special issue paper 171 cavitation and contrast: the...

Cavitation and contrast: the use of bubbles inultrasound imaging and therapyE P Stride1* and C C Coussios2

1Department of Mechanical Engineering, University College London, London, UK2Institute of Biomedical Engineering, Department of Engineering Science, University of Oxford, Oxford, UK

The manuscript was received on 25 March 2009 and was accepted after revision for publication on 26 October 2009.

DOI: 10.1243/09544119JEIM622

Abstract: Microbubbles and cavitation are playing an increasingly significant role in bothdiagnostic and therapeutic applications of ultrasound. Microbubble ultrasound contrast agentshave been in clinical use now for more than two decades, stimulating the development of arange of new contrast-specific imaging techniques which offer substantial benefits inechocardiography, microcirculatory imaging, and more recently, quantitative and molecularimaging. In drug delivery and gene therapy, microbubbles are being investigated/developed asvehicles which can be loaded with the required therapeutic agent, traced to the target site usingdiagnostic ultrasound, and then destroyed with ultrasound of higher intensity energy burst torelease the material locally, thus avoiding side effects associated with systemic administration,e.g. of toxic chemotherapy. It has moreover been shown that the motion of the microbubblesincreases the permeability of both individual cell membranes and the endothelium, thusenhancing therapeutic uptake, and can locally increase the activity of drugs by enhancing theirtransport across biologically inaccessible interfaces such as blood clots or solid tumours. Inhigh-intensity focused ultrasound (HIFU) surgery and lithotripsy, controlled cavitation is beinginvestigated as a means of increasing the speed and efficacy of the treatment. The aim of thispaper is both to describe the key features of the physical behaviour of acoustically drivenbubbles which underlie their effectiveness in biomedical applications and to review the currentstate of the art.

Keywords: bubbles, cavitation, ultrasound, contrast agents

1 INTRODUCTION

It is well known that the presence of gas bubbles in

vivo and particularly in the bloodstream can be

highly undesirable, the most familiar example being

the potentially fatal decompression sickness (‘the

bends’) and pulmonary barotraumas suffered by

underwater divers when surfacing too quickly [1].

There are, however, a rapidly growing number of

biomedical applications in which bubbles can offer

significant benefits, particularly in the context of

diagnostic and therapeutic ultrasound.

There are several mechanisms by which bubbles

may be formed or occur in vivo. First, they can

simply be injected intravenously in the form of a

suspension of stabilized microbubbles or liquid

droplets, which subsequently vaporize to form

bubbles. Microbubble agents of this type have been

in clinical use now for more than two decades as

contrast agents for ultrasound imaging [2, 3]. They

are also being extensively investigated for therapeu-

tic applications, in particular drug delivery, gene

therapy, and thrombolysis [4–7]. Second, bubbles

may be formed as the result of a reduction in

pressure in a given region. In general, the solubility

of a gas in a liquid falls with pressure; thus, reducing

the pressure will drive gas out of solution. Similarly,

if the pressure falls below the liquid vapour pressure,

vapour-filled cavities or bubbles will also be formed.

Further reductions in pressure will promote further

diffusion of gas and/or vapour into existing bubbles

and also growth of the bubbles according to the

*Corresponding author: Department of Mechanical Engineering,

UCL, Torrington Place, London WC1E 7JE, UK.

email: [email protected]

SPECIAL ISSUE PAPER 171

JEIM622 Proc. IMechE Vol. 224 Part H: J. Engineering in Medicine

appropriate gas law. Third, bubbles may be formed

as a result of an increase in temperature. Again, the

solubility of most gases falls with increasing liquid

temperature while the vapour pressure increases, in

both cases promoting the formation and growth of

gas- and/or vapour-filled cavities.

The formation and growth of bubbles in tissue or

in vivo as a result of ultrasound exposure normally

involves a combination of changes in both pressure

and temperature. By definition, ultrasound is the

propagation of a pressure disturbance through a

medium at a particular frequency, and the pressure

at a given location will therefore fluctuate at that

frequency. Similarly, part of the energy carried by

the propagating wave will be converted into heat by

viscous absorption, resulting in a local temperature

rise. The formation of and subsequent interaction

between bubbles and ultrasound are the key to their

exploitation in diagnostic and therapeutic appli-

cations. The aim of this paper is to describe the

features of the physical behaviour of bubbles that

underlie their effectiveness in these applications,

and to review the current state of the art.

2 BASIC PHYSICS OF MICROBUBBLES

2.1 Nucleation

In the case of bubbles formed in vivo, a further

question arises as to precisely where in the tissue the

bubbles originate or ‘nucleate’ from. The pressures

and/or temperatures required to overcome the

theoretical tensile strength of a liquid are much

larger than those at which bubbles are formed in

practice [8]. This implies that there must be some

form of defect present in the liquid which reduces its

effective strength and provides sites from which

bubbles can nucleate [9]. The precise nature of these

nuclei in vivo, or indeed in any liquid, is a somewhat

contentious subject [10], but there are two main

theories. First, it is possible that very small (, 1 mm

diameter) gas bubbles may be stabilized by adsorp-

tion of surfactant molecules onto their surfaces,

allowing them to persist indefinitely [11] (these

surfactant molecules such as fatty acids or proteins

would be expected to be present to some degree as

impurities in water and certainly in tissue). Second,

gas may become trapped in crevices on the surfaces

of particles or boundaries that are not wetted by the

surrounding liquid [12]. Studies of decompression

sickness have indicated that there are a number of

potential sites for such crevices in vivo, including

mitochondrial membranes and discontinuities in the

endothelium [13–15]. It has been demonstrated

experimentally that both surfactant stabilized micro-

bubbles in the form of ultrasound contrast agents,

and hydrophobic nanoparticles with rough surfaces,

can dramatically reduce the pressures required to

produce cavitation bubbles both in vitro and in vivo

[15, 16].

