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.

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.

On the threshold of 1/2 order subharmonic emissions in the oscillations of ultrasonically excited bubbles

The pressure threshold for 1/2 order subharmonic (SH) emissions and period doubling during the oscillations of ultrasonically excited bubbles is thought to be minimum when the bubble is sonicated with twice its resonance frequency (f_r). This estimate is based on studies that simplified or neglected the effects of thermal damping. In this work, the nonlinear dynamics of ultrasonically excited bubbles is investigated accounting for the thermal dissipation. Results are visualized using bifurcation diagrams as a function of pressure. Here we show that, and depending on the gas, the pressure threshold for 1/2 order SHs can be minimum at a frequency between 0.5f_r≤f≤0.6f_r. In this frequency range, the generation of 1/2 order SHs are due to the occurrence of 5/2 order ultra-harmonic resonance. The stability of such oscillations is size dependent. For an air bubble immersed in water, only bubbles bigger than 1 μm in diameter are able to emit non-destructive SHs in these frequency ranges.

Nonlinear dynamics and bifurcation structure of ultrasonically excited lipid coated microbubbles

In many applications, microbubbles (MBs) are encapsulated by a lipid coating to increase their stability. However, the complex behavior of the lipid coating including buckling and rupture sophisticates the dynamics of the MBs and as a result the dynamics of the lipid coated MBs (LCMBs) are not well understood. Here, we investigate the nonlinear behavior of the LCMBs by analyzing their bifurcation structure as a function of acoustic pressure. We show that, the LC can enhance the generation of period 2 (P2), P3, higher order subharmonics (SH), superharmonics and chaos at very low excitation pressures (e.g. 1 kPa). For LCMBs sonicated by their SH resonance frequency and in line with experimental observations with increasing pressure, P2 oscillations exhibit three stages: generation at low acoustic pressures, disappearance and re-generation. Within non-destructive oscillation regimes and by pressure amplitude increase, LCMBs can also exhibit two saddle node (SN) bifurcations resulting in possible abrupt enhancement of the scattered pressure. The first SN resembles the pressure dependent resonance phenomenon in uncoated MBs and the second SN resembles the pressure dependent SH resonance. Depending on the initial surface tension of the LCMBs, the nonlinear behavior may also be suppressed for a wide range of excitation pressures.

Critical corrections to models of nonlinear power dissipation of ultrasonically excited bubbles

Current models for calculating nonlinear power dissipation during the oscillations of acoustically excited bubbles generate non-physical values for the radiation damping (D_rd) term for some frequency and pressure regions that include near resonance oscillations. Moreover, the ratio of the dissipated powers significantly deviate from the values that are calculated by the linear model at low amplitude oscillations (acoustic excitation pressure of P_A=1 kPa and expansion ratio of <≊1.01). In high amplitude oscillation regimes (P_a⩾20 kPa), the dissipated power due to D_rd deviates largely from the dissipated power as calculated by the widely accepted approach that uses the scattered power by the bubbles. We provide critical corrections to the present models. The validity of the results was examined in regimes of low amplitude oscillations and high amplitude oscillations. In the low amplitude regime, the ratio of the dissipated powers as calculated by the current and proposed model were compared with the linear model predictions. At higher amplitude oscillations, the dissipated power by radiation loss as calculated by the current and the proposed models were compared with the dissipated power calculated using the scattered power by the bubbles. We show that non-physical values are absent in the proposed model. Moreover, predictions of the proposed approach are identical to the predictions of the linear model and the dissipated power estimated using the scattered pressure by the bubble. We show that damping due to thermal effects, liquid viscosity and radiation heavily depend on the excitation pressure and that the linear model estimations are not valid even at pressures as low as 20 kPa.

