Introduction

The growth, oscillation, and collapse of bubbles in the medium under ultrasonic irradiation is known as the cavitation effect. This phenomenon can either expand gradually and eventually collapse, or oscillate steadily, which is known as steady-state cavitation [ 1 ]. Transient cavitation, which is commonly referred to as cavitation, is influenced by a number of variables, such as the ultrasonic frequency, bubble size, and the properties of the surrounding medium [ 2 ]. Ultrasonic cavitation generates high temperatures and pressures, which are widely used in the medical field for tumor and thrombosis treatment [ 3 - 5 ]. Compared with conventional single-frequency ultrasound, multi-frequency ultrasound demonstrates two unique advantages in enhancing the cavitation effect, such as lowering the cavitation threshold and increasing response efficiency .

Ultrasound has been the subject of much research in recent years as a way to introduce energy into a medium and cause chemical and physical changes through the process of bubble collapse . A dual-frequency acoustic processor, which has increased cavitation yield, was quantitatively examined by Mohalkar et al. [ 11 ], who proposed that the processor’s more uniform acoustic field was the cause of the rise in cavitation yield. They also offered a way to achieve homogeneous cavitation intensity in the reactor by optimizing the dual-frequency acoustic field settings. According to Tatake et al.’s numerical analysis [ 12 ], the greatest cavitation enhancement is anticipated when a second acoustic wave with the same frequency as the first one is introduced. In their numerical modeling, Servent et al. [ 13 ] took into account the spatiotemporal dynamics of cavitation bubbles and nucleation spots in the three-dimensional geometry of an acoustic chemical reactor. They also concluded that multi-frequency systems have a greater cavitation bubble volume fraction (indicating that more cavitation events may occur with a dual-frequency source than with a single-frequency source).

Dual- or multi-frequency ultrasonography has been widely used in recent years to boost sonochemical yield and cavitation intensity [ 14 ]. Multi-frequency systems encourage cavitation nucleation more than single-frequency ultrasonic cavitation. Furthermore, bubbles have a higher growth rate with a collapse pressure. Brotchie et al. [ 15 ] showed that simultaneous ultrasonic excitation at 20 kHz and 355 kHz greatly increased both acoustoluminescence and sonochemistry. According to Saletes et al. [ 16 ], lowering the cavitation threshold for multi-frequency ultrasonic excitation makes it easier for transient cavitation to start. Koufaki et al. [ 17 ] significantly shortened the reaction time for the synthesis of 3,5 dimethylisoxazole by dual-frequency ultrasound at 20 kHz and 40 kHz. The transient cavitation threshold of microbubbles under multi-frequency ultrasonic irradiation was examined by Suo et al. [ 18 ], who also examined the correlation between the cavitation threshold and microbubble size at various frequencies and media. They put forth a theoretical framework for lowering the cavitation threshold using dual-frequency ultrasonography.

The process is still unclear, despite numerous researchers’ experimental confirmation that dual-frequency and multi-frequency ultrasound can enhance cavitation intensity and sonochemical yield . Information on cavitation can be efficiently obtained through numerical analysis of cavitation kinetics [ 19 ]. More variables, including the frequency and amplitude of each ultrasonic beam, their phase difference, and the power distribution, are present in multi-frequency ultrasonic systems. Cavitation phenomena and bubble oscillations become more intricate as a result.

