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.     

Dual-high-frequency ultrasound excitation on microbubble destruction volume

Objective and motivation

The goal of this work was to test experimentally that exposing air bubbles or ultrasound contrast agents in water to amplitude modulated wave allows control of inertial cavitation affected volume and hence could limit the undesirable bioeffects.

Methods

Focused transducer operating at the center frequency of 10 MHz and having about 65% fractional bandwidth was excited by 3 μs 8.5 and 11.5 MHz tone-bursts to produce 3 MHz envelope signal. The 3 MHz frequency was selected because it corresponds to the resonance frequency of the microbubbles used in the experiment. Another 5 MHz transducer was used as a receiver to produce B-mode image. Peak negative acoustic pressure was adjusted in the range from 0.5 to 3.5 MPa. The spectrum amplitudes obtained from the imaging of SonoVue^TM contrast agent when using the envelope and a separate 3 MHz transducer were compared to determine their cross-section at the - 6 dB level.

Results

The conventional 3 MHz tone-burst excitation resulted in the region of interest (ROI) cross-section of 2.47 mm while amplitude modulated, dual-frequency excitation with difference frequency of 3 MHz produced cross-section equal to 1.2 mm.

Conclusion

These results corroborate our hypothesis that, in addition to the considerably higher penetration depth of dual-frequency excitation due to the lower attenuation at 3 MHz than that at 8.5 and 11.5 MHz, the sample volume of dual-frequency excitation is also smaller than that of linear 3-MHz method for more spatially confined destruction of microbubbles.     

Discussion of the application of finite Volterra series for the modeling of the oscillation behavior of ultrasound contrast agents

Ultrasound contrast agents consist of microbubbles with diameters in the micrometer range. Excited by ultrasound, these bubbles exhibit highly nonlinear oscillation. While well developed physical models for microbubble oscillation exist, the efficiency of pulse sequences for sensitive microbubble detection is discussed based on simple mathematical models of general nonlinearity. Typically, Taylor series are used to model microbubble nonlinearity for the development of detection schemes. Recently, pulse sequences were proposed which exploit nonlinear memory of microbubbles, a property that cannot be modeled by a Taylor series but can be explained using a Volterra series. Therefore, this paper discusses and evaluates the usage of Volterra series for the modeling of the scattering behavior of contrast agent microbubbles. A numerically stable linear estimation algorithm is implemented to determine a third order Volterra model for a free gas bubble with a resting radius . For insonification pressures up to 100 kPa, the identified model allowed for a mean-square error of less than −16 dB with respect to the reference signal. Analysis of the response to narrowband signals showed that the achievable mean-square error is further reduced for the bandwidth available to typical ultrasound transducers used for clinical diagnostics.     

调频激励增强微气泡的次谐波理论和实验研究

超声造影剂的次谐波成像可以提高造影组织比，提供更好的图像质量. 提出一种利用调频脉冲激励以增强造影剂微气泡产生的次谐波新方法. 基于修正的Church方程，从理论上讨论了次谐波的产生与调频激励声压的关系及产生阈值，并且实验证实了优化调频信号的带宽及调频时间可以提高次谐波信号幅度及改善主瓣和旁瓣特性. 理论与实验表明，与传统脉冲信号激励相比，调频信号激励产生的次谐波幅度可提高约22dB. 关键词： 调频激励 超声造影剂 微气泡 次谐波     

Ultrasound-assisted microbubbles gene transfer in tendons for gene therapy

Our study aimed at evaluating the use of ultrasound-assisted microbubbles gene transfer in mice Achilles tendons. Using a plasmid encoding luciferase gene, it was found that an efficient and stable gene expression for more than two weeks was obtained when tendons were injected with 10 μg of plasmid in the presence of 5 × 10⁵ BR14 microbubbles with the following acoustic parameters: 1 MHz, 200 kPa, 40% duty cycle and 10 min of exposure time. The rate of gene expression was 100-fold higher than that obtained with naked plasmid injected alone without ultrasound or with ultrasound in absence of microbubbles. The long term expression of transgene makes ultrasound-assisted microbubble a suitable method for gene therapy in tendons.     

