Weakly nonlinear theory on ultrasound propagation in liquids containing many microbubbles encapsulated by visco-elastic shell

Abstract

Aimed towards an application of ultrasound diagnosis using contrast agents, the dynamics of encapsulated bubbles has been theoretically investigated under the restriction of a single bubble. In this paper, we extend the theory for single bubble or some bubbles to that for many bubbles, and theoretically investigate weakly nonlinear propagation of ultrasound in an initially quiescent incompressible liquid, uniformly containing many microbubbles encapsulated by the shell as a viscoelastic body (Kelvin–Voigt model). As a result, we derived the Korteweg–de Vries–Burgers equation for a low-frequency long wave and clarified that the shell affects the advection, nonlinear, and dissipation (not dispersion) effects of ultrasound propagation. In particular, shell rigidity, surface tension, and shell viscosity increased the advection, nonlinear, and dissipation effects, respectively.

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.…”

Influence of microphone characteristics on demodulated sound measurement in near field of parametric loudspeaker

“…To evaluate the measurement accuracy, we use theoretical prediction as a reference. Many studies 12,15,[33][34][35][36] have carried out nonlinear sound propagation analyses for generating harmonics and parametric sound. Although the Khokhlov-Zabolotskaya-Kuznetsov equation is often used for theoretical prediction, 15) this equation is only applicable in a field relatively distant from the emitter due to the parabolic approximation of nonlinear sound wave equation.…”