2.2 Stability in a stationary liquid

The requirement for cavitation nuclei to be stabi-

lized, as above, stems from the fact that an uncoated

gas bubble suspended in a liquid will dissolve away

very rapidly if its radius is less than 1 mm [17]. This is

due first to the fact that the pressure difference

across the bubble surface (the Laplace pressure)

produced by interfacial/surface tension s is inversely

proportional to the bubble radius R

pG{po~2s

Rð1Þ

where pG is the pressure of the gas inside the bubble

and po is the ambient pressure in the liquid.

Second, there is normally a relatively large con-

centration gradient produced by the difference

between the initial dissolved gas concentration in

the liquid surrounding the bubble (ci) and the

dissolved gas concentration at the bubble surface

(cs), the latter of which will also depend on pG 2 po

according to Henry’s law and hence s. The third

factor affecting bubble stability is the rate at which

gas is able to diffuse out into the liquid, i.e. the

effective diffusivity of the interface D, which will

itself relate to the molar mass of the gas M.

These effects can be represented by the well-

known differential equation developed by Epstein

and Plesset [17] which equates the rate of gas

diffusion at the bubble surface to the rate of change

of mass inside the bubble

_RR~D ci{csð ÞBT

M poz4s=3Rð Þ1

Rz

1ffiffiffiffiffiffiffiffiffipDtp

� �ð2Þ

where R is the rate of change of bubble radius, B

is the universal gas constant, T is the absolute

temperature, and t is time.

Both D and s will depend on the specific gas and

liquid in question and also the presence of any

coating material at the bubble surface, e.g. an

adsorbed surfactant layer which may reduce surface

tension and significantly increase the resistance to

gas diffusion [18]. These quantities will also be

affected by the temperature and pressure at the

172 E P Stride and C C Coussios

Proc. IMechE Vol. 224 Part H: J. Engineering in Medicine JEIM622

bubble surface, as will the value of the dissolved gas

concentration (cs) there. In addition, it should be

noted that convective effects are ignored in equation

(2). The effects of buoyancy are also neglected in the

above consideration of bubble stability. Clearly, an

uncoated bubble will be destroyed by rising to the

surface of the liquid in which it is suspended. A

coated bubble on the other hand will survive

flotation and, in vivo, the ability of bubbles to

translate is significantly restricted by the surround-

ing tissue. Hence, for the purposes of this review,

buoyancy is not an important factor. The fully

coupled mass transport problem was solved by

Readey and Cooper [19] and Weinberg [20],

although the effects of convection were found to be

relatively small, particularly for bubbles with dia-

meters smaller than 1 mm and/or where the gas

concentration in the surrounding fluid was close to

its saturation value.

2.3 Response to an acoustic field

On account of their high compressibility, gas- and/

or vapour-filled bubbles will expand and contract in

response to the locally varying pressure produced by

an ultrasound field (Fig. 1). It is these volumetric

oscillations that are the key to their effectiveness

both as ultrasound contrast agents and also in

therapeutic applications.

2.3.1 Equation of motion

In deriving equation (2), the motion of the bubble wall

and the surrounding liquid was ignored. When con-

sidering the oscillations of a microbubble in a sound

field, however, these must be taken into account.

For the most general case of bubble motion the

three conservation equations for mass, momentum,

and energy must be solved simultaneously. Useful

physical insights can nevertheless be obtained by

applying certain assumptions in order to simplify the

modelling. For example, if it is assumed that the

bubble remains spherical, the conservation equa-

tions may be expressed in spherical polar coordi-

nates respectively as

dr

dtzr+:u~0 ð3Þ

rLu

Ltzu

Lu

Lr

� �z

Lp

Lr~+ �SS ð4Þ

Fig. 1 Variation with time of the radius of a sphericalgas bubble in water with initial radius 2 mmexposed to continuous sinusoidal excitation: (a)coated and uncoated microbubbles excited atresonance (at 3.0 and 1.6 MHz respectively)with amplitude 5 kPa; (b) uncoated bubbleexcited at 15, 50, and 150 kPa; (c) uncoatedbubble excited at 300 kPa. Simulation para-meters are given in Appendix 1

The use of bubbles in ultrasound imaging and therapy 173


and in the gas

+: c+Tð Þ~ c

c{1ð Þp

T

LT

Ltzu

LT

Lr

� �{

dp

dtð5Þ

where r is density, r is the radial coordinate mea-

sured from the bubble centre, u is radial velocity, p is

pressure, �SS is the stress tensor, C is thermal con-

ductivity, and c is the ratio of specific heats [21].

If it is further assumed that there is negligible heat

transfer and diffusion of gas across the bubble

surface then equation (5) is no longer required.

The influence of the liquid inertia upon the bubble

motion is significant on a length scale, which is small

compared with the wavelength of the incident

ultrasound field. Thus the effect of density changes

in the surrounding liquid may be neglected and so

equation (3) reduces to

ur2~ _RRR2 ð6Þ

where R and _RR are the instantaneous radius and

radial velocity as above. While this greatly simplifies

the analysis, treating the liquid as incompressible

does, however, require that the energy dissipation

due to reradiation of sound by the bubble is

reintroduced into the equation of motion for the

bubble via a correction term as described below.

