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

Highlights

• •Nonlinear effect of dual-frequency ultrasound on ADV nucleation was studied.
• •Nucleation rate was decreased due to nonlinear waveform distortion.
• •Non-elliptical nucleation region distribution was produced and could be regulated.
• •Dual-frequency combinations like 1.5 MHz + 3 MHz could enhance nonlinear effects.

Abstract

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.

1. Introduction

In the field of biomedical ultrasound, microbubbles could contribute to increasing the heating rate [1], cavitation [2] and mechanical effects [3] and have been widely employed in the applications including high intensity focused ultrasound (HIFU) ablation [4], boiling histotripsy [5], drug delivery [6], thrombus dissolution [7], etc. However, the large size and short lifespan of microbubbles limit their clinical application. Recently, as an alternative to conventional microbubbles, nanosized acoustically sensitive droplets, such as perfluorocarbon (PFC) nanodroplets, have been developed and employed [8]. Triggered by the acoustic wave, droplets experience a phase-shift process from liquid to gas named as acoustic droplet vaporization (ADV) [9], which plays an increasingly indispensable role in enhanced HIFU treatment, histotripsy and ultrasound monitoring imaging [10], [11].

ADV was normally triggered by single-frequency ultrasound, which has stronger mechanical effect at higher frequency ultrasound [12] and enhanced cavitation-induced bio-effects at lower frequency ultrasound [13]. However, it has been reported that the high ADV threshold and low cavitation efficiency when using single-frequency excitation lead to unsatisfactory therapeutic effects and limit the ADV applications in theranostics [14], [15]. To overcome the limitations, dual-frequency excitation has been employed as a promising method and investigated in several experimental and numerical studies. Xu et al. found that the dual-frequency ultrasound could reduce the ADV threshold and improve the ADV efficiency in vitro experiments [16]. Tatake and Pandit demonstrated that the dual-frequency ultrasound had better control over bubble oscillation mode and the spatial distribution in both modelling and experimental studies [17]. The possible mechanisms were usually investigated in numerical studies for the universe applications under various conditions and convenience in conducting a large number of experiments involving varied parameters. It has been reported that the inertial cavitation threshold could be lowered by dual-frequency ultrasound in the following ways including the intensified Bjerknes force, combination resonance and simultaneous resonance of bubble oscillation, the increase mass transfer through the bubble interface, the increased peak negative pressure, etc. [18], [19] Our previous study reported that the on-demand spatiotemporal control of ADV nucleation can be achieved due to independent control of both the acoustic and temperature fields using dual-frequency ultrasound [20].

In recent years, when it comes to the cavitation-related numerical studies using dual-frequency ultrasound, linear acoustic model is commonly used [21], [22] because it simplifies the complexity and reduces computational burden of the model. It makes sense to a certain extent if low intensity ultrasound excitation is applied [23]. However, in many practical application scenarios like boiling histotripsy and HIFU ablation, high acoustic intensity is required and linear model is not the best fit for particular situations. Nonlinear effects have a significant impact on almost all fields of biomedical ultrasound including ADV especially in high intensity conditions [24]. The change of peak negative pressure caused by waveform distortion exercises considerable influence over ADV threshold and subsequent vapor bubble growth and dynamics [25]. Moreover, higher temperature caused by high-order harmonic components according to power-law acoustic absorption is instrumental in the biological activity of tissue and drugs [26]. Furthermore, some studies illuminate the dependence of the nucleation threshold on the ultrasound frequency and suggest that superharmonic focusing effects of ultrasonic nonlinearity strongly contribute to the ADV nucleation [27]. It is necessary and profit to study the nonlinear effects of dual-frequency ultrasound on the regulation of ADV.

Researchers have already conducted some studies on the nonlinear effects of dual-frequency ultrasound. In these studies, only specific pairs of dual-frequency combination were investigated (e.g.,3.16 MHz + 3.20 MHz used in Reference [28] and 1.09 MHz + 1.11 MHz used in Reference [29]) and it is necessary to extend the frequency range to have a comprehensive understanding of characteristics of dual-frequency ultrasound. Moreover, the effects of nonlinearity on the temperature raise caused by ultrasound which play an important role in the biomedical ultrasound applications should be taken into consideration as well. Added to that, KZK equation was commonly used to model the nonlinear propagation of dual-frequency ultrasound wave nowadays [28], [29], [30], which was a simplification based on the parabolic approximation. The generalized Westervelt equation in the time domain will be better to simulate the acoustic field more accurately especially when severe waveform distortion is produced [31]. Few studies have been focused on the effects of ultrasonic nonlinearity on the waveforms, acoustic fields and temperature fields using dual-frequency ultrasound in a wide frequency range currently. The specific impacts of dual-frequency ultrasound parameters including acoustic powers and frequencies on the on-demand regulation of ADV considering nonlinear effects are still unclear and further research is needed.

In this work, the impacts of ultrasonic nonlinear effects on the on-demand regulation of ADV nucleation of PFC nanodroplets in a wide parameter range were numerically investigated. The ultrasonic nonlinear effects on pressure and temperature were considered and modelled by the generalized Westervelt and Pennes’ Bioheat Transfer equations, respectively. The ADV nucleation was calculated with a modified classical nucleation theory (CNT) based on the pressure and temperature simulations. In addition, parametric studies were conducted to investigate the patterns of degree of nonlinearity (DoN), nucleation rate and dimensions in the focal area under different dual-frequency focused ultrasound conditions. Furthermore, the comparisons between the results of nonlinear and linear models were made to analyze and discuss the nonlinear effects on ADV nucleation. It is expected to give a more scientific and comprehensive instruction for on-demand regulation of ADV in the dual-frequency focused ultrasound for theranostics considering nonlinear effects of ultrasound.

