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Ultrasonics - Sonochemistry

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Numerical investigation of the inertial cavitation threshold by dual-frequency excitation in the fluid and tissue

Mingjun Wanga,b,⁎, Yufeng Zhoua

a School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Ave., 639798, SingaporebMotor Group, R&D, ASM Pacific Technology Ltd, 3/F, Watson Centre, 16-22 Kung Yip St, Kwai Chung, Hong Kong, PR China

A R T I C L E I N F O

Keywords:Acoustic cavitationInertial cavitation thresholdDual-frequency excitation

A B S T R A C T

Inertial cavitation thresholds, which are defined as bubble growth by 2-fold from the equilibrium radius, by twotypes of ultrasonic excitation (at the classical single-frequency mode and dual-frequency mode) were calculated.The effect of the dual-frequency excitation on the inertial cavitation threshold in the different surrounding media(fluid and tissue) was studied, and the paramount parameters (driving frequency, amplitude ratio, phase dif-ference, and frequency ratio) were also optimized to maximize the inertial cavitation. The numerical predictionconfirms the previous experimental results that the dual-frequency excitation is capable of reducing the inertialcavitation threshold in comparison to the single-frequency one at the same output power. The dual-frequencyexcitation at the high frequency (i.e., 3.1+3.5MHz vs. 1.1+ 1.3MHz) is preferred in this study. The simulationresults suggest that the same amplitudes of individual components, zero phase difference, and large frequencydifference are beneficial for enhancing the bubble cavitation. Overall, this work may provide a theoretical modelfor further investigation of dual-frequency excitation and guidance of its applications for a better outcome.

1. Introduction

Acoustic cavitation induced by ultrasonic irritation, stable and in-ertial cavitation, is known to play a key role in a wide range of ther-apeutic ultrasound applications [1,2]. Bubble oscillation about itsequilibrium radius at the low acoustic pressure is termed as stable ca-vitation whereas inertial cavitation releases high energy, increases thetemperature of the vapor to several thousand kelvins, and generateshigh pressure and/or high-speed jets at the bubble collapse, which mayproduce mechanical damages to the nearby media [3,4]. Increasing thedriving acoustic pressure could result in the transition from stable ca-vitation to inertial cavitation. Examples of successful ultrasoundtherapies utilizing acoustic cavitation include the tissue heating inwhich stable cavitation causes high shear stress between the bubble andsurrounding medium and inertial cavitation produces the broadbandnoise emission for the substantially promoted and enhanced thermaldeposition [5,6]; histotripsy in which inertial cavitation releasesshockwaves capable of destroying cell membranes, mechanically frac-tionating the soft tissues, and in many cases even completely liquefyingthem into subcellular components [7,8]; thrombolysis in which stablecavitation in the vicinity of blood clots promotes the uptake of lyticagent and inertial cavitation produces micro-jet and pitting to directlyand mechanically damage the clot’s surface [9,10]; blood-brain barrier

(BBB) opening in which stable cavitation temporarily affects the in-tegration of the tight junction and reduces the blood flow. It could beseen that stable cavitation and inertial cavitation function distinctivelydifferent in the aforementioned cavitation facilitated therapeutic pro-cesses. Therefore, it is of a great concern to determine the thresholdpressure for the transition from stable cavitation to inertial cavitation inorder to assess the likelihood of different ultrasound-induced bio-ef-fects.

In contrast to the conventional single-frequency acoustic excitation,the dual- or multiple-frequency excitation shows some remarkablemerits in sonoluminescence and sonochemistry, such as low cavitationthreshold, more chemical activities, and high reaction efficiency, ef-fective elimination of standing wave, and better control over the cavi-tational activity in the acoustic field in terms of bubble oscillation modeand the spatial distribution [11]. The maximum photocurrent could beboosted up to 300% using double harmonic driving compared withsingle harmonic driving [12]. In the context of biomedicine, some in-itial investigations of enhancing the inertial cavitation by the dual-frequency have also been carried out. Dual-frequency excitation in bothkHz (525+ 565 kHz) and MHz (∼1.5MHz) improved the in vitrothrombolysis efficiency [13,14]. Guo et al. [15] showed ∼5% to 10%improvement in the temperature elevation in the chicken breast by thedual-frequency excitation. Kuang et al. [16] and He et Al. [17] have

https://doi.org/10.1016/j.ultsonch.2017.11.045Received 27 June 2017; Received in revised form 18 October 2017; Accepted 29 November 2017

⁎ Corresponding author at: School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Ave., 639798, Singapore.E-mail address: wang0683@e.ntu.edu.sg (M. Wang).