Finally, for continuity of stress at the surface

(r 5 R)

pG Rð Þzpv Rð ÞzrLRfS~pL Rð Þ{Srr, L Rð Þ ð7Þ

then equation (4) may be integrated with respect to r

to give an equation of motion for the bubble

€RR~{3 _RR

2

2Rz

1

rLRpG R, tð Þzpv R, tð Þ{p‘ tð Þ½ �zfLzfS

ð8Þ

where rL is the density of the surrounding liquid, €RR is

the acceleration of the bubble surface, pv is the

vapour pressure inside the bubble, p‘ is the far field

pressure in the liquid, pL(R) is the pressure in the

liquid at the bubble surface, Srr, L(R) is the corre-

sponding stress in the liquid, and fS and fL represent

the resistance to motion provided by the surface and

the surrounding liquid respectively.

For a Newtonian liquid with viscosity mL

fL~{4mL

_RR

rLR2ð9Þ

and for an uncoated bubble

fs~{2so

rLR2ð10Þ

where so is the interfacial tension for the particular

gas/liquid combination in the absence of any surface

contamination.

Various mathematically equivalent definitions for

fs have been derived [11, 22–26] and are reviewed in

reference [27]. For a bubble coated with a layer of

adsorbed molecules

fs~ {4 _RR

rLR3gso eZR2

x= R2{R2xð Þ

" #

{2

rLR2soz

QCxz1o

xz1ð Þ 1{Ro

R

� �2 xz1ð Þ" #( )

ð11Þ

where gso is the effective surface viscosity, Z is a

power law exponent [28], Rx is the radius at which

the surface buckles, Co is the initial molecular con-

centration at the bubble surface, Ro is the bubble’s

initial radius, and Q and x are constants characteriz-

ing the relationship between surface tension and

adsorbed molecular concentration [29]. (Similar

terms may also be derived for a coating of finite

thickness provided that additional terms are in-

cluded in equation (8) to account for its inertia [11].)

Additional terms may also be included to take

into account energy dissipation due to heat conduc-

tion and acoustic reradiation. Equation (8) then

becomes

€RR~{3 _RR

2

2Rz

1

rLRpG R, tð Þzpv R, tð Þ{p‘ tð Þ½ �

zfLzfSzfThzfRad ð12Þ

The appropriate form of these terms depends on the

relative significance of these additional damping

mechanisms in a particular situation. For example,

for coated microbubbles excited at medical diag-

nostic frequencies and low acoustic pressures, fTh

and fRad may be reasonably neglected [30, 25]. For

uncoated bubbles, with larger initial radii and/or

excited at moderate amplitudes of oscillation, a basic

approximation for fTh and fRad may be made using

the terms given by Prosperetti [31]

fTh~{4mTh

_RR

rLR2ð13Þ



fRad~{R2

o_RR

R2

v2Ro

�c

1z vRo=cð Þ2

" #ð14Þ

where mTh is the effective thermal viscosity and c is

the speed of sound in the surrounding liquid. For

uncoated bubbles undergoing large amplitudes of

oscillation, a more rigorous form is required for fRad,

such as that derived by Keller and modified by

Prosperetti [21]

fRad~_RR

c€RRz

_RR2

2Rz

1

rLRpG R, tð Þzpv R, tð Þ{p‘ tð Þ½ �

(

zfLzfS z1

c

d

dt

pG R, tð ÞrL

zfLRzfSR

� �

ð15Þ

If the bubble’s initial radius is also large (. 50mm)

and/or the bubble collapses very rapidly then it is

necessary to solve equations (3), (4), and (5) simulta-

neously and also to account for non-spherical motion

of the bubble surface as discussed below.

2.3.2 Linear response

At very low excitation pressures both coated and

uncoated bubbles will undergo oscillations that are

approximately linear (Fig. 1(a)). Hence an analytical

expression for the linear resonance frequency of a

gas bubble (vR) can be derived by reducing equation

(12) to an ordinary second-order differential equa-

tion (see Appendix 2) (Note that vapour pressure and

thermal and acoustic damping have been neglected

as they are likely to be negligible in the regime in

which equation (16) is valid.)

v2R~

1

rLR2o

� � 3poz4so

Roz

4QCo

Ro

� �{

8 mLzgso=Roð Þ2

rLR2o

� �2

ð16Þ

As may be seen from equation (16), vR is strongly

dependent on the size of the bubble (Ro) and also the

coating parameters. The presence of a surface coating

can significantly increase vR in addition to reducing

the amplitude of oscillation as indicated in Fig. 1(a).

2.3.3 Inertial and non-inertial collapse

As the amplitude of oscillation increases, the be-

haviour of the bubble becomes increasingly non-

linear (Fig. 1(b)). Equation (16) is then no longer

strictly valid as the radial amplitude in expansion

and contraction may differ significantly, and the

frequency at which the volume amplitude is max-

imized becomes dependent on pressure [10, 32].

Under these conditions the bubble will still undergo

repetitive oscillations, but periodicity may only be

observed over several cycles (Fig. 1(b) dot–dash

curve). This is commonly referred to as non-inertial

or stable cavitation [33].