2. Theory and methods

2.1. ADV nucleation model for a PFP nanodroplet activated by dual-frequency ultrasound

Fig. 1 illustrates the dual-frequency focused ultrasound system employed for simulating the ADV nucleation of PFP nanodroplets, which is based on the model developed and implemented in our previous work [20]. The ultrasound transducer used in this study is identical to that utilized in prior experimental investigations [32], consisting of two confocal spherical annular elements. The inner ring, stimulated at a high frequency, has a radius of r₁ (25 mm), while the outer ring, stimulated at a low frequency, has a radius of r₂ (47 mm). The transducer has a central hole with a radius of r₀ (11 mm) and a focal length of R_FU (60 mm). A square tissue phantom with a side length of 5 cm was submerged in water, 5 cm away from the bottom of the transducers. The PFP nanodroplets were assumed to be uniformly distributed within the tissue phantom, with a volume fraction $V_f$ of 10⁻⁴.

Fig. 1.

Schematic diagram of dual-frequency focused ultrasound model (figure not to scale). The R_FU,r₀, r₁, and r₂ represent the focal distance, the radius of central hole, inner ring for the high-frequency ultrasound, and outer ring for the low-frequency ultrasound, respectively. The d₀ is the distance from the bottom of transducers to the tissue phantom and a₀ is the side length of the tissue phantom.

To model the ADV nucleation of PFP nanodroplets activated by dual-frequency ultrasound, a modified CNT [33], [34] was employed. The nucleation rate J, which represents the formation of critical nuclei per unit time and volume, is a vital measure of the nucleation outcome. Its logarithmic form can be calculated using the following equation [34]:

$$\mathit{lg}J\left(T_l,p_l\right)={\mathit{log}}_{10}\left(\sqrt{\frac{3{\sigma }_r\left(T_l,p_l\right){\rho }_l^2}{\pi m^3}}·\mathit{exp}\left(-\frac{W^{\ast }\left(T_l,p_l\right)}{k_B{T}_l}\right)\right)$$ (1)

where $T_l$ and $p_l$ are the temperature and negative pressure of the PFP liquid, which can be obtained from the simulation results to be discussed later. ${\sigma }_r$ is the size-dependent surface tension, accounting for the influence of nucleus curvature through a scaling function [34], [35]. ${\rho }_l$ represents the density of PFP liquid, $m$ represents the mass of a single molecule, $k_B$ is the Boltzmann constant, and $W^{\ast }$ represents the critical work for nucleation. Further detailed information on the modified CNT can be found in our previous work [20].

To accurately model and understand the nonlinear propagation of the dual-frequency focused ultrasound waves in heterogeneous media, the generalized Westervelt equation [36] was employed:

$${\mathrm{\nabla }}^2{p}_a-\frac{1}{c_0^2}\frac{{\partial }^2{p}_a}{\partial t^2}-\frac{1}{{\rho }_0}\mathrm{\nabla }{\rho }_0·\mathrm{\nabla }{p}_a+\frac{\beta }{{\rho }_0{c}_0^4}\frac{{\partial }^2{p}_a^2}{\partial t^2}-L{\mathrm{\nabla }}^2{p}_a=0$$ (2)

where $p_a$ is the acoustic pressure, $c_0$ is the isentropic sound speed, t is propagation time,${\rho }_0$ is ambient density, $\beta$ is the coefficient of nonlinearity, $L$ is a linear integro-differential operator that accounts for acoustic absorption and dispersion and has two terms both dependent on a fractional Laplacian:

$$\mathrm{L}=\tau \frac{\partial }{\partial t}{(-{\mathrm{\nabla }}^2)}^{\frac{y}{2}-1}+\eta {(-{\mathrm{\nabla }}^2)}^{\frac{y+1}{2}-1}$$ (3)

Where $\tau =-2{\alpha }_0{c}_0^{y-1}$ and $\eta =2{\alpha }_0{c}_0^y\mathrm{t}\mathrm{a}\mathrm{n}(\pi y/2)$ are absorption and dispersion proportionality coefficients. Here, ${\mathrm{\alpha }}_0$ is the power law prefactor and $y$ is the power law exponent. The absorption coefficient $\mathrm{\alpha }={\mathrm{\alpha }}_0·f^y$ follows a frequency power law [37] where $f$ is the ultrasound frequency. To highlight the role of nonlinearity, a linear model was also simulated under the same conditions and compared with the nonlinear model as a control group. In the linear model, the Helmholtz equation, an equivalent form of Eq. (2) after ignoring the nonlinear terms, was used [38]:

$${\mathrm{\nabla }}^2{p}_a-\frac{1}{c_0^2}\frac{{\partial }^2{p}_a}{\partial t^2}=0$$ (4)

