Search

With promising applications in medical diagnosis and therapy, the behavior of shell-encapsula-ted ultrasound contrast agents (UCAs) has attracted considerable attention. Currently, second-generation contrast agents stabilized by a phospholipid membrane are widely used and studies have focused on the dynamics of single phospholipid shell-encapsulated microbubbles. To improve the safety and the efficiency of the methods using the propagation or targeted ultrasound, a better understanding of the propagation of ultrasound in liquids containing multiple encapsulated microbubbles is required. By incorporating the Marmottant–Gompertz model into the multiple scale analysis of two-phase model, this study derived a Korteweg–de Vries–Burgers equation as a weakly nonlinear wave equation for one-dimensional ultrasound in bubbly liquids. It was found that the wave propagation characteristics changed with the initial surface tension, highlighting two notable features of the phospholipid shell: buckling and rupture. These results may provide insights into the suitable state of microbubbles, and better control of ultrasound for medical applications, particularly those that require high precision.

Using microbubbles coated by a thin shell as ultrasound contrast agents for ultrasound diagnosis improves image resolution. Since numerous microbubbles are used in clinical practice, understanding the acoustic properties of liquids containingmultiplemicrobubbles is important. However, interactions between ultrasound and numerous coated microbubbles have not been fully investigated theoretically. Additionally, ultrasound contrast agents with shells made of various materials have been developed. Recently, an equation of motion that considers the anisotropy of the shell was proposed [Chabouh et al., J. Acoust. Soc. Am., 149 (2021), 1240], and the effect of shell anisotropy on the resonance of the oscillating bubbles was reported. In this study, we derived a nonlinear wave equation describing ultrasound propagation in liquids containing numerous coated microbubbles based on the method of multiple scales by expanding Chabouh's equation of motion for a single bubble. This was achieved by considering shell anisotropy in the volumetric average equation for the liquid and gas phases. Shell anisotropy was observed to affect the advection, nonlinearity, attenuation, and dispersion of ultrasound. In particular, the attenuation effects increased or decreased depending on the anisotropic shell elasticity.

The importance of viscoelasticity of biological media that are used in medical ultrasounds has been discussed in the literature. Furthermore, the use of microbubbles in biological media drastically improves the efficiency of both diagnostic and therapeutic ultrasounds. Weakly nonlinear wave equations for ultrasound propagation in liquids containing microbubbles have long been studied, although the viscoelasticity of the liquid phase has been ignored for simplicity. In this study, we derived a nonlinear wave equation for ultrasound propagation in a viscoelastic liquid containing microbubbles by considering the effect of the elasticity of the liquid. Additionally, we evaluated how the elasticity of the liquid modifies the nonlinear, dissipation, and dispersion effects of the ultrasound in a few tissue models (i.e., liver, muscle, breast cancer, fat, and skin models and that without shear elasticity). The results revealed that liquid shear elasticity decreases the nonlinear and dissipation effects and increases the dispersion effect, and this tendency is more significantly observed in the breast cancer tissue compared with other tissues. Furthermore, we numerically solved the nonlinear wave equation and investigated the changes in ultrasonic wave evolution with and without shear elasticity.

Professionals in the energy sector are required to understand the interactions among technology, society, and the environment to tackle complex trade-offs among policy issues. The fostering of such professionals with interdisciplinary perspectives has been recognized as a significant target for engineering education in universities. Several studies have shown that game-based learning is suitable for teaching interdisciplinary aspects of sustainability-related issues; however, only a few studies have quantitatively evaluated the percentage of participants that learn something relevant to the predetermined learning targets. This study quantitatively evaluates the effectiveness of a game-based class designed to teach energy policy issues in an engineering course. This game-based class aims to develop a perspective for overcoming the complex trade-offs among policy issues and for developing the abilities and attitudes necessary to build society-wide consensus. Content analysis was adopted as a research method; the contents of free-form reflection reports submitted by 128 students were categorized into 6 topics through careful reading and in-depth discussions, and the proportion of students who mentioned topics relevant to the learning targets was calculated. The results show that 83% of the students learned something relevant to the learning targets, and 37% of them made proposals to overcome the trade-offs among policy issues and conflicts among stakeholders. Further, some students actively deepened their understanding through voluntary surveys, presentations of other students, and by comparing the game and reality. These results suggest that game-based learning is an effective method for interdisciplinary education regarding energy policy issues.

Microbubbles coated by visco-elastic shells are important for ultrasound diagnosis using contrast agents, and the dynamics of single coated bubbles has been investigated in the literature. However, although a high number of contrast agents are used in practical situations, there has long been an absence of a nonlinear acoustic theory for multiple coated bubbles, except for our recent work by Kikuchi and Kanagawa [“Weakly nonlinear theory on ultrasound propagation in liquids containing many microbubbles encapsulated by visco-elastic shell,” Jpn. J. Appl. Phys. 60, SDDD14 (2021)], 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 bubbles 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. The 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.