For a given bubble size and driving frequency v,

there is a critical excitation pressure above which the

periodic nature of the oscillation is effectively lost

and the bubble collapses very violently (Fig. 1(c)),

often resulting in its fragmentation into smaller

bubbles. This is referred to as inertial, unstable, or

transient cavitation. (To refer to inertial cavitation as

transient or unstable is slightly misleading as inertial

collapse can be a repetitive process [34], and

ultimately the process of a bubble collapsing and

fragmenting to form new bubble nuclei which

subsequently grow and collapse can also be regarded

as cyclical.) The terms ‘inertial’ and ‘non-inertial’

derive from the analysis by Flynn [35] in which it was

shown that an approximate criterion for the trans-

ition between the two types of behaviour can be

obtained by comparing the magnitude of two

different terms on the right-hand side of equation

(12), fI and fP

fI~{3

2

_RR2

Rð17Þ

fP~1

rLRpG Rð Þ{ 2so

R{p‘

� �ð18Þ

Inertial cavitation is said to occur if, at the point

when fP reaches a minimum, the magnitude of fI is

greater than the magnitude of fP (Fig. 2). As will

be discussed in later sections, this marked transition

in behaviour has significant implications for the

physical consequences of cavitation and also the

acoustic signals generated. It should be noted,

however, that this analysis only provides a useful

indication of the conditions under which violent

bubble collapse will occur, and that the accurate

prediction of cavitation dynamics remains an active

area of research.

Flynn’s analysis neglected the remaining terms on

the right-hand side of equation (12) (fL, fS, fTh, and

fRad) but plots similar to those shown in Fig. 2 can be

produced which include them [27]. At low ampli-

tudes of oscillation these further demonstrate the

(



significant effect a coating may have on the

behaviour of a bubble, reducing the amplitude of

oscillation and increasing the resonance frequency

compared with an uncoated bubble (cf. Fig. 1(a)).

Beyond a certain driving pressure, however, nor-

mally in the non-inertial cavitation regime, the

coating will no longer have a significant effect on

the oscillations, and the fS term in equation (12) can

be neglected. This is because either the coating has

physically broken down or the concentration of

adsorbed molecules is simply too low during expan-

sion to influence the motion of the surface.

2.3.4 Diffusion

As mentioned above, in the absence of significant

changes in the temperature and pressure of its

surroundings, a small bubble would be expected to

dissolve away according to equation (2). In the

presence of a sound field, however, there are a

number of mechanisms by which the bubble may

grow, and indeed this process is necessary for the

development of cavitation bubbles from nuclei. First,

as described in section 1, even if diffusion is

neglected, heating of the surroundings and/or a

reduction in pressure will cause a bubble of volume

V containing a mass m of gas to expand according to

the ideal gas law [36]

pV ~m

MBT ð19Þ

Second, a rise in temperature will also increase the

vapour pressure in the bubble and promote inwards

diffusion of gas from the surrounding liquid, again

resulting in expansion. Third, although gas diffusion

will occur in both directions across the bubble

surface owing to the varying pressure gradient, there

may be a net increase in the mass of gas contained in

the bubble on each cycle. This is because the surface

area of the bubble will be smaller during compres-

sion than during expansion. Thus the degree to

which gas diffuses into the bubble may exceed that

to which it diffuses outwards into the liquid. In

addition, the local concentration of gas in the liquid

close to the bubble surface will be higher during

bubble expansion than during compression, and this

again encourages inwards diffusion [37]. Whether

or not a bubble undergoes this ‘rectified-diffusion’

process depends on its initial size, the frequency and

pressure of the incident field, and the solubility and

concentration of the gas in the surrounding liquid.

Approximate thresholds for rectified diffusion have

been derived for uncoated bubbles (e.g., see refer-

ences [37] and [38]). The effect of an adsorbed

surfactant layer has been investigated by Fyrillas and

Szeri [39] who found that a soluble surfactant could

either enhance or inhibit bubble growth by rectified

diffusion, which corresponded with the experimen-

tal findings by Crum [40]. It should be noted that

rectified diffusion can also refer to the transfer of

heat across the bubble surface [33] and that this

process will in turn affect the rate of change of

bubble size.

Fig. 2 Inertia and pressure factors (equations (17) and(18)) for uncoated spherical gas bubbles inwater with initial radii of 2mm, exposed tocontinuous sinusoidal excitation at 1.6 MHz atamplitudes of: (a) 15 kPa (non-inertial cavita-tion IF . PF when PF is at its minimum value);and (b) 250 kPa (inertial cavitation IF , PFwhen PF is at its minimum value). Accelerationis non-dimensionalized with respect to theinitial bubble radius Ro and excitation fre-quency v



2.4 Scattering

Bubbles are extremely strong scatterers of ultra-

sound because of their high compressibility and the

negligible inertia of the encapsulated gas compared

to the surrounding liquid. As will be described in

section 3, it is this, together with their highly non-

linear behaviour, which makes them so effective as

ultrasound contrast agents and also as a means of

monitoring the progress of treatments such as HIFU

surgery and shock wave lithotripsy.

The pressure scattered or reradiated by a bubble at

a distance r from its centre can be predicted from

potential flow theory [41] as

pscat r, tð Þ~rL

1

rR2 €RRz2R _RR

2

{R4 _RR

2

2r4

" #{p‘ tð Þ

ð20Þ

where the radial terms (R, €RR, _RR) must be determined

from the solution to equation (12). This treatment

neglects attenuation of the scattered field due to

absorption in the surrounding liquid and/or the effect

of any other scatterers present, although the latter will

normally be small compared with scattering from the

bubble. It also neglects the retardation effect due to

the finite speed of propagation, which can be

compensated for as described in reference [42].

