Resonance frequency of microbubbles: effect of viscosity

The transmitted frequency at which a gas bubble of millimeter or submillimeter size oscillates resonantly in a low-viscosity liquid is approximately equal to the undamped natural frequency (referred to as the Minnaert frequency if surface tension effects are disregarded). Based on a theoretical analysis of bubble oscillation, this paper shows that such an approximation cannot be validated for microbubbles used in contrast-enhanced ultrasound imaging. The contrast-agent microbubbles represent either encapsulated bubbles of size less than 10 microm or free (nonencapsulated) bubbles of submicron size. The resonance frequency of the microbubbles deviates significantly from the undamped natural frequency over the whole range of microbubble sizes due to the increased viscous damping coefficient. The difference between these two frequencies is shown to have a tremendous impact on the resonant backscatter by the microbubbles. In particular, the first and second harmonics of the backscattered signal from the microbubbles are characterized by their own resonance frequencies, equal to neither the microbubble resonance frequency nor the undamped natural frequency.     

Optimization of acoustic scattering from dual-frequency driven microbubbles at the difference frequency

The second harmonic radiation of acoustically driven bubbles is a useful discriminant for their presence in clinical ultrasound applications. It is useful because the scatter from a bubble at a frequency different from the driving can have a contrast-to-tissue ratio better than at the drive frequency. In this work a technique is developed to optimize the scattering from a microbubble at a frequency different from the driving. This is accomplished by adjusting the relative phase and amplitudes of the components of a dual-frequency incident ultrasound wave form. The investigation is focused primarily on the example of dual-mode driving at frequencies of 1 MHz and 3 MHz, with the scattering optimized at 2 MHz. Bubble radii of primary interest are 0.5 to 2 microm and driving amplitudes to 0.5 atm. Bubbles in this size range are sensitive to modulation of driving. It is shown that an optimal forcing scheme can increase the target response eightfold or more. This suggests new applications in imaging and in bubble detection.     

Resonance frequencies of lipid-shelled microbubbles in the regime of nonlinear oscillations

Knowledge of resonant frequencies of contrast microbubbles is important for the optimization of ultrasound contrast imaging and therapeutic techniques. To date, however, there are estimates of resonance frequencies of contrast microbubbles only for the regime of linear oscillation. The present paper proposes an approach for evaluating resonance frequencies of contrast agent microbubbles in the regime of nonlinear oscillation. The approach is based on the calculation of the time-averaged oscillation power of the radial bubble oscillation. The proposed procedure was verified for free bubbles in the frequency range 1-4 MHz and then applied to lipid-shelled microbubbles insonified with a single 20-cycle acoustic pulse at two values of the acoustic pressure amplitude, 100 kPa and 200 kPa, and at four frequencies: 1.5, 2.0, 2.5, and 3.0 MHz. It is shown that, as the acoustic pressure amplitude is increased, the resonance frequency of a lipid-shelled microbubble tends to decrease in comparison with its linear resonance frequency. Analysis of existing shell models reveals that models that treat the lipid shell as a linear viscoelastic solid appear may be challenged to provide the observed tendency in the behavior of the resonance frequency at increasing acoustic pressure. The conclusion is drawn that the further development of shell models could be improved by the consideration of nonlinear rheological laws.     

Numerical modeling of microbubble backscatter to optimize ultrasound particle image velocimetry imaging: initial studies

We have developed a promising non-invasive ultrasound-based method for performing particle image velocimetry (PIV) in vivo. This method, termed echo PIV, provides multi-component blood velocity data with good (2 ms) temporal resolution. The method takes advantage of the non-linear ultrasound backscatter characteristics of small gas-filled microbubbles (ultrasound contrast) that are seeded into the blood stream. In this study, we use a numerical model to explore potential areas to focus future work in echo PIV.