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demonstrated that using two confocal transducers driven at twoneighboring frequencies (1.495+1.505MHz and 1.563+ 1.573MHz)a large lesion could be achieved. Liu et al. [18] and Chen et al. [19]showed that lesions created by the dual-frequency excitation were moreuniform and closer to the desired target region. Dual-frequency(500 kHz+3MHz) excitation in histotripsy has been shown to pre-cisely tailor the bubble expansion [20,21]. Sokka et al. [22] have suc-cessfully manipulated the cavitation distribution and preferentiallylowered the cavitation threshold at the focus by the means of dual-frequency phased array. The first in vivo dual-frequency sonodynamictherapy performed on mice showed more effective in terms of sup-pressing tumor growth and reducing the tumor volumes [23].

The mechanism for the enhancement of bubble cavitation by dual-or multiple-frequency excitation has not been fully understood al-though some unique characteristics were found in some preliminaryinvestigations. The modeling developed by Zhang et al. [24] shows thatbubble cavitation induced by the dual-frequency excitation has a re-duced critical radius for the generation of inertial cavitation comparingwith that by the single-frequency one. Acoustical scattering cross sec-tion by the dual-frequency excitation displayed more resonances for amuch broader range of bubbles [25]. In addition, dual-frequency ex-citation can lead to a more rapid growth of a wide range of bubblesthrough the rectified mass diffusion as long as the acoustic pressure isabove a certain threshold [26].

So far, plenty of experimental work has confirmed that the dual-frequency excitation could significantly reduce the inertial cavitationthreshold [27,28], but few theoretical studies have been devoted to thissubject. Numerical modeling shows promise towards understanding themechanisms due to its universe applications under various conditions.A theoretical prediction of the inertial cavitation threshold by dual-frequency excitation in the various media, such as fluid (i.e., water andblood) and viscoelastic tissue (i.e., kidney and liver), was studied here,and the paramount parameters that influence the inertial cavitationthreshold (i.e., amplitude ratio, phase difference, frequency ratio) werethoroughly investigated in order to optimize the bubble dynamics.

2. Methods

For the single-frequency excitation, the driving signal is describedas:

=P P sin πft(2 )a A (1)

where f is the driving frequency and PA is the acoustic pressure. For thedual-frequency excitation, the driving signal is described as:

= + +P P sin πf t P sin πf t φ(2 ) (2 )a 1 1 2 2 (2)

where f1and f2 (usually <f f1 2) and P1 and P2 are the driving fre-quencies and acoustic pressures of two harmonics, respectively, and φ isthe phase difference between them. The common setting is

= =P P P / 2A1 2 , where the coefficient 1/ 2 is to ensure the sameacoustic output power delivered as that of the single-frequency ex-citation.

Since the bubble behaves differently in various media, two theore-tical models are employed to calculate its dynamics in the fluid andtissue. For simplicity, several important assumptions are made[6,29,30]. 1) The medium is homogeneous and isotropic; 2) The airbubble is always spherical and initially at rest ( =R 0) so that bubbledynamics can be calculated in one-dimensional space; 3) The gas insidethe bubble is ideally adiabatic (γ=1.4); 4) There is no gas or vaporexchange between the gas and the surrounding medium; 5) The pres-sure inside the bubble is spatially uniform, and the vapor pressure isconstant throughout the bubble oscillation.

For the fluid (i.e., water and blood), the Gilmore-Akuliches equationwas utilized [31].

⎛⎝

− ⎞⎠

+ ⎛⎝

− ⎞⎠

= ⎛⎝

+ ⎞⎠

+ ⎛⎝

− ⎞⎠

R UC

dUdt

UC

U UC

HC

UC

RdHdt

1 32

13

1 1 12(3)

where R is the bubble radius, U (=dR/dt) is the velocity at the bubblewall, and C and H are the speed of sound in the liquid at the bubble walland enthalpy difference between the pressure at the bubble wall P(R)and the pressure at infinity ∞P , respectively, which are given by

∫=∞

H dPρ

P

P R( )

(4)

= + −C C m H[ ( 1) ]l2 1/2 (5)

where P is the time-varying pressure, ρ is the medium density, Cl is theinfinitesimal speed of sound in the liquid, and m is a constant normallyset to 7. The state equation of a compressible fluid and P(R) in relationto the gas pressure inside the bubble of Pg, medium viscosity of µ, andsurface tension of σ in the liquid are given by

= −P A ρ ρ B( / )m0 (6)

= − −P R P σR

μR

U( ) 2 4g (7)

where ρ0 is the equilibrium liquid density, =A C ρ P/l2

0 and B=A−1are two constants.