The frequency spectra for the scattered fields from

the bubbles in Fig. 1 are shown in Fig. 3. As may be

seen, the non-linear components become increasingly

pronounced with increasing excitation pressure am-

plitude, progressively generating whole and then

fractional harmonics. Eventually, broadband noise is

generated with the onset of inertial cavitation.

The scattering cross-section sscat presented by a

bubble to an ultrasound field is defined as the ratio

of the power scattered by the bubble Pscat to the

intensity of the incident field Iinc

sscat~Pscat

Iincð21Þ

Similarly the absorption cross-section is the ratio of

absorbed power to incident intensity

sabs~Pabs

Iincð22Þ

and the extinction cross-section is the sum of these

two

sext~sscatzsabs ð23Þ

For small-amplitude oscillations, once again linear-

ized terms can be derived which relate sscat and sabs

to the properties of the bubble (Appendix 2). As

discussed in detail by Hilgenfeldt et al. [43], the form

of the expressions for the scattering, absorption, and

hence extinction cross-sections for single bubbles

will depend on the mechanisms of energy dissi-

pation being considered in the bubble model.

2.5 Bubble populations

2.5.1 Linear bubble response at low bubbleconcentrations

Clearly, it is the response of a population of bubbles

which is most frequently of interest in both diag-

Fig. 3 Frequency spectra for the scattered powergenerated by the bubbles shown in Figs 1(a)and (b): (a) coated and uncoated microbubblesexcited at resonance (at 3.0 and 1.6 MHzrespectively) with amplitude 5 kPa; (b) un-coated bubble excited at 15, 50, and 150 kPa



nostic and therapeutic applications. For example,

microbubble contrast agents are injected with in-

itial concentrations of ,109 microbubbles/ml. For

low-concentration bubble suspensions, the linear

attenuation and scattering coefficients (a and b) may

be estimated by performing a linear summation over

the extinction and scattering cross-sections (sext and

sscat) respectively for individual bubbles over the

population [44]

a vð Þ~10 log10 e

ðRmax

Rmin

sext Ro1, vð Þn Ro1ð ÞdR ð24Þ

b vð Þ~10 log10 e

ðRmax

Rmin

sscat Ro1, vð Þn Ro1ð ÞdR ð25Þ

where n represents the size distribution of the

population. (It should be noted that an integration

may also be performed with respect to the micro-

bubble coating parameters, which may not be

consistent across the population.) In this case the

speed of sound in the suspension may be assumed

to be unchanged from that in the liquid in the

absence of microbubbles. At higher concentrations,

energy is not only dissipated because of scattering or

absorption by individual bubbles, but also as a result

of interactions between multiple bubbles. Further

details may be found in references [45], [46], and

[47].

2.5.2 Non-linear bubble response

For acoustic pressure amplitudes at which it is no

longer valid to assume linear bubble behaviour, the

extinction and scattering cross-sections in equations

(24) and (25) must be determined from a numerical

solution of equation (12) for low-concentration

suspensions. For higher concentrations, it is neces-

sary to solve the wave equation in inhomogeneous

form assuming propagation through an effective

medium, where the inhomogeneous term describes

the microbubble dynamics

1

rLc2L

L2p

Lt2{+2p~

L2b

Lt2ð26Þ

where b is the bubble concentration at time t and

location y in the liquid

b~4

3p

ð‘0

R31 Ro1, y, tð Þn Ro1, yð ÞdRo1 ð27Þ

which is determined by solving equation (12)

simultaneously with equation (26). Details of this

treatment may be found in references [47] and [48].

Equation (26) can describe both linear and non-

linear bubble oscillations but is still limited in terms

of the maximum bubble concentration for which it

can be used. For the assumption of an effective

medium to be valid, it is necessary to assume that

the average pressure field incident upon any one

bubble is large compared with that radiated by its

immediate neighbour. Taking N21/3 as the average

distance between bubbles (where N is the equivalent

number density for a monodisperse suspension

containing the same gas volume fraction), then for

linear bubble oscillations equation (26) is only valid

if the following condition is satisfied [48]

vRo1

N{1=3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2

o{v2� �2

z4d2dv2

h ir vv1 ð28Þ

The equivalent condition for non-linear bubble

oscillations cannot be so readily defined but equa-

tion (28) may still be used as an approximate gauge

for the validity of equation (26). The condition

defined by equation (28) is least likely to be met

for excitation frequencies close to the undamped

natural frequency (vo) for a given bubble size, and

when the effective damping coefficient dd is small

(see Appendix 2).

2.6 Non-spherical behaviour

There may be a number of causes for non-spherical

bubble behaviour, such as the bubble being large

compared to the acoustic wavelength, being in the

vicinity of another object, or, for a coated bubble, as

a result of a surface inhomogeneity.

If the condition imposed by equation (28) is not

satisfied, such as at concentrations for which the

average bubble separation is only a few bubble

diameters, secondary radiation (Bjerknes) forces

between the bubbles become important [49]. Not

only will these produce translation of the bubbles,

they will also cause them to oscillate non-spheri-

cally. Similarly, the presence of a surface, e.g. a tissue

boundary such as a blood vessel wall, will also

disrupt the symmetry of an oscillating bubble owing

to the asymmetry in the motion of the surrounding

liquid. The effect will become more pronounced

with decreasing distance between the bubble and

the surface (Fig. 4). At large amplitudes of oscillation

a bubble collapsing close to a rigid surface may ‘turn



in’ upon itself, producing a high-speed micro-jet

travelling towards the surface [50] (‘Rigid’ means

rigid compared with the surrounding liquid; near a

free boundary such as gas/liquid interface, the jet

will travel in the opposite direction). The implica-

tions of this phenomenon in vivo are discussed

further in section 4. Details of the modelling of non-

spherical bubble oscillations are outside the scope of

this review and require the use of computationally

intensive techniques such as boundary and finite

element methods [51–55].