The temperature (see Eq. (1)) in the tissue phantom produced by the ultrasound thermal effect was calculated by the Pennes’ Bioheat Transfer equation through ignoring the vascular distribution and blood perfusion [39]:

$${\rho }_0{C}_p\frac{\partial T}{\partial t}=\mathrm{\nabla }\bullet \left(k\mathrm{\nabla }T\right)+Q$$ (5)

where $C_p$ is the specific heat capacity, $T$ is temperature, $k$ is thermal conductivity and $Q$ represents the volume rate of heat deposition, and it is defined by the equation $Q={\sum }_{i=1}^n(2·{\mathrm{\alpha }}_i·I_i)$. In this equation, the subscript i denotes the i^th harmonic which could be obtained in the spectrum of the waveform signal after performing Fourier transform. ${\mathrm{\alpha }}_i$ could be calculated according to the frequency power law and n was determined by the maximum frequency supported in the simulation. Additionally, $I$ stands for the acoustic intensity, which can be calculated using the formula $I=\frac{{p_a}^2}{2·{\rho }_0·c_0}$ based on the plane wave hypothesis [40]. In the context of a linear ultrasound field, as described by the Helmholtz equation, only the fundamental frequency is taken into account when calculating $Q$. In addition, when solving Eq. (5), it was assumed that the temperature in the tissue and water was initialed as 25 °C and there was no heat flux at the boundary of tissue.

2.2. Numerical simulations of dual- frequency ultrasound pressure and temperature field

To simulate ADV nucleation of the PFP nanodroplets activated by a confocal dual-frequency ultrasound, an open-source MATLAB toolbox called k-Wave Version 1.4 was used. k-Wave is an advanced numerical model that can account for both linear and nonlinear wave propagation, an arbitrary distribution of heterogeneous material parameters, and power law acoustic absorption. Quantitative validation of the k -Wave toolbox was performed when using a bowl-shaped transducer as a source and the results demonstrated that acoustic nonlinearity and absorption have been modeled correctly [41]. k -Wave solves a system of coupled partial differential equations considering the momentum conservation, mass conservation, and pressure-density relation respectively [42]. These equations can be combined in a form of Eq. (2) in nonlinear situations or Eq. (4) in linear situations.

To reduce the computational load and enhance the computing efficiency, a 2D axisymmetric coordinate system was used instead of a 3D Cartesian coordinate system. The size of the computational domain was 98 mm*49 mm which was divided into 2¹² * 2¹¹ grid points. (Fig. 1 has been mirror flipped.) A Courant-Friedrichs-Lewy (CFL) number, which is defined as $\mathit{CFL}={c_0}_{\mathit{max}}\mathrm{\Delta }t/\mathrm{\Delta }x$ to represent the numerical stability of the simulation, was set to 0.05 in the present simulation. This means a temporal step $\mathrm{\Delta }\mathrm{t}$ of 0.774 ns and a grid spacing $\mathrm{\Delta }\mathrm{x}$ of 24.28 μm in both the axial and lateral directions were employed in the simulations and the maximum frequency supported in the simulation is 30.54 MHz.

The simulation also accounted for heat diffusion caused by the focused ultrasound. The thermodynamic parameters of the ambient water were found to have no significant effect on the temperature increase due to few ultrasound absorptions compared to the tissue phantom. The physical properties of the heterogeneous acoustic medium, including water and a tissue phantom, used in the simulations are listed in Table 1.

Table 1.

Physical properties used in the simulations.^*

	Water [43]	Tissue phantom [37]
Speed of sound (m/s)	1483	1568
Density (kg/$m^3$)	998.09	1044
Attenuation coefficient (Np/MHz/m)	0.025	8.55
Power law exponent	2	1
Coefficient of nonlinearity	5.2	7.4
Specific heat capacity (J/(kg⋅K))	–	3710
Thermal conductivity (W/(m·K))	–	0.59

^*All parameters are taken as values at a temperature of 298.15 K (i.e. 25 °C).

2.3. Simulation conditions and data analysis

Parametric studies were conducted to investigate the effects of ultrasonic nonlinearity on the on-demand regulation of ADV using dual-frequency focused ultrasound. A dual-frequency ultrasound consisting of a continuous low-frequency ultrasound and a pulsed high-frequency ultrasound with a 1 % duty cycle was employed here. The pulse-repetition frequency (PRF) and irradiation time of the focused ultrasound were set at 100 Hz and 3 s, respectively. Considering clinical applications [44], we selected three different driving frequencies for the low-frequency ultrasound (LF = 0.5, 1.0, and 1.5 MHz) and eight different acoustic powers (LW ranging from 0 to 70 W with increment of 10 W). Nonlinear effects of ultrasound become more obvious as the increase of frequency and acoustic power [23]. Given this, for the high-frequency ultrasound, we determined a parameter group with driving frequencies ranging from 3 MHz to 5 MHz with increment of 1 MHz (HF = 3, 4 and 5 MHz) and acoustic powers HW ranging from 0 W to 25 W with increment of 5 W. The parameter range we selected are based on clinical practice and the latest research hotspots on dual-frequency ultrasound. The intensity applied in the target area is neither too high to be unsafe to normal tissue nor too small to be unable to produce therapeutic effects. In both linear and nonlinear models, the acoustic power (LW and HW) was implemented using the acoustic pressure $p_a$ on the surface of the transducers, i.e. $p_a$ = $\sqrt{2\rho c\frac{W}{S_t}}$, where $\rho$ and c is the density and sound velocity of the material, $S_t$ is the area of the transducer surface which can be calculated by the formula for the area of a frustum. Finally, a total of 702 conditions including both nonlinear and linear dual-frequency focused ultrasound models were simulated. All simulations were performed on the high-performance computing platform (HPC) of Xi’an Jiaotong University with 2.4 GHz CPU (Intel Xeon 6258R), 384 GB of RAM and NVIDIA A100 (40 GB) GPU. Each simulation took around 2.5 h to complete.