Khanna et al. [ 20 ] explained the spatial distribution of cavitation intensity and the dynamic phase diagram of cavitation using the dual-frequency ultrasound model. They also observed that for the same ultrasonic pressure amplitude, the cavitation strength increased as the dual-frequency ultrasound frequency ratio droped. The impact of dual-frequency ultrasound’s frequency and pressure amplitude on heat transfer at the bubble-liquid interface was examined numerically by Zhang et al. [ 21 ]. They proposed that cavitation may be more successfully promoted by boosting the power of dual-frequency ultrasound’s low-frequency component. The wider frequency spectrum created by nonlinear mixing of dual-frequency ultrasound was the reason why Guédra et al. [ 22 ] used the asymptotic method to analyze the nonlinear vibration of bubbles generated by dual-frequency ultrasound excitation. They discovered that at high enough ultrasound pressures and a specific frequency difference, the response amplitude of the bubbles was larger than that of single-frequency ultrasound excitation. In order to investigate the dynamic evolution of bubbles in single- and dual-frequency ultrasound modes, Ye et al. [ 23 ] created a dual-frequency ultrasound cavitation dynamics model. The findings demonstrated that cavitation bubbles oscillate more unstably and collapse more readily in a dual-frequency ultrasound field than in a single-frequency ultrasound field. The cavitation impact is lessened as the ultrasonic frequency rises because of the shorter oscillation period. Azam et al. [ 24 ] used multi-frequency (20 kHz, 40 kHz, and 60 kHz) combined ultrasonic modes to remove residual pesticides, such as methomyl from the surface of fresh lettuce and found that the triple-frequency combined modes were more effective in removing the residual pesticides. Liao et al. [ 25 ] showed an acoustic cavitation under ultrasonic irradiation is affected by dual-frequency coupling factors, including frequency difference, phase difference, and power distribution ratio. The acoustic cavitation was shown to be significantly impacted by every coupling parameter. While the combination of high and low frequencies easily created an attenuation effect, the mixing of low and low frequencies was discovered to have an augmentation impact for various frequency combinations. In order to promote mass transfer or diffusion, Wang et al. [ 26 ] numerically examined the mass transfer behavior of microbubbles in liquids under multi-frequency acoustic excitation with different frequencies, pressure amplitudes, and amplitude ratios. They proposed that the low-frequency component of the multi-frequency acoustic excitation should receive the power because low-frequency acoustic excitation has a more significant effect than high frequency in the growth region.

It is clear from the above studies that multi-frequency ultrasound has an important effect on bubble dynamics, cavitation intensity, and acoustic chemical yield in the ultrasound field. However, the research in this area still needs to be further strengthened. In this paper, a multi-frequency ultrasound bubble dynamics model was developed for four mediums: water, urine, kidney, and muscle. The bubble dynamics equations were used to simulate and analyze the effects of multi-frequency ultrasound combination and frequency difference on the maximum and minimum values of bubble radius and bubble collapse time.

Material and Methods

For this theoretical analysis of fluid (i.e., Water and Urine), the Gilmore-Akulichev formulation of the bubble dynamics is shown in equation (1) below [ 27 ]. The computation makes the assumptions that the bubbles stay spherical during the simulation and that nonlinear propagation aberrations do not taint the ultrasonic waves.

$\begin{array}{cc}R(1-\frac{U}{C})\frac{\mathrm{dU}}{\mathrm{dt}}+\frac{3}{2}(1-\frac{U}{\mathrm{3C}})U^2=(1-\frac{U}{C})H+\frac{U}{C}(1-\frac{U}{C})R\frac{\mathrm{dH}}{\mathrm{dR}}& \mathrm{(1)}\end{array}$

The reaction of a single bubble in relation to time is shown by equation (1). The initial radius is represented by R, the first-order derivative with respect to time by U, the speed of sound of the liquid containing the bubble by C, and the enthalpy of that liquid by H. The calculation of equation (1) and its related parameters (such as density ρ, the multivariate exponent κ, the shear coefficient viscosity μ, the surface tension σ) are related to five sub-equations [ 28 - 30 ].

In contrast, in tissues, such as kidney and muscle, a modified Keller-Miksis equation that takes into account the surrounding tissue’s compressibility and viscoelasticity is used to explain the bubble dynamics [ 31 ].

$\begin{array}{cc}(1-\frac{\stackrel{.}{R}}{C})R\stackrel{\mathrm{..}}{R}+\frac{2}{3}(1-\frac{\stackrel{.}{R}}{\mathrm{3C}})R^2=(1-\frac{\stackrel{.}{R}}{C})[\frac{P_B-P_{\infty }\left(t\right)}{\rho }-\frac{\mathrm{4\mu R}}{R}-\frac{\mathrm{2\sigma }}{\mathrm{\rho R}}-E_{\mathrm{NT}}]+\frac{R}{\mathrm{\rho c}}\frac{d}{\mathrm{dt}}(P_a-P_{\infty })& \mathrm{(1)}\end{array}$

Where c is the constant speed of sound in the tissue, “.” denotes the time derivative, and $\stackrel{\mathrm{..}}{R}$ is the bubble wall acceleration. The tissue’s viscoelasticity is modeled using the Kelvin-Voigt method. The hyperelastic constitutive relation is used to obtain the elastic term while taking into account massive deformations during bubble growth [ 32 ].