Oscillations may also be set up on the surface of

the bubble [33]. The direct effect of non-spherical

oscillations on the scattered field from the bubble is

unclear. According to linear analysis the higher-

order components of the scattered field should

decay rapidly with distance from the bubble and

thus should not be detected at the distances relevant

to biomedical ultrasound [10]. However, Longuet-

Higgins has shown that under certain conditions

there may be coupling between the surface and

radial modes so that the former will contribute to the

harmonic content of the latter [56, 57]. To date there

is insufficient experimental evidence both to confirm

this theory and to test its applicability to the various

frequencies and bubble sizes encountered in diag-

nostic and therapeutic applications. It is clear,

however, that the presence of an interface such as

a boundary or another bubble will affect the

characteristic frequency of microbubble oscillation

[58] and can have significant effects on other aspects

including microstreaming (cf. section 4), the pres-

sure and temperature inside the bubble during

collapse, and whether or not it will fragment [33].

3 DIAGNOSTIC APPLICATIONS

3.1 Types of microbubble agent

The development of microbubbles as ultrasound

contrast agents came about as the result of an

accidental discovery by Dr Claude Joyner in the late

1960s [59]. He was conducting a study of cardiac

output by periodically injecting indocyanine green

dye into the patient’s left ventricle while simulta-

neously performing an M-mode echocardiogram. He

observed that each injection of dye produced a

temporary increase in the ultrasound echo from the

ventricle. Initially it was thought that the contrast

enhancement was due to the nature of the dye, but

further investigation by Gramiak and Shah [60] and

Kremkau et al. [61] demonstrated that it was not the

dye itself that was the source of the effect, but rather

the formation of gas bubbles at the catheter tip. It

was subsequently discovered that more prolonged

contrast enhancement could be achieved using

saline containing a small amount of a patient’s

blood [62], and this led to the development of one of

the first commercial contrast agents, AlbunexH(Mallinckrodt Inc., Hazelwood, MO, USA), which

consisted of air microbubbles coated with a thin

stabilizing layer of cross-linked human serum albu-

min.

While Albunex microbubbles were stable in com-

parison to uncoated bubbles, they were unable to

provide prolonged contrast enhancement in vivo,

and new agents containing higher-molecular-weight

gases were consequently developed. OptisonTM (GE

Healthcare Inc., Princeton, NJ, USA), for example,

contains perfluoropropane with a relative molecular

mass approximately six times that of air (188 as

opposed to 29), resulting in decreased diffusivity of

the gas across the bubble wall. In addition, many of

the new agents were packaged as freeze-dried

powders, which could be stored and resuspended

in saline as required, rather than being prepared

immediately prior to injection. Alternative means of

administering coated microbubbles were also ex-

plored; e.g. EchovistH and its successor, LevovistH(Schering AG, Berlin, Germany), consisted of sus-

pensions of galactose microcrystals, which dissolved

in the blood following injection, releasing air micro-

bubbles from defects on the crystal surfaces.

Levovist also contained palmitic acid to provide

additional stability.

Levovist microbubbles were found to be more

echogenic than either Albunex or Optison, but

considerably less stable on account of the higher

diffusivity of their surfactant coatings. Hence, Levo-

Fig. 4 Non-spherical oscillations of an uncoated gasbubble, initial radius 1.5 mm, collapsing at adistance of 3 mm from a rigid boundary (simu-lation was performed using finite elementsoftware Comsol Multiphysics v.3.4, ComsolLtd, Hatfield, Herts, UK)



vist was soon superseded by agents containing

bubbles stablized by phospholipid monolayers that

provided a better compromise between longevity

and echogenicity. These included SonoVueH (Bracco

International BV, The Netherlands), DefinityH (Bris-

tol-Myers Squibb Medical Imaging Inc., USA) and

SonozoidTM (GE Healthcare Inc., Princeton, NJ,

USA). Other coating materials, including polymers

such as cyanoacrylates and polycaprolactone, have

also been used, and there are currently numerous

experimental agents in various stages of develop-

ment offering greater stability, improved acoustic

response, and multiple layers to enable loading of

therapeutic components and/or attachment of tar-

geting species (see below). However, SonoVue and

Optison are currently the only microbubble agents

approved for clinical use worldwide.

Another approach has been to use a stabilized

emulsion of volatile liquid droplets which vaporize

to form microbubbles either upon injection, or

following exposure to ultrasound of sufficient in-

tensity. This type of agent has a number of

advantages, in terms of both stability during storage

and administration, and also its ability to diffuse into

the surrounding tissue. Preformed gas bubbles will

remain predominantly within the blood pool, but by

keeping the agent in the form of liquid nanopart-

icles, they may be sufficiently small to undergo

extravasation before they are vaporized to form

microbubbles under the effect of an acoustic field.

This facilitates imaging and/or treatment, e.g. within

a tumour mass [63], particularly if vaporization is

suppressed by using a stabilizing coating to confine

a superheated droplet until sufficient energy has

been absorbed from the ultrasound field to vaporize

the droplet and rupture the shell. Examples include

EchoGenH (Sonus Pharmaceuticals Inc., Bothell,

USA), which consists of surfactant-coated droplets

of perfluoropentane that have a boiling point coin-

ciding with normal body temperature (37 uC) [64].