Pulse compression technique for simultaneous HIFU surgery and ultrasonic imaging: a preliminary study

In an ultrasound image-guided High Intensity Focused Ultrasound (HIFU) surgery, reflected HIFU waves received by an imaging transducer should be suppressed for real-time simultaneous imaging and therapy. In this paper, we investigate the feasibility of pulse compression scheme combined with notch filtering in order to minimize these HIFU interference signals. A chirp signal modulated by the Dolph-Chebyshev window with 3-9 MHz frequency sweep range is used for B-mode imaging and 4 MHz continuous wave is used for HIFU. The second order infinite impulse response notch filters are employed to suppress reflected HIFU waves whose center frequencies are 4 MHz and 8 MHz. The prototype integrated HIFU/imaging transducer that composed of three rectangular elements with a spherically con-focused aperture was fabricated. The center element has the ability to transmit and receive 6 MHz imaging signals and two outer elements are only used for transmitting 4 MHz continuous HIFU wave. When the chirp signal and 4 MHz HIFU wave are simultaneously transmitted to the target, the reflected chirp signals mixed with 4 MHz and 8 MHz HIFU waves are detected by the imaging transducer. After the application of notch filtering with pulse compression process, HIFU interference waves in this mixed signal are significantly reduced while maintaining original imaging signal. In the single scanline test using a strong reflector, the amplitude of the reflected HIFU wave is reduced to −45 dB. In vitro test, with a sliced porcine muscle shows that the speckle pattern of the restored B-mode image is close to that of the original image. These preliminary results demonstrate the potential for the pulse compression scheme with notch filtering to achieve real-time ultrasound image-guided HIFU surgery.     

Influence of nesting shell size on brightness longevity and resistance to ultrasound-induced dissolution during enhanced B-mode contrast imaging

This study aims to bridge the gap between transport mechanisms of an improved ultrasound contrast agent (UCA) and its resulting behavior in a clinical imaging study. Phospholipid-shelled microbubbles nested within the aqueous core of a polymer microcapsule are examined for their use and feasibility as an improved UCA. The nested formulation provides contrast comparable to traditional formulations, specifically an SF₆ microbubble coated by a DSPC PEG-3000 monolayer, with the advantage that contrast persists at least nine times longer in a mock clinical, in vitro setting. The effectiveness of the sample was measured using a contrast ratio in units of decibels (dB) which compares the brightness of the nested microbubbles to a reference value of a phantom tissue mimic. During a 40 min imaging study, six nesting formulations with average outer capsule diameters of 1.95, 2.53, 5.55, 9.95, 14.95, and 20.51 μm reached final contrast ratio values of 0.25, 2.35, 3.68, 4.51, 5.93, and 8.00 dB, respectively. The starting contrast ratio in each case was approximately 8 dB and accounts for the brightness attributed to the nesting shell. As compared with empty microcapsules (no microbubbles nested within), enhancement of the initial contrast ratio increased systematically with decreasing microcapsule size. The time required to reach a steady state in the temporal contrast ratio profile also varied with microcapsule diameter and was found to be 420 s for each of the four smallest shell diameters and 210 s and 150 s, respectively, for the largest two shell diameters. All nested formulations were longer-lived and gave higher final contrast ratios than a control sample comprising un-nested, but otherwise equivalent, microbubbles. Specifically, the contrast ratio of the un-nested microbubbles decreased to a negative value after 4 min of continuous ultrasound exposure with complete disappearance of the microbubbles after 15 min whereas all nested formulations maintained positive contrast ratio values for the duration of the 40 min trial. The results are consistent with two distinct stages of gas transport: in the first stage, passive diffusion occurs under ambient conditions across the microbubble monolayer within the first few minutes after formulation until the aqueous interior of the microcapsule is saturated with gas; in the second stage ultrasound drives additional gas dissolution even further due to pressure modulation. It is important to understand the chemistry and transport mechanisms of this contrast agent under the influence of ultrasound to attain better perspicacity for enhanced applications in imaging. Results from this study will facilitate future preclinical studies and clinical applications of nested microbubbles for therapeutic and diagnostic imaging.     

