Keywords

1 Introduction

Medical ultrasound is widely used for imaging purposes (Szabo 2004). It is an effective, mobile, inexpensive method and has the ability to provide high-resolution real-time images of tissue (Shung 2006). Ultrasound imaging is performed by propagating waves through tissue and evaluating the echo that is returned. Due to the different scattering properties of the different tissues, the ultrasound receiver can evaluate the echo and construct an acoustic image.

The ultrasound wave is transmitted by an ultrasound transducer. It consists of piezoelectric crystals, which have the property of changing their volume when a voltage is applied. Applying an alternating current across piezoelectric crystals causes them to volumetrically oscillate at frequencies (∼MHz) that cause mechanical stress on the surrounding medium, thereby converting electric energy into a mechanical wave, which is then transmitted into the body. Analogously, upon receiving the echo the transducer turns the mechanical sound waves back into electrical energy, which can be measured and displayed. The transmit signal consists of a short ultrasound burst. After each burst, the electronics measure the return signal within a small window of time corresponding to the time it takes for the energy to pass through the tissue.

Blood is a poor ultrasound scatterer and individual blood vessels are almost invisible to ultrasound. To increase the contrast of the blood pool, microbubbles can be injected into the bloodstream. The microbubbles scatter ultrasound much more efficiently, allowing very good contrast on the echo image. The contrast ability was discovered accidentally more than 40 years ago during an intravenous injection of a saline solution (Gramiak and Shah 1968). Saline, when injected intravenously, generates tiny microbubbles within the patient’s blood vessels, thus creating an echo on the acoustic image. Since then, the second and third generations of ultrasound contrast agents were developed. Nowadays, commercially available microbubbles are small spheres (typically 1–5 μm in diameter) of gas encapsulated in a biocompatible shell. This size is similar to that of red blood cells, allowing them to circulate inside the bloodstream. The resonance frequency is directly related to the size of the bubbles (1–10 μm diameter) and coincides with the optimum imaging frequencies used in medical ultrasound imaging, 1–10 MHz.

Microbubbles are also widely used for therapy. They can enhance high intensity focused ultrasound (HIFU) therapy (Unger et al. 2004). The bubbles increase heat uptake by the tissue and can reduce the time necessary for an ultrasound therapeutic procedure. They are sufficiently stable for time periods of approximately 15 min following injection (Klibanov 2006). Bubble oscillations and disruptions close to cells create reversible pores within the cell membrane that can enhance drug uptake (Karshafian et al. 2009). Microbubbles may also be used as potential carriers for selective drug delivery (Unger et al. 2009) and for non-invasive molecular imaging (Lindner 2004; Klibanov 2006). They can be covered with targeting ligands, such as antibodies, which bind specifically to target cells at the blood vessel wall.

A novel approach is the use of liquid-based agents rather than gas bubbles. Ultrasound can be used to phase-transition these liquid droplets into gas bubbles; a process known as acoustic droplet vaporization (ADV). Droplets are composed of a volatile perfluorocarbon (PFC), such as perfluoropentane (PFP, 29 °C boiling point). A PFP emulsion does not spontaneously vaporize when injected in-vivo at 37 °C. However, upon exposure to ultrasound above certain acoustic pressure amplitudes, the PFP within the emulsion is vaporized. This opens up possibilities in a wide variety of diagnostic and therapeutic applications, such as embolotherapy (Zhang et al. 2010), aberration correction (Carneal et al. 2011) and drug delivery (Fabiilli et al. 2010a, b). Single and double emulsions of PFC-in-water and oil-in-PFC-in-water can be prepared, for instance, to encapsulate oil soluble drugs (Fig. 9.1).

Fig. 9.1
figure 1

PFC-in-water, PFC-in-oil-in-water and oil-in-PFC-in-water emulsions under the microscope

Full size image

PFC liquids are known for their use in medicine due to their biocompatibility and inertness (Biro et al. 1987). PFC nanodroplet emulsions can be utilized for selective extravasation in tumor regions (Long et al. 1978). Due to their biocompatibility and suggested ability to passively target regions of cancer growth, PFC droplets represent an attractive tool for cancer diagnosis. PFC droplets may also extravasate and be retained in the extravascular space due to the enhanced permeability and retention effect in tumors (Rapoport et al. 2007; Zhang and Porter 2010). Extravasated droplets may be acoustically converted into gas bubbles allowing for ultrasound tumor imaging. At the same time PFC droplets are rich in fluorine, which makes them potential candidates as a contrast agent for MRI imaging. The availability of both intravascular contrast agents (microbubbles), and tumor-specific extravascular contrast agents (nanodroplets), would significantly increase diagnostic and therapeutic capabilities. Moreover, the droplets may be used to deliver chemotherapeutic agents to tumor regions, and locally release them upon exposure to triggered ultrasound (Rapoport et al. 2009).

2 Nonlinear Propagation

The amplitude of the acoustic pressure that is required to nucleate droplets in ADV turns out to be very high (Kripfgans et al. 2000). To obtain a sufficiently high pressure, a focused ultrasound transducer is applied and the droplet is placed in the focal area of the emitted beam. Moreover, the frequency of the emitted ultrasound wave is several MHz. In a typical ADV experiment, the ultrasound wave travels a few centimeters (Kripfgans et al. 2000; Reznik et al. 2013; Shpak et al. 2013a, b; Giesecke and Hynynen 2003; Schad and Hynynen 2010; Williams et al. 2013) before impinging on the droplet. The high pressure, high frequency, applied focusing and long propagation distances are all factors that strengthen the nonlinear behavior of the ultrasound wave (Blackstock 1964; Bacon 1984). As a result, the wave that impinges on the droplet will be a highly deformed version of the one that is emitted by the transducer (Fig. 9.2). This has important consequences for the focusing inside the droplet, as will be demonstrated in Sect. 9.4.2.2.

