## 3 Simulation Results  
### Influence of Multifrequency Ultrasonic Combination on Transient Cavitation Threshold  
Figure 1 shows the relationship between the corresponding transient cavitation threshold and the initial bubble radius in water, blood, brain, and liver under different triple-frequency combined ultrasound irradiation. It can be seen from Figure 1a-d that the transient cavitation threshold is different under different triple-frequency combined ultrasound irradiation. The transient cavitation threshold decreases first and then increases with the increase of the initial bubble radius. In addition, a \"groove\" appears in Figure 1, and this indicates that the ultrasonic frequency is close to the resonant frequency of the bubble or several times of the resonant frequency of the bubble. The resonant frequency of the bubble in fluid [28] and tissue [29; 30] can be described by the following equations, respectively: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Medium**} & **Speed/\(c\)** & **Density/\(\rho\)** & **Surface Tension/\(\sigma\)** & **Viscosity/\(\mu\)** & **Shear Modulus/\(G\)** \\  & **(m/s)** & **(g/m\({}^{3}\))** & **(mN/m)** & **(mPa.s)** & **(MPa)** \\ \hline Water & 1500 & 1000 & 68 & 1 & 0 \\ Blood & 1570 & 1050 & 56 & 5 & 0 \\ Brain & 1540 & 1050 & 56 & 9 & 0.012 \\ Liver & 1549 & 1100 & 56 & 9 & 0.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of media [27]. \[f_{l}=\frac{1}{2\pi R_{0}\sqrt{\rho}}\sqrt{3\kappa(p_{0}-p_{g}+\frac{2\sigma}{R_{0} })-\frac{2\sigma}{R_{0}}-\frac{4\mu^{2}}{\rho R_{0}^{2}}} \tag{14}\]  
\[f_{t}=\frac{1}{2\pi R_{0}\sqrt{\rho}}\sqrt{3\kappa(p_{0}-p_{g})+(3\kappa-1)\frac {2\sigma}{R_{0}}-\frac{8\mu^{2}}{\rho R_{0}^{2}}+4G} \tag{15}\]  
Because the transient cavitation threshold is closely related to the initial bubble size, the effectiveness of this study mainly depends on the existence of stable small bubble nuclei. It has been reported that the radius of cavitation nuclei in water was about 1 \(\upmu\)m or less, and distribution density decreased exponentially with the radius of the nuclei [31; 32]. However, the nuclei radius in vivo was estimated to be less than 0.3 \(\upmu\)m [33; 34]. Therefore, the bubble size in the range of 0.1 \(\upmu\)m to 10 \(\upmu\)m used here is close to the actual situation. Taking the triple-frequency combination of \(1.4+2.0+2.6\) MHz in Figure 1 as an example, when the initial bubble radius is 10 \(\upmu\)m, the corresponding transient cavitation threshold from large to small in the media are: liver, brain, blood, and water. Therefore, for the high-frequency combination, the higher the viscosity and shear modulus of the medium, the higher the change trend of the transient cavitation threshold under the same and larger initial bubble radius. For any combination of triple frequency in Figure 1a-d, the transient cavitation threshold of the combination of triple frequency with higher frequency (such as \(1.4+1.41+1.42\) MHz) is higher than that of the combination of triple frequency with lower frequency (such as \(0.7+0.71+0.72\) MHz) and when the frequency difference is the same, the higher combination frequency and viscosity of triple frequency will increase the transient cavitation threshold. The higher frequency results in an increase of the cavitation threshold. The inertia of large expansion ratios leads to the suppression of bubble collapse in the positive half cycle, and then the bubble continues to grow in the following negative  
Figure 1: Transient cavitation threshold corresponding to different triple-frequency combinations in different media: (**a**) water; (**b**) blood; (**c**) brain; (**d**) liver. half cycle, resulting in the higher transient cavitation threshold [35].