Whereas in the tissue, the bubble dynamics was modeled by themodified Keller-Miksis equation that includes the compressibility andviscoelasticity of the surrounding medium [29].

⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎛⎝

− ⎞⎠

+ ⎛⎝

− ⎞⎠

= ⎛⎝

+ ⎞⎠

⎡⎣⎢

−− − − ⎤

⎦⎥

+ −

∞

∞

Rc

RR Rc

R Rc

p p tρ

μRR

SρR

E

Rρc

ddt

p p

1 ¨ 3

21

3

1 ( ) 4 2

( )

BNT

a

2

(8)

where c is the constant speed of sound in the medium, and S is thesurface tension coefficient. Kelvin–Voigt approach was used to modelthe viscoelasticity of tissue. Considering the large deformations duringthe bubble growth, a hyperelastic (Neo-Hookean) constitutive relationwas utilized to yield the elastic term [32].

= ⎡⎣⎢

− ⎛⎝

⎞⎠

−⎛⎝

⎞⎠

⎤⎦⎥

E G RR

RR2

5 4NT0 0

4

(9)

where R0 is the initial bubble radius, and G is the linear shear modulus.There are many definitions of the inertial cavitation threshold

[30,33–35]. The maximum bubble expansion to twice of the initialbubble radius is commonly used in the investigation of cavitation andchosen as the onset of inertial cavitation in this paper. The Gilmore andKeller-Miksis equations were numerically solved using the fifth-orderRunge-Kutta-Fehlberg method with a step-size control algorithm inMATLAB (MathWorks, Naticks, MA, USA) with an absolute and relativetolerance of 1e-12 and 1e-7, respectively, to obtain the radius-timeprofiles with satisfactory precision. The sonication period is 60 µs,which is sufficiently long for bubble dynamics. Throughout this study,the threshold value was determined by varying the acoustic pressure ina step size of 2 kPa for a bubble radius ranging from 0.1 µm to 10 µm.The driving frequencies (f1 and f2) were set to 1.1 MHz and 1.3MHz,3.1 MHz, and 3.5MHz, which are within the -6 dB bandwidth of acommercial HIFU transducer (H102, Sonic Concepts, Woodinville, WA,USA) at its fundamental and third harmonic. Water, blood, kidney, andliver are commonly used for the study of sonochemistry reaction,thrombolysis, blood-brain barrier opening, and ultrasound-inducedbiological effects, respectively, and their physical parameters are listedin Table 1 [30,36]. Vapor pressure in the tissue and ambient pressurewere assumed to be 6 kPa and 101.3 kPa, respectively.

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3. Results

3.1. Bubble dynamics

With the introduction of another acoustic wave in the dual-fre-quency excitation, the driving signal is in a beat mode with repetitiveconstruction and destruction. The representative bubble dynamicsdriven by the single- and dual-frequency excitation (1.1+1.3MHz and3.1+3.5MHz) are shown in Figs. 1 and 2, respectively. Initial bubblesize was set to 1 µm while the acoustic pressure to 5 bars. It is foundthat the bubble dynamics relies on both the surrounding medium andthe excitation source. The dual-frequency excitation at the high fre-quency (3.1+ 3.5MHz) has smaller bubble radius than that at the lowfrequency (1.1+ 1.3MHz). The damping time for a bubble in the waterand blood was around 3 cycles for the single-frequency excitation.However, the bubble got a steady oscillation immediately for the dual-frequency excitation. The viscoelasticity of tissue restricts the bubbleoscillation, reducing the amplitude and nonlinearity [30]. Subse-quently, the bubble oscillation in the tissue was immediately stabilizedregardless of the excitation conditions. Furthermore, the bubble ex-pansion by the dual-frequency excitation is clearly larger than that ofthe single-frequency one, particularly in the tissue and at the high

frequency. The maximum bubble radius under the dual-frequency ex-citation (3.1+ 3.5MHz) is 1.22 fold of 3.1MHz-excitation in the liver,but only 1.1 fold in the water.