Fig. 9.2
figure 2

Schematics of nonlinear propagation of an ultrasound wave. T and f are the period of oscillation and frequency of an ultrasound wave, respectively. Two upper plots are the transmitted and propagated wave, and two lower plots are their frequency domains. –fft– stands for the Fast Fourier Transform

Full size image

2.1 Basic Equations for the Nonlinear Ultrasound Beam

Similar to most cases involving nonlinear medical ultrasound, the description of the beam that hits the droplet can be based on the Westervelt equation (Westervelt 1963; Hamilton and Morfey 2008):

$$ {\nabla}^2p-\frac{1}{c_0^2}\frac{\partial^2p}{\partial {t}^2}+\frac{d}{c_0^4}\frac{\partial^3p}{\partial {t}^3}=-\frac{b}{r_0{c}_0^4}\frac{\partial^2{p}^2}{\partial {t}^2} $$
(9.1)

where \( {\nabla}^2={\partial}^2/\partial {x}^2+{\partial}^2/\partial {y}^2+{\partial}^2/\partial {z}^2 \) is the Laplace operator and \( p=p\left(x,y,z,t\right) \) denotes the acoustic pressure. The medium in which the ultrasound wave propagates is characterized by the ambient speed of sound \( {c}_0 \), the ambient density of mass \( {\rho}_0 \), the diffusivity of sound \( \delta \) and the coefficient of nonlinearity \( \beta \). Unfortunately, closed-form analytical solutions of this equation do not exist and its numerical solution generally requires considerable computational effort. However, in the present case of a narrow, focused beam and a homogeneous medium, some simplifying assumptions can be made. Firstly, it may be assumed that the predominant direction of propagation is along the transducer axis, which is taken in the z -direction. In this case, we can replace the ordinary time coordinate \( t \) by the retarded time coordinate \( t=\left(t-{t}_0\right)-\left(z-{z}_0\right)/{c}_0 \), which keeps the same value when traveling along with the wave. Here, \( {t}_0 \) is the time at which the transducer emits the pressure wave, and \( {z}_0 \) is the axial position of the transducer. The equivalent of Eq. 9.1 in the co-moving time frame is:

$$ {\nabla}^2\overline{p}-\frac{2}{c_0}\frac{\partial^2\overline{p}}{\partial z\partial \tau }+\frac{d}{c_0^3}\frac{\partial^3\overline{p}}{\partial {\tau}^3}=-\frac{b}{r_0{c}_0^4}\frac{\partial^2{\overline{p}}^2}{\partial {\tau}^2} $$
(9.2)

with \( \overline{p}=\overline{p}\left(x,y,z,t\right) \) denoting the acoustic pressure in the co-moving time frame. Secondly, it may be assumed that in the retarded time frame the axial derivative \( {\partial}^2\overline{p}/\partial {z}^2 \) is much smaller than the lateral derivatives \( {\partial}^2\overline{p}/\partial {x}^2 \) and \( {\partial}^2\overline{p}/\partial {y}^2 \). This motivates the use of the parabolic approximation \( {\nabla}^2\overline{p}\approx {\nabla}_{\perp}^2\overline{p} \), where \( {\nabla}_{\perp}^2={\partial}^2/\partial {x}^2+{\partial}^2/\partial {y}^2 \) is the Laplace operator in the lateral plane. This approximation is valid for waves propagating under at most 20° of the transducer axis (Lee and Pierce 1995). Applying the parabolic approximation to Eq. 9.2 and rearranging terms results in the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation (Zabolotskaya and Khokhlov 1969; Kuznetsov 1971):

$$ \frac{\partial^2\overline{p}}{\partial z\partial \tau }=\frac{c_0}{2}{\nabla}_{\perp}^2\overline{p}+\frac{\delta }{2{c}_0^3}\frac{\partial^3\overline{p}}{\partial {\tau}^3}+\frac{b}{2{r}_0{c}_0^3}\frac{\partial^2{\overline{p}}^2}{\partial {\tau}^2} $$
(9.3)

Dedicated coordinate transformations may be applied to improve the numerical solution in the far field (Hamilton et al. 1985; Hart and Hamilton 1988) or to adapt to specific forms of focused beams (Kamakura et al. 2000), but these will not be discussed here.

2.2 Numerical Solution for the Nonlinear Ultrasound Beam

We will follow a well-known numerical solution strategy (Lee and Hamilton 1995; Cleveland et al. 1996) that is based on the time integrated version of Eq. 9.3:

$$ \frac{\partial \overline{p}}{\partial z}=\frac{c_0}{2}\underset{-\infty }{\overset{t}{{\displaystyle \int }}}{\nabla}_{\perp}^2\overline{p}\left({\tau}^{\prime}\right)d{\tau}^{\prime }+\frac{\delta }{2{c}_0^3}\frac{\partial^2\overline{p}}{\partial {\tau}^2}+\frac{b\overline{p}}{r_0{c}_0^3}\frac{\partial \overline{p}}{\partial \tau } $$
(9.4)

The first term at the right-hand side of this equation accounts for the diffraction of the beam, the second term for its attenuation, and the third term for its nonlinear distortion. Further, the solution strategy is based on the split-step approach. This means that the field \( \overline{p} \) is stepped forward over a succession of parallel planes with mutual distance \( \varDelta \kern0.1em z \), where the field \( \overline{p}\left(x,y,{z}_0,\tau \right) \) in the transducer plane acts as the starting plane. The stepsize \( \varDelta z \) is taken sufficiently small, allowing that each of the above phenomena may be accounted for in separate sub steps (Varslot and Taraldsen 2005). Therefore, the total step \( z\to z+\varDelta z \) involves the numerical solution of the separate equations:

$$ \frac{\partial \overline{p}}{\partial z}=\frac{c_0}{2}\underset{-\infty }{\overset{t}{{\displaystyle \int }}}{\nabla}_{\perp}^2\overline{p}\left(\tau^{\prime}\right)d\tau^{\prime } $$
(9.5)
$$ \frac{\partial \overline{p}}{\partial z}=\frac{\delta }{2{c}_0^3}\frac{\partial^2\overline{p}}{\partial {\tau}^2} $$
(9.6)
$$ \frac{\partial \overline{p}}{\partial z}=\frac{b\overline{p}}{\rho_0{c}_0^3}\frac{\partial \overline{p}}{\partial \tau } $$
(9.7)

over the same interval, where the result of solving one equation is used as the input for solving the next one. A numerical implementation of the above process is used to step the acoustic pressure from the transducer to the focus of the beam, i.e. the location of the droplet. For convenience, it is now assumed that the droplet is located at the origin of the coordinate system and that the source emits the pressure wave at \( {t}_0={z}_0/{c}_0 \). This makes \( \tau =t \) at the position of the droplet. For ease of notation, the bar and the coordinates of the droplet will be suppressed, and the pressure at the droplet position, as obtained from the numerical solution of the KZK-equation, will simply be indicated by \( {p}_{KZK}(t) \).

2.3 Nonlinear Pressure Field at the Focus of the Beam

The nonlinear pressure field at the focus of the beam can be expanded in a Fourier series:

$$ \begin{array}{c}{p}_{KZK}(t)={\displaystyle \sum}_{n=0}^{\infty }{a}_n \cos \left(nwt+{f}_n\right)\\ {}=Re\left[{\displaystyle \sum}_{n=0}^{\infty }{a}_n{e}^{i\left(nwt+{f}_n\right)}\right]\end{array} $$
(9.8)

where \( {a}_n \) and \( {f}_n \) are the amplitudes and the phases of the \( n-th \) harmonic component of the ultrasound wave. For convenience, all the subsequent derivations will be given in the complex representation, so we will omit taking the real part and simply write:

$$ {p}_{KZK}(t)={\displaystyle \sum}_{n=0}^{\infty }{a}_n{e}^{i\left(n\omega t+{\phi}_n\right)} $$
(9.9)

Given that nonlinear deformation of the waveform builds up over distance and the droplet is four orders of magnitude smaller in size than the distance to the transducer, the additional nonlinear distortion inside the droplet is neglected. This implies that wave propagation inside the droplet is considered linear, so the superposition theorem holds, and the focusing of each harmonic component in the droplet may be analyzed on an individual basis, as will be done in Sect. 9.4.2.2.