Echogenic liposomes represent a further class of

agent which consist of phospholipid bilayers en-

capsulating a mixture of liquid and gas [65, 66].

These are also more stable than monolayer-coated

microbubbles and are particularly attractive for drug

delivery applications, as larger quantities of either

aqueous or non-aqueous material can be encapsu-

lated. Larger doses of echogenic liposomes (i.e.

particles per unit volume) are required to obtain

equivalent levels of contrast enhancement during

imaging, on account of their lower gas content per

particle, but such high concentrations are well

tolerated physiologically [67]. Similarly, specific

pulse regimes are required to initiate drug release

[68]. In terms of scattering efficiency (i.e. proportion

of the incident field which is scattered rather than

absorbed), however, it has been reported that the

acoustic properties of echogenic liposomes may be

superior to those of microbubbles [69].

3.2 Contrast-specific imaging

Extensive reviews of the clinical applications of

microbubbles may be found in references [2] and

[70]. The following is intended only to provide a brief

overview.

A wide range of new ultrasound imaging tech-

niques have been developed which aim to maximize

both sensitivity to contrast agents and the ratio of

echoes from bubbles to those from tissue. Initial

trials of microbubble agents using conventional

ultrasound scanners identified a number of prob-

lems. While the brightness of microbubble-per-

fused regions was seen to be considerably enhanced,

the concentrations required to produce a noticeable

increase in Doppler signals also produced significant

shadowing of underlying structures. The bubbles

moreover reduced the ability to differentiate be-

tween blood flow and other tissue motion in Doppler

scanning [71]. The first contrast-specific imaging

method was a modification of the existing colour

Doppler mode, which had been shown to cause

microbubble destruction. ‘Loss of correlation’ im-

aging exploits the large change in signal strength

resulting from microbubble destruction, and pro-

vides a very sensitive means of bubble detection

[72].

Loss of correlation imaging cannot, however,

provide images of bubbles in real time; harmonic

imaging was developed as a means of overcoming

this limitation [73]. This exploits the ability of

microbubbles to generate signals at harmonics of

the frequency at which they are excited (Fig. 1). By

frequency filtering the echoes received from the

region of interest, it is possible both to differentiate

between the predominantly linear signals from

tissue and those from microbubbles and to construct

contrast-specific images accordingly. Originally the

second harmonic was used [74] but other compo-

nents, most notably subharmonic signals, have also

been investigated [75]. Unfortunately, the spatial

resolution of this technique was limited by the need

to use relatively long pulses in order to minimize the

overlap between the frequency spectra of the

transmitted and scattered signals. There were also

restrictions owing to the finite bandwidth of the



transducer, which meant that the sensitivity to the

received signal was inevitably lower than for con-

ventional imaging at a single frequency.

The solution to this problem came with the

development of modulated pulse sequence imaging

[76–79]. In this method a pair or longer sequence of

pulses is transmitted with alternating amplitude

and/or phase. If the pulses are scattered linearly by

tissue, then when they are combined upon ‘receive’

they should cancel one another out, producing a

zero signal. If, on the other hand, they are scattered

non-linearly by microbubbles, there will be a

residual signal. Thus a contrast-specific image can

be obtained without the need for frequency filtering.

A basic example of this type of pulse sequence is

where two pulses of opposite phase are transmitted

sequentially. It has been shown that by combining

phase and amplitude modulation contrast specificity

can be substantially increased [78], and the high

sensitivity to non-linear signals removes the need for

large-amplitude pulses. Hence the risk of micro-

bubble destruction is reduced and images can be

produced in real time, although the use of longer

pulse sequences will inevitably reduce the maximum

frame rate. An alternative method is ‘dual frequency

excitation’, also known as ‘radial modulation im-

aging’, in which a low-frequency ‘pumping’ signal is

used to stimulate bubble oscillations, and the

scattering of a high-frequency ‘imaging’ signal is

used to produce the image [80]. Incompressible

scatterers produce effectively identical scatter during

both the compression and rarefaction phases of the

pumping signal, while there is significant decorrel-

ation in the scatter from bubbles, thus providing a

high contrast to tissue ratio.

3.3 Quantitative imaging

In general, microbubbles will remain in the vascular

system and potentially therefore can be used as a

means of quantifying tissue perfusion and other

physiologically relevant parameters such as relative

vascular volume and flow velocity. These measure-

ments are particularly significant for examination of

myocardial function, kidney, and tumour vascula-

ture. Flash replenishment imaging is the most

commonly employed technique [81, 82], whereby

the microbubbles in the image plane are initially

destroyed by a high-amplitude ultrasound ‘release

burst’, and the rate at which the image plane is

replenished with bubbles is then monitored in real

time using low-amplitude pulses. Fitting the results

to an appropriate flow model provides estimates for

the blood volume and tissue perfusion. To date,

however, clinical implementation of quantitative

imaging procedures has been hindered by poor

characterization of the complex relationship be-

tween microbubble concentration, scattering, and

image intensity. Experimental measurements of

microbubble suspensions have demonstrated the

pressure and frequency dependence of both scatter-

ing and attenuation [83]. Consequently, image

intensity does not necessarily correspond to micro-

bubble concentration, and at present there is a lack

of methodology for properly calibrating received

echoes for microbubble quantification, although this

is actively being addressed in current research [84].