The bubble dynamics in the tissue under the dual-frequency ex-citation also shows some unique characteristics in comparison to that inthe fluid. The bubble oscillation exhibits not only larger maximumexpansion but also smaller minimum compression, implying a higherinternal temperature. In addition, the bubble collapse time, tc, the in-terval time between the bubble reaching its maximum size and be-coming less than 1% of the initial size [11], is not severely altered bythe dual-frequency excitation. The increased maximum bubble radius,reduced minimum bubble radius, and the unaffected bubble collapsetime (characterized by R t/max c

3 and R R/max min) lead to the enhancedstrength and temperature of bubble collapse, which is particularlypreferred for the induced bio-effects [37,38]. The simulation results aresummarized in Tables 2 and 3. The increased viscoelasticity of sur-rounding medium (tissue vs. fluid) reduces the violence of bubblegrowth and collapse significantly. The value of R R/max min in the waterunder the ultrasound exposure at the frequency of 3.1MHz, 3.5 MHz,and 3.1+ 3.5MHz was 23.92, 19.47, and 28.05, respectively; whereasthe corresponding values in the liver are 3.77, 3.37, and 7.72, respec-tively. The enhancement of dual-frequency excitation to the single-frequency one in the liver is about 2.3 fold, but only 1.4 fold in thewater.

3.2. Inertial cavitation threshold

The inertial cavitation thresholds using the single- and dual-fre-quency excitations were calculated in all the aforementioned mediawith the bubble radius ranging from 0.1 µm to 10 µm in a step size of0.1 µm and illustrated in Fig. 3. It is found that the inertial cavitationthreshold initially drops to the global minimum and then graduallyincreases with the initial bubble radius. The dual-frequency excitation

Table 1Material properties.

Material Soundspeed (m/s)

Density(kg/m3)

Surfacetension (mN/m)

Viscosity(mPa·s)

Rigidity(MPa)

Water 1500 1000 68 1.0 0.00Blood 1570 1050 56 5.0 0.00Kidney 1561 1100 56 5.0 0.18Liver 1549 1100 56 9.0 0.04

Fig. 1. The instantaneous dynamics of a 1 µm bubble driven by the single- (1.1 MHz and 1.3 MHz) and dual-frequency (1.1+ 1.3MHz) at the acoustic pressure of 5 bars in the a) water, b)blood, c) kidney, and d) liver.

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could significantly lower the inertial cavitation threshold, particularlyin the blood and water. Both the high driving frequency and viscoe-lasticity lead to an increase of the threshold. The inertia of the sur-rounding medium suppresses the bubble collapse during the half-cycleof the compressive wave, and then bubble continues to grow during thefollowing half-cycle of the tensile wave, hence leading to a higher

cavitation threshold [39]. However, the inertial cavitation thresholds inthe water and blood have small variations comparing with those in thekidney and liver, particularly for the large initial bubble radii at thehigh frequency. This occurrence may be due to the onset of chaoticoscillations, where bubble collapses at irregular intervals [40].

The optimal bubble size and the corresponding inertial cavitation

Fig. 2. The instantaneous dynamics of a 1 µm bubble driven by the single- (3.1 MHz and 3.5MHz) and dual-frequency excitation (3.1+ 3.5MHz) at the acoustic pressure of 5 bars in thea) water, b) blood, c) kidney, and d) liver.

Table 2Comparison of the bubble dynamics in water and blood.

Material Water Blood

Frequency(MHz)

Rmax(µm)

Rmin(µm)

R R/max min tc (µs) R t/max c3 Rmax

(µm)Rmin(µm)

R R/max min tc (µs) R t/max c3

3.1 3.9996 0.1672 23.92 0.306 209.09 2.8126 0.2678 10.50 0.123 180.893.5 3.6892 0.1895 19.47 0.278 180.61 2.8742 0.2675 10.74 0.111 213.913.1+ 3.5 4.3679 0.1557 28.05 0.358 232.77 3.2586 0.2183 14.93 0.125 276.811.1 8.2643 0.0657 125.79 0.327 1726.12 5.4786 0.0887 61.77 0.219 750.871.3 7.1055 0.0762 93.25 0.289 1241.33 4.6883 0.1086 43.17 0.186 554.031.1+ 1.3 9.2253 0.0678 136.07 0.386 2032.96 7.0516 0.0834 84.55 0.26 1348.62

Table 3Comparison of the bubble dynamics in the kidney and liver.