3 Bubble Dynamics

3.1 Dynamics of a Gas Bubble

Bubble radial oscillations are governed by the Rayleigh-Plesset equation:

$$ \ddot{R}R+\frac{3}{2}{\dot{R}}^2=\frac{\varDelta P}{\rho } $$
(9.10)

where \( R \), \( \dot{R} \), and \( \ddot{R} \) are the radius, the velocity and the acceleration of the bubble wall, respectively, and \( \rho \) is the density of the liquid. \( \varDelta P={P}_L(R)-{P}_{\infty } \) is the pressure difference between the liquid at the bubble wall \( {P}_L(R) \) and the external pressure infinitely far from the bubble \( {p}_{\infty } \). Equation 9.10 was first described by Lord Rayleigh (1917) for the case \( \varDelta P=0 \) and was later refined (Plesset 1949; Noltingk and Neppiras 1950; Neppiras and Noltingk 1951; Poritsky 1952). It is derived for a spherically symmetric bubble, and follows from the Bernoulli’s equation and the continuity equation (Leighton 1994). Equation 9.10 assumes spherical symmetry of the bubble, and the motion of the liquid around the bubbles is considered to be spherically symmetric. The liquid is incompressible.

The bubble is assumed to be much smaller than the acoustic wavelength, such that acoustic pressure is considered to be uniform. Thus, the pressure at infinity is the sum of the acoustic forcing \( P(t) \) and the ambient pressure \( {P}_0 \):

$$ {p}_{\infty }=P(t)+{P}_0 $$
(9.11)

The interfacial pressure acting on the liquid at the bubble wall consists of the Laplace pressure \( 2\sigma /{R}_0 \), viscous pressure \( 4\mu \dot{R}/R \) and the gas pressure \( {P}_g \). Neglecting the vapor pressure of the liquid, the gas pressure inside the bubble as a function of the bubble radius \( R \) can be described by the ideal gas relation \( {P}_g{V}^{\gamma }= const \), where \( \gamma \) is the polytropic constant and \( V\propto {R}^3 \) is the bubble volume. For this derivation we first neglect the gas diffusion. Thus, the total number of gas molecules inside the bubble is constant. In equilibrium, the pressure inside the bubble \( {P}_{eq} \) is equal to the sum of the ambient pressure \( {P}_0 \) and the Laplace pressure:

$$ {P}_{eq}={P}_0+\frac{2\sigma }{R_0} $$
(9.12)

where \( \sigma \) and \( {R}_0 \) are the surface tension and the equilibrium radius, respectively. In combination with the ideal gas law, the dependence of the gas pressure as a function of the bubble radius can be written as \( {P}_g=\left({P}_0+\frac{2\sigma }{R_0}\right){\left(\frac{R_0}{R}\right)}^{3\gamma } \). The right-hand side of Eq. 9.10 can then be written as:

$$ \begin{array}{c}DP=\left({P}_0+\frac{2s}{R_0}\right){\left(\frac{R_0}{R}\right)}^{3\gamma }-{P}_0-\frac{2s}{R}\\ {}-4\mu \frac{\dot{R}}{R}-P(t)\end{array} $$
(9.13)

which gives the final form of the bubble dynamic equation:

$$ \begin{array}{c}r\left(\ddot{R}R+\frac{3}{2}{\dot{R}}^2\right)=\left({P}_0+\frac{2s}{R_0}\right){\left(\frac{R_0}{R}\right)}^{3g}\\ {}-{P}_0-\frac{2s}{R}-4m\frac{\dot{R}}{R}-P(t)\end{array} $$
(9.14)

The microbubbles in the ultrasound contrast agents can be encapsulated with a phospholipid, protein or polymer coating, thus preventing bubbles from dissolution. For more details please see (Marmottant et al. 2005; de Jong et al. 2007; Church 1995). The viscoelastic coating also contributes to an increased stiffness and to additional viscous damping (Overvelde et al. 2010).

3.2 Linearization

The acoustic pressure typically has the form of a sinusoidal oscillation \( P(t)={P}_A \sin \left(\omega t\right) \), with \( {P}_A \) being the driving pressure amplitude, and \( \omega \) the driving pressure angular frequency. With relatively small oscillation amplitudes Eq. 9.10 can be linearized. To rewrite Eq. 9.10 in linear terms we express the bubble radius \( R \) as:

$$ R={R}_0\left(1+x\right) $$
(9.15)

with \( {R}_0 \) the equilibrium radius, as before, and \( x\ll 1 \) a small dimensionless perturbation to the radius. Substituting Eq. 9.15 into Eq. 9.10 and retaining only first-order terms, \( x \), \( \dot{x} \) and \( \ddot{x} \) gives:

$$ \ddot{x}+2b\dot{x}+{w}_0^2x=\frac{P_A}{r{R}_0^2} \sin \left(w\kern0.1em t\right) $$
(9.16)

where

$$ {w}_0=\sqrt{\frac{1}{r{R}_0^2}\left[3g\left({P}_0+\frac{2s}{R_0}\right)-\frac{2s}{R_0}\right]} $$
(9.17)

the eigenfrequency of bubble oscillations, and:

$$ b=\frac{2m}{r{R}_0^2} $$
(9.18)

the damping due to viscosity. The damping has the dimensions of the reversed time \( \left[{s}^{-1}\right] \) and represents how fast the amplitude of oscillations is decaying in time due to the energy loss.

The solution to the equation Eq. 9.16 is:

$$ x(t)={X}_t{e}^{-bt} \cos \left({w}_1t\right)+{X}_s \cos \left( wt+{f}_1\right) $$
(9.19)

with the \( {\phi}_1 \) being the phase shift between the two terms:

$$ {f}_1= \arctan \left(\frac{w_0^2-{w}^2}{2bw}\right) $$
(9.20)

The first term of Eq. 9.19 is the transient solution. Its amplitude dampens out in time as \( {X}_t{e}^{-bt} \), where \( {X}_t \) is the amplitude of transient oscillations at time \( {t}_0=0 \). Not only the viscosity of water can contribute to the damping, but also the acoustic reradiation and the viscosity of the coating shell and thermal damping. For more details please see (Overvelde et al. 2010). The frequency of the transient solution is equal to \( {w}_1=\sqrt{w_0^2-{b}^2} \). The amplitude of the transient solution \( {X}_t \) depends strongly on the initial conditions.

