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.     

超声微泡低频动力学特性及其生物效应

理论上利用有耗散函数的Lagrange方程,建立了有壳微泡的R(t)运动方程,开展了自由空间中有壳微泡动力学特性的研究,表明微泡内外半径增量随声压的增大、超声频率的降低、初始内径的增大及壳厚的减薄而迅速增大。实验上,利用Mie散射技术在80°散射角和前向Mie散射检测新技术实验测量了微泡R(t)曲线;利用体视显微镜,实时观察了超声微泡对动物活体微血管损伤,开展了超声微泡生物效应的动物和细胞试验研究。结果表明:(1)超声作用下,微泡引起肿瘤中微血管壁周期性膨胀收缩而发生管壁破裂,形成血栓和微血管栓塞,抑制了肿瘤生长;(2)超声联合微泡可以破坏微血管内皮生长因子(VEGF)和肝癌细胞,可以减少肿瘤血管和癌细胞再生,因此,低频超声联合微泡技术是一种值得探索抑制肿瘤生长的新技术。     

调频超声脉冲驱动微气泡运动偏移的研究

提出调频超声辐射力技术驱动微泡群,以加强微泡的吸附效率.基于改进的RP方程及粒子轨迹方程研究了微泡群整体的运动位移与调频信号的中心频率、调频范围、信号声压,以及微泡半径分布关系.研究结果表明调频信号在驱动半径具有宽泛分布的气泡群,以及半径分布远离谐振半径的气泡群时,作用效果好于传统正弦波信号.例如中心频率1 MHz、调频范围0.75 MHz的调频脉冲作用高斯分布（平均半径3.5 μm、均方差为1）的微泡群200 μs,可比同等声压的正弦波多约12%的微气泡产生位移30 μm. 关键词： 超声辐射力 调频波 高斯分布     

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.     

超声对近壁微气泡溃灭过程的影响*

近壁微气泡溃灭特性的深入研究对靶向给药和基因治疗等技术具有较好的指导作用。该文基于数值模拟技术,采用有限体积法结合流体体积模型对超声作用下的近壁微气泡溃灭特性进行了研究,分析了超声对近壁微气泡溃灭动力学过程的影响。结果表明气泡溃灭最大射流速度与近壁距离无量纲参数在1.1~1.6范围内时成正比,与超声频率在10~60 Hz范围内时成正比,与气泡初始半径在50~100μm范围内时成反比;近壁气泡的二次溃灭最大射流速度大于一次溃灭,二次溃灭的作用更加明显。超声参数对近壁气泡溃灭过程存在较大影响,该研究为超声在医学上的应用提供了依据。     

Effect of static pressure on acoustic energy radiated by cavitation bubbles in viscous liquids under ultrasound

The effect of static pressure on acoustic emissions including shock-wave emissions from cavitation bubbles in viscous liquids under ultrasound has been studied by numerical simulations in order to investigate the effect of static pressure on dispersion of nano-particles in liquids by ultrasound. The results of the numerical simulations for bubbles of 5 μm in equilibrium radius at 20 kHz have indicated that the optimal static pressure which maximizes the energy of acoustic waves radiated by a bubble per acoustic cycle increases as the acoustic pressure amplitude increases or the viscosity of the solution decreases. It qualitatively agrees with the experimental results by Sauter et al. [Ultrason. Sonochem. 15, 517 (2008)]. In liquids with relatively high viscosity (～200 mPa s), a bubble collapses more violently than in pure water when the acoustic pressure amplitude is relatively large (～20 bar). In a mixture of bubbles of different equilibrium radius (3 and 5 μm), the acoustic energy radiated by a 5 μm bubble is much larger than that by a 3 μm bubble due to the interaction with bubbles of different equilibrium radius. The acoustic energy radiated by a 5 μm bubble is substantially increased by the interaction with 3 μm bubbles.     

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.     