$\begin{array}{cc}{E}_{\mathrm{NT}}=\frac{G}{2}[5-4(\frac{R_0}{R})-{\left(\frac{R_0}{R}\right)}^4]& \mathrm{(3)}\end{array}$

Where G is the linear shear modulus. For multi-frequency ultrasonic irradiation, the ultrasonic excitation signal can be expressed as:

$\begin{array}{cc}{P}_s=p_1\sin \left(\mathrm{2\pi }{f}_{1}t\right)+\phi p_2\sin (\mathrm{2\pi }{f}_{2}t+\alpha )\varphi p_3\sin (\mathrm{2\pi }{f}_{3}t+\beta )& \mathrm{(4)}\end{array}$

Where f₁, f₂ and f₃ are the excitation signal frequencies, p₁, p₂ and p₃ are the sound pressure amplitudes, 0≤φ≤1, 0≤ϕ≤1. α and β are the excitation signal phase angles, 0≤α≤2π and 0≤β≤2π. The ordinary differential equation solver ode45 was used in MATLAB R2018b (MathWorks, Naticks, MA, USA) to determine the transient bubble radius R. The Gilmore-Akulichev and modified Keller-Miksis equations were numerically solved in MATLAB (MathWorks, Naticks, MA, USA) using the fifth-order Runge-Kutta-Fehlberg method and the step control algorithm [ 27 , 33 , 34 ] with absolute and relative tolerances of 2e-12 and 1e-8, respectively, yielding radius-time parameters of satisfactory accuracy. R_max=2R₀ can be used as a condition for the occurrence of transient cavitation in order to solve the transient cavitation threshold in the medium [ 35 ]. For transient cavitation, the bubble collapse time is defined as the period of time between the bubble’s maximum size and its reduction to less than 1% of its initial size [ 12 ].

(a)When φ=ϕ=0, i.e., a single-frequency (f₁) excitation, p₁=p₀,

$\begin{array}{cc}{p}_s=p_0\sin \left(\mathrm{2\pi }{f}_{1}t\right)& \mathrm{(5)}\end{array}$

(b)When φ=1, ϕ=0, α=0, i.e., a dual-frequency (f₁≠f₂) excitation, p₁ =p₂ = $\frac{1}{\sqrt{2}}$p₀,

$\begin{array}{cc}{p}_s=\frac{1}{\sqrt{2}}{p}_0\sin \left(\mathrm{2\pi }{f}_{1}t\right)+\frac{1}{\sqrt{2}}\sin \left(\mathrm{2\pi }{f}_{2}t\right)& \mathrm{(6)}\end{array}$

(c)When φ=ϕ=1, α=β=0, i.e., a triple-frequency (f₁≠f₂≠f₃) excitation, p₁ = p₂ = p₃ = $\frac{1}{\sqrt{3}}$p₀,

$\begin{array}{cc}{p}_s=\frac{1}{\sqrt{3}}{p}_0\sin \left(\mathrm{2\pi }{f}_{1}t\right)+\frac{1}{\sqrt{3}}\sin \left(\mathrm{2\pi }{f}_{2}t\right)+\frac{1}{\sqrt{3}}\sin \left(\mathrm{2\pi }{f}_{3}t\right)& \mathrm{(7)}\end{array}$

To guarantee that the acoustic power for dual- and triple-frequency excitation is equal to that for single-frequency excitation, the aforementioned factors $\sqrt{2}$ and $\sqrt{3}$ are utilized. Assuming zero velocity at the interface between the first bubble and the surrounding medium ($\stackrel{.}{R}$=0), the bubble’s initial radius ranges from R₀=0.1 μm to 30 μm. Single-frequency, dual-frequency, and triple-frequency ultrasonography are the irradiation modes in the following, unless otherwise noted, and κ is assumed to be 1.4.The physical parameters of urine, water, kidney, and muscle are used for the simulation study of the bubble radius variation and collapse time for all the four mediums as shown in Table 1.