The second term of Eq. 9.19 is the steady-state solution. The amplitude of the steady-state response depends on the driving frequency as:

$$ {X}_s=\frac{P_A}{r\kern0.1em {R}_0^2}\frac{1}{\sqrt{{\left({w}_0^2-{w}^2\right)}^2+4{w}^2{b}^2}} $$
(9.21)

The resonance frequency \( {\omega}_{\mathrm{res}} \) of the system, by definition, corresponds to the maximal amplitude of the steady-state solution. \( {X}_s \) is at maximum, when the denominator in the Eq. 9.21 is at minimum. Thus, the resonance frequency relates to the eigenfrequency \( {\omega}_0 \) as:

$$ {w}_{\mathrm{res}}=\sqrt{w_0^2-2{b}^2} $$
(9.22)

The smaller the damping \( b, \) the closer the resonance frequency to the eigenfrequency of the bubble oscillations. Additionally, for large bubbles, when the Laplace pressure is small compared to the ambient pressure Eq. 9.17 simplifies to:

$$ {w}_M=2p{f}_M=\sqrt{3g\frac{P_0}{r{R}_0^2}} $$
(9.23)

with \( {f}_M \) the Minnaert eigenfrequency, resonance frequency of the bubble (Minnaert 1933). Relation Eq. 9.23 tells us that the resonance frequency can be estimated directly from the bubble radius \( {R}_0 \). For a bubble in water at standard pressure (\( {P}_0=100kPa,\;r=1000 kg/{m}^3 \)), the equation becomes \( {f}_M{R}_0\approx 3.26\;m\kern0.1em m. MHz \). The smaller the bubble radius, the higher the resonance frequency becomes.

It is insightful to make the analogy to the classical mass-spring system. The dynamics of the classical mass spring system is governed by the equation:

$$ \ddot{x}+2\frac{b^{\prime }}{m}\dot{x}+{w}_0^{\prime 2}x=\frac{F_0}{m} \sin (wt) $$
(9.24)

where \( w{\prime}_0=\sqrt{k/m} \) is the eigenfrequency, \( {\beta}^{\prime } \) is the damping constant, \( k \) is the spring stiffness, \( {F}_0 \) is the driving force and \( m \) is the mass. Equation 9.24 has the same form as Eq. 9.16. Thus, a gas inside the bubble, represented by the polytropic constant \( g \), acts as the restoring force, the liquid around the bubble acts as a mass \( \left(4p{R}_0^3r\right) \), and ultrasound is acting as a driving force \( \left(12 pg\kern0.1em {R}_0{P}_0\right) \).

3.3 Pressure Emitted by the Bubble

Far from the bubble wall, at a distance \( r \), the velocity of the liquid \( {v}_r \) can be calculated from the continuity equation (Prosperetti 2011):

$$ 4p{r}^2{v}_r=4p{R}^2\dot{R} $$
(9.25)
$$ {v}_r=\frac{R^2}{r^2}\dot{R} $$
(9.26)

The liquid is incompressible and the bubble wall and the liquid motion around the bubble are spherically symmetric. \( \dot{R} \) is the velocity of the bubble wall in the radial direction, as before.

The pressure field, generated by the radial bubble wall oscillations, can be calculated from the Euler equation (Prosperetti 2011):

$$ r\frac{\partial v}{\partial t}+\frac{\partial p}{\partial r}=0 $$
(9.27)

where \( p \) is the pressure emitted by the bubble. In Eq. 9.27 we omit the nonlinear convective term. Substituting the expression of the velocity field (Eq. 9.26) into Eq. 9.27 gives the pressure gradient:

$$ \frac{\partial p}{\partial r}=-\frac{r}{r^2}\frac{d}{dt}\left({R}^2\dot{R}\right) $$
(9.28)

and the pressure emitted by the bubble:

$$ p=\frac{r}{r}\frac{d}{dt}\left({R}^2\dot{R}\right)=r\left(\frac{R^2\ddot{R}+2R{\ddot{R}}^2}{r}\right) $$
(9.29)

3.4 Secondary Bjerknes Force

Let us now consider two interacting gas bubbles, separated by a distance \( l \). The distance between the bubbles 1 and 2 is much larger that their radii \( {R}_1(t) \) and \( {R}_2(t) \). Thus, we can consider the motion of the liquid around the bubbles to be spherically symmetric. Bubble 2 with volume \( {V}_2=\frac{4}{3}p{R}_2^3 \) experiences a force \( {F}_{12} \) as a result of the pressure emitted by bubble 1, \( {p}_1 \) (Leighton 1994):

$$ {F}_{12}=-{V}_2\nabla {p}_1 $$
(9.30)

The force is directed along the line, which connects the centers of the two interacting bubbles. Substitution of Eq. 9.28 (expression for the pressure gradient generated by the first bubble) into Eq. 9.30 yields the force of the first bubble on the second one at a distance \( l \):

$$ \begin{array}{c}{F}_{12}=-{V}_2\frac{\partial {p}_1}{\partial r}\Big|{}_{r=l}={V}_2\frac{r}{l^2}\frac{d}{dt}\left({R}_1^2{\dot{R}}_1\right)\\ {}=\frac{r}{4\pi {l}^2}{V}_2\frac{d^2{V}_1}{d{t}^2}\end{array} $$
(9.31)

where \( {V}_1=\frac{4}{3}p{R}_1^3 \) the volume of bubble 1.

The net radiation force acting on a neighboring bubble is called the secondary Bjerknes force \( {F}_B \) after Bjerknes (Bjerknes 1906). The time averaged equation \( \left\langle {F}_{12}\right\rangle \) is obtained by integrating Eq. 9.31 over the period of volume oscillations by partial integration:

$$ {F}_B=\left\langle {F}_{12}\right\rangle =-\frac{r}{4p{d}^2}\left\langle {\dot{V}}_1{\dot{V}}_2\right\rangle $$
(9.32)

A positive value of \( \left\langle {\dot{V}}_1{\dot{V}}_2\right\rangle \) corresponds to attraction of the bubbles, and a negative value to repulsion. This means that the bubbles that oscillate with the same phase will attract each other. Note also the symmetry of Eq. 9.32. To calculate the force of the second bubble on the first one \( \left\langle {F}_{21}\right\rangle \), one just needs to exchange indexes 1 ↔ 2. \( \left\langle {F}_{21}\right\rangle \) has the same magnitude, but an opposite direction as \( \left\langle {F}_{12}\right\rangle \).

