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Kagami S, Kanagawa T. Weakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles. Ultrason Sonochem 2023; 97:106455. [PMID: 37271029 DOI: 10.1016/j.ultsonch.2023.106455] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2023] [Revised: 05/12/2023] [Accepted: 05/23/2023] [Indexed: 06/06/2023]
Abstract
To facilitate practical medical applications such as cancer treatment utilizing focused ultrasound and bubbles, a mathematical model that can describe the soft viscoelasticity of human body, the nonlinear propagation of focused ultrasound, and the nonlinear oscillations of multiple bubbles is theoretically derived and numerically solved. The Zener viscoelastic model and Keller-Miksis bubble equation, which have been used for analyses of single or few bubbles in viscoelastic liquid, are used to model the liquid containing multiple bubbles. From the theoretical analysis based on the perturbation expansion with the multiple-scales method, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, which has been used as a mathematical model of weakly nonlinear propagation in single phase liquid, is extended to viscoelastic liquid containing multiple bubbles. The results show that liquid elasticity decreases the magnitudes of the nonlinearity, dissipation, and dispersion of ultrasound and increases the phase velocity of the ultrasound and linear natural frequency of the bubble oscillation. From the numerical calculation of resultant KZK equation, the spatial distribution of the liquid pressure fluctuation for the focused ultrasound is obtained for cases in which the liquid is water or liver tissue. In addition, frequency analysis is carried out using the fast Fourier transform, and the generation of higher harmonic components is compared for water and liver tissue. The elasticity supresses the generation of higher harmonic components and promotes the remnant of the fundamental frequency components. This indicates that the elasticity of liquid suppresses shock wave formation in practical applications.
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Affiliation(s)
- Shunsuke Kagami
- Department of Engineering Mechanics and Energy, Degree Program of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan.
| | - Tetsuya Kanagawa
- Department of Engineering Mechanics and Energy, Degree Program of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan.
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Massaad J, van Neer PLMJ, van Willigen DM, de Jong N, Pertijs MAP, Verweij MD. Exploiting nonlinear wave propagation to improve the precision of ultrasonic flow meters. Ultrasonics 2021; 116:106476. [PMID: 34098419 DOI: 10.1016/j.ultras.2021.106476] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2020] [Revised: 04/16/2021] [Accepted: 05/23/2021] [Indexed: 06/12/2023]
Abstract
Acoustic wave propagation in ultrasonic flow measurements is typically assumed to be linear and reciprocal. However, if the transmitting transducer generates a sufficiently high pressure, nonlinear wave propagation effects become significant. In flow measurements, this would translate into more information to estimate the flow and therefore a higher precision relative to the linear case. In this work, we investigate how the generated harmonics can be used to measure flow. Measurements in a custom-made flow loop and simulations using the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation will show that the second harmonic component provides similar transit time differences to those obtained from the fundamental component, their linear combination results in more precise flow measurements compared to the estimations with the fundamental component alone.
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Affiliation(s)
- Jack Massaad
- Delft University of Technology, Department of Imaging Physics, Laboratory of Medical Imaging, Lorentzweg 1, 2628CJ, Delft, The Netherlands.
| | - Paul L M J van Neer
- Delft University of Technology, Department of Imaging Physics, Laboratory of Medical Imaging, Lorentzweg 1, 2628CJ, Delft, The Netherlands; TNO, Department of Acoustics and Sonar, Oude Waalsdorperweg 63, 2597AK, The Hague, The Netherlands
| | - Douwe M van Willigen
- Delft University of Technology, Department of Microelectronics, Electronic Instrumentation Laboratory, Mekelweg 4, 2628CD, Delft, The Netherlands
| | - Nicolaas de Jong
- Delft University of Technology, Department of Imaging Physics, Laboratory of Medical Imaging, Lorentzweg 1, 2628CJ, Delft, The Netherlands; Erasmus MC, Thorax Center, Department of Biomedical Engineering, Doctor Molewaterplein 40, 3015GD, Rotterdam, The Netherlands
| | - Michiel A P Pertijs
- Delft University of Technology, Department of Microelectronics, Electronic Instrumentation Laboratory, Mekelweg 4, 2628CD, Delft, The Netherlands
| | - Martin D Verweij
- Delft University of Technology, Department of Imaging Physics, Laboratory of Medical Imaging, Lorentzweg 1, 2628CJ, Delft, The Netherlands; Erasmus MC, Thorax Center, Department of Biomedical Engineering, Doctor Molewaterplein 40, 3015GD, Rotterdam, The Netherlands
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Jafarzadeh E, Amini MH, Sinclair AN. Determination of the Ultrasonic Non-linearity Parameter B/A versus Frequency for Water. Ultrasound Med Biol 2021; 47:809-819. [PMID: 33353785 DOI: 10.1016/j.ultrasmedbio.2020.11.027] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/11/2020] [Revised: 11/19/2020] [Accepted: 11/21/2020] [Indexed: 06/12/2023]
Abstract
For tissue characterization, it is desirable to determine B/A using high-frequency transducers. Moreover, an accurate estimate of B/A at elevated frequencies (or at least the assumption of frequency independence of B/A) is required to evaluate the safety of high-frequency systems. However, common finite-amplitude approaches become increasingly inaccurate at high frequencies. In this article, a practical variation of the finite-amplitude method is proposed which combines experiments with numerical simulations of the Khokhlov-Zabolotskaya-Kuznetsov equation and can be used at elevated frequencies. The results at low frequencies show that the proposed approach is accurate with lower uncertainties compared with the finite-amplitude method because it avoids assumptions and approximations. The measured values of B/A versus frequency for water at 2.25-20 MHz show that there is no statistically significant variation in B/A values with frequency, and therefore the assumption of frequency independence of B/A is realistic.
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Affiliation(s)
- Ehsan Jafarzadeh
- Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada.
| | | | - Anthony N Sinclair
- Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada
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