Dynamical model of encapsulated gas microbubble under ultrasound based on elastic mechanics

Based on the theory of elastic mechanics, a dynamical model of an encapsulated gas microbubble under ultrasound is presented. The dynamical motion of the microbubble is divided into three states: buckled, elastic, and ruptured. The model describes the compression-only behavior appropriately and derives the transient variation of the resonance frequency of the damped oscillation and the relation between the critical rupture radius and initial outer radius. The normal stress in the tangential direction plays the principal role in the rupture and buckling of the encapsulating shell, resulting in likely rupture for a larger microbubble and resistance to rupture for a thicker-shell microbubble. Comparison of proposed dynamical model with Marmottant’s model has been given. The dynamical model can be employed in ultrasound medical diagnostics and therapy of drug incorporation or extravasation through further understanding the influence of the encapsulating shell.     

The dynamics of cavitation bubbles in a sealed vessel

A model of cavitation bubbles is derived in liquid confined in an elastic sealed vessel driven by ultrasound. In this model, an assumption that the pressure acting on the sealed vessel due to bubble pulsations is proportional to total volume change of bubbles is made. Numerical simulations are carried out for a single bubble and for bubbles. The results show that the pulsation of a single bubble can be suppressed to a large extent in sealed vessel, and that of two matched bubbles with same ambient radius can be further suppressed. However, when two mismatched bubbles have the same ambient radii, an interesting breathing phenomenon takes place, where one bubble pulsates inversely with the other one. Due to this breathing phenomenon the suppression effect becomes weak, so the maximum radii of two mismatched bubbles can be larger than that of a single bubble or that of two matched bubbles in sealed vessel. Besides that, for two mismatched bubbles with different ambient radii, the small one in sealed vessel under some certain parameters can pulsate as strong as or even stronger than that of a single bubble in an open vessel.     

Interaction of microbubbles with high intensity pulsed ultrasound

High intensity pulsed ultrasound, interacting with microbubble contrast agents, is potentially useful for drug delivery, cancer treatment, and tissue ablation, among other applications. To establish the fundamental understanding on the interaction of a microbubble (in an infinite volume of water) with an ultrasound pressure field, a numerical study is performed using the boundary element method. The response of the bubble, in terms of its shape at different times, the maximum bubble radius obtained, the oscillation time, the jet velocity, and its translational movement, is studied. The effect of ultrasound intensity and initial bubble size is examined as well. One important outcome is the determination of the conditions under which a clear jet will be formed in a microbubble in its interaction with a specific sound wave. The high speed jet is crucial for the aforementioned intended applications.     

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.     

Microflow-induced shear stress on biomaterial wall by ultrasound-induced encapsulated microbubble oscillation

A model of an ultrasound-driven encapsulated microbubble(EMB) oscillation near biomaterial wall is presented and used for describing the microflow-induced shear stress on the wall by means of a numerical method. The characteristic of the model lies in the explicit treatment of different types of wall for the EMB responses. The simulation results show that the radius-time change trends obtained by our model are consistent with the existing models and experimental results. In addition, the effect of the elastic wall on the acoustic EMB response is stronger than that of the rigid wall, and the shear stress on the elastic wall is larger than that of the rigid wall. The closer the EMB to the wall, the greater the shear stress on the wall. The substantial shear stress on the wall surface occurs inside a circular zone with a radius about two-thirds of the bubble radius. This paper may be of interest in the study of potential damage mechanisms to the microvessel for drug and gene delivery due to sonoporation.     