4 Droplet Dynamics

4.1 Oscillatory Translations

The typical pressure amplitudes, which are used to activate perfluorocarbon droplets, are two orders of magnitude higher that those used to drive ultrasound contrast agents. Water itself always experiences periodic compression as a result of ultrasound forcing (Leighton 1994). Let us express such oscillations as:

$$ \epsilon ={\epsilon}_0 \sin \left( wt-kx\right) $$
(9.33)

where \( \epsilon \) is the fluid particle displacement. The acoustic impedance, by definition, is the ratio of the driving pressure to the fluid particle velocity (Leighton 1994):

$$ Z={P}_A/{\dot{\epsilon}}_0 $$
(9.34)

where \( {\dot{\epsilon}}_0 \) is the particle displacement velocity amplitude, and \( {P}_A \) is the acoustic pressure amplitude. The \( {\dot{\epsilon}}_0 \) particle velocity amplitude relates to the particle displacement amplitude \( {\epsilon}_0 \) as \( {\dot{\epsilon}}_0=w\kern0.1em {\epsilon}_0 \), which follows from Eq. 9.33 by taking its time derivative. The \( dP \) pressure change with respect to the equilibrium value is related to a \( dV \) volume change by the bulk modulus \( B, \) defined by (Leighton 1994):

$$ dP=-B\frac{dV}{V} $$
(9.35)

Equation 9.35 can be used to calculate the acoustic pressure \( P \) at any given spatial point \( {x}_0 \) as \( P\left({x}_0\right)=-B\frac{\partial \epsilon }{\partial x}\Big|{}_{x={x}_0} \). Applying this relation to Eq. 9.33 gives \( {P}_A=Bk{\epsilon}_0 \), or \( {P}_A=B\frac{k}{w}{\dot{\epsilon}}_0 \). The acoustic impedance Eq. 9.34 can then be written as:

$$ Z=B\frac{k}{w} $$
(9.36)

or by using the equation for the wave speed \( c=w/k=\sqrt{B/r} \):

$$ Z=rc $$
(9.37)

With the relations given above, one can now estimate the oscillatory translational amplitude. For the case of \( f=3.5 MHz, \) \( {P}_A=8MPa \) and \( {c}_w=1522m\kern0.1em /\kern0.1em s, \) the speed of sound in water at 37 °C, the amplitude is \( {\epsilon}_0={P}_A/2prf{c}_w=210 nm. \)

The acoustic impedance \( Z=rc \) has the analogy with a refractive index \( n \) in optics. The ultrasound wave at the interface of two substances with different acoustic impedances \( {Z}_1 \) and \( {Z}_2 \) will experience a reflection and a refraction, similarly as light would experience at the interface with two different refractive indexes \( {n}_1 \) and \( {n}_1 \).

Let us now denote \( {q}_I \), \( {q}_R \), and \( {q}_T \) the incident, the translated and the reflected angles, respectively (Fig. 9.3). One can derive the relation between these angles by considering continuity of the normal displacement at the interface. This gives Snell’s reflection law \( \sin \left({q}_I\right)= \sin {q}_R \) and \( {c}_2 \sin {q}_I={c}_1 \sin {q}_T \), where \( {c}_1 \) and \( {c}_2 \) are the speed of sound of the first and the second medium, respectively (Leighton 1994).

Fig. 9.3
figure 3

Schematics of reflection and transmission of a wave of displacement

Full size image

4.2 Focusing inside a Spherical Droplet

When an interface between two acoustic media has a finite curvature \( R, \) as in the case of a spherical droplet, acoustic focusing is observed. This is a similar effect as the focusing of light by an optical lens. First, the case of large droplets is considered, i.e. when the acoustic wavelength \( \lambda \) is much smaller than the droplet radius \( R. \) Next, the case where \( \lambda \) is of the order of \( R, \) or even larger, is considered.

4.2.1 Case 1: Droplets much Larger in Size than the Wavelength

When \( l\ll R, \) the refraction formulas provided by the theory of geometrical scattering apply. When a parallel beam of light travels in a medium with refractive index \( {n}_1 \) and encounters a spherical interface between this medium and a second medium with refractive index \( {n}_2 \), either the transmitted or the reflected wave focuses in a point at a distance:

$$ f=R\frac{n_2}{n_2-{n}_1} $$
(9.38)

This distance is measured from the intersection point of the interface and the beam axis, which crosses the center of the curvature (Fig. 9.4).

Fig. 9.4
figure 4

Schematics of the focusing of an acoustic wave on the droplet sphere when the acoustic wavelength is much larger than the droplet radius

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In analogy with the optical focus, an acoustic focus can be calculated for the case \( l\ll R \) by simply replacing \( {n}_1/{n}_2 \) with \( {c}_1/{c}_2 \) in the equation above. This gives:

$$ f=R\frac{c_1}{c_1-{c}_2} $$
(9.39)

For instance, when a large spherical perfluoropentane droplet with \( {c}_2=406m/s \) is immersed in water with \( {c}_1=1522m/s \) (at 37 °C), the acoustic focus is at \( f=1.36R \). This means that the acoustic wave focuses on a distal side, \( 0.36R \) away from the geometrical droplet center.

4.2.2 Case 2: Droplets Similar or Smaller in Size than the Wavelength

The situation complicates when the radius of the droplet is of the same order of magnitude as the wavelength, or even smaller, i.e. when \( l\sim Rorl\gg R \). Experimental data obtained with small droplets shows that the ultrasound beam focuses on the proximal side of the droplet, which is not in agreement with the prediction above. Obviously, in this case geometrical considerations can no longer be applied, and a full wave theory must be applied. Figure 9.5 shows the configuration of the acoustic diffraction problem that will be solved here. Throughout the derivations, the parameters of the surrounding medium are labeled with a subscript 1, and the parameters of the medium inside the droplet are labeled with a subscript 2.