Ultrasound backscatter from encapsulated microbubbles was modeled using a modified Rayleigh–Plesset equation (Church model, 1995), taking into account the protein/lipid shell layer as a thick, mass-conserving incompressible fluid surrounded by incompressible blood-like fluid. The equation of motion was solved numerically to characterize the fundamental and second harmonic components of the backscattered pressure. Results show a significant advantage in using the second harmonic component for echo PIV, especially for small bubble sizes less than 3 μm in diameter at 2.2 MHz frequency. The effect of the shell thickness ranging from 10 to 500 nm on the vibration amplitude of the bubble was examined and it is shown that the presence of the shell requires mechanical index (MI)>0.2 of incident pressure amplitude to improve bubble detectability. Analysis of the effect of pulse length shows a tradeoff between axial resolution (short pulse length) and bubble detectability (longer pulse length) will most likely be required. The effect of varying MI between 0.1 and 0.6 was also studied at a center frequency of 2.2 MHz and the results indicate that the resonance of the second harmonic is maximized for bubbles with diameter of approximately 2.75 μm. Bubble non-linearities at MI>0.2 induced a resonant frequency shift away from the integer multiple of the incident frequency in the second harmonic backscatter. For a given bubble size, there is a combination of optimal incident frequency and mechanical index range that maximizes the ratio of the second harmonic compared to the fundamental. This resonant frequency decreases with increasing bubble radius. Further, a narrow bandwidth pulse is shown to increase signal strength. Both of these effects may cause conflict with factors governing spatial resolution. Optimization of the incident frequency, microbubble size and mechanical index to enhance bubble detectability will depend on the particular clinical application. These theoretical predictions provide further understanding of the physics behind our echo PIV technique, and should be useful for guiding the design of echo PIV systems.     

High frequency attenuation measurements of lipid encapsulated contrast agents

A number of recent studies have indicated the potential of ultrasound contrast agent imaging at high ultrasound frequencies. However, the acoustic properties of microbubbles at frequencies above 10 MHz remain poorly understood at present. In this study we characterize the high frequency attenuation properties of (1) BR14, (2) BR14 that has been mechanically filtered (1 and 2 microm pore sizes) to exclude larger bubbles, and (3) the micron to submicron agent BG2423. A narrowband pulse-echo substitution method is employed with a series of four transducers covering the frequency range from 2 to 50 MHz. For BR14, attenuation decreases rapidly from 2 to 10 MHz and then more gradually from 10 to 50 MHz. For 2 microm filtration, the attenuation peaks between 10 and 15 MHz. For 1 microm filtration, attenuation continues to rise until 50 MHz. The agent BG2423 exhibits a diffuse attenuation peak in the range of 15-25 MHz and remains high until 50 MHz. These results demonstrate a strong influence of bubble size on high frequency attenuation curves, with bubble diameters of 1-2 microm and below having more pronounced acoustic activity at frequencies above 10 MHz.     

Forced linear oscillations of microbubbles in blood capillaries

A theoretical investigation of the forced linear oscillations of a gas microbubble in a blood capillary, whose radius is comparable in size to the bubble radius is presented. The natural frequency of oscillation, the thermal and viscous damping coefficients, the amplitude resonance, the energy resonance, as well as the average energy absorbed by the system, bubble plus vessel, have been computed for different kinds of gas microbubbles, containing air, octafluropropane, and perflurobutane as a function of the bubble radius and applied frequency. It has been found that the bubble behavior is isothermal at low frequencies and for small bubbles and between isothermal and adiabatic for larger bubbles and higher frequencies, with the viscous damping dominating over the thermal damping. Furthermore, the width of the energy resonance is strongly dependent on the bubble size and the natural frequency of oscillation is affected by the presence of the vessel wall and position of the bubble in the vessel. Therefore, the presence of the blood vessel affects the way in which the bubble absorbs energy from the ultrasonic field. The motivation of this study lies in the possibility of using gas microbubbles as an aid to therapeutic focused ultrasound treatments.     

双泡模型共振频率的超声空化动力学研究

该研究旨在研究双泡模型的自然共振频率对超声空化的影响,通过理论计算研究了自然共振频率的影响因素,以及单频超声和双频超声与自然共振频率的关系。研究结果表明：气泡初始半径是影响自然共振频率的主要因素;低频驱动下的非线性波动程度会比高频的更加剧烈,当驱动频率等于气泡自然共振频率时,超声空化的效果更好;双频超声取气泡自然共振频率时超声空化效果远远优于单频超声驱动。该研究在超声医学和理解超声空化特性方面有着重要的意义。     

Effect of an elastic wall on the dynamics of an encapsulated microbubble: A simulation study