Material kidney liver

Frequency(MHz)

Rmax(µm)

Rmin(µm)

R R/max min tc (µs) R t/max c3 Rmax

(µm)Rmin(µm)

R R/max min tc (µs) R t/max c3

3.1 1.6611 0.4409 3.77 0.078 58.76 1.6147 0.5302 3.05 0.1 42.103.5 1.569 0.4658 3.37 0.073 52.91 1.5391 0.5451 2.82 0.084 43.403.1+ 3.5 2.0212 0.2617 7.72 0.073 113.11 1.9835 0.3875 5.12 0.082 95.171.1 2.2224 0.4802 4.63 0.133 82.53 3.0928 0.2737 11.30 0.161 183.751.3 2.0866 0.4168 5.01 0.122 74.47 2.6916 0.3386 7.95 0.143 136.361.1+ 1.3 3.7296 0.1322 28.21 0.14 370.56 4.5703 0.1548 29.52 0.172 555.02

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threshold are summarized in Table 4. It could be seen that the dual-frequency excitation does not significantly change the optimal bubblesize but only reduce the inertial cavitation threshold. The optimalbubble size is a function of the driving frequency and acoustic pressuregiven by [41]:

=+

R p fG f K p

( , ) 1( ) · 2 (10)

where G(f) is a function of frequency independent of the acousticpressure, K is an unknown constant, and p is the acoustic pressure. Thelinear resonant frequency for a specific bubble size in the fluid [42] andtissue [32,43] can be described by the following equations, respec-tively.

⎜ ⎟= ⎛⎝

− + ⎞⎠

+ − −fπR ρ

γ p p SR

p SR

μρR

12

3 2 2 4v v

00

0 0

2

02 (11)

= − + − − +fπR ρ

γ p p γ SR

μρR

G12

3 ( ) (3 1) 2 84v

00

0

2

02 (12)

The resonant structure is related to the (sub) fractional-order sub-harmonic resonance minima as described previously [44] and shown byarrows in Fig. 3. Another noticeable trend in the threshold is the shift oflocal resonance minima by the dual-frequency excitation. Because ofthe interference of individual components, bubble oscillation by thedual-frequency excitation shows a combination resonance and si-multaneous resonance and hence is initially assumed to have more localminima [45]. Nevertheless, our calculation results have a much smalleramount of local minima for the dual-frequency excitation, 4 localminima for the dual-frequency excitation vs. 7 or 8 for the single-fre-quency excitation in Fig. 3b, which further testifies the previous con-clusion that the dual-frequency excitation may lead to a stabilizedbubble oscillation [46]. Furthermore, it is worth noticing that the in-ertial cavitation threshold by the dual-frequency excitation at the highfrequency is not only lower than that of the individual component(3.1MHz and 3.5MHz) but also lower than that of 1.1MHz and1.3MHz when the bubble size is no larger than about 0.5 µm, which islikely related to the bubble resonance size (see the inset plot in Fig. 3).It is well known that the resonant bubble size is inversely proportional

Fig. 3. The predicted inertial cavitation threshold (MPa) by the single- (1.1 MHz, 1.3MHz, 3.1MHz, and 3.5MHz) and dual-frequency excitations (1.1+ 1.3MHz and 3.1+ 3.5MHz) forbubble ranging from 0.1 µm to 10 µm in the a) water, b) blood, c) kidney, and d) liver. The inset plots show the results of bubble ranged from 0.1 µm to 1 µm.

Table 4Comparison of the optimal seed bubble size and the corresponding inertial cavitation threshold.