Fig. 9.5
figure 5

Configuration of the droplet and the incident, transmitted, reflected ultrasound waves

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At the location of the droplet, the incident ultrasound wave is considered to be planar. In view of Eq. 9.9, it is written as:

$$ {p}_i\left(x,y,z,t\right)={\displaystyle \sum}_{n=0}^{\infty }{a}_n{e}^{i\left(nwt-n{k}_1z+{f}_n\right)} $$
(9.40)

where \( {k}_1=w/{c}_1 \) is the wave number outside the droplet. To keep the derivations simple, the diffraction problem will first be solved for one spectral component:

$$ {p}_i\left(x,y,z,t\right)=a{e}^{i\left( wt-{k}_1z+f\right)} $$
(9.41)

In view of the spherical symmetry of the configuration, it is convenient to apply a coordinate transformation from cartesian coordinates \( \left(x,y,z\right) \) to spherical coordinates \( \left(r,\;q,\;f\right) \), where \( f \) is the azimuthal angle measured with respect to the positive \( z \)-axis, and \( f \) is the elevation angle in the \( xy \)-plane. Due to the waves having rotational symmetry with respect to the \( z \)-axis, there will be no dependence on \( \phi, \) and this coordinate will be omitted. In spherical coordinates, the incident pressure wave can be written as a summation of spherical harmonics:

$$ {p}_i\left(r,\;q,\;t\right)=a{e}^{i\left(w\kern0.1em t+f\right)}{\displaystyle \sum}_{m=0}^{\infty }{g}_m{j}_m\left({k}_1r\right){P}_m\left( \cos q\right) $$
(9.42)

Here, \( {g}_m=\left(2m+1\right){\left(-i\right)}^m \), \( {j}_m \) is the spherical Bessel function of the first kind and order \( m \), and \( {P}_m \) is the \( m- \)th order Legendre polynomial. The spherical Bessel function \( {j}_m \) is related to the ordinary Bessel function \( {J}_m \) according to \( {j}_m(x)=\sqrt{\left(p/2x\right)}{J}_{m+1/2}(x) \).

When the incident wave encounters the droplet, it gives rise to a transmitted wave inside the droplet:

$$ {p}_t\left(r,\;q,\;t\right)=a{e}^{i\left( wt+f\right)}{a}_m{j}_m\left({k}_2r\right){P}_m\left( \cos q\right) $$
(9.43)

and a reflected wave outside the droplet:

$$ {p}_r\left(r,\;q,\;t\right)=a{e}^{i\left( wt+f\right)}{b}_m{h}_m^{(2)}\left({k}_1r\right){P}_m\left( \cos q\right) $$
(9.44)

In these equations, \( {k}_2=w/{c}_2 \) is the wave number inside the droplet, and \( {h}_m^{(2)} \) is the spherical Hankel function of the second kind and order \( m \). The spherical Hankel function \( {h}_m^{(2)} \) is related to the ordinary Hankel function \( {H}_m^{(2)} \) following \( {h}_m^{(2)}(x)=\sqrt{\left(p/2x\right)}{H}_{m+1/2}^{(2)}(x) \). At the spherical interface between the outside and the inside of the droplet, the pressure and the radial particle velocity should be continuous. The latter requirement can be translated into a condition on the radial derivative of the pressure. In mathematical form, the boundary conditions at the interface are:

$$ \underset{r\downarrow R}{ \lim}\left[{p}_i\left(r,\;q,\;t\right)+{p}_r\left(r,\;q,\;t\right)\right]=\underset{r\uparrow R}{ \lim }{p}_t\left(r,\;q,\;t\right) $$
(9.45)
$$ \begin{array}{l}\underset{r\downarrow R}{ \lim}\frac{1}{r_1}\frac{\partial }{\partial r}\left[{p}_i\left(r,\;q,\;t\right)+{p}_r\left(r,\;q,\;t\right)\right]\\ {}=\underset{r\uparrow R}{ \lim}\frac{1}{r_2}\frac{\partial }{\partial r}{p}_t\left(r,\;q,\;t\right)\end{array} $$
(9.46)

which should hold for all \( \theta \) and \( t \). Substitution of Eqs. 9.42, 9.43, and 9.44 into these boundary conditions results in a system of two equations for \( {a}_m \) and \( {b}_m \). Solution of this system yields:

$$ {a}_m={g}_m\frac{Z_2{j}_m\left({k}_1R\right){h}_m^{(2)}\left({k}_1R\right)-{Z}_2{h}_m^{(2)}\left({k}_1R\right)j{\prime}_m\left({k}_1R\right)}{Z_2{j}_m\left({k}_2R\right){h}_m^{(2)}\left({k}_1R\right)-{Z}_1{h}_m^{(2)}\left({k}_1R\right)j{\prime}_m\left({k}_2R\right)} $$
(9.47)
$$ {b}_m={g}_m\frac{Z_1{j}_m\left({k}_1R\right)j{\prime}_m\left({k}_2R\right)-{Z}_2{j}_m\left({k}_2R\right)j{\prime}_m\left({k}_1R\right)}{Z_2{j}_m\left({k}_2R\right){h}_m^{(2)}\left({k}_1R\right)-{Z}_1{h}_m^{(2)}\left({k}_1R\right)j{\prime}_m\left({k}_2R\right)} $$
(9.48)

where \( {Z}_1={r}_1{c}_1 \) and \( {Z}_2={r}_2{c}_2 \) are the acoustic impedances of the media outside and inside the droplet, respectively. The prime indicates the derivative of a function. The constant \( {a}_m \) can be considered as the transmission coefficient of the droplet interface for spherical harmonics of order \( m \), and the constant \( {b}_m \) can be considered as the corresponding reflection coefficient. At this stage, the problem of finding the wave inside the droplet due to a single sinusoidal component of the incident wave is solved.

To find the wave that is formed inside the droplet by the nonlinear incident wave, all the transmitted waves caused by the individual components of the incident wave must be added. The result is:

$$ {p}_{inside}\left(r,\;q,\;t\right)={p}_t\left(r,\;q,\;t\right)={\displaystyle \sum}_{n=0}^{\infty }{\displaystyle \sum}_{m=0}^{\infty }{a}_n{e}^{i\left(nwt+{f}_n\right)}{a}_{n,m}{j}_m\left(n{k}_1r\right){P}_m\left( \cos q\right) $$
(9.49)

where \( {a}_{m,n} \) follows from Eq. 9.47 by replacing \( {k}_1 \) by \( n{k}_1 \) and \( {k}_2 \) by \( n{k}_2 \). This equation can be used to calculate the pressure in any position \( \left(r,\;q,\;j\right) \) at any time \( t \). Numerical implementation requires that both summations involve a finite number of terms. This forms no significant limitation, because in practice only a limited number of \( N \) harmonics will give a significant contribution to the ultrasound field inside the droplet, and only a limited number of \( M \) spherical harmonics is required to accurately represent this field. However, another numerical issue arises when the radius of the droplet is much smaller than the wavelength. In this case, the numerical results for the spherical Bessel and Hankel functions may contain large errors. This problem may be eliminated by first approximating these functions by their series expansion around zero. With the full pressure field determined both in space and time we can now find the local maximum of pressure - the focus.

The pressure amplification factor in the focusing spot, as well as its location, depend on the input parameter values, i.e. the pressure amplitude, the frequency and the transducer geometry and size, which prescribe the focusing strength and the propagation distance to the focal point. For instance, in case of a \( R=10 mm \) perfluoropentane droplet immersed in water and insonified with an incoming ultrasound wave with a peak negative pressure \( {P}_i^{-}=-4.5MPa \) and frequency \( f=3.5 MHz \) (\( l=430m\mathrm{m} \) in water at 37 °C) coming from a transducer with a 3.81 cm focal distance, a focused peak negative pressure of \( {P}_{inside}^{-}=-26MPa \) is achieved within the droplet (Fig. 9.6). Thus, a near six-fold increase in the peak negative pressure amplitude is observed in a concentrated region on the proximal side around z = −0.4R.