In this study, the weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid was theoretically investigated. This liquid contains several small uniformly distributed spherical polydisperse gas bubbles. The polydispersity considered here represents various types of initial bubble radii, and the liquid contains multiple bubbles, each with an initial radius. Using the method of multiple scales, we first derived the Korteweg–de Vries–Burgers (KdVB) equation with a correction term as a nonlinear wave equation. This equation describes the long-range wave propagation with weak nonlinearity, low frequency, and long wavelength in the polydisperse bubbly liquid using the basic equations in a two-fluid model. The utilization of the two-fluid model incorporates the dependence of an initial void fraction on each coefficient in the nonlinear, dissipation, and dispersion terms in the KdVB equation. Furthermore, unlike previous studies on waves in polydisperse bubbly liquids, we achieved the formulation without assuming an explicit form of the polydispersity function. Consequently, we discovered the contribution of polydispersity to the various effects of wave propagation, that is, the nonlinear, dissipation, and dispersion effects. In particular, the dispersion effect of the waves was found to be strongly influenced by polydispersity.

Weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid uniformly containing many spherical microbubbles is theoretically investigated, especially focusing on an initial small polydispersity of both the bubble radius and the number density of bubbles (i.e., void fraction), which appears in a field far from the sound source. Nonlinear waves in polydispersed bubbly liquids are classified into a form of three cases of nonlinear wave equation describing long-range propagation of waves. Using the method of multiple scales with perturbation expansions and the scaling relations of some nondimensional ratios, from the set of basic equations based on a two-fluid model, (i) for a low-frequency long wave, the Korteweg–de Vries–Burgers (KdVB) equation is derived and (ii) for a moderately high-frequency short wave, (ii-a) the NLS (nonlinear Schrodinger)-I (or LG (Landau–Ginzburg)-I) equation for a weak polydisperse medium and (ii-b) the NLS-II (or LG-II) equation in a strong polydisperse medium are derived in a unified manner. For all cases, polydispersity contributes to the advection effect of waves and induces variable coefficients into the KdVB, NLS-I, and NLS-II equations. Furthermore, the KdVB equation includes an inhomogeneous term owing to the polydispersity and the NLS-II equation includes second-order nonlinearity of polydispersity. The polydisperse effect is finally clarified quantitatively by focusing on the advection coefficients with a help of numerical examples.

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.

The present study theoretically carries out a derivation of the Korteweg–de Vries–Burgers (KdVB) equation and the nonlinear Schrödinger (NLS) equation for weakly nonlinear propagation of plane (i.e., one-dimensional) progressive waves in water flows containing many spherical gas bubbles that oscillate due to the pressure wave approaching the bubble. Main assumptions are as follows: (i) bubbly liquids are not at rest initially; (ii) the bubble does not coalesce, break up, extinct, and appear; (iii) the viscosity of the liquid phase is taken into account only at the bubble–liquid interface, although that of the gas phase is omitted; (iv) the thermal conductivities of the gas and liquid phases are dismissed. The basic equations for bubbly flows are composed of conservation equations for mass and momentum for the gas and liquid phases in a two-fluid model, the Keller-Miksis equation (i.e., the equation for radial oscillations as the expansion and contraction), and so on. By using the method of multiple scales and the determination of size of three nondimensional ratios that are wavelength, propagation speed and incident wave frequency, we can derive two types of nonlinear wave equations describing long range propagation of plane waves. One is the KdVB equation for a low frequency long wave, and the other is the NLS equation for an envelope wave for a moderately high frequency short carrier wave.

The present study theoretically elucidates an effect of the viscosity and the thermal conductivity on the propagation process of finite amplitude disturbance in bubbly liquids by deriving two types of weakly nonlinear wave equations. Appropriate choices of a set of scaling relations of physical parameters characterizing waves, that is, the wavelength, incident wave frequency, propagation speed, yield the derivation systematically. From the combination of appropriate scaling relations and the method of multiple scales, we can derive the Korteweg–de Vries–Burgers equation for the low frequency long wave and the nonlinear Schrödinger equation for slowly varying envelope wave of the quasi-monochromatic short carrier wave. As a result, the incorporation of conservation equation of energy affects nonlinear, dispersion, and dissipation terms for both long and short waves. Especially, the viscosity and the thermal conductivity lead to change considerably the form of coefficient of dissipation term.

The present paper theoretically treats weakly nonlinear propagation of plane progressive waves in an initially quiescent compressible liquid containing a tremendously large number of spherical gas bubbles, focusing on the derivation of an amplitude evolution equation (i.e., nonlinear wave equation). We emphasize the following points: (i) the compressibility of the liquid phase, which has long been neglected, is considered; (ii) the wave propagates with a large phase velocity exceeding the speed of sound in pure water; (iii) bubbles are not created or annihilated. From the method of multiple scales with an appropriate scaling of three nondimensional parameters, we can derive an attenuated nonlinear Schrödinger (NLS) equation, where the phase velocity is larger than the speed of sound in a pure liquid.