The purpose of the present simulation study is to reveal how confining surfaces with different mechanical properties affect the acoustic response of a contrast agent microbubble. To this end, numerical simulations are carried out for three types of walls: a plastic (OptiCell) wall, an aluminium wall, and a biological tissue. For each wall, the behaviour of contrast microbubbles of three sizes is investigated. The spectral characteristics of the scattered pressure produced by the microbubbles are compared for two cases: the bubble oscillates far away from the wall and the same bubble oscillates in the immediate vicinity of the wall. The results of the simulations allow one to make the following main conclusions. The effect of the OptiCell wall on the acoustic bubble response is stronger than that of the aluminium and tissue walls. Changes in the bubble response near the wall are stronger when bubbles are excited above their resonance frequency. Considering changes in the fundamental and the 2nd harmonic with respect to the peak values of these components at different bubble radii, it is found that the changes are stronger for smaller bubbles and that the changes in the 2nd harmonic are stronger than those in the fundamental. These results allow one to gain an insight into conditions under which the effect of an elastic wall on the acoustic response of a contrast agent microbubble is easier to be detected.     

Ultrasonic bubbles in medicine: influence of the shell

Ultrasound contrast agents consist of microscopically small bubbles encapsulated by an elastic shell. These microbubbles oscillate upon ultrasound insonification, and demonstrate highly nonlinear behavior, ameliorating their detectability. (Potential) medical applications involving the ultrasonic disruption of contrast agent microbubble shells include release-burst imaging, localized drug delivery, and noninvasive blood pressure measurement. To develop and enhance these techniques, predicting the cracking behavior of ultrasound-insonified encapsulated microbubbles has been of importance. In this paper, we explore microbubble behavior in an ultrasound field, with special attention to the influence of the bubble shell. A bubble in a sound field can be considered a forced damped harmonic oscillator. For encapsulated microbubbles, the presence of a shell has to be taken into account. In models, an extra damping parameter and a shell stiffness parameter have been included, assuming that Hooke's Law holds for the bubble shell. At high acoustic amplitudes, disruptive phenomena have been observed, such as microbubble fragmentation and ultrasonic cracking. We analyzed the occurrence of ultrasound contrast agent fragmentation, by simulating the oscillating behavior of encapsulated microbubbles with various sizes in a harmonic acoustic field. Fragmentation occurs exclusively during the collapse phase and occurs if the kinetic energy of the collapsing microbubble is greater than the instantaneous bubble surface energy, provided that surface instabilities have grown big enough to allow for break-up. From our simulations it follows that the Blake critical radius is not a good approximation for a fragmentation threshold. We demonstrated how the phase angle differences between a damped radially oscillating bubble and an incident sound field depend on shell parameters.     

Visualization and optimization of cavitation activity at a solid surface in high frequency ultrasound fields

Despite the increasing use of high frequency ultrasound in heterogeneous reactions, knowledge about the spatial distribution of cavitation bubbles at the irradiated solid surface is still lacking. This gap hinders controllable surface sonoreactions. Here we present an optimization study of the cavitation bubble distribution at a solid sample using sonoluminescence and sonochemiluminescence imaging. The experiments were performed at three ultrasound frequencies, namely 580, 860 and 1142 kHz. We found that position and orientation of the sample to the transducer, as well as its material properties influence the distribution of active cavitation bubbles at the sample surface in the reactor. The reason is a significant modification of the acoustic field due to reflections and absorption of the ultrasonic wave by the solid. This is retraced by numerical simulations employing the Finite Element Method, yielding reasonable agreement of luminescent zones and high acoustic pressure amplitudes in 2D simulations. A homogeneous coverage of the test sample surface with cavitation is finally reached at nearly vertical inclination with respect to the incident wave.     