In the field of single frequency nonlinear ultrasound studies, various criteria have been used to assess the degree of nonlinearity (DoN) of ultrasound signals. These criteria include the F-number ($F_{\#}$) of the transducers, the ratio of peak pressures ($p_{+}/p_{-}$), and the ratio of durations of the rarefaction and compression phases ($t_{-}/t_{+}$) [45]. In our preparatory work, however, evidence suggests that these criteria may not accurately reflect the degree of nonlinearity in the area of dual-frequency nonlinear ultrasound because no positive relation with ultrasound parameters had been presented. To overcome this limitation and provide a more quantitative assessment of nonlinearity under different conditions of dual-frequency ultrasound, a new DoN index was calculated by considering the frequency component at the sum of two fundamental frequencies (DoN = $A_{f=LF+HF}$). This index considers the interaction between the ultrasonic waves of different frequencies, which causes the failure of the criteria mentioned above to be applicable to dual-frequency ultrasound. Essentially, DoN is the energy transferred from fundamental frequencies (LF and HF) to their sum frequency (LF + HF). It could be obtained from the frequency spectrum by applying Fast Fourier Transform (FFT) to the waveform signal.

To comprehensively evaluate the outcomes of nucleation, studies on the nucleation rate J have been conducted and discussed. However, it has been observed that using the nucleation rate J alone is challenging for studying the spatial dimensions of ADV nucleation. This is primarily due to the significantly smaller scale of the effective action time of negative pressure and the volume of the action site compared to the unit time(1 s) and volume (1 $m^3$). To address this limitation, another parameter called the number of critical nuclei $N_J$ was introduced. The logarithmic form of relationship between $N_J$ and J was expressed by the equation $\lg N_J={\mathit{log}}_{10}(J\ast V_0\ast V_f\ast {\tau }_0)$. Here, $V_0$ = 1*10⁻⁶ mm³ and ${\tau }_0$ is calculated as ${\tau }_0=$ 1/(10f) [46]. In addition, the spatial dimensions of the ADV nucleation site were also studied. Here, the nucleation site was recognized as the place where at least one critical nucleus was formed (i.e. $\lg N_J$ ≥0) and evaluated by measuring the length a, width b and area S.

3. Results

3.1. Acoustic pressure and distribution

Fig. 2 illustrates the waveform distortion and the presence of high harmonics caused by nonlinear propagation of ultrasound. It showcased the typical waveforms of nonlinear acoustic pressures for both single- and dual-frequency ultrasound at the focal point [Fig. 2(a)]. In dual-frequency mode, the waveform distortion was more pronounced, with the ratio of peak positive pressure to peak negative pressure ($p_{+}/p_{-}$) being considerably greater than that in single-frequency mode [23]. In Fig. 2(a), p+/p− in low-frequency excitation was 12.11 MPa/8.79 MPa = 1.38, which in high-frequency excitation was 12.31 MPa/6.34 MPa = 1.94. And in dual-frequency excitation, p+/p− = 28.17 MPa/12.71 MPa = 2.22, which was greater than that in single-frequency mode. In the frequency domain, the spectrum [Fig. 2(b)] revealed not only the fundamental frequencies ($A_{f=LF}$, $A_{f=HF}$) and multiple frequencies of the low-frequency and high-frequency components ($A_{f=n\ast LF}$, $A_{f=m\ast HF}$), but also the presence of sum and difference frequency signals derived from these components ($A_{f=n\ast LF\pm m\ast HF}$) [47]. In linear model, $p_{+}$ was equal to $p_{-}$ (10.15 MPa) in single-frequency mode, and they are different from that in nonlinear model. For example, in low-frequency mode with the same frequency and acoustic power, p+ was 12.11 MPa and p- was 8.79 MPa in nonlinear case shown in Fig. 2(a) while p+ and p− were both equal to 10.15 MPa in linear case shown in Fig. 2(c). Due to the ultrasound nonlinear effects, p+ will be much higher than that in linear case and p− will be lower. In addition, the dual-frequency signal could be gotten by directly adding low-frequency and high-frequency signals together [Fig. 2(c)]. Besides, only the components at fundamental frequencies ($A_{f=LF}$, $A_{f=HF}$) could be observed in the spectrum in Fig. 2(d).

Fig. 2.

Typical waveforms and spectra of nonlinear (a, b) and linear (c, d) acoustic pressures of single- and dual-frequency ultrasound at the focal point with LF = 1 MHz, LW = 40 W, HF = 4 MHz and HW = 15 W respectively.