Results

The bubble radii generated by cavitation in the fluid and tissue are shown in Figures 1 and 2 by triple-frequency (1.2+1.8+2.4 MHz, 1.2+1.3+1.4 MHz, 1.2+1.21+1.22 MHz, 0.5+1.1+1.7 MHz, 0.5+0.6+0.7 MHz, and 0.5+0.51+0.52 MHz), dual-frequency (1.2+1.8 MHz and 1.8+2.4 MHz), and single-frequency (1.2 MHz and 2.4 MHz) irradiation of four different medium, the bubble radius generated by cavitation versus time is shown in Figures 1 and 2, and the variation of the bubble radius with time in the fluid and tissues exhibits nonlinear oscillations.

Figure 1. Radius of oscillation of bubbles at different combinations of triple frequency (a) Urine; (b) Water; (c) Kidney; (d) Muscle

Figure 2. Radius of oscillation of bubbles for single, dual, and triple frequency combinations (a) Urine; (b) Water; (c) Kidney; (d) Muscle

The simulation results show that the bubble radius oscillations are related not only to the surrounding medium, but also to the irradiation frequency. The simulation results of bubble expansion radius maximum (R_max), bubble expansion radius minimum (R_min), R_max/R_min, and bubble collapse time (t_c) for different medium under different triple-frequency combinations of ultrasonic irradiation are shown in Tables 2 and 3.

The simulation results of bubble expansion radius maximum (R_max), bubble expansion radius minimum (R_min), R_max/R_min, and bubble collapse time (t_c) for different medium under different single-frequency, dual-frequency and triple-frequency combinations of ultrasonic irradiation are shown in Tables 4 and 5.

In the same medium, the triple-frequency combinations corresponding to the maximum value of bubble radius from small to large are 1.2+1.8+2.4 MHz, 1.2+1.3+1.4 MHz, 1.2+1.21+1.22 MHz, 0.5+1.1+1.7 MHz, 0.5+0.6+0.7 MHz, and 0.5+0.51+0.52 MHz, while the triple-frequency combinations corresponding to the minimum value of bubble radius from small to large are just the opposite. For the single-frequency, dual-frequency and triple-frequency combinations, the irradiation frequencies corresponding to the maximum values of the bubble radius in descending order are 2.4 MHz, 1.8+2.4 MHz, 1.2+1.8+2.4 MHz, 1.2+1.8 MHz and 1.2 MHz, whereas the minimum values of the bubble radius corresponding to the smallest values of the bubble radius in descending order are just the opposite of the irradiation frequencies. The viscosity of the medium limits the bubble oscillations to some extent, lowering the nonlinearity and amplitude [ 32 ]. In addition, the bubble expansion corresponding to the triple-frequency combination with lower frequency and smaller frequency difference is significantly larger than that corresponding to the triple-frequency combination with higher frequency and larger frequency difference, and the maximal bubble radius in muscle generated by the 0.5+0.51+0.52 MHz combination is 1.10 times greater than that produced by the 1.2+1.8+2.4 MHz combination, and 1.09 times higher than that in urine. The maximum value of the bubble radius produced by the 1.2+1.8+2.4 MHz combination is 1.06 times that under the 1.8+2.4 MHz combination in the urine, and the bubble expansion under the triple-frequency combination is significantly larger than that under the dual-frequency combination when the highest frequency of the dual-frequency combination is the same as that of the triple-frequency combination, which is 1.06 times that in the muscle. When the dual-frequency combination’s lowest frequency and the triple-frequency combination’s lowest frequency are identical, the bubble expansion under the dual-frequency combination is significantly larger than that under the triple-frequency combination, and the maximum value of the bubble radius produced by the 1.2+1.8 MHz combination in the urine is 1.11 times as much as that under the 1.2+1.8+2.4 MHz combination, and 1.06 times as much as that in the muscle. When the single-frequency ultrasound frequency was the same as the highest frequency of the dual-frequency combination, the bubble expansion under the dual combination was significantly greater than that under the single-frequency irradiation, and the maximum value of the bubble radius produced by the 1.8+2.4 MHz combination in the urine was 1.02 times higher than that under the 2.4 MHz irradiation, which was 1.05 times higher than that in muscle, and when the ultrasound’s single-frequency matched the dual combination’s lowest frequency, the double combination of bubble expansion was significantly smaller than that under single-frequency irradiation, with 1.2 MHz irradiation producing a maximum bubble radius in urine that was 1.04 times greater than that under the 1.2+1.8 MHz combination, and 1.02 times greater in muscle. For single-frequency ultrasound irradiation, the bubble expansion corresponding to the higher frequency was significantly smaller than that corresponding to the lower frequency, with the maximum bubble radius in urine produced by 1.2 MHz irradiation being 1.25 times that of 2.4 MHz irradiation, and only 1.17 times that of muscle.