Ambient pressure dependence of the ultra-harmonic response from contrast microbubbles

Sub-harmonic response from ultrasound contrast agent microbubbles has been demonstrated to be an effective modality for noninvasive pressure measurement. In the present study, the dependence of ultra-harmonic response on the ambient overpressure was investigated by both experimental measurements and simulations. In the measurements, the microbubbles were exposed to Gaussian pulses with varied driving frequencies and pulse lengths, at an acoustic pressure of 0.3 MPa. The amplitudes of sub- and ultra-harmonic components were measured when the ambient overpressures varied from 0-25 kPa. At the driving frequency of 1.33 MHz, the ultra-harmonic energy decreased but the sub-harmonic energy increased with the increasing overpressure; while at the driving frequency of 4 MHz, both the sub- and ultra-harmonic components showed the same tendency that the corresponding energy decreased as the overpressure was increased. A 4-MHz Gaussian pulse with 64 cycles could provide an ultra-harmonic response with both good ambient pressure sensitivity and high linearity. Furthermore, the effects of shell parameters of a microbubble on the generation of ultra- and sub-harmonic responses were discussed based on simulations using Marmottant's model. This study suggests that the ultra-harmonic response from contrast microbubbles might be applicable for noninvasive pressure measurement.     

Phantom investigation of phase-inversion-based dual-frequency excitation imaging for improved contrast display

Objective

The goal of this work is to examine the effects of pulse-inversion (PI) technique in combination with dual-frequency (DF) excitation method to separate the high-order nonlinear responses from microbubble contrast agents for improvement of image contrast. DF excitation method has been previously developed to induce the low-frequency ultrasound nonlinear responses from bubbles by using the composition of two high-frequency sinusoids (f₁ and f₂).

Motivation

Although the simple filtering was conventionally utilized to provide signal separation, the PI approach is better in the sense that it minimizes the mutual interferences among these high-order nonlinear responses in the presence of spectral overlap. The novelty of the work is that, in addition to the common PI summation, the PI subtraction was also applied in DF excitation method.

Methods

DF excitation pulses having an envelope frequency of 3 MHz (i.e., f₁ = 8.5 MHz and f₂ = 11.5 MHz) with pulse lengths of 3-10 μs and the pressure amplitudes from 0.5 to 1.5 MPa were used to interrogate the nonlinear responses of SonoVue™ microbubbles in the phantom experiments. The high-order nonlinear responses in the DF excitation were extracted for contrast imaging using PI summation for even-order nonlinear components or PI subtraction for odd-order nonlinear ones.

Results

Our results indicated that, as compared to the conventional filtering technique, the PI processing effectively increases the contrast-to-tissue ratio (CTR) of the third-order nonlinear response at 5.5 MHz and the fourth-order nonlinear response at 6 MHz by 2-5 dB. For these high-order nonlinear components, the CTR increase varies with the transmission pressures from 0.5 to 1.5 MPa due to the microbubbles’ displacement induced by the radiation force of DF excitation.

Conclusions

For DF excitation technique, the PI processing can help to extract either the odd-order or the even-order nonlinear components for higher CTR estimates.     

Scaling of the viscoelastic shell properties of phospholipid encapsulated microbubbles with ultrasound frequency

Phospholipid encapsulated microbubbles are widely employed as clinical diagnostic ultrasound contrast agents in the 1–5 MHz range, and are increasingly employed at higher ultrasound transmit frequencies. The stiffness and viscosity of the encapsulating “shells” have been shown to play a central role in determining both the linear and nonlinear response of microbubbles to ultrasound. At lower frequencies, recent studies have suggested that shell properties can be frequency dependent. At present, there is only limited knowledge of how the viscoelastic properties of phospholipid shells scale at higher frequencies. In this study, four batches of in-house phospholipid encapsulated microbubbles were fabricated with decreasing volume-weighted mean diameters of 3.20, 2.07, 1.82 and 1.61 μm. Attenuation experiments were conducted in order to assess the frequency-dependent response of each batch, resulting in resonant peaks in response at 4.2, 8.9, 12.6 and 19.5 MHz, respectively. With knowledge of the size measurements, the attenuation spectra were then fitted with a standard linearized bubble model in order to estimate the microbubble shell stiffness S_p and shell viscosity S_f, resulting in a slight increase in S_p (1.53–1.76 N/m) and a substantial decrease in S_f (0.29 × 10⁻⁶–0.08 × 10⁻⁶ kg/s) with increasing frequency. These results performed on a single phospholipid agent show that frequency dependent shell properties persist at high frequencies (up to 19.5 MHz).     

