# Effects of Nonlinear Propagation of Focused Ultrasound on the Stable Cavitation of a Single Bubble

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models

#### 2.1. Nonlinear Acoustic Pressure from FU Transducer

^{®}(2016a, Mathworks, Natick, MA, USA). In this method, the linear parts (Equations (3) and (4)) and nonlinear term (Equation (7)) are integrated separately. At each integration, the linear parts are calculated in the frequency-domain and then the solution is converted to the time-domain using a fast Fourier transform algorithm to solve the nonlinear part of the KZK equation. After integration, the result is again transformed into the frequency-domain. This method is repeated for each of the harmonics used in the solution.

#### 2.2. Single-Bubble Dynamics in a FU Field

^{®}, we obtain the Fourier coefficients in Equation (31) for the linear and nonlinear pressure fields.

## 3. Numerical Results

#### 3.1. Effects of Excitation Frequency on Bubble Dynamics

#### 3.2. Effects of Acoustic Pressure (Input Power to the Transducer) on Bubble Dynamics

## 4. Conclusions

- The results show the dynamical motions of the bubble in the FU field generated by the focused transducer at three different range of frequencies (including the resonance and off-resonance cases). Effects of the change of excitation frequency are investigated and it is shown that in all the frequency ranges, the nonlinearity expedites and amplifies the radial growth and translational motion of the bubble where the indication of the translational instability is observed by a steep rise in the translational velocity of the bubble. It is seen that the initiation of the erratic dancing motion of the bubble in the acoustic field is well-matched with the time of instability of other modes. Moreover, acoustic nonlinearity amplifies almost all the shape oscillations and the instability of the modes. Among the three ranges of frequency investigated in this study, the results show that the changes in the dynamical behavior of the bubble and the translational instability due to nonlinear effects are more significant for higher frequencies, which must be taken into account while investigating an erratic translational (dancing) motion of the bubble. However, a momentous effect of nonlinearity on the translational motion of the bubble is seen at resonance frequency where the bubble initially moves away from the pressure antinode and then makes the translation jumps around the antinode. It is seen that as the excitation frequency increases, the inertial (unstable) cavitation occurs earlier in the nonlinear acoustic field; the nonlinear excitation causes more energy deposition at higher harmonics at the focal point (where the bubble is located); therefore, the radius growth of the bubble is more pronounced at higher frequencies. Accordingly, studying nonlinear acoustics in bubble dynamics is essential for inertial cavitation prediction in biomedically enhanced delivery of drugs into tissue [65] as well as the prediction of the induced impact pressure as a shock loading to a structure, which might even cause mechanical destruction [68].
- As the input power is increased, the amplitudes of all the harmonics increase. Comparing the bubble dynamics for two equal pressure amplitudes in linear and nonlinear cases show significant differences in the shape modes and translational instability of the bubble. This confirms that the radial growth and onset of the erratic motion of the bubble in a focused ultrasound field are reliant on both pressure amplitude and the harmonic components of the excitation. It is seen that when the applied pressure increases, the results of the linear model give up to 56% error for predicting the threshold for unstable cavitation (maximum radius of the bubble). In nonlinear acoustics, when the input power increases, the bubble may collapse and shows unstable cavitation as well as the amplified and accelerated translational instability.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Prolonged dynamical motions of the bubble in the HIFU field (${R}_{0}=7\text{}\mathsf{\mu}\mathrm{m}$) and blood domain excited by the focused transducer ($f=3\text{}\mathrm{MHz}$, $P=2\text{}\mathrm{W}$): (

**a**) radial pulsation; (

**b**) second mode; (

**c**) third mode, and (

**d**) fourth mode amplitude; (

**e**) Translational velocity and (

**f**) translational motion $({x}_{0}=0.52\text{}\mathsf{\mu}\mathrm{m})$.

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**Figure 1.**Schematic representation of acoustic waves from a focused source to a bubble at the focal point.