Fig. 3 illustrates the distributions of acoustic fields under different dual-frequency focused ultrasound conditions and their relationships with the waveforms at the focus. The results indicated that the distributions of acoustic fields primarily vary with the dual-frequency combination rather than the acoustic powers. The variations of the acoustic distribution were mainly caused by the interaction between waves of different frequencies, which is a specific nonlinear effect occurred in dual-frequency excitations. Typical dual-frequency combinations containing all the 3 variations of LF (0.5,1.0 and 1.5 MHz) and HF (3,4 and 5 MHz) are selected and presented in Fig. 3. In contrast to the elliptical distribution obtained through the linear superposition of two single-frequency acoustic fields [20], the results obtained from the nonlinear model exhibited greater complexity and irregular cloud-like spatial distributions were produced as shown in Fig. 3(a). Taking the combination of 1 MHz + 4 MHz as an example, there are 3 sub focal points in the distribution of $p_{+}$ field and 2 sub focal points in the distribution of $p_{-}$ field. Which is consistent with the distributions of $p_{+}$ and $p_{-}$ fields in the axial direction as shown in Fig. 3(b). Interestingly, we found that the number of sub focal points was associated with the crests and troughs of the waveforms at the focus [Fig. 3(c)]. As marked with black dots, there are 3 crests within the positive phase and 2 troughs within the negative phase, corresponding to the number of sub focal points in the distribution of $p_{+}$ and $p_{-}$ fields. Moreover, the number of crests and troughs of the waveform in a single cycle is mainly determined by the frequency of high- frequency ultrasound. With the increase of HF, the number of sub focal points became larger and their distribution become more concentrated. If too many sub focal points concentrated in a narrow space, it can be equivalent to recognized as main focus, as shown in Fig. 3 at the condition of 0.5 MHz + 5 MHz.

Fig. 3.

The typical distributions of the acoustic pressure and the relationship between the distributions and waveforms at the focus. (a) The distributions of the peak positive and negative pressures at focal area. (b) Peak positive and negative pressures along the axis of the transducers. (c) The waveforms within a single cycle. The black dots in the figures represent sub focal points of the pressure fields as well as crests and troughs of the waveforms. For each dual-frequency combination, LW = 40 W and HW = 15 W.

The explanation for the relationship between the pressure distributions and waveforms may be that as the time prolongs, the ultrasound wave has traveled a certain distance in space. The variation of waveform is the reflection of the ultrasound wave on time and the distribution of acoustic pressure field is the reflection in space. They are essentially consistent because they are characteristics of the same wave observed from different perspectives. Our findings can also explain why in single-frequency ultrasound excitation, the increase of frequency will result in a more focused pressure field. A high frequency ultrasound which has a minor cycle travels a short distance within a cycle reflected in space. Therefore, a small focal area is observed. In practical application, it is much easier to measure the waveform at the focus that the distribution. Utilizing the principles above, the approximate shape of the distribution can be estimated according to the signal at the focus.

The variation of $p_{-}$ in the nonlinear model with frequency and acoustic power is presented in Fig. 4(a). As expected, a positive correlation between the pressure and ultrasound parameters was observed, similar to the findings based on linear model in our previous work [20]. However, it is noteworthy that the increase in peak negative pressure with HF was not as pronounced as that in the linear model. This could be attributed to the stronger nonlinear effects induced by higher frequency ultrasound which made more contributions to the increase of $p_{+}$ rather than $p_{-}$.

Fig. 4.

Peak negative pressure under different dual-frequency focused ultrasound conditions of nonlinear model (a) and the ratio of the results of nonlinear model to that of linear model (b).

Fig. 4(b) illustrates the ratio of the peak negative pressure in the nonlinear model to that in the linear model (rp). It is worth noting that $p_{-}$ in the nonlinear model was generally smaller (rp were less than 1 at all conditions). For example, the rp was 0.85 at the condition with LF = 1 MHz, LW = 10 W, HF = 3 MHz, HW = 5 W, while the rp became smaller (0.79) at the condition with LF = 1 MHz, LW = 50 W, HF = 5 MHz, HW = 5 W, which means that the gap between the results of linear and nonlinear models became more larger with increasing ultrasound frequency and acoustic power. Furthermore, the ratio rp was found to be less sensitive to HF, especially in the conditions with LF = 1.5 MHz in Fig. 4(b), indicating that low frequency ultrasound had a greater impact on $p_{-}$. Additionally, it is noteworthy that rp for the case of 1.5 MHz + 3 MHz was found to be smaller than that for the case of 1.5 MHz + 4 MHz. For example, when LW = 10 W and HW = 5 W, rp was 0.77 with 1.5 MHz + 3 MHz but was 0.82 with 1.5 MHz + 4 MHz. The reason for the phenomenal is related to the degree of nonlinearity which will be discussed in the subsequent section.

3.2. Degree of nonlinearity

Fig. 5 illustrates the influence of driving frequency and acoustic power on DoN in dual-frequency focused ultrasound. For example, when LF = 1 MHz, LW = 40 W, HF = 4 MHz and HW = 15 W (the frequency spectrum of it was shown in the third picture of Fig. 2(b) with a legend of DF), the frequency component at $f=LF+HF=$ 5 MHz is 2.99 MPa, so DoN in this case equals 2.99 * 10⁶ which means that so much energy has been transferred to the sum frequency. In general, an increase in the driving frequency and acoustic power of either the low- or high-frequency ultrasound resulted in a corresponding increase in DoN. Among the dual-frequency ultrasound parameters, HF had a greater impact on DoN, which aligned with the phenomenon observed in single-frequency scenarios [23]. For instance, it can be observed that the degree of nonlinearity when LF = 0.5 MHz, LW = 70 W, HF = 3 MHz, and HW = 25 W (DoN = 2.53*10⁶) was roughly equal to that when LF = 0.5 MHz, LW = 30 W, HF = 5 MHz, and HW = 15 W (DoN = 2.52*10⁶). By increasing HF, acoustic power can be reduced while ensuring the same level of DoN, thus raising the efficiency of energy utilization. Interestingly, a similar phenomenon was observed once again that the DoN was higher at 1.5 MHz + 3 MHz compared to the case of 1.5 MHz + 4 MHz. When LW = 70 W and HW = 25 W, DoN was 4.97*10⁶ for the case of 1.5 MHz + 3 MHz and 4.69*10⁶ for the case of 1.5 MHz + 4 MHz, showing a decrease of DoN as increasing HF. This can be attributed to the fact that under these specific conditions, the sum frequency coincides with the third harmonic of the low frequency. As a result, the interaction between the two frequencies is strengthened, leading to a significant enhancement of the nonlinear effect. This provides a plausible explanation for the exceptions mentioned earlier shown in Fig. 4(b).