Impact of acoustic pressure on ambient pressure estimation using ultrasound contrast agent

Local blood pressure measurements provide important information on the state of health of organs in the body and can be used to diagnose diseases in the heart, lungs, and kidneys. This paper presents an approach for investigating the ambient pressure sensitivity of a contrast agent using diagnostic ultrasound. The experimental setup resembles a realistic clinical setup utilizing a single array transducer for transmit and receive. The ambient pressure sensitivity of SonoVue (Bracco, Milano, Italy) was measured twice using two different acoustic driving pressures, which were selected based on a preliminary experiment. To compensate for variations in bubble response and to make the estimates more robust, the relation between the energy of the subharmonic and the fundamental component was chosen as a measure over the subharmonic peak amplitude. The preliminary study revealed the growth stage of the subharmonic component to occur at acoustic driving pressures between 300 and 500 kPa. Based on this, the pressure sensitivity was investigated using a driving pressure of 485 and 500 kPa. At 485 kPa, a linear pressure sensitivity of 0.42 dB/kPa was found having a linear correlation coefficient of 0.94. The second measurement series at 485 kPa showed a sensitivity of 0.41 dB/kPa with a correlation coefficient of 0.89. Based on the measurements at 500 kPa, this acoustic driving pressure was concluded to be too high causing the bubbles to be destroyed. The pressure sensitivity for these two measurement series were 0.42 and 0.25 dB/kPa with linear correlation coefficients of 0.98 and 0.93, respectively.     

In vitro measurement of attenuation and nonlinear scattering from echogenic liposomes

Echogenic liposomes (ELIP) are an excellent candidate for concurrent imaging and drug delivery applications. They combine the advantages of liposomes-biocompatibility and ability to encapsulate both hydrophobic and hydrophilic drugs-with strong reflections of ultrasound. The objective of this study is to perform a detailed in vitro acoustic characterization - including nonlinear scattering that has not been studied before - along with an investigation of the primary mechanism of echogenicity. Both components are critical for developing viable clinical applications of ELIP. Mannitol, a cryoprotectant, added during the preparation of ELIP is commonly believed to be critical in making them echogenic. Accordingly, here ELIP prepared with varying amount of mannitol concentration are investigated for their pressure dependent linear and non-linear scattered responses. The average diameter of these liposomes is measured to be 125-185 nm. But they have a broad size distribution including liposomes with diameters over a micro-meter as observed by TEM and AFM. These larger liposomes are critical for the overall echogenicity. Attenuation through liposomal solution is measured with four different transducers (central frequencies 2.25, 3.5, 5, 10 MHz). Measured attenuation increases linearly with liposome concentration indicating absence of acoustic interactions between liposomes. Due to the broad size distribution, the attenuation shows a flat response without a distinct peak in the range of frequencies (1-12 MHz) investigated. A 15-20 dB enhancement with 1.67 μg/ml of lipids is observed both for the scattered fundamental and the second harmonic responses at 3.5 MHz excitation frequency and 50-800 kPa amplitude. It demonstrates the efficacy of ELIP for fundamental as well as harmonic ultrasound imaging. The scattered response however does not show any distinct subharmonic peak for the acoustic excitation parameters studied. Small amount of mannitol proves critical for echogenicity. However, mannitol concentration above 100 mM shows no effect.     

Acoustic characterization of a new trisacryl contrast agent. Part I: In vitro study

The objective of the study was to acoustically characterize trisacryl polymeric microparticles (TMP), which are derived from biocompatible embolic agents.With significant acoustic properties, these polymeric particles could be potentially used as targeted ultrasound contrast agents, directed towards a specific site, with ligands conjugation on the polymeric network surface. In the in vitro study, a pulser/receiver (PRF of 1 kHz), associated to different transducers (5, 10 and 15 MHz), was used to measure the acoustic properties of the TMP inserted in a Couette flow device. Acoustic characterization according to TMP concentration (0.12-15.63 mg/ml), frequency (4.5-17 MHz, defined by each transducer bandwidth), ultrasound pressure (137-378 kPa) and exposure time (0-30 min) was conducted. Particle attenuation was also evaluated according to TMP concentration and emission frequency. Backscattering increased non linearly with concentration and maximum enhancement was of 16.4 dB ± 0.89 dB above 7.8 mg/ml. This parameter was found non-linear with increasing applied pressure and no harmonic oscillation could be noticed. Attenuation reached approximately 1.4 dB/cm at 15 MHz and for the 15.6 mg/ml suspension.The TMP have revealed in vitro ultrasound properties comparable to those observed with known contrast agents, studied in similar in vitro systems. However, such set-ups combined with a rather aqueous suspending medium, have some limitations and further investigations need now to be conducted to approach in vivo conditions in terms of flow and blood environment.     

