Nonlinear acoustic theory on flowing liquid containing multiple microbubbles coated by a compressible visco-elastic shell: Low and high frequency cases

Abstract

Microbubbles coated by visco-elastic shells are important for ultrasound diagnosis using contrast agents, and dynamics of single coated bubble has been investigated in literature. However, although a high number of contrast agents are used in practical situations, there has long been an absense of nonlinear acoustic theory for multiple coated bubbles, except for our recent work (Kikuchi & Kanagawa, Jpn. J. Appl. Phys., 60 (2021), SDDD14), under several assumptions to be excluded. Aiming for generalization, in this study, we theoretically investigate weakly nonlinear propagation of ultrasound in liquid containing multiple bubbles coated by a visco-elastic shell with compressibility. Leveraging the method of multiple scales, both the Korteweg-de Vries-Burgers (KdVB) equation for a low-frequency long wave and nonlinear Schrödinger (NLS) equation for a high-frequency short wave, are derived from the volumetric averaged equations for bubbly liquids based on a two-fluid model and the up-to-date model for single coated bubble with shell compressibility. Neglected factors in our previous paper, i.e., compressibility of the shell and liquid, drag force acting on bubbles, bubble translation, and thermal conduction, are incorporated in the present KdVB and NLS equations; the proposed model will be regarded as a generic physico-mathematical model. Results show that shell compressibility attenuated ultrasound strongly and decreased nonlinearity of ultrasound. Finally, we compared the magnitudes of six dissipation factors (shell compressibility, shell viscosity, liquid compressibility, liquid viscosity, thermal effect, and drag force) for five typical ultrasound contrast agents, and a similar tendency between KdVB and NLS equations was revealed.

Weakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles

“…In the theoretical analysis, the following assumptions are introduced for simplicity. (a-1)The elasticity of the liquid phase is considered; however, the viscoelastic shells [30] , [31] , [23] , [27] , [25] , [26] , [28] , [29] coating the bubbles are not. (a-2)The viscosity and elasticity of the gas phase are neglected because they are significantly smaller than those of the liquid phase.…”

“…(a-1)The elasticity of the liquid phase is considered; however, the viscoelastic shells [30] , [31] , [23] , [27] , [25] , [26] , [28] , [29] coating the bubbles are not.…”

“…The dispersion effect is represented in the form of the third derivative with coefficient in the resultant KZK Eq. (102) and obtained by introducing the bubble oscillation [10] , [93] , [25] , [26] , [28] , [29] , [27] , [115] , [88] , [114] , [95] , [89] , [90] , [91] , [92] , [94] , [102] , [103] , [113] . By the dispersion effect, the propagation speed of waves shifts depending on the frequency of each wave.…”

“…Ultrasound also induces the oscillation or collapse of bubbles injected into blood vessels during drug delivery to brain cancer cells, which opens the blood–brain barrier [21] . Moreover, bubbles encapsulated by lipid shells improve the resolution of images [22] , [30] , [31] , [23] , [24] , [27] , [25] , [26] , [28] , [29] for ultrasound imaging as a real-time diagnostic technique.…”

High-speed photography of gas release from bioactive glass

“…If the bubble is subjected to an ultrasound pulse, its radial response can be expressed by a Rayleigh-Plesset equation. [5][6][7] For the purposes of this study, we may ignore nonspherical terms and multiple scattering, [8][9][10] as we concentrate on the first pulsation cycles. A simplified equation for ultrasound-driven microbubble pulsation has been derived in a prior publication, 11) and modified to incorporate viscous, reradiation, and thermal damping terms.…”