In-situ synchrotron X-ray imaging of ultrasound (US)-generated bubbles: Influence of US frequency on microbubble cavitation for membrane fouling remediation

Gaining an in-depth understanding of the characteristics and dynamics of ultrasound (US)--generated bubbles is crucial to effectively remediate membrane fouling. The goal of present study is to conduct in-situ visualization of US-generated microbubbles in water to examine the influence of US frequency on the dynamics of microbubbles. This study utilized synchrotron in-line phase contrast imaging (In-line PCI) available at the biomedical imaging and therapy (BMIT) beamlines at the Canadian Light Source (CLS) to enhance the contrast of liquid/air interfaces at different US frequencies of 20, 28 and 40 KHz at 60 Watts. A high-speed camera was used to capture 2,000 frames per second of the bubble cavitation generated in water under the ultrasound influence. Key parameters at the polychromatic beamlines were optimized to maximize the phase contrast of gas/liquid of the microbubbles with a minimum size of 5.5 µm. ImageJ software was used to analyze the bubble characteristics and their behavior under the US exposure including the microbubble number, size, and fraction of the total area occupied by the bubbles at each US frequency. Furthermore, the bubble characteristics over the US exposure time and at different distances from the transducer were studied. The qualitative and quantitative data analyses showed that the microbubble number or size did not change over time; however, it was observed that most bubbles were created at the middle of the frames and close to the US field. The number of bubbles created under the US exposure increased with the frequency from 20 kHz to 40 kHz (about 4.6 times). However, larger bubbles were generated at 20 kHz such that the average bubble radius at 20 kHz was about 6.8 times of that at 40 kHz. Microbubble movement/traveling through water was monitored, and it was observed that the bubble velocity increased as the frequency was increased from 20 kHz to 40 kHz. The small bubbles moved faster, and the majority of them traveled upward towards the US transducer location. The growth pattern (a correlation between the mean growth ratio and the exposure time) of bubbles at 20 kHz and 60 W was obtained by tracking the oscillation of 22 representative microbubbles over the 700 ms of imaging. The mean growth ratio model was also obtained.     

A numerical investigation of the resonance of gas-filled microbubbles: resonance dependence on acoustic pressure amplitude

The general Keller-Herring equation for free gas bubbles is augmented by specific terms to describe the elasticity, viscosity and thickness of the encapsulating shell in ultrasound contrast agent microbubbles. A numerical investigation that analyses the acoustic backscatter from bubbles is employed to identify resonance frequencies that can be compared, for increasing driving pressure amplitude, with linear approximations obtained via analytical considerations. Calculations for bubbles of the size employed in diagnostic ultrasound, between 2 and 6 mum diameter, that are immersed in water and blood and exposed to monochromatic insonation, causing the bubbles to undergo stable cavitation, reveal that the resonance frequency diverges from the linear approximation as the pressure amplitude is increased. The shift in resonance, to lower frequency values, is found to be more pronounced for larger bubbles with the calculated value differing by up to 40% from the linear approximation. The results of this simulation might be potentially useful in preparation of formulations of ultrasound contrast agents with the specifically desired features, such as for instance resonance frequency.     

A numerical investigation of the resonance of gas-filled microbubbles: resonance dependence on acoustic pressure amplitude

The general Keller–Herring equation for free gas bubbles is augmented by specific terms to describe the elasticity, viscosity and thickness of the encapsulating shell in ultrasound contrast agent microbubbles. A numerical investigation that analyses the acoustic backscatter from bubbles is employed to identify resonance frequencies that can be compared, for increasing driving pressure amplitude, with linear approximations obtained via analytical considerations. Calculations for bubbles of the size employed in diagnostic ultrasound, between 2 and 6 μm diameter, that are immersed in water and blood and exposed to monochromatic insonation, causing the bubbles to undergo stable cavitation, reveal that the resonance frequency diverges from the linear approximation as the pressure amplitude is increased. The shift in resonance, to lower frequency values, is found to be more pronounced for larger bubbles with the calculated value differing by up to 40% from the linear approximation. The results of this simulation might be potentially useful in preparation of formulations of ultrasound contrast agents with the specifically desired features, such as for instance resonance frequency.     

Influence of ultrasonic frequency on multibubble sonoluminescence

Computer simulations of bubble oscillations are performed under conditions of multibubble sonoluminescence (MBSL) in water for various ultrasonic frequencies. The range of the ambient bubble radius for sonoluminescing bubbles narrows as the ultrasonic frequency increases; at 20 kHz it is 0.1-100 microm while at 1 MHz it is 0.1-3 microm. At 1 MHz, any sonoluminescing bubble disintegrates into a mass of smaller bubbles in a few or a few tens of acoustic cycles, while at 20 kHz and 140 kHz some sonoluminescing bubbles are shape stable. The mechanism of the light emission also depends on the ultrasonic frequency. As the ultrasonic frequency increases, the amount of water vapor trapped inside bubbles at the collapse decreases. As a result, MBSL originates mainly in plasma emissions at 1 MHz while it originates in chemiluminescence of OH radicals and plasma emissions at 20 kHz.     