Fig. 5.

Degree of nonlinearity under different dual-frequency focused ultrasound conditions.

3.3. Temperature rise and distribution

Due to the small duty cycle (1 %) of high-frequency ultrasound, it is difficult to form effective thermal effects, so the temperature rise was determined by the low-frequency ultrasound. A considerable amount of relevant research has been conducted on the thermal effects of the single-frequency considering nonlinear effects, and we share the same views on this matter [48], [49]. An oval-like distribution of the temperature field under both nonlinear and linear models is observed in Fig. 6(a and b). The raised temperature produced in nonlinear model in Fig. 6(a) (maximum rise was 26.51 °C after 3 s) was higher than that in linear model Fig. 6(b) (maximum rise was 25.83 °C after 3 s) due to the higher absorption coefficient corresponding to high harmonics generated by the nonlinear propagation of ultrasound ($\mathrm{\alpha }={\mathrm{\alpha }}_0\bullet f^y$). And the difference between nonlinear and linear models increased as the driving frequency and acoustic power increased, which could be observed in Fig. 6(c and d). When the driving frequency was low (i.e. LF = 0.5 MHz), there was almost no visual distinction between the results of two models.

Fig. 6.

Typical time-dependent temperature variation around the focal region (a) under linear condition and (b) under nonlinear condition with LF = 1 M, LW = 40 W. (c) Time-dependent temperature variation at the focus under different focused ultrasound conditions. (d) Temperature rise under different dual-frequency focused ultrasound conditions after 3 s irradiation.

3.4. Nucleation rates

Fig. 7(a) demonstrates the nucleation rates of nonlinear model under different dual-frequency focused ultrasound conditions after 3 s. In the nonlinear model, the nucleation rate showed a similar change with the ultrasound parameters as in the linear model, i.e. the nucleation rate went up as increasing the driving frequency and acoustic power, which is consistent with our previous study [20]. The ratio of the nucleation rate of the nonlinear model to the linear model (rJ) is shown in Fig. 7(b). It can be observed that the values of rJ were less than 1, which was the same as rp, indicating that fewer critical nuclei are formed when considering the nonlinear effects of dual-frequency ultrasound. According to the modified CNT model, the decrease of $p_{-}\mathrm{a}\mathrm{n}\mathrm{d}$ increase of temperature will cause completely opposite impacts on the nucleation rate, and the results that rJ at all conditions were less than 1 indicated the predominant influence of $p_{-}$ on nucleation compared to temperature. The work of de Andrade et al. [50] has shown that acoustic pressure is the ultimate trigger for nucleation, and HIFU heat deposition helps lower nucleation pressure thresholds, which could confirm our findings to a certain degree. Additionally, in Fig. 7(b), it can be observed that rJ had a positive correlation with ultrasound frequency and power, indicating that the difference in nucleation rates between the nonlinear and linear models was slight in high degrees of nonlinearity situations. This outcome can be explained by the superior limit of the nucleation rate (the maximum of lgJ ≈ 38). In high degrees of nonlinearity situations, both the linear and nonlinear results approach the limit value and are nearly the same.

Fig. 7.

Nucleation rates under different dual-frequency focused ultrasound conditions of nonlinear model (a) and the ratio of the results of nonlinear model to that of linear model (b) after 3 s irradiation.

3.5. Spatial distribution of nucleation

As shown in Fig. 8, the nucleation area was also irregular in the nonlinear model, unlike oval region in the linear model [20]. The distribution of the nucleation area was closely related to the pressure distribution shown in Fig. 3 but was also affected by the temperature field shown in Fig. 6(a). When LF = 0.5 MHz, due to the low temperature, the nucleation area was concentrated in the focal area where $p_{-}$ was high, presenting a slender shape. With the increase of LF, the nucleation area expanded due to a more significant role of temperature, and the shape could be seen as the superposition and overlap of the fields of $p_{-}$ and temperature. When both LF and HF were raised, a more focused nucleation site was produced, and all sub focal points were so concentrated that they could be seen as one main focal point. Therefore, a familiar oval-like nucleation area appeared with LF = 1.5 MHz and HF = 5 MHz.

Fig. 8.

Distribution of nucleation area under different dual-frequency focused ultrasound conditions with LW = 40 W and HW = 15 W.

Fig. 9 illustrates the trends of the nucleation region change in length, width, and area over time. The nucleation region expanded with the increase of irradiation time and the growth rates of length, width, and area decreased which shared the similar trend of raised temperature shown in Fig. 6(c). However, sometimes (e.g., at 0.9 s when LF = 1 MHz, LW = 30 W, HF = 4 MHz, and HW = 15 W) there was a bulge in the growth of the length of the nucleation site. The reason may be that at that time, the nucleation region happened to expand to the position where $p_{-}$ was lower than the surrounding area as it located at the edge of the region dominated by a particular sub focal point. As the temperature increased over time, the nucleation threshold was reduced, allowing the nucleation region to continue expanding. There was no bulge in the growth of width, because all sub focal points were distributed in axial direction rather than radial direction and the expansion of the nucleation region in the radial direction was largely dependent on the thermal effect. Additionally, the area of the nucleation site also increased steadily, which facilitates the control and regulation of ADV nucleation in the dual-frequency focused ultrasound.