Subharmonic behavior of phospholipid-coated ultrasound contrast agent microbubbles

Coated microbubbles, unlike tissue are able to scatter sound subharmonically. Therefore, the subharmonic behavior of coated microbubbles can be used to enhance the contrast in ultrasound contrast imaging. Theoretically, a threshold amplitude of the driving pressure can be calculated above which subharmonic oscillations of microbubbles are initiated. Interestingly, earlier experimental studies on coated microbubbles demonstrated that the threshold for these bubbles is much lower than predicted by the traditional linear viscoelastic shell models. This paper presents an experimental study on the subharmonic behavior of differently sized individual phospholipid coated microbubbles. The radial subharmonic response of the microbubbles was recorded with the Brandaris ultra high-speed camera as a function of both the amplitude and the frequency of the driving pulse. Threshold pressures for subharmonic generation as low as 5 kPa were found near a driving frequency equal to twice the resonance frequency of the bubble. An explanation for this low threshold pressure is provided by the shell buckling model proposed by Marmottant et al. [J. Acoust. Soc. Am. 118, 3499-3505 (2005)]. It is shown that the change in the elasticity of the bubble shell as a function of bubble radius as proposed in this model, enhances the subharmonic behavior of the microbubbles.     

Preparation and in vitro evaluation of an ultrasound-triggered drug delivery system: 10-hydroxycamptothecin loaded PLA microbubbles

10-Hydroxycamptothecin (HCPT) loaded PLA microbubbles, used as an ultrasound-triggered drug delivery system, were fabricated by a double emulsion-solvent evaporation method. The obtained microbubbles were characterized by scanning electron microscope (SEM), transmission electron microscope (TEM), X-ray diffraction (XRD), differential scanning calorimetry (DSC) and confocal laser scanning microscope (CLSM). In addition, the effect of diagnostic ultrasound exposure on BEL-7402 cells combined with HCPT-loaded PLA microbubbles was evaluated using cytotoxicity assay, CLSM and flow cytometry (FCM). It was found that the HCPT-loaded PLA microbubbles showed smooth surface and spherical shape, and the drug was amorphously dispersed within the shell and the drug loading content reached up to 1.69%. Nearly 20% of HCPT was released upon exposure to diagnostic ultrasound at frequency of 3.5 MHz for 10 min. Moreover, HCPT fluorescence in the cells treated only with the HCPT-loaded PLA microbubbles was discernible, but less intense, while those treated with the microbubbles in conjunction with ultrasound exposure was evident and intense, indicating an increased cellular uptake of HCPT by ultrasound exposure. Cytotoxicity test on BEL-7402 cells indicated that the HCPT-loaded PLA microbubbles combined with ultrasound exposure were more cytotoxic than the microbubbles alone. The results suggest that the combination of drug loaded PLA microbubbles and diagnostic ultrasound exposure exhibit an effective intracellular drug uptake by tumor cells, indicating their great potential for antitumor therapy.     

Pulse sequences for uniform perfluorocarbon droplet vaporization and ultrasound imaging

Phase-change contrast agents (PCCAs) consist of liquid perfluorocarbon droplets that can be vaporized into gas-filled microbubbles by pulsed ultrasound waves at diagnostic pressures and frequencies. These activatable contrast agents provide benefits of longer circulating times and smaller sizes relative to conventional microbubble contrast agents. However, optimizing ultrasound-induced activation of these agents requires coordinated pulse sequences not found on current clinical systems, in order to both initiate droplet vaporization and image the resulting microbubble population. Specifically, the activation process must provide a spatially uniform distribution of microbubbles and needs to occur quickly enough to image the vaporized agents before they migrate out of the imaging field of view. The development and evaluation of protocols for PCCA-enhanced ultrasound imaging using a commercial array transducer are described. The developed pulse sequences consist of three states: (1) initial imaging at sub-activation pressures, (2) activating droplets within a selected region of interest, and (3) imaging the resulting microbubbles. Bubble clouds produced by the vaporization of decafluorobutane and octafluoropropane droplets were characterized as a function of focused pulse parameters and acoustic field location. Pulse sequences were designed to manipulate the geometries of discrete microbubble clouds using electronic steering, and cloud spacing was tailored to build a uniform vaporization field. The complete pulse sequence was demonstrated in the water bath and then in vivo in a rodent kidney. The resulting contrast provided a significant increase (>15 dB) in signal intensity.     