**Figure 2.**Acoustic pressure generated by the focused transducer as the input parameter to bubble dynamics at $P=2\text{}\mathrm{W}$ (

**a**) for various excitation frequency; temporal waveform and (

**b**) the corresponding Fourier spectra for $\beta =4.1$.

**Figure 3.**Dynamical motions of the bubble in the focused ultrasound (FU) field (${R}_{0}=7\text{}\mathsf{\mu}\mathrm{m}$) and blood domain excited by the focused transducer ($f=0.3\text{}\mathrm{MHz}$, $P=2\text{}\mathrm{W}$): (

**a**) radial pulsation; (

**b**) second mode; (

**c**) third mode; and (

**d**) fourth mode amplitude; (

**e**) Translational velocity and (

**f**) translational motion $({x}_{0}=5.2\text{}\mathsf{\mu}\mathrm{m})$.

**Figure 4.**Dynamical motions of the bubble in the FU field (${R}_{0}=7\text{}\mathsf{\mu}\mathrm{m}$) and blood domain excited by the focused transducer ($f=0.5\text{}\mathrm{MHz}$, $P=2\text{}\mathrm{W}$): (

**a**) radial pulsation; (

**b**) second mode; (

**c**) third mode; and (

**d**) fourth mode amplitude; (

**e**) Translational velocity and (

**f**) translational motion $({x}_{0}=3.1\text{}\mathsf{\mu}\mathrm{m})$.

**Figure 5.**Dynamical motions of the bubble in the FU field (${R}_{0}=7\text{}\mathsf{\mu}\mathrm{m}$) and blood domain excited by the focused transducer ($f=3\text{}\mathrm{MHz}$, $P=2\text{}\mathrm{W}$): (

**a**) radial pulsation; (

**b**) second mode; (

**c**) third mode; and (

**d**) fourth mode amplitude; (

**e**) Translational velocity and (

**f**) translational motion $({x}_{0}=0.52\text{}\mathsf{\mu}\mathrm{m})$.

**Figure 6.**Fourier spectra of the radial oscillation of the bubble in blood domain shown in Figure A1a. The curve is on a log scale.

**Figure 7.**Acoustic pressure generated by the focused transducer as the input parameter to bubble dynamics at $f=3\text{\hspace{0.17em}}\mathrm{MHz}$ (

**a**) for various input power; temporal waveform and (

**b**) the corresponding Fourier spectra for $\beta =4.1$. (

**c**) Pressure amplitude vs. transducer input power in linear and nonlinear cases.

**Figure 8.**Dynamical motions of the bubble in the FU field (${R}_{0}=7\text{}\mathsf{\mu}\mathrm{m}$) and blood domain excited by the focused transducer ($f=3\text{}\mathrm{MHz}$, $P=6\text{}\mathrm{W}$): (

**a**) radial pulsation; (

**b**) second mode; (

**c**) third mode; and (

**d**) fourth mode amplitude; (

**e**) Translational velocity and (

**f**) translational motion $({x}_{0}=0.52\text{}\mathsf{\mu}\mathrm{m})$

**Figure 9.**Radius versus time plot of a $7-\mathsf{\mu}\mathrm{m}-\mathrm{radius}$ bubble at $f=3\text{}\mathrm{MHz}$ for various input power in (

**a**) linear and (

**b**) nonlinear acoustic field; (

**c**) A comparison of the linear (dashed line) and nonlinear (solid line) model predictions for varying input power (black line) and acoustic pressure (red line).

**Figure 10.**Dynamics of translation of a $7-\mathsf{\mu}\mathrm{m}-\mathrm{radius}$ bubble at $f=3\text{}\mathrm{MHz}$ for various input power: (

**a**) Translational motion and (

**b**) translational velocity (the corresponding acoustic pressures are given in Figure 7).

Attenuation Parameter, ${\alpha}_{0}$(dB/(m·MHz)) | 20 |

Attenuation parameter, $\nu $ | 1.3 |

Coefficient of nonlinearity, $\beta $ | 4.1 |

Density, $\rho $ (kg/m^{3}) | 1055 |

Sound speed, $c$ (m/s) | 1570 |

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**MDPI and ACS Style**

Bakhtiari-Nejad, M.; Shahab, S. Effects of Nonlinear Propagation of Focused Ultrasound on the Stable Cavitation of a Single Bubble. *Acoustics* **2019**, *1*, 14-34.
https://doi.org/10.3390/acoustics1010003

**AMA Style**

Bakhtiari-Nejad M, Shahab S. Effects of Nonlinear Propagation of Focused Ultrasound on the Stable Cavitation of a Single Bubble. *Acoustics*. 2019; 1(1):14-34.
https://doi.org/10.3390/acoustics1010003

**Chicago/Turabian Style**

Bakhtiari-Nejad, Marjan, and Shima Shahab. 2019. "Effects of Nonlinear Propagation of Focused Ultrasound on the Stable Cavitation of a Single Bubble" *Acoustics* 1, no. 1: 14-34.
https://doi.org/10.3390/acoustics1010003