Fig. 9.

Dimensions of the nucleation site over time with LF = 1 MHz, HF = 4 MHz, and HW = 15 W. (a) The length of the nucleation site. (b) The width of the nucleation site. (c) The area of the nucleation site.

The areas of the nucleation regions under nonlinear conditions after 3 s irradiation are shown in Fig. 10(a). For each case under different dual-frequency ultrasound parameters, the nucleation region like which was shown in Fig. 8 could be obtained. Each pixel point represents a square with a side length of Δx = 24.28 μm. By counting the number of the pixel points where$\lg N_J$ ≥0, the length, width and area of the nucleation region could be obtained. It was observed that the nucleation areas increased with the acoustic power, as expected. The patterns of area changing with frequency was quite complex. On one hand, an increase in frequency results in higher pressure, leading to a larger nucleation area. On the other hand, higher frequency ultrasound has better focusing characteristics [51], which causes a smaller nucleation area. Consequently, the largest nucleation area was observed at LF = 1 MHz, as shown in Fig. 10(a). It is worth noting that the increase in peak negative pressure with high frequency is not as remarkable. Instead, high-frequency ultrasound has a greater effect on the nucleation area under conditions with lower LF such as 0.5 MHz.

Fig. 10.

The area of nucleation site under different dual-frequency focused ultrasound conditions of nonlinear model (a) and the ratio of the results of nonlinear model to that of linear model (b) after 3 s irradiation.

The ratio of the area of the nucleation site under the nonlinear model to the linear model (rS) is shown in Fig. 10(b). Generally, similar to rJ, rS also exhibited a positive correlation between ultrasound frequency and power, which was more pronounced with LF = 0.5 MHz. However, as the frequency increases, various influencing factors such as temperature, $p_{-}$ and focusing characteristic mutually constrain each other and they do not change synchronously or proportionally with ultrasound parameters in the linear and nonlinear models. Especially, the nucleation area distributions changed with different dual-frequency combinations, which made it more difficult to summarize a concise universal rule. In most cases, the effects of ultrasonic nonlinearity resulted in smaller nucleation areas (i.e. rS <1), however, the complex mechanism could lead to a larger nucleation area under the nonlinear model, such as rS = 1.0109 under the conditions of LF = 1.5 MHz, LW = 50 W, HF = 4 MHz, and HW = 5 W. Such exceptions did not appear in the results of rp or rJ and they were all less than 1. It reflects the complexity and difficulty in controlling the nucleation sites using dual-frequency ultrasound.

4. Discussion

This work aimed to investigate the effects of ultrasonic nonlinearity on the on-demand regulation of ADV using dual-frequency focused ultrasound. A modified CNT was applied based on time-domain simulation of nonlinear dual-frequency acoustic wave fields and temperature fields to study how the dual-frequency ultrasound parameters influence the nucleation rate and region in the focal space. Significant differences were observed compared to the results of linear propagation in previous studies, highlighting the significant role of ultrasonic nonlinear effects in the control of ADV nucleation activated by dual-frequency ultrasound.

Like single-frequency ultrasound, the nonlinearity of dual-frequency ultrasound results in lower $p_{-}$ compared to that in the linear model due to the waveform distortion. In addition, the interaction of the two ultrasound beams, which is the unique characteristic of dual-frequency ultrasound, determines that $p_{-}\mathrm{i}\mathrm{n}$ dual-frequency ultrasound excitation is not the sum of that in two single-frequency ultrasound excitations. Especially in high HF conditions where DoN is large, nonlinearity gives rise to a sharp increase in $p_{+}$ rather than $p_{-}$. Since $p_{-}$ is the direct trigger of nucleation, it means that it is not so efficient to raise the nucleation rate by increase HF. Compared to acoustic power, ultrasound central frequency plays a more significant role in the context of nonlinearity. In dual-frequency ultrasound mode, different dual-frequency combinations also affect the ultrasound nonlinearity. In practical applications, considering nonlinearity, low-frequency ultrasound parameters should be determined first based on the desired temperature and approximate range of nucleation rate. And then, choose appropriate high-frequency ultrasound parameters allowing for other aspects such as the mechanical effects of ultrasound, ADV bubble dynamics, and feasibility of hardware systems. For example, a lower HF would be better for ultrasound neuromodulation and thrombolysis to avoid damage to nerves and blood vessels [52], while a higher HF may be suitable for histotripsy to make full use of shocks at the focus [53].

The comparations between the nucleation results of linear and nonlinear models were conducted. Due to the high computational consumption in the nonlinear time-domain simulation of dual-frequency acoustic wave propagation, the results of linear model could be introduced as a reference. Note that under low DoN conditions, the nucleation rates and areas get from linear model are larger than the true state of affairs. And the distributions of nucleation sites are influenced by the dual-frequency combination (see Fig. 8). Under high DoN conditions, though their nucleation rates and areas are similar (for the superior limit of the nucleation rate has been reached), the ADV bubble dynamics and related cavitation mechanical effect will be greatly affected by the presence of high harmonics due to ultrasonic nonlinearity [54] which needs to be investigated in the future work. The main focus of this work is the dual-frequency ultrasound nonlinear effects on the ADV nucleation and whether the nonlinear effects could be ignored. The results showcased the great significance of dual-frequency ultrasound nonlinearity on the spatiotemporal control of ADV nucleation in nucleation rates and spatial distribution of nucleation. It is impossible to output ADV nucleation of the linear model similar as those of the nonlinear model in all aspects by adjusting the parameters of linear model. However, if the requirements for the accuracy of the results are not so strict and nonlinear effects are not so severe, the linear model is still reasonable and convenient because of the low computational burden and high computational efficiency.