Study of ultrasound stiffness imaging methods using tissue mimicking phantoms

A pilot study was carried out to investigate the performance of ultrasound stiffness imaging methods namely Ultrasound Elastography Imaging (UEI) and Acoustic Radiation Force Impulse (ARFI) Imaging. Specifically their potential for characterizing different classes of solid mass lesions was analyzed using agar based tissue mimicking phantoms. Composite tissue mimicking phantom was prepared with embedded inclusions of varying stiffness from 50 kPa to 450 kPa to represent different stages of cancer. Acoustic properties such as sound speed, attenuation coefficient and acoustic impedance were characterized by pulse echo ultrasound test at 5 MHz frequency and they are ranged from (1564 ± 88 to 1671 ± 124 m/s), (0.6915 ± 0.123 to 0.8268 ± 0.755 db cm^-1 MHz^-1) and (1.61×10⁶ ± 0.127 to 1.76 × 10⁶ ± 0.045 kg m^-2 s^-1) respectively. The elastic property Young’s Modulus of the prepared samples was measured by conducting quasi static uni axial compression test under a strain rate of 0.5 mm/min upto 10 % strain, and the values are from 50 kPa to 450 kPa for a variation of agar concentration from 1.7% to 6.6% by weight. The composite phantoms were imaged by Siemens Acuson S2000 (Siemens, Erlangen, Germany) machine using linear array transducer 9L4 at 8 MHz frequency; strain and displacement images were collected by UEI and ARFI. Shear wave velocity 4.43 ± 0.35 m/s was also measured for high modulus contrast (18 dB) inclusion and X.XX m/s was found for all other inclusions. The images were pre processed and parameters such as Contrast Transfer Efficiency and lateral image profile were computed and reported. The results indicate that both ARFI and UEI represent the abnormalities better than conventional US B mode imaging whereas UEI enhances the underlying modulus contrast into improved strain contrast. The results are corroborated with literature and also with clinical patient images.     

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     

Modeling of nonlinear viscous stress in encapsulating shells of lipid-coated contrast agent microbubbles

A general theoretical approach to the development of zero-thickness encapsulation models for contrast microbubbles is proposed. The approach describes a procedure that allows one to recast available rheological laws from the bulk form to a surface form which is used in a modified Rayleigh-Plesset equation governing the radial dynamics of a contrast microbubble. By the use of the proposed procedure, the testing of different rheological laws for encapsulation can be carried out. Challenges of existing shell models for lipid-encapsulated microbubbles, such as the dependence of shell parameters on the initial bubble radius and the “compression-only” behavior, are discussed. Analysis of the rheological behavior of lipid encapsulation is made by using experimental radius-time curves for lipid-coated microbubbles with radii in the range 1.2-2.5 μm. The curves were acquired for a research phospholipid-coated contrast agent insonified with a 20 cycle, 3.0 MHz, 100 kPa acoustic pulse. The fitting of the experimental data by a model which treats the shell as a viscoelastic solid gives the values of the shell surface viscosity increasing from 0.30 × 10⁻⁸ kg/s to 2.63 × 10⁻⁸ kg/s for the range of bubble radii, indicated above. The shell surface elastic modulus increases from 0.054 N/m to 0.37 N/m. It is proposed that this increase may be a result of the lipid coating possessing the properties of both a shear-thinning and a strain-softening material. We hypothesize that these complicated rheological properties do not allow the existing shell models to satisfactorily describe the dynamics of lipid encapsulation. In the existing shell models, the viscous and the elastic shell terms have the linear form which assumes that the viscous and the elastic stresses acting inside the lipid shell are proportional to the shell shear rate and the shell strain, respectively, with constant coefficients of proportionality. The analysis performed in the present paper suggests that a more general, nonlinear theory may be more appropriate. It is shown that the use of the nonlinear theory for shell viscosity allows one to model the “compression-only” behavior. As an example, the results of the simulation for a 2.03 μm radius bubble insonified with a 6 cycle, 1.8 MHz, 100 kPa acoustic pulse are given. These parameters correspond to the acoustic conditions under which the “compression-only” behavior was observed by de Jong et al. [Ultrasound Med. Biol. 33 (2007) 653-656]. It is also shown that the use of the Cross law for the modeling of the shear-thinning behavior of shell viscosity reduces the variance of experimentally estimated values of the shell viscosity and its dependence on the initial bubble radius.