Fig. 9.6
figure 6

Schematics of the superharmonic focusing effect within a perfluoropentane spherical droplet. The black line represents the acoustic pressure waveform on the axis of symmetry (θ = 0) as a function of the z-coordinate in the absence of a droplet. The red solid line is the focused pressure in presence of the droplet. The snapshot represents the moment of minimum focused pressure both in time and space. The blue shaded region depicts the position of the droplet

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From Eq. 9.47 it follows that the pressure inside the droplet, due to a single incident wave component, depends on the dimensionless product \( wR \). When for two droplets with different radii \( {R}_1 \) and \( {R}_2 \) the relation \( {w}_1{R}_1={w}_2{R}_2 \) holds, an incident wave with frequency \( {f}_1 \) encountering a droplet with radius \( {R}_1 \) is focused at the same relative position within the droplet as an incident wave with frequency \( {f}_2 \) that hits a droplet with radius \( {R}_2. \) This implies that when larger droplets turn out to vaporize more easily than smaller droplets at the same frequency, it also follows that for the same radius droplets are easier to evaporate at high frequencies than at low frequencies, and vice-versa.

However, nonlinear propagation makes this picture more complex. First, the higher the acoustic pressure amplitude, the more nonlinear the wave becomes as the amplitudes of the higher harmonics build up roughly as \( {\left({P}_{surface}\right)}^{n-1} \), where \( {P}_{surface} \) is the pressure amplitude at the transducer surface and \( n \) is the number of the particular harmonic. Second, the nonlinear propagation depends on the frequency. Additionally, of course, the nonlinear beam is focused differently from the linear one, with different pressure amplification factors and focusing positions for each harmonic. Finally, the shape of the nonlinearly distorted wave is strongly dependent on the parameters of the propagating media. For human tissue the Goldberg ratio is lower than for water (Szabo et al. 1999). This indicates that nonlinear distortion is easier to achieve in water compared to tissue. Therefore the experiments performed in-vivo are expected to have different nucleation patterns, with a higher nucleation threshold compared to the in-vitro experiments.

Knowledge of the physics of acoustic focusing in small droplets is important for the optimization of acoustic droplet vaporization for therapeutic applications. This is particularly the case for attaining activation at low acoustic pressures, thereby minimizing the negative bio-effects associated with the use of high-intensity ultrasound. Moreover, it helps in the design of droplets: by mixing liquids with different physical properties, the acoustic impedance may be tuned through a change of the density of mass and/or the speed of sound. Using dedicated waveforms, the amplitudes and phases of the nonlinear wave at the focus of the beam can be optimized to obtain maximal constructive interference within the droplets and obtain maximal focusing strength at any particular acoustic input pressure. Moreover, the knowledge of consecutive droplet vaporization dynamics is important because it affects the surrounding tissue and may cause damage. It is not only the acoustic impedance mismatch between the droplet and the surrounding media that determines the interior pressure, but also the exterior of the droplet. Here we have only considered single droplets, but clouds of droplets may cause complicated pressure scattering patterns and may lead to different focusing spots. One can also think of periodic arrangements of monodisperse droplets to observe similar diffraction relations as we have with light passing through crystals.

4.3 Radial Vapor Bubble Expansion

There are three main physical mechanisms that govern the vapor bubble growth process: phase-change, heat transfer and inertia. There are also two phenomena, which can limit vapor bubble growth. Firstly, the vapor bubble pushes the surrounding liquid as it grows. The force by which the liquid is pushed is determined by the pressure which acts on the bubble wall. The surrounding liquid has inertia, and the vapor bubble growth rate will be limited by this inertia. Secondly, the phase-change from liquid to vapor is an endothermic process, requiring heat absorption. The required heat for vaporization is transferred from the liquid around the bubble by cooling the surroundings. The rate of this process is limited by heat transfer.

Let us now first have a closer look at inertial growth limitation. Here we assume that the heat transfer is high enough to supply the required energy for the endothermic phase-transition. In this case the Rayleigh-Plesset equation can be written as:

$$ \ddot{R}R+\frac{3}{2}{\dot{R}}^2=\frac{P_v-{P}_{\infty }}{r} $$
(9.50)

where \( {P}_v \) is the vapor pressure and \( {P}_{\infty } \) is the pressure far away from the bubble wall. We disregard the surface tension, the sound reradiation and the viscosity. The boiling temperature of the liquid is \( {T}_b \) and the ambient temperature is \( {T}_{\infty } \). The liquid is superheated (\( {T}_{\infty } \) > \( {T}_b \)) so that \( {P}_v \) > \( {P}_{\infty } \). The vapor pressure \( {P}_v \) is a function of the temperature and assumed to be constant during vapor bubble growth. Initially the velocity of the bubble wall \( \dot{R} \) is small, and the first term on the left hand side of Eq. 9.50 is dominant. After approximately a few nanoseconds at \( {P}_v \) = \( 1.4{P}_{\infty }, \) the bubble wall velocity reaches its terminal value and the second term on the right hand side of Eq. 9.50 becomes dominant.

Terminal velocity is reached at the condition \( \ddot{R} \) → 0. Substituting this into Eq. 9.50 and integrating with the initial condition \( R\left(t=0\right)=0 \) gives the radius-time dependency of the inertially limited vapor bubble growth:

$$ R(t)={\left(\frac{2}{3}\frac{\left({P}_v-{P}_{\infty}\right)}{r}\right)}^{1/2}t $$
(9.51)

Equation 9.51 is linear with time and is faster for higher vapor pressures \( {P}_v \), thus at higher ambient temperatures \( {T}_{\infty } \).

Let us now have a look at the second case, where we focus on heat transfer and where inertial limitations are neglected. Contrary to the solution of the inertial problem, the heat transfer is complicated by the temperature distribution outside the vapor bubble (Fig. 9.7).