Under the same irradiation combination of single-frequency or multi-frequency (dual and triple frequency), the corresponding R_max/R_min values corresponded to the following medium in descending order: urine, water, kidney, and muscle. For the triple-frequency combination, the corresponding R_max/R_min values of the four mediums gradually increased from small to large as the combination frequency varied from high to low, and the frequency difference varied from large into small. Among them, the maximum values of R_max/R_min in urine and muscle were 17.88 and 9.41 for the 0.5+0.51+0.52 MHz combination, respectively, and the minimum values of R_max/R_min were 8.80 and 4.75 for the 1.2+1.8+2.4 MHz combination in urine and muscle, respectively. A total of 2.4 MHz combination had a 2.03-fold increase in R_max/R_min values in urine and about 1.98-fold in muscle. Regarding the combinations of single, dual, and triple frequencies, the corresponding R_max/R_min values in the four mediums in ascending order corresponding to the irradiation frequencies of 2.4 MHz, 1.8+2.4 MHz, 1.2+1.8+2.4 MHz, 1.2+1.8 MHz, and 1.2 MHz; among them, the R_max/R_min values of the 2.4 MHz combination were 5.93 and 5.93 and 1.98 times higher in the urine and the muscle, respectively. The maximum values of the 2.4 MHz combination in urine and muscle were 5.93 and 3.72, respectively, and the minimum values of R_max/R_min for the 1.2 MHz combination in urine and muscle were 14.16 and 7.61, respectively; the 1.2 MHz combination showed a 2.39-fold increase in the R_max/R_min value in urine relative to that of the 2.4 MHz combination, and the value in muscle was about 2.05-fold.

Discussion

This work presents a numerical investigation and comparison of the maximum and minimum values of bubble radius and bubble collapse time utilizing multi-frequency excitation in liquids and tissues with the values obtained from single-frequency excitation. The final equilibrium bubble radius grows rapidly due to the rapid expansion of bubbles created by rectified mass double diffusion under multi-frequency excitation [ 37 ]. Since it is doubtful that the transient cavity will exist for more than one or two excitation cycles, this omission is not important. Additionally, under the adiabatic process assumption, heat transmission is disregarded. It is easy to account for this effect by changing the multidirectional exponent’s value at maximal bubble expansion from 1.4 to 1.0, or an isothermal process. The maximal radius of the bubble is demonstrated to be overestimated by the isothermal assumption and underestimated by the adiabatic assumption [ 38 ]. Variations in temperature also have a significant impact on cavitation. The cavitation threshold in liquids decreases with increasing ambient temperature, although it increases somewhat in tissues [ 39 ]. Because the bubbles do not have enough time to undergo a thermal reaction before transient cavitation occurs, this effect was not considered in this work. In pulsed ultrasound applications like ultrasonic thrombolysis, the temperature rise is negligible even though multi-frequency excitation can raise it during tissue ablation due to greater transient cavitation [ 40 ]. Furthermore, because multi-frequency excitation reduces waveform distortion, it produces less sound absorption at the same strength as single-frequency modes [ 41 ].

Under single-frequency and combined multi-frequency irradiation, the oscillations of the bubbles exhibit both maximum expansion and minimum compression, suggesting that higher temperatures are generated inside the bubbles. Longer bubble collapse times, a lowering minimum bubble radius, and an increasing maximum bubble radius all contribute to an increase in the warmth and intensity of the bubble collapse (as indicated by R_max/R_min and t_c), which is especially advantageous for cavitation effects brought on by ultrasonic irradiation [ 42 , 43 ]. Furthermore, there are variations in the impact of irradiation frequencies and different media on the bubble collapse time. The longer the R_max/R_min ratio for the same medium, regardless of single-frequency or multi-frequency combined irradiation, and vice versa, the shorter the bubble collapse time. For the four types of medium, the longest bubble collapse times under the same single-frequency or multi-frequency combination of irradiation are urine, water, kidney, and muscle. The intensity of bubble growth and collapse reduced as the medium surrounding the bubbles became more viscous.