Material Water Blood Kidney Liver

Frequency(MHz)

Bubble size(µm)

Pressure(MPa)

Bubble size(µm)

Pressure(MPa)

Bubble size(µm)

Pressure(MPa)

Bubble size(µm)

Pressure(MPa)

3.1 1.15 0.0970 1.04 0.3080 1.62 0.4394 1.34 0.61103.5 1.04 0.1050 0.97 0.3383 1.46 0.4848 1.23 0.69183.1+ 3.5 1.13 0.0959 1.07 0.2323 1.62 0.3484 1.32 0.47471.1 2.83 0.0505 2.54 0.1263 4.20 0.2020 3.12 0.24751.3 2.46 0.0505 2.18 0.1465 3.56 0.2273 2.76 0.28281.1+ 1.3 2.63 0.0454 2.51 0.1111 4.04 0.1919 3.13 0.2071

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to the driving frequency and the growth of small bubble is highlysensitive around the resonance regime so that even a small frequencychange results in a large variation of resonant bubble radius. Therefore,bubble oscillation is more significant with more occurrence of bubblecollapse by the dual-frequency excitation at the high frequency becauseof broad frequency span range covering more resonant bubble sizes.Since inertial cavitation threshold is strongly dependent on the initialbubble size, the validity of this study essentially relies on the pre-ex-istence of stabilized small bubble nuclei. It has been reported that theradius of cavitation nuclei in the water is in the order of 1 µm or lessand the distribution density decreases exponentially with the radius[47,48]. However, the nuclei radii in vivo were estimated to be less than0.3 μm [49,50]. Therefore, the bubble size in the range of 0.1 to 10 µmused here is close to the real cases. In summary, the dual-frequencyexcitation at the high frequency is preferable for the small bubbles inthe tissue.

3.3. Parameter optimization

In this sub-section, the influences of the paramount parameters (i.e.amplitude ratio, phase difference, and frequency ratio) of the dual-frequency excitation on the inertial cavitation threshold were numeri-cally investigated. Each individual parameter was varied by keeping therest constant.

3.3.1. Amplitude ratioFor the convenient discussion, the ratio of two acoustic pressure

amplitudes is defined as =N P P/p 2 1 and was varied from 0.1 to 10 in astep size of 0.1, implying the gradual shift of energy from the low-fre-quency component to the high-frequency one. The frequencies werekept as 1.1+ 1.3MHz and 3.1+ 3.5MHz, and the phase differencewas set to zero. Generally, the influence of the amplitude ratio is moresignificant for a large bubble (see Figs. 4and 5). Small bubbles are re-sistant to the change of amplitude ratio, which may be due to the large

surface tension [51]. The inertial cavitation threshold with the ampli-tude ratio of Np=0.1, 0.5, 1.0, 2.0, and 5.0 was compared in Fig. 6.Because of the onset of chaotic oscillation, the inertial cavitationthreshold varies significantly over the range of bubble radius in-vestigated, and the optimal amplitude ratio depends on both the bubbleradius and the surrounding medium. The inertial cavitation threshold atNp=1 is found always lower than the other ratios under all simulationconditions. Finally, the dual-frequency excitation at the high frequencyfavors a reduced threshold, particularly in the fluid. Overall, the op-timal amplitude ratio is suggested to be close to 1 for all bubble sizesinvestigated.

3.3.2. Phase differenceThe phase difference between two individual components affects

the acoustic pressure distribution, the energy dissipation, the radicalproduction, and even the onset of chaotic oscillation [46,52]. Here thephase difference increased from 0 to 2π in a step of π/18 (10°), and itsinfluence on the inertial cavitation threshold is shown in Figs. 7 and 8.The inertial cavitation threshold is the lowest when the two compo-nents are in phase (φ=0 or 2π), but the highest for out of phase(φ= π). Furthermore, a large bubble is more sensitive to the phasedifference, which is more apparent in the fluid at the high frequency.Few variations of the threshold are found for bubbles in the tissue at thelow-frequency excitation (see Fig. 9c and 9d). The inertial cavitationthreshold in the fluid is initially not affected much by the phase dif-ference (see Fig. 9). But the discrepancy becomes significant when theinitial bubble radius is larger than 2.5 µm by the excitation of3.1+ 3.5MHz and 6.8 µm by the excitation of 1.1+ 1.3MHz in thewater, and the corresponding values are 1.0 µm and 6.2 µm in theblood, respectively. The critical bubble radius for the significant impactof the phase difference at the high frequency is 5.8 µm and 5 µm for theliver and the kidney, respectively. However, the phase difference seemsto be more insensitive to the bubble radius at the low frequency, whichmay be due to the reduced nonlinearity of bubble oscillation as a

Fig. 4. The inertial cavitation threshold (MPa) at the excitation of 1.1+ 1.3MHz as a function of bubble size ranging from 0.1 µm to 10 µm and the amplitude ratio from 0.1 to 10 in astep size of 0.1 in the a) water, b) blood, c) kidney, and d) liver.

