarize recent research on the formulation and use of NBs as imaging agents and as therapeutic vehicles. We discuss the ongoing debates in the field and reluctance to accepting NBs as an acoustically active construct and a potentially impactful clinical tool that can help shape the future of medical ultrasound. We hope that the overview of key experimental and theoretical findings in the NB field presented in this paper provides a fundamental framework that will help clarify NB-ultrasound interactions and inspire engagement in the field.

Toward Precisely Controllable Acoustic Response of Shell-Stabilized Nanobubbles: High Yield and Narrow Dispersity

Understanding the pressure dependence of the nonlinear behavior of ultrasonically excited phospholipid-stabilized nanobubbles (NBs) is important for optimizing ultrasound exposure parameters for implementations of contrast enhanced ultrasound, critical to molecular imaging. The viscoelastic properties of the shell can be controlled by the introduction of membrane additives, such as propylene glycol as a membrane softener or glycerol as a membrane stiffener. We report on the production of high-yield NBs with narrow dispersity and different shell properties. Through precise control over size and shell structure, we show how these shell components interact with the phospholipid membrane, change their structure, affect their viscoelastic properties, and consequently change their acoustic response. A two-photon microscopy technique through a polarity-sensitive fluorescent dye, C-laurdan, was utilized to gain insights on the effect of membrane additives to the membrane structure. We report how the shell stiffness of NBs affects the pressure threshold (P_t) for the sudden amplification in the scattered acoustic signal from NBs. For narrow size NBs with 200 nm mean size, we find P_t to be between 123 and 245 kPa for the NBs with the most flexible membrane as assessed using C-Laurdan, 465-588 kPa for the NBs with intermediate stiffness, and 588-710 kPa for the NBs with stiff membranes. Numerical simulations of the NB dynamics are in good agreement with the experimental observations, confirming the dependence of acoustic response to shell properties, thereby substantiating further the development in engineering the shell of ultrasound contrast agents. The viscoelastic-dependent threshold behavior can be utilized for significantly and selectively enhancing the diagnostic and therapeutic ultrasound applications of potent narrow size NBs.