Department of Engineering Mechanics and Energy, Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan.
Department of Engineering Mechanics and Energy, Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan.
Ultrason Sonochem. 2022 Aug;88:105911. doi: 10.1016/j.ultsonch.2022.105911. Epub 2022 Jan 11.
A physico-mathematical model composed of a single equation that consistently describes nonlinear focused ultrasound, bubble oscillations, and temperature fluctuations is theoretically proposed for microbubble-enhanced medical applications. The Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation that has been widely used as a simplified model for nonlinear propagation of focused ultrasound in pure liquid is extended to that in liquid containing many spherical microbubbles, by applying the method of multiple scales to the volumetric averaged basic equations for bubbly liquids. As a result, for two-dimensional and three-dimensional cases, KZK equations composed of the linear combination of nonlinear, dissipation, dispersion, and focusing terms are derived. Especially, the dissipation term depends on three factors, i.e., interfacial liquid viscosity, liquid compressibility, and thermal conductivity of gas inside bubbles; the thermal conduction is evaluated by using four types of temperature gradient models. Finally, we numerically solve the derived KZK equation and show a moderate temperature rise appropriate to medical applications.
理论上提出了一个由单个方程组成的物理数学模型，该模型一致地描述了非线性聚焦超声、气泡振荡和温度波动，用于微泡增强的医学应用。Khokhlov-Zabolotskaya-Kuznetsov（KZK）方程已被广泛用作纯液中聚焦超声非线性传播的简化模型，通过对含多球形微泡的液相间的体积平均基本方程应用多尺度方法，将其扩展到含多球形微泡的液相间的体积平均基本方程中。结果，对于二维和三维情况，导出了由非线性、耗散、色散和聚焦项线性组合构成的 KZK 方程。特别是，耗散项取决于三个因素，即界面液体粘度、液体压缩性和气泡内气体的热导率；热传导通过四种类型的温度梯度模型进行评估。最后，我们数值求解了导出的 KZK 方程，并显示了适合医学应用的适度温升。
