Investigation of transition frequencies of two acoustically coupled bubbles using a direct numerical simulation technique

The theoretical results regarding the ``transition frequencies'' of two acoustically interacting bubbles have been verified numerically. The theory provided by Ida [Phys. Lett. A 297 (2002) 210] predicted the existence of three transition frequencies per bubble, each of which has the phase difference of $\pi /2$ between a bubble's pulsation and the external sound field, while previous theories predicted only two natural frequencies which cause such phase shifts. Namely, two of the three transition frequencies correspond to the natural frequencies, while the remaining does not. In a subsequent paper [M. Ida, Phys. Rev. E 67 (2003) 056617], it was shown theoretically that transition frequencies other than the natural frequencies may cause the sign reversal of the secondary Bjerknes force acting between pulsating bubbles. In the present study, we employ a direct numerical simulation technique that uses the compressible Navier-Stokes equations with a surface-tension term as the governing equations to investigate the transition frequencies of two coupled bubbles by observing their pulsation amplitudes and directions of translational motion, both of which change as the driving frequency changes. The numerical results reproduce the recent theoretical predictions, validating the existence of the transition frequencies not corresponding to the natural frequency.Comment: 18 pages, 8 figures, in pres

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

Modeling of the acoustic response from contrast agent microbubbles near a rigid wall

In ultrasonic targeted imaging, specially designed encapsulated microbubbles are used, which are capable of selectively adhering to the target site in the body. A challenging problem is to distinguish the echoes from such adherent agents from echoes produced by freely circulating agents. In the present paper, an equation of radial oscillation for an encapsulated bubble near a plane rigid wall is derived. The equation is then used to simulate the echo from a layer of contrast agents localized on a wall. The echo spectrum of adherent microbubbles is compared to that of free, randomly distributed microbubbles inside a vessel, in order to examine differences between the acoustic responses of free and adherent agents. It is shown that the fundamental spectral component of adherent bubbles is perceptibly stronger than that of free bubbles. This increase is accounted for by a more coherent summation of echoes from adherent agents and the acoustic interaction between the agents and the wall. For cases tested, the increase of the fundamental component caused by the above two effects is on the order of 8-9 dB. Bubble aggregates, which are observed experimentally to form near a wall due to secondary Bjerknes forces, increase the intensity of the fundamental component only if they are formed by bubbles whose radii are well below the resonant radius. If the formation of aggregates contributes to the growth of the fundamental component, the increase can exceed 17 dB. Statistical analysis for the comparison between adhering and free bubbles, performed over random space bubble distributions, gives p-values much smaller than 0.05

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

Vaporization dynamics of volatile perfluorocarbon droplets: A theoretical model and in vitro validation

Perfluorocarbon (PFC) microdroplets, called phase-change contrast agents (PCCAs), are a promising tool in ultrasound imaging and therapy. Interest in PCCAs is motivated by the fact that they can be triggered to transition from the liquid state to the gas state by an externally applied acoustic pulse. This property opens up new approaches to applications in ultrasound medicine. Insight into the physics of vaporization of PFC droplets is vital for effective use of PCCAs and for anticipating bioeffects. PCCAs composed of volatile PFCs (with low boiling point) exhibit complex dynamic behavior: after vaporization by a short acoustic pulse, a PFC droplet turns into a vapor bubble which undergoes overexpansion and damped radial oscillation until settling to a final diameter. This behavior has not been well described theoretically so far. The purpose of our study is to develop an improved theoretical model that describes the vaporization dynamics of volatile PFC droplets and to validate this model by comparison with in vitro experimental data

Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer

Nonlinear waves are studied in a mixture of liquid and gas bubbles. Influence of viscosity and heat transfer is taken into consideration on propagation of the pressure waves. Nonlinear evolution equations of the second and the third order for describing nonlinear waves in gas-liquid mixtures are derived. Exact solutions of these nonlinear evolution equations are found. Properties of nonlinear waves in a liquid with gas bubbles are discussed.Comment: Physics Letters A, Volume 374, Issues 19-20, Pages 2011-201

Characterisation of Liposome-Loaded Microbubble Populations for Subharmonic Imaging

Therapeutic microbubbles could make an important contribution to the diagnosis and treatment of cancer. Acoustic characterisation was performed on microfluidic generated microbubble populations that either were bare or had liposomes attached. Through the use of broadband attenuation techniques (3–8 MHz), the shell stiffness was measured to be 0.72 ± 0.01 and 0.78 ± 0.05 N/m and shell friction was 0.37 ± 0.05 and 0.74 ± 0.05 × 10−6 kg/s for bare and liposome-loaded microbubbles, respectively. Acoustic scatter revealed that liposome-loaded microbubbles had a lower subharmonic threshold, occurring from a peak negative pressure of 50 kPa, compared with 200 kPa for equivalent bare microbubbles. It was found that liposome loading had a negligible effect on the destruction threshold for this microbubble type, because at a mechanical index >0.4 (570 kPa), 80% of both populations were destroyed

Standing waves for acoustic levitation

Standing waves are the most popular method to achieve acoustic trapping. Particles with greater acoustic impedance than the propagation medium will be trapped at the pressure nodes of a standing wave. Acoustic trapping can be used to hold particles of various materials and sizes, without the need of a close-loop controlling system. Acoustic levitation is a helpful and versatile tool for biomaterials and chemistry, with applications in spectroscopy and lab-on-a-droplet procedures. In this chapter, multiple methods are presented to simulate the acoustic field generated by one or multiple emitters. From the acoustic field, models such as the Gor'kov potential or the Flux Integral are applied to calculate the force exerted on the levitated particles. The position and angle of the acoustic emitters play a fundamental role, thus we analyse commonly used configurations such as emitter and reflector, two opposed emitters, or arrangements using phased arrays