Nonlinear power loss in the oscillations of coated and uncoated bubbles: Role of thermal, radiation and encapsulating shell damping at various excitation pressures

This study presents the fundamental equations governing the pressure dependent disipation mechanisms in the oscillations of coated bubbles. A simple generalized model (GM) for coated bubbles accounting for the effect of compressibility of the liquid is presented. The GM was then coupled with nonlinear ODEs that account for the thermal effects. Starting with mass and momentum conservation equations for a bubbly liquid and using the GM, nonlinear pressure dependent terms were derived for power dissipation due to thermal damping (Td), radiation damping (Rd) and dissipation due to the viscosity of liquid (Ld) and coating (Cd). The pressure dependence of the dissipation mechanisms of the coated bubble have been analyzed. The dissipated energies were solved for uncoated and coated 2-20 μm in bubbles over a frequency range of 0.25f_r-2.5f_r (f_r is the bubble resonance) and for various acoustic pressures (1 kPa-300 kPa). Thermal effects were examined for air and C3F8 gas cores. In the case of air bubbles, as pressure increases, the linear thermal model looses accuracy and accurate modeling requires inclusion of the full thermal model. However, for coated C3F8 bubbles of diameter 1-8 μm, which are typically used in medical ultrasound, thermal effects maybe neglected even at higher pressures. For uncoated bubbles, when pressure increases, the contributions of Rd grow faster and become the dominant damping mechanism for pressure dependent resonance frequencies (e.g. fundamental and super harmonic resonances). For coated bubbles, Cd is the strongest damping mechanism. As pressure increases, Rd contributes more to damping compared to Ld and Td. For coated bubbles, the often neglected compressibility of the liquid has a strong effect on the oscillations and should be incorporated in models. We show that the scattering to damping ratio (STDR), a measure of the effectiveness of the bubble as contrast agent, is pressure dependent and can be maximized for specific frequency ranges and pressures.

Collective nonlinear behavior of interacting polydisperse microbubble clusters

Acoustically excited microbubbles (MBs) have shown to exhibit rich dynamics, enabling them to be employed in various applications ranging from chemistry to medicine. Exploiting the full potential of MBs for applications requires a good understanding of their complex dynamics. Improved understanding of MB oscillations can lead to further enhancement in optimizing their efficacy in many applications and also invent new ones. Oscillating MBs have been shown to generate secondary pressure waves that modify the dynamics of the MBs in their proximity. A modified Keller-Miksis equation is used to account for inter-bubble interactions. The oscillatory dynamics of each MB within clusters was computed by numerically solving the resulting system of coupled nonlinear second order differential equations in potential fluid flow. Frequency response analysis and bifurcation diagrams were employed to track the dynamics of interacting MBs. We start with investigating the effect of inter-bubble interactions for cases of three and four MBs over a wide range of acoustic and geometric parameters. Emergent collective behavior was observed which are dominated by the dynamics of the largest MB within the cluster. The emergent dynamics of smaller MBs within clusters can be characterized by constructive and destructive inter-bubble interactions. In constructive interactions, the radial oscillations of smaller MBs matched those of the largest MB and their oscillations are amplified. In destructive interactions, the oscillations of smaller bubbles are suppressed so that their oscillations match those of the largest MB. Furthermore, a special case of constructive interactions is presented where dominant MB (largest) can force smaller MBs into period doubling and subharmonic oscillations. The collective behavior is further investigated in large MB cluster and it is shown that largest MBs, even in small numbers can force smaller ones into period doubling and subharmonic oscillations.

A simple method to analyze the super-harmonic and ultra-harmonic behavior of the acoustically excited bubble oscillator

The bubble oscillator is a highly nonlinear system, which makes it difficult to generate a comprehensive understanding of its oscillatory behavior. One method used to investigate such complex dynamical systems is the bifurcation analysis. Numerous investigations have employed the method of bifurcation diagrams to study the effect of different control parameters on the bubble behavior. These studies, however, focused mainly on investigating the subharmonic (SH) and chaotic oscillations of the bubbles. Super-harmonic (SuH) and ultra-harmonic (UH) bubble oscillations remain under-investigated. One reason is that the conventional method used for generating bifurcation diagrams cannot reliably identify features that are responsible for the identification of SuH and UH oscillations. Additionally, the conventional method cannot distinguish between the UHs and SHs. We introduce a simple procedure to address this shortcoming. In this method, the maxima of the bubble oscillatory response were selected and plotted alongside the traditional bifurcation points for the corresponding control parameter. Results show that depending on the control parameters the conventional method or the method of maxima may miss intricate details of the oscillations. In order to have a comprehensive knowledge on the rich dynamics of the system, the two methods should be employed side by side. Through plotting the two bifurcation structures in tandem, the oscillatory behavior of the bubble was analyzed with more detail, and stable SuH and UH bubble oscillations were investigated. Based on this new analysis, the conditions for the generation and amplification of UH and SuH regimes are discussed.