This study theoretically investigates plane progressive quasi-monochromatic waves in an initially quiescent compressible liquid containing many spherical gas bubbles, on the basis of the derivation of a nonlinear wave equation that represents waves propagating at a high phase velocity induced by taking the effect of liquid compressibility in consideration. The governing equations for bubbly flows are composed of the conservation equations of mass and momentum, the equation of bubble dynamics as radial oscillations, and so on. By using the method of multiple scales with an appropriate choice of set of scaling relations of nondimensional parameters, the nonlinear Schrödinger (NLS) equation with an attenuation term and some correction terms can be derived from the governing equations. The decrease in the group velocity in a far field is then clarified. The dependence of waveform on wavenumber is implied.

Weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid uniformly containing many spherical gas bubbles is theoretically studied. Our especial focus is an initial polydispersity of bubble radius (i.e., an initial nonuniform distribution of bubble size); an initial number density is uniform. By using the method of multiple scales, the derivation of the KdV-Burgers equation including the polydisperse effect from the basic equations is succeeded. Then, an advection term of waves appears as a new term and the coefficient of advection term is varied from the constant coefficient in the previous study to the variable coefficient in the present study. The polydispersity does not affect the nonlinear, dissipation, and dispersion effects. For the case of monodispersity (i.e., without the advection term), the present KdVB equation coincides with the previously derived KdVB equation for the monodisperse case.

A KdV-Burgers (KdVB) equation describing an weakly nonlinear evolution of pressure waves in liquids uniformly containing many spherical microbubbles has been derived by many researchers and our group under the assumption that the initial void fraction is of the order of unity. However, an effective range of initial void fraction has not been clarified quantitatively. In this paper, we numerically solve the KdVB equation via a split step Fourier method. As a result, we find that (i) pressure wave evolves into an attenuated soliton due to a balance among the nonlinear, dissipation, and dispersion effects; (ii) it is implied that an upper limit of the effective range of the initial void fraction is from 0.69 to 0.7 for the case of the initial bubble radius is 1 mm.

Weakly nonlinear propagation of plane progressive pressure waves with a low frequency and a long wavelength in a compressible water flow uniformly containing many spherical gas bubbles acting a drag force is theoretically investigated. Both a drag force and a virtual mass force are considered as an interfacial transport in the laws of momentum conservation in a two-fluid model. The following results are obtained: (i) By using the method of multiples scales, a KdV-Burgers equation with the term due to the drag force is derived from the basic equations for bubbly flows. (ii) The drag force acting bubbles affects the dissipation effect of waves. (iii) The size of a dissipation due to the drag force is comparable with a dissipation due to the acoustic radiation from oscillating bubbles and is considerably larger than a dissipation due to the viscosity.

Toward medical applications such as a tumor ablation therapy by High-Intensity Focused Ultrasound (HIFU), weakly nonlinear propagation of quasi-planar focused ultrasound in a liquid containing many spherical microbubbles is theoretically investigated, especially focusing on thermal effects of microbubble oscillations. We shall derive a Khokhlov–Zabolotskaya–Kuznetsov (KZK) type equation incorporating the thermal effects of the microbubble oscillations by introducing the energy equation proposed by Prosperetti [J. Fluid Mech. (1991)] into the set of basic equations based on mixture and two-fluid models. Moreover, the phase difference between the temperature gradient and the average temperature inside bubble is considered in the energy equation. As a result, we found that (i) the thermal conduction of gas inside bubbles strongly affects the dissipation of ultrasound and (ii) the phase difference affects the advection of ultrasound.

Weakly nonlinear propagation of plane pressure waves in initially quiescent liquids containing spherical bubbles with a small initial polydispersity of the bubble radius and bubble number density is theoretically studied to clarify the effect of the polydispersity on wave propagation. The wavelength is comparable with the bubble radius and the incident frequency of waves is with an eigenfrequency of the bubble. We consider two different cases of the size of the polydispersity (i.e., small and large polydispersities) and assume that the polydispersity appears at a field far from a sound source. The polydispersity is formulated into a set of perturbation expansions. Bubble oscillations are spherically symmetric, and bubbles do not coalesce, break up, appear, and disappear. The basic set is composed of the conservation laws of mass and momentum for gas and liquid phases based on a two-fluid model, equation of motion for bubble wall, and so on. From the perturbation analysis up to the third order of approximation, two cases of nonlinear wave equation for a long-range propagation of waves were derived. We concluded that the polydispersity affects the advection of waves and is expressed as a variable coefficient.