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.     

超声波作用下泡群的共振声响应

从球状泡群气泡动力学方程出发, 考虑泡群间次级声辐射的影响, 得到了声场中两泡群共同存在时气泡振动的动力学方程, 并以此为基础探讨声波驱动下双泡群振动系统的共振响应特征. 由于泡群间气泡间的相互作用, 系统存在低频共振和高频共振现象, 两不同共振频率的数值与泡群内气泡的本征频率相关. 泡群内气泡的本征频率又受到初始半径、泡群大小和泡群内气泡数量的影响. 气泡自由振动和驱动声波的耦合激起泡群内气泡的受迫振动, 气泡初始半径、气泡数密度和驱动声波频率等都会影响泡群内气泡的振动幅值和初相位. 关键词： 气泡群 共振 声响应 超声空化     

弹性微管内气泡的非线性受迫振动

将弹性管壁视为膜弹性结构, 探索在外部声场作用下弹性微管内液柱-气泡-管壁构成耦合振动系统的非线性特征. 利用逐级近似法对系统非线性共振频率、基频和三倍频振动幅值响应、 分频激励共振机理等进行了理论分析. 基频和三倍频振动的幅-频响应数值结果表明: 气泡的轴向共振和管壁共振不能同时出现; 两垂直方向的振动均表现出幅值响应多值性, 进而可能引起系统的不稳定声响应; 三倍频振动在低频区的声响应强于高频区. 关键词： 弹性微管 受迫振动 非线性振动 气泡声响应     

Influence of frequency sweep on sonochemiluminescence and sonoluminescence

Bubbles generated by acoustic cavitation may be efficient in light production by direct emission (sonoluminescence) or indirect emission (sonochemiluminescence) depending on operating parameters such as acoustic pressure and surface tension. These conditions are quite difficult to reach at very high frequencies, even by concentrating the acoustic power at a given location via focusing the acoustic field thanks to the transducer shape (High Intensity Focused Ultrasound). The current work aims at probing the cavitation bubble behaviour under short frequency sweeps by monitoring sonochemiluminescence and sonoluminescence activities. When the frequency was swept in reverse (negative sweep), an enhancement in the SCL, relative to the SCL observed under a single frequency irradiation, was observed. Conversely, a positive frequency sweep resulted in the quenching of SCL intensity. The degree of SCL enhancement and quenching was also dependent on the rate at which the frequency was being swept and on the change in the size of cavitation bubbles. The size of cavitation bubbles varied with varying starting sweep frequency (3.4, 3.6 and 4.2 MHz), affecting both SCL and sonoluminescence (SL) emissions. The addition of a surfactant (sodium dodecyl sulphate) affected the observed results, possibly due to its influence on coalescence between cavitation bubbles. The results suggest that the enhancement and quenching are related to the response of bubbles generated by the starting frequency to the direction of the frequency sweep and the influence of the sweep rate on growth and coalescence of bubbles, which affected the population of the active bubbles.     

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.     

Difference-frequency ultrasound generation from microbubbles under dual-frequency excitation

The difference-frequency (DF) ultrasound generated by using parametric effect promises to improve detection depth owing to its low attenuation, which is beneficial for deep tissue imaging. With ultrasound contrast agents infusion, the harmonic components scattered from the microbubbles, including DF, can be generated due to the nonlinear vibration. A theoretical study on the DF generation from microbubbles under the dual-frequency excitation is proposed in formula based on the solution of the RPNNP equation. The optimisation of the DF generation is discussed associated with the applied acoustic pressure, frequency, and the microbubble size. Experiments are performed to validate the theoretical predictions by using a dual-frequency signal to excite microbubbles. Both the numerical and experimental results demonstrate that the optimised DF ultrasound can be achieved as the difference frequency is close to the resonance frequency of the microbubble and improve the contrast-to-tissue ratio in imaging.