The dynamics of bubbles are also strongly influenced by their interactions with one another. Neighboring bubbles are attracted or repelled by the radiation pressure produced by bubble oscillations when several bubbles of various radii are present. The acoustic wave’s velocity and amplitude alter when bubbles spread through a liquid, and its attenuation and scattering are significantly increased [ 44 - 46 ]. Additionally, during multi-frequency excitation, intricate patterns can be observed in the sign and magnitude of the Bjerknes force between two bubbles [ 47 , 48 ].

Since bubble dynamics are nonlinear, the emergence of shock fronts or waveform distortions could be quite important. Because the bubble’s response frequency is much lower than its resonance frequency, higher-order harmonics introduced to the distorted waveform have been shown with minimal impact on the occurrence of transient cavitation. The fundamental harmonic is thought to be the primary cause of transient cavitation [ 14 , 49 , 50 ]. Nevertheless, recent research indicates that taking nonlinear waveform distortions into account lowers the maximum bubble size at which transient cavitation occurs , suggesting that our simulations using perfect sinusoids may have overstated the maximum bubble radius. However, because of the decreased waveform distortion, the difference is negligible for multi-frequency excitation. This is particularly true at low frequencies and sound pressure, the bubble radius maximum anticipated by multi-frequency stimulation is more accurate than that predicted by single-frequency excitation.

Conclusion

In this paper, the nonlinear vibration of bubbles in the different mediums was simulated by using Gilmore-Akulichev and modified Keller-Miksis bubble dynamics equations and the dynamic change of bubble radius with time in the medium during multi-frequency combined ultrasonic irradiation was analyzed, and it was investigated how the frequency difference and multi-frequency ultrasonic combination affected the bubble collapse time as well as the bubble radius’s maximum and minimum values. The effects of multi-frequency ultrasound combination and frequency difference on the maximum and minimum values of bubble radius and bubble extinction time in the different mediums were investigated, and this served as a guide for improving the multi-frequency ultrasonic combination to lower the medium’s transient threshold value. The simulation results demonstrate that the combination of the irradiation frequency and frequency difference is related to the bubble radius’s maximum and lowest values in addition to the medium’s characteristics. Where the highest ultrasound frequency is the same for single-frequency, dual-frequency, and triple-frequency combinations, the triple-frequency combination of irradiation under the bubble expansion is the largest and single-frequency irradiation under the bubble expansion is the smallest; when the lowest frequency combination of single-frequency, dual-frequency, and triple-frequency ultrasonography is the same, the triple-frequency combination of irradiation under the bubble expansion is the smallest, and single-frequency irradiation under the bubble expansion is the largest, and the bubble routing time is exactly proportional to the bubble collapse time, with the proportionality coefficient defined as the dimensionless radius ratio R_max/R_min (maximum to minimum bubble radius). This ratio characterizes the oscillation amplitude and correlates with nonlinear energy conversion efficiency . The triple-frequency combination with lower frequency and smaller frequency difference clearly has a larger bubble expansion radius maximum value than the triple-frequency combination with higher frequency and larger frequency difference maximum value. However, the minimum value of the bubble expansion radius is the exact opposite. The bubble collapse time has a positive relationship with the ratio of the maximum to minimum bubble expansion radius. In order to better control cavitation in the medium, the article’s findings can help doctors choose the right frequency combination and frequency difference for multi-frequency ultrasound. They can also serve as a guide for designing tissue ablation treatment protocols and adjusting treatment parameters and irradiation dose during HIFU surgery.

Acknowledgment

The anonymous reviewers’ insightful remarks and recommendations are much appreciated by the authors.

Authors’ Contribution

H. Dong conceived and designed the research idea and framework; G. Liu performed the simulations; G. Peng analyzed the simulated data and wrote the manuscript. H. Dong revised the paper. All authors have read and agreed to the published version of the manuscript.

Ethical Approval

The School of Information Science and Engineering at Changsha Normal University has approved the research and use of necessary resources.

Funding

The Changsha Natural Science Foundation Project (grant No. kq2202313) and the Ministry of Education’s Youth Fund Project of Humanities and Social Sciences Research (22YJCZH025) provided funding for this work.

Conflict of Interest

None

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