超声药物释放空化动力学行为研究

以直径1 μm的脂质体为空化研究对象,从修正的Rayleigh空化方程入手,研究机械系数(MI)对300 kHz和1 MHz超声作用时空化效应的影响。脂质体的药物释放以超声作用前后脂质体中钙黄绿素的荧光强度为量度。模拟结果表明:在微泡振荡过程中,由超声波驱动产生的负向最大泡壁运动速度促使微泡半径从最大快速减小接近于零,微泡积聚到最大能量。对于300 kHz和1 MHz的激励超声,存在一个拐点(MI)值,当MI小于接近0.4时,1 MHz微泡半径变化幅度强于300 kHz;当MI>0.4时,300 kHz微泡半径变化幅度强于1 MHz。这一结果预示在此范围内,300 kHz的药物释放效果好于1 MHz。本研究为超声空化效应研究及超声药物释放应用提供了理论依据。     

Attenuation and scattering of axisymmetrical modes in a fluid-filled round pipe with internally rough walls

The attenuation of axisymmetric eigenmodes in a cylindrical, elastic, fluid-filled waveguide with a statistically rough elastic wall is studied. It is shown that small perturbation theory can be used to relate explicitly the statistical characteristics of the internal wall surface roughness of an elastic pipe to the attenuation and scattering coefficients of the acoustic modes in the filling fluid. Analytical expressions for modal attenuation coefficients are obtained. The analysis of the frequency dependent attenuation coefficients and the ratio between the roughness correlation length and the inner radius of the pipe is made for different correlation functions of the roughness. It is shown that two scale parameters control the overall behavior of the modal attenuation coefficients. These are the ratios of the roughness correlation length and the inner pipe radius to the acoustic wavelength. The numerical results for sound propagation in a pipe and in a borehole with statistically rough, elastic walls are obtained and discussed.     

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.     

Limited damage of tissue mimic caused by a collapsing bubble under low-frequency ultrasound exposure

In this study, we investigated the bubble induced serious damage to tissue mimic exposed to 27-kHz ultrasound. The initial bubble radius ranged from 80 to 100 μm, which corresponded approximately to the experimentally-evaluated resonant radius of the given ultrasound frequency. The tissue mimic consisted of 10 wt% gelatine gel covered with cultured canine kidney epithelial cells. The collapsing bubble behaviour during the ultrasound exposure with negative peak pressures of several hundred kPa was captured by a high-speed camera system. After ultrasound exposure, a cell viability test was conducted based on microscopic bright-field images and fluorescence images for living and dead cells. In the viability test, cells played a role in indicating the damaged area. The bubble oscillations killed the cells, and on occasion detached layers of cultured cells from the gel. The damaged area was comparable or slightly larger than the initial bubble size, and smaller than the maximum bubble size. We concluded that only a small area in close proximity to the bubble could be damaged even above transient cavitation threshold.     

Effect of surfactant addition on removal of microbubbles using ultrasound

It is difficult to control the bubble in a liquid by the external operation, because the behavior of the bubble is controlled in buoyancy and flow of liquid. On the other hand, microbubbles, whose diameter is several decades μm, stably disperse in static liquid because of their small buoyancy and electrical repulsion. When an ultrasound, whose frequency was 2.4 MHz, was irradiated, the milky white microbubbles suspended solution became rapidly clear. In this study, the effects of surfactant addition on the removal of microbubbles from a liquid in an ultrasonic field were investigated. The efficiency of removal of microbubbles decreased with surfactant addition. Surfactant type influenced the size of agglomerated microbubbles, and the efficiency of removal of microbubbles changed. The surface of microbubble was modified by surfactant adsorption, and the steric inhibition influenced the removal of microbubbles.     

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.     

Microbubble spectroscopy of ultrasound contrast agents

A new optical characterization of the behavior of single ultrasound contrast bubbles is presented. The method consists of insonifying individual bubbles several times successively sweeping the applied frequency, and to record movies of the bubble response up to 25 million frames/s with an ultrahigh speed camera operated in a segmented mode. The method, termed microbubble spectroscopy, enables to reconstruct a resonance curve in a single run. The data is analyzed through a linearized model for coated bubbles. The results confirm the significant influence of the shell on the bubble dynamics: shell elasticity increases the resonance frequency by about 50%, and shell viscosity is responsible for about 70% of the total damping. The obtained value for shell elasticity is in quantative agreement with previously reported values. The shell viscosity increases significantly with the radius, revealing a new nonlinear behavior of the phospholipid coating.