GPU-accelerated study of the inertial cavitation threshold in viscoelastic soft tissue using a dual-frequency driving signal

Pulsation of bubbles in the viscoelastic medium under dual-frequency ultrasound

The pulsation of bubbles under dual-frequency ultrasound is closely associated with their application in medical diagnostics. In this study, we derive the analytical solution of the bubble dynamic equations in viscoelastic media under dual-frequency ultrasound and compare it with numerical solutions. The effects of the surrounding medium's viscoelasticity, bubble equilibrium radius, incident acoustic frequency, and incident sound pressure ratio on the bubble pulsation and its scattered acoustic waves are analyzed. The results demonstrate consistency between the analytical and numerical solutions. Moreover, our analytical solution reveals that an increase in medium viscosity leads to a decrease in the amplitude of bubble pulsation, even at resonance frequency. Additionally, when the sound pressure ratio is 1, both the difference frequency and sum frequency exhibit maximum amplitude in the spectrum of instantaneous bubble radius. Furthermore, for resonance approximation in sparse bubble cluster detection using sum and difference frequencies, it is necessary for incident frequency f1 to be greater than twice the resonance frequency of the smallest bubble within the cluster.

Numerical investigation of acoustic cavitation characteristics of a single gas-vapor bubble in soft tissue under dual-frequency ultrasound

The viscoelastic tissue under dual-frequency ultrasound excitation affects the acoustic cavitation of a single gas-vapor bubble. To investigate the effect of the cavitation dynamics, the Gilmore-Akulichev-Zener (GAZ) model is coupled with the Peng-Robinson equation of state (PR EOS). Results indicate that the GAZ-PR EOS model can accurately estimate the bubble dynamics by comparing with the Gilmore PR EOS and GAZ-Van der Waals (VDW) EOS model. Furthermore, the acoustic cavitation effect in different viscoelastic tissues is investigated, including the radial stress at the bubble wall, the temperature, pressure, and the number of water molecules inside the bubble. Results show that the creep recovery and the relaxation of the stress caused by viscoelasticity can affect the acoustic cavitation of the bubble, which could inhibit the bubble's expansion and reduce the internal temperature and pressure within the bubble. Moreover, the effect of dual-frequency ultrasound on the cavitation of single gas-vapor bubbles is studied. Results suggest that dual-frequency ultrasound could increase the internal temperature of bubbles, the internal pressure of bubbles, and the radial stress at the bubble wall. More importantly, there is a specific optimal combination of frequencies for particular viscoelasticity by exploring the impact of different dual-frequency ultrasound combinations and tissue viscoelasticity on the acoustic cavitation of a single gas-vapor bubble. In conclusion, this study helps to provide theoretical guidance for dual-frequency ultrasound to improve acoustic chemical and mechanical effects, and further optimize its application in acoustic sonochemistry and ultrasound therapy.

Enhancing sonothrombolysis outcomes with dual-frequency ultrasound: Insights from an in silico microbubble dynamics study

Sonothrombolysis is a technique that employs the ultrasound waves to break down the clot. Recent studies have demonstrated significant improvement in the treatment efficacy when combining two ultrasound waves of different frequencies. Nevertheless, the findings remain conflicted on the ideal frequency pairing that leads to an optimal treatment outcome. Existing experimental studies are constrained by the limited range of frequencies that can be investigated, while numerical studies are typically confined to spherical microbubble dynamics, thereby restricting the scope of the analysis. To overcome this, the present study investigated the microbubble dynamics caused by the different combinations of ultrasound frequencies. This was carried out using computational modelling as it enables the visualisation of the microbubble behaviour, which is difficult in experimental studies due to the opacity of blood. The results showed that the pairings of two ultrasound waves with low frequencies generally produced stronger cavitation and higher flow-induced shear stress on the clot surface. However, one should avoid the frequency pairings that are integer multipliers of each other, i.e., frequency ratio of 1/3, 1/2 and 2, as they led to resultant wave with low pressure amplitude that weakened the cavitation. At 0.5 + 0.85 MHz, the microbubble caused the highest shear stress of 60.5 kPa, due to its large translational distance towards the clot. Although the pressure threshold for inertial cavitation was reduced using dual-frequency ultrasound, the impact of the high-speed jet can only be realised when the microbubble travelled close to the clot. The results obtained from the present study provide groundwork for deeper understanding on the microbubble dynamics during dual-frequency sonothrombolysis, which is of paramount importance for its optimisations and the subsequent clinical translation.

A new model for bubble cluster dynamics in a viscoelastic media

Bubble cluster dynamics in viscoelastic media is instructive for ultrasound diagnosis and therapy. In this paper, we propose a statistical model for bubble cluster dynamics in viscoelastic media considering the radius distribution of bubble nuclei. By investigating and comparing the response for a bubble in three conditions: single bubble; multi-bubble with the same radius; multi-bubble with different radius, the following rules are found: The promotion or suppression of the bubble cluster on the bubble vibration is not monotonous with the increase of the number of bubbles. The promotion or suppression of the bubble cluster on the bubble vibration varies alternately with the frequency. The effect of bubble cluster on bubble vibration is mostly suppressed when the driving acoustic pressure amplitude pa is high (5000 kPa). Usually, the bubble cluster promotes the vibration of the large bubbles (R0 = 10 μm) more, or suppresses it less.