Unlike an oval-shaped nucleation area distribution shown in linear model, cloud-like spatial distributions of nucleation were produced considering the nonlinear interaction between waves of different frequencies. The distributions are determined mainly by the acoustic pressure fields which are related to the focus waveforms. In order to realize the regulation of nucleation distribution, an optimal choice of ultrasound parameters could be applied based on the prediction of the focus waveform. It is noteworthy that if a non-elliptical nucleation site is wanted, high acoustic power (both LW and HW) and HF should be avoided for the severe waveform distortion at high DoN conditions. In this case, there are so many crests and troughs of the waveform in a single cycle and, correspondingly, so many sub focal points in the pressure filed that it can be equivalent to recognized as main focus. At low DoN conditions, the focus waveform of dual-frequency ultrasound is similar to the superposition of that of low- and high-frequency ultrasound. An arbitrary nucleation distribution could be produced as long as the suitable ultrasound parameters are found based on the basic regularity of the relationship between the variation of waveform and the distribution of acoustic pressure field we discussed above on Section 3.1. And such an arbitrary nucleation distribution could be utilized for customized tumor treatment protocols.

Furthermore, the dual-frequency combination of 1.5 MHz + 3 MHz presented some special phenomenon, such as relatively lower rp and higher DoN, indicating that this superposition of fundamental and second harmonic may be a unique type of dual-frequency combination. In fact, there have been numerous recent studies focusing on this specific superposition type. For example, Ma et al. [55] designed and characterized a 1.5 MHz + 3 MHz dual-frequency HIFU transducer, which proved to be a more effective tissue ablation method. Li et al. [56] investigated the feasibility of histotripsy liquefaction of large hematomas in vitro using a 1.1 MHz + 2.2 MHz dual-frequency pulse ultrasound. And the lesion pattern in the gel phantom is similar to the nucleation area using a 1.5 MHz + 3 MHz dual-frequency ultrasound shown in Fig. 8. However, the reason for choosing such a combination was not clearly explained. Our research indicates that this combination of fundamental and second harmonic can effectively enhance ultrasonic nonlinear effects, resulting in relatively higher peak positive pressure and lower peak negative pressure. Additionally, all frequency components in the spectrum are integer multiples of the fundamental frequency, making it easier to control and analyze subsequent bubble dynamics behavior.

One limitation of this study is that the effects of initial phase differences between dual-frequency ultrasound on the nucleation were not investigated. Related researches have been investigated that the $p_{-}$ of dual-frequency ultrasound can be regulated by initial phase differences [57]. In this work, the initial phase difference for each dual-frequency combination was set to zero in the nonlinear model and the effects of phase differences on the modulation of nucleation rate and distribution remain unknown. Furthermore, only the nucleation process was investigated in this work and the ADV bubble dynamics have not been studied yet. And the dynamic behaviors of bubbles are more affected by ultrasonic nonlinearity than nucleation. Future work will be focused on the effects of dual-frequency nonlinear ultrasound waves on the bubble dynamics under different excitation conditions.

5. Conclusions

In this work, the effects of ultrasonic nonlinearity on the regulation of ADV nucleation in the dual-frequency focused ultrasound was numerically simulated and the effects of the ultrasound parameters on the pressure, temperature, degree of nonlinearity, nucleation rate and nucleation dimension were investigated. The simulations showed that considering waveform distortion and nonlinear heating, lower peak negative pressure and higher temperature were produced and $p_{-}$ played a dominant role in nucleation. Furthermore, lower nucleation rates were observed compared to that in the linear model, which is noteworthy when linear model is used as a reference for convenience. More importantly, irregular cloud-like spatial distributions of nucleation regions were produced, which were mainly determined by the acoustic pressure fields and could be regulated by adjusting the dual-frequency ultrasound parameters. In addition, like 1.5 MHz + 3 MHz, such a dual-frequency combination of fundamental and second harmonic frequency showed special characteristics such as relatively higher DoN and lower rp, and had great application potentials in ultrasound theranostics. In summary, considering ultrasonic nonlinearity, the on-demand control of ADV nucleation using dual-frequency focused ultrasound became more complicated but meanwhile more promising. Our method provides a more comprehensive and scientific ADV nucleation model which could potentially lay the foundation for achieving customized treatment protocols and on-demand ultrasound theranostics in clinical applications.

CRediT authorship contribution statement

Yubo Zhao: Writing – original draft, Software, Resources, Methodology, Conceptualization. Yi Feng: Writing – review & editing, Supervision, Project administration, Funding acquisition, Conceptualization. Liang Wu: Writing – review & editing, Validation, Supervision, Software, Methodology, Funding acquisition, Conceptualization.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Yi Feng reports financial support was provided by National Natural Science Foundation of China. Liang Wu reports financial support was provided by National Natural Science Foundation of China. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.