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consequence of the long rarefaction duration [53].

3.3.3. Frequency ratioThe frequency ratio is defined as =N f f/f 1 2 and varies from 0.1 to 10

in a step size of 0.1 (setting φ=0, Np=1, and =f 1.12 MHz or3.1 MHz). The influences of the frequency ratio are shown inFigs. 10–12. The inertial cavitation threshold first increases rapidlywith the frequency ratio and then remains quite stable. Large initial

Fig. 5. The inertial cavitation threshold (MPa) at the excitation of 3.1+ 3.5MHz as a function of bubble size ranging from 0.1 µm to 10 µm and the amplitude ratio from 0.1 to 10 in astep size of 0.1 in the a) water, b) blood, c) kidney, and d) liver.

Fig. 6. The dependence of inertial cavitation threshold (MPa) on the bubble ranging from 0.1 µm to 10 µm at the amplitude ratio of 0.1, 0.5, 1, 2, and 5 driven by the excitations of a)1.1+1.3MHz and b) 3.1+ 3.5MHz in the water, blood, kidney, and liver (from left to right). The inset plot compares the results of bubble ranging from 0.1 µm to 1 µm.

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Fig. 7. The inertial cavitation threshold (MPa) at the excitation of 3.1+ 3.5MHz as a function of bubble size ranging from 0.1 µm to 10 µm and the phase difference from 0 to 2π in a stepsize of π/18 in the a) water, b) blood, c) kidney, and d) liver.

Fig. 8. The inertial cavitation threshold (MPa) at the excitation of 1.1+ 1.3MHz as a function of bubble size ranging from 0.1 µm to 10 µm and the phase difference from 0 to 2π in a stepsize of π/18 in the a) water, b) blood, c) kidney, and d) liver.

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Fig. 9. The inertial cavitation threshold (MPa) of bubble ranging from 0.1 µm to 10 µm at the phase difference of 0, π/2, and π by the excitations of 1.1+ 1.3MHz and 3.1+3.5MHz inthe a) water, b) blood, c) kidney, and d) liver.

Fig. 10. The inertial cavitation threshold (MPa) by the low dual-frequency excitation (f2=1.1MHz) as a function of bubble ranging from 0.1 µm to 10 µm and the frequency ratio from0.1 to 10 in a step size of 0.1 in the a) water, b) blood, c) kidney, and d) liver.

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bubbles are insensitive to the change of frequency ratio, which is con-trary to the cases of the amplitude ratio and phase difference. Thefluctuation of the inertial cavitation threshold at the high frequency issmooth and small, but large and sharp at the low frequency, particu-larly in the water and blood (see Fig. 12). The larger the frequencydifference, the lower of the cavitation threshold. Overall, the results

suggest that large frequency difference is favorable for the dual-fre-quency excitation. However, the practice is usually limited by thebandwidth of a single transducer; otherwise, two or more separatedtransducers are required.

Fig. 11. The inertial cavitation threshold (MPa) by thehigh dual-frequency excitation (f2=3.1MHz) as afunction of bubble ranging from 0.1 µm to 10 µm andthe frequency ratio from 0.1 to 10 in a step size of 0.1in the a) water, b) blood, c) kidney, and d) liver.

Fig. 12. The inertial cavitation threshold of the bubble ranging from 0.1 µm to 10 µm at the frequency ratio of 0.1, 0.5, 1.0, 2.0, 5.0, and 10.0 driven by a) low dual-frequency excitation(f2=1.1MHz) and b) high dual-frequency excitation (f2=3.1MHz) in the water, blood, kidney and liver (from left to right).