Nonlinear effects of dual-frequency focused ultrasound on the on-demand regulation of acoustic droplet vaporization

Dual-frequency ultrasound has been widely employed to enhance and regulate acoustic droplet vaporization (ADV) but the role of ultrasonic nonlinear effects on it remains unclear. The main objective of this study is to investigate the influence of nonlinear effects on the control of ADV nucleation under different dual-frequency focused ultrasound conditions. ADV nucleation of PFC nanodroplets activated by nonlinear dual-frequency ultrasound was modeled and parametric studies were conducted to investigate the influence of dual-frequency ultrasound frequency and acoustic power on the degree of nonlinearity (DoN), nucleation rates and dimensions of the nucleation region in a wide parameter range. The results showed that the ultrasonic nonlinearity caused a significant decrease in peak negative pressure due to waveform distortion, which leads to a lower nucleation rate in the nonlinear model compared to that in the linear model. Furthermore, the distributions of nucleation regions were also affected by the interaction between waves of different frequencies and cloud-like spatial distributions were produced, which could be modulated by the dual-frequency ultrasound parameters and have great potentials in the spatial regulation of the ADV and customized treatment protocols in clinical applications. In addition, represented by 1.5 MHz + 3 MHz, such a dual-frequency combination of fundamental and second harmonic could effectively enhance ultrasonic nonlinear effects with relatively lower peak negative pressure and higher DoN. Therefore, nonlinear effect of the dual-frequency ultrasound plays an important role in the ADV regulation, which should be considered in the numerical model and practical applications.

Investigation of interaction effects on dual-frequency driven cavitation dynamics in a two-bubble system

The cavitation dynamics of a two-bubble system in viscoelastic media excited by dual-frequency ultrasound is studied numerically with a focus on the effects of inter-bubble interactions. Compared to the isolated bubble cases, the enhancement or suppression effects can be exerted on the amplitude and nonlinearity of the bubble oscillations to different degrees. Moreover, the interaction effects are found to be highly sensitive to multiple paramount parameters related to the two-bubble system, the dual-frequency ultrasound and the medium viscoelasticity. Specifically, the larger bubble of a two-bubble system shows a stronger effect on the smaller one, and this effect becomes more pronounced when the larger bubble undergoes harmonic and/or subharmonic resonances as well as the two bubbles get closer (e.g., d0 < 100 μm). For the influences of the dual-frequency excitation, the results show that the bubbles can achieve enhanced harmonic and/or subharmonic oscillations as the frequency combinations with small frequency differences (e.g., Δf < 0.2 MHz) close to the corresponding resonance frequencies of bubbles, and the interaction effects are consequently intensified. Similarly, the bubble oscillations and the interaction effects can also be enhanced as the acoustic pressure amplitude of each frequency component is equal and the pressure amplitude pA increases. Above a pressure threshold (pA = 215 kPa), a larger bubble undergoes period 2 (P2) oscillations, which can force a smaller bubble to change its oscillation pattern from period 1 (P1) into P2 oscillations. In addition, it is found that the medium viscosity dampens the bubble oscillations while the medium elasticity affects the bubble resonances, accordingly exhibiting stronger interaction effects at smaller viscosities (e.g., μ < 4 mPa·s) or certain elasticities (approximately G = 70-120 kPa, G = 160-200 kPa and G = 640-780 kPa) at which the bubble resonances occur. The study can contribute to a better understanding of the complex dynamic behaviors of interacting cavitation bubbles in viscoelastic tissues for high efficient cavitation-mediated biomedical applications using dual-frequency ultrasound.

Weakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles

To facilitate practical medical applications such as cancer treatment utilizing focused ultrasound and bubbles, a mathematical model that can describe the soft viscoelasticity of human body, the nonlinear propagation of focused ultrasound, and the nonlinear oscillations of multiple bubbles is theoretically derived and numerically solved. The Zener viscoelastic model and Keller-Miksis bubble equation, which have been used for analyses of single or few bubbles in viscoelastic liquid, are used to model the liquid containing multiple bubbles. From the theoretical analysis based on the perturbation expansion with the multiple-scales method, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, which has been used as a mathematical model of weakly nonlinear propagation in single phase liquid, is extended to viscoelastic liquid containing multiple bubbles. The results show that liquid elasticity decreases the magnitudes of the nonlinearity, dissipation, and dispersion of ultrasound and increases the phase velocity of the ultrasound and linear natural frequency of the bubble oscillation. From the numerical calculation of resultant KZK equation, the spatial distribution of the liquid pressure fluctuation for the focused ultrasound is obtained for cases in which the liquid is water or liver tissue. In addition, frequency analysis is carried out using the fast Fourier transform, and the generation of higher harmonic components is compared for water and liver tissue. The elasticity suppresses the generation of higher harmonic components and promotes the remnant of the fundamental frequency components. This indicates that the elasticity of liquid suppresses shock wave formation in practical applications.

Numerical investigation of the translational motion of bubbles: The comparison of capabilities of the time-resolved and the time-averaged methods

In the present study, the accuracies of two different numerical approaches used to model the translational motion of acoustic cavitational bubble in a standing acoustic field are compared. The less accurate but less computational demanding approach is to decouple the equation of translational motion from the radial oscillation, and solve it by calculating the time-averaged forces exerted on the bubble for one acoustic cycle. The second approach is to solve the coupled ordinary differential equations directly, which provides more accurate results with higher computational effort. The investigations are carried out in the parameter space of the driving frequency, pressure amplitude and equilibrium radius. Results showed that both models are capable to reveal stable equilibrium positions; however, in the case of oscillatory solutions, the difference in terms of translational frequency may be more than three fold, and the amplitude of translational motion obtained by the time-averaged method is roughly 1.5 times higher compared to the time-resolved solution at particular sets of parameters. This observation implies that where the transient behaviour is important, the time-resolved approach is the proper choice for reliable results.