Fig. 9.7
figure 7

Schematics of temperature distribution during the vaporization of a superheated perfluorocarbon droplet immersed in water. T b is boiling temperature of perfluorocarbon, T∞ is ambient temperature and δeff is effective thermal boundary layer around the vapor bubble of the radius R

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The temperature distribution changes with time due to thermal diffusion. In addition, it is also affected by the expansion of the bubble, as described by the continuity equation. The effective thermal boundary layer around the vapor bubble is determined by (Prosperetti 2011):

$$ {d}_{eff}=\sqrt{Dt} $$
(9.52)

where \( D \) is the thermal diffusivity of the liquid. This estimation follows from the thermal diffusion equation and shows that the thermal boundary layer diffuses with time as √t. On the vapor side of the thermal boundary layer the temperature is \( {T}_b \), and on the liquid side of the thermal boundary layer the temperature is \( {T}_{\infty } \). The effective temperature gradient over the thermal boundary layer is \( \varDelta T/{d}_{eff} \), where \( \varDelta T={T}_{\infty }-{T}_b \) is the temperature difference. The heat flow \( {W}_1 \) inside the vapor bubble from the surrounding liquid caused by the temperature mismatch can be estimated as follows:

$$ {W}_1=4p{R}^2k\frac{\varDelta T}{\sqrt{Dt}} $$
(9.53)

where \( k \) is the heat transfer coefficient and \( 4p{R}^2 \) is the interfacial area.

The latent heat energy per unit time \( {W}_2 \) required to supply the vapor bubble growth is:

$$ {W}_2=4p{R}^2L{r}_v\frac{dR}{dt} $$
(9.54)

with \( L \) the latent heat, and \( 4p{R}^2{r}_v\frac{dR}{dt} \) the derivative of the mass, with \( {r}_v \) the density of the vapor. Equalizing Eqs. 9.53 and 9.54, and integrating with the initial condition \( R\left(t=0\right)=0, \) gives the radial dynamics of the heat transfer limited vapor bubble growth:

$$ R(t)=2\frac{k\varDelta T}{L{r}_v\sqrt{D}}\sqrt{t} $$
(9.55)

It is dependent on time \( t \) following a square root behavior, and eventually will become slower than the inertia limited vapor bubble growth expressed by the linear dependence by Eq. 9.51. Thus, initially the vapor bubble growth is limited by the inertia, and then the vapor bubble growth becomes limited by the heat transfer. One can estimate the radius and the time when the transition of the two regimes occurs by calculating the intersection of the two curves expressed by the Eqs. 9.51 and 9.55. For typical parameters of acoustic perfluorocarbon droplet vaporization, the vapor bubble growth is heat transfer limited for a typical timescale longer than 1 microsecond (Fig. 9.8).

Fig. 9.8
figure 8

Radius time dynamics of the inertial and the heat transfer limited vapor bubble growth in superheated liquid. The curves are calculated for perfluoropentane liquid (Boiling point T b  = 29 °C) at 35 °C ambient temperature employing the Eqs. 9.51 and 9.55. At t = 0.52 μs, the growth becomes limited by the heat transfer as it becomes slower than the growth limited by the inertia

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When the bubble growth is accompanied by bubble oscillations due to the ultrasound forcing, one can observe a phenomenon called rectified heat transfer. Rectified heat transfer is the net effect of the decrease of the heat transfer during the ultrasound half cycle when the vapor bubble surface contracts, and which is lower than the increase of the heat transfer during the second half cycle when the surface expands. Here, two effects come into play; the increment of the bubble wall area during the expansion cycle, and the increment of the temperature gradient. The increment of the temperature gradient can be understood in the following way. Let us consider that the radius of the vapor bubble changes from \( {R}_0 \) to \( R \). The change of the thin thermal boundary layer from \( {\delta}_0 \) to \( \delta \) is then calculated from continuity: \( 4p{R}^2d=4p{R}_0^2{d}_0 \). This gives (Prosperetti 2011):

$$ d={d}_0\frac{R_0^2}{R^2} $$
(9.56)

and from the reciprocal relation for the temperature gradient:

$$ \frac{\varDelta T}{d}=\frac{\varDelta T}{d_0}\frac{R^2}{R_0^2} $$
(9.57)

it can be seen that the temperature gradient increases with the radius squared, \( {R}^2 \), i.e. with the bubble wall area \( 4p{R}^2 \). Thus, the bubble wall area and the temperature gradient will both decrease with \( {R}^2 \) when the radius decreases. However, the net effect is typically positive, meaning that bubble wall oscillations due to the interaction with ultrasound will pump additional heat into the bubble, thereby promoting the phase-conversion process; the larger the bubble oscillation amplitude, the stronger the pumping of additional heat.

4.4 Activation Below Boiling Point

After the initiation of droplet vaporization by the focused ultrasound pulse, gas diffuses into the nucleus/vapor bubble during vapor bubble growth as perfluorocarbon droplets dissolve air by an order of magnitude more than water. As was shown before, vapor bubble growth strongly depends on temperature. From both Eq. 9.51 and Eq. 9.55 it follows that the vapor bubble growth is slower when the ambient temperature is lower, whereas the dependence of air diffusion on the temperature is much less pronounced. This means that at low ambient temperatures (\( {T}_{\infty}\le {T}_b \)), the air diffusion dynamics becomes comparable to the evaporation processes.

Here, for simplicity, we only show the bubble growth dynamics due to gas diffusion, disregarding the evaporation processes and oscillations of the bubble due to ultrasound forcing. The partial pressure of gas \( {P}_g \) which is in equilibrium with the saturated gas concentration \( {c}_s \) dissolved in the liquid is given by Henry’s law:

$$ {P}_g=H{c}_s $$
(9.58)

We assume that the liquid is at a uniform supersaturated concentration \( i \). The mass flow of gas into the bubble per unit time is:

$$ \frac{dm}{dt}=4p{R}^2k\frac{\partial c}{\partial r}\Big|{}_{r=R} $$
(9.59)

where \( \kappa \) is the coefficient of diffusivity of the gas in the liquid. If \( {r}_g \) is the density of the gas in the bubble, the mass flow can be written as follows:

$$ \frac{dm}{dt}=4p{R}^2r\frac{dR}{dt} $$
(9.60)

One can use the reasonable physical approximation to calculate the gradient of the concentration for a bubble interface, which changes in time by diffusion (Epstein and Plesset 1950):

$$ \frac{\partial c}{\partial r}\Big|{}_{r=R}=\left({c}_i-{c}_s\right)\left(\frac{1}{R}+\frac{1}{\sqrt{pk\kern0.1em t}}\right) $$
(9.61)

Substitution of Eqs. 9.60 and 9.61 into Eq. 9.59 gives the radial time dynamics equation for the gas diffusion:

$$ \frac{dR}{dt}=\frac{k\left({c}_i-{c}_s\right)}{r}\left(\frac{1}{R}+\frac{1}{\sqrt{pk\kern0.1em t}}\right) $$
(9.62)

The gas bubble shrinks when \( {c}_i<{c}_s, \) and grows when \( {c}_i>{c}_s \). Similar to the rectified heat transfer problem, gas diffusion into the bubble can be promoted due to interaction with ultrasound. This phenomenon is called rectified diffusion and similar relations can be derived as was shown in the previous subsection.