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4. Discussion

In this study, inertial cavitation threshold using the dual-frequencyexcitation in the fluid and tissue was investigated numerically andcompared with that using the single-frequency one, and paramountparameters were optimized. The bubble growth through the rectifiedmass diffusion by the dual-frequency excitation is so fast that the finalequilibrium bubble radius becomes larger [26]. However, the modelsused here did not account for the gas diffusion across the bubble wall.Such an omission is not critical because the inertial cavities are notlikely to survive more than one or two cycles of excitation. Our simu-lation involved in the mass transfer shows that the difference of max-imum bubble radius is less than 1% (data not included), but the com-putation time is 10 times longer. Under the assumption of the adiabaticprocess, heat transfer was also neglected. To include its influence, thevalue of polytropic exponent can simply be changed from 1.4 to 1.0 atthe maximum bubble expansion, which means the isothermal process.It has been shown that the adiabatic assumption underestimated themaximum bubble radius while the isothermal assumption over-estimated it [54]. The error between these two assumptions was ap-proximately 10% and acceptable in the simulation. The temperaturevariation also plays a significant role in the inertial cavitationthreshold. Higher ambient temperature leads to a lower cavitationthreshold in the fluid, but a slightly higher one in the tissue [39]. Thisstudy did not consider this effect because the inertial cavitation occursin such a short time that the bubble does not have sufficient time for thethermal response. Although multiple-frequency excitations were able toenhance the temperature elevation in the tissue ablation because of theenhanced inertial cavitation, the temperature elevation in the pulsedultrasound application, such as sonothrombolysis, is insignificant [55].Furthermore, dual-frequency excitation has less acoustic absorptionthan the single-frequency mode at the same power because of the re-duced waveform distortion [56].

The interaction between bubbles also has a great impact on theirdynamics. When numerous bubbles with different radii are presented,the radiation pressure generated by the bubble oscillation attracts orrepulses the neighboring bubbles mutually. The velocity and amplitudeof the acoustic waves will be modified when traveling through thebubbly liquids with much higher acoustic attenuation and scattering[57,58]. It has been shown that the inertial cavitation threshold of anencapsulated microbubble increases with its concentration [59]. Inaddition, the sign and strength of the Bjerknes force between twobubbles under the dual-frequency excitation showed a complicatedpattern [20]. Therefore, the determination of inertial cavitation at thepresence of bubble cloud is quite challenging and needs continuousinvestigations.

Nonlinear distortion in the acoustic wave propagation was not takeninto account in this study, which limits the accuracy of the prediction,especially for the applications at the high acoustic intensity [60]. Due tothe nonlinear nature of bubble dynamics, waveform distortion or theformation of shock front is likely to have a great influence. It has beenshown that higher-order harmonics introduced in the distorted wave-form have little effect on the occurrence of inertial cavitation becausebubbles respond well below their resonant frequencies and the funda-mental harmonic is suggested to be the major contributor for the in-ertial cavitation [54,61]. Nevertheless, the recent study revealed thatthe maximum bubble size undergoing inertial cavitation was reducedwith the consideration of the nonlinear waveform distortion [62],which implies that our simulation utilizing an ideal sinusoidal wavemay overestimate the inertial cavitation threshold. However, the dis-crepancy is less at the dual-frequency excitation because of less wave-form distortion. Therefore, the predicted inertial cavitation threshold ofthe dual-frequency excitation is more reliable than that of the single-frequency one, especially at the low frequency and acoustic pressure.

Although the simulation has confirmed the experimental observa-tion that dual frequency can reduce the inertial cavitation threshold,

the underlying mechanisms are complex and still not very clear now. Sofar, several possibilities have been presented. Firstly, the dual-fre-quency irritation can produce more bubble nucleation as the result ofthe low-frequency stimulation and the intensified Bjerknes force [20].Secondly, bubble oscillation by the dual-frequency excitation shows acombination resonance and simultaneous resonance, which could causecavitation over a wider range of bubble radii [45]. Lastly, the increasedpeak negative pressure by the dual-frequency excitation diminishes thecavitation threshold and increases the number of bubbles. This quasi-static pressure contributes to an increase of the collapse rate during thepositive pressure [63,64].

5. Conclusion

In this study, the effect of dual-frequency excitation on the inertialcavitation threshold was calculated in the both fluid and tissue andcompared with the single-frequency excitation. It is found that the in-duction of another acoustic component into the excitation can sub-stantially reduce the inertial cavitation threshold. The effect of para-mount parameters (amplitude ratio, phase difference, and frequencyratio) of the dual-frequency excitation on the inertial cavitationthreshold was also studied to optimize the outcome. It is shown that theamplitude ratio of 1 (same amplitude for each individual component),the phase difference of 0 or 2π (in phase), and the large frequencydifference would achieve the best performance, the low inertial cavi-tation threshold. Experimental measurement is required to confirm ourconclusions. Further investigation, such as increasing the acoustic in-tensity, would extend our work for a wide range of practical applica-tions.

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