3.4 Targeted and molecular imaging

Some commercial contrast agents, e.g. Levovist, have

been found to have a tissue (liver and spleen)-

specific late phase [85]. True tissue specificity,

however, e.g. for targeted imaging and therapy,

requires some form of functionalization of the

microbubble surface. Microbubbles coated with

material carrying a charge have been shown to

locate preferentially to inflamed tissue [86], but a

more effective method is to attach ligands to the

bubble surface that will bind to receptors on

particular types of cell. A detailed discussion of this

topic is outside the scope of this paper, and more

comprehensive reviews may be found in reference

[87], but examples include: targeting to activated

leucocytes by incorporating phosphatidylserine in

the microbubble coating [88]; angiogenic markers

[89, 90]; and attaching antibodies to microbubbles

targeted to receptors expressed during inflammation

(e.g. anti-P-selectin monoclonal antibody, anti-

ICAM antibody, anti-VCAM antibody) [91–93]. An

alternative method for localizing microbubbles in

vivo is to load them with magnetic nanoparticles.

This enables the microbubbles to be guided into the

target region using an externally applied magnetic

field [94] either as an alternative to biochemical

targeting, or as a means of slowing the microbubbles

down sufficiently to facilitate binding.

4 THERAPEUTIC APPLICATIONS

Detailed reviews of the therapeutic applications of

microbubbles may be found in other articles in this

special issue; the aim of this section is therefore to

review the relevant physical phenomena in relation

to the aspects of microbubble behaviour discussed

in section 2.



4.1 Bubbles as delivery vehicles

As mentioned above, there has been increasing

interest in the use of microbubbles for drug and

gene delivery. Microbubbles represent excellent

vehicles for this type of application, as drugs or

DNA can be incorporated into the microbubble

coating, traced through the body using low-intensity

ultrasound, and then released by destroying the

microbubbles with high-intensity ultrasound at a

target site such as a tumour. The ability of ultra-

sound to be tightly focused into small tissue volumes

(, 5 mm3) provides a physical means of localizing

microbubble activity, which can be further enhanced

by the use of targeting strategies as described in

section 3.6. Hence, the risk of harmful side effects

from therapeutic agents can be substantially re-

duced.

A variety of strategies for loading microbubbles

with therapeutic components are being actively

investigated. With some polymeric microbubbles it

is possible to dissolve or disperse the drug directly in

the coating, provided that it is of sufficient thickness

for the required dose [95, 96]. The drug is released as

the coating breaks down when the microbubble

oscillates at sufficient amplitude. More frequently,

an additional layer of oil is included between the gas

core and an outer thin polymer shell or surfactant

coating into which the drug is dissolved [97, 98]. As

above, echogenic liposomes provide a means of

encapsulating larger quantities of both hydrophobic

and aqueous material, and some therapeutic com-

ponents may also be attached to the outside of the

microbubbles, e.g. by biochemical (ligand/receptor)

or electrostatic binding [99, 100]. An alternative

method, which has recently been demonstrated

successfully in vivo, is to use a suspension of

microbubbles or a phase shift emulsion mixed with

drug-filled particles/micelles without any form of

physical or chemical binding. Oscillation of the

microbubbles upon exposure to ultrasound disrupts

the micelles to release the drug [37].

4.2 Microjetting

In addition to providing a means of encapsulating

therapeutic material, there are a number of physical

phenomena produced by microbubble oscillations

which may contribute to the therapeutic effect, e.g.

by enhancing the rate of uptake of drugs/DNA by

cells and/or increasing the rate of erosion or

denaturing of diseased tissue. The formation of

microjets described in section 2.6 is an example of

one of these (this effect is frequently referred to as

sonoporation or sonophoresis; in this paper, how-

ever, the authors have elected to refer to enhanced

uptake to avoid any confusion as regards the

underlying mechanisms, which are poorly under-

stood and may not involve the formation of ‘pores’).

It has been shown in vitro that a microjet can

easily puncture a cell membrane, and this effect has

been observed extensively, including with contrast

agent microbubbles near cell monolayers [101]. It

has been suggested that microjetting could poten-

tially be the cause of the enhanced cell uptake

generated by microbubbles [101]. It is less clear,

however, whether microjetting occurs in vivo,

particularly at lower acoustic pressures, on account

of the lack of rigid surfaces occurring in tissue.

Moreover, the relatively high ultrasound pressures

required and consequent risk of permanent cell

damage suggest that reversible permeability en-

hancement is more likely to be associated with

non-inertial cavitation phenomena, such as micro-

streaming described in the next section [102], or

simply the stimulation of uptake mechanisms in the

cell membrane by physical contact or ‘poking’ of the

cell by the oscillating bubble [103].

4.3 Microstreaming

When a sound wave propagates through a liquid,

steady currents are set up in the direction of the

beam as a result of momentum transfer from the

wave to the liquid [104, 105]. This acoustic stream-

ing also occurs, albeit on a much smaller scale,

around microbubbles undergoing stable oscillations

[106, 107]. These eddying flows may, in turn, impose

shear stresses on nearby surfaces, such as cell

membranes, and it is thought that this may promote

the uptake of therapeutic components [108, 109].

Microstreaming has also been shown to cause

significant damage to cells [110–112] at higher am-

plitudes of oscillation. In addition, microstreaming

will contribute to circulating therapeutic agents in

the target region, which is also likely to be important

in the context of both sonothrombolysis and drug

uptake [113]. Recent work by Tho et al. [114] has

demonstrated that the streaming velocities induced

by bubbles undergoing shape oscillations appear to

be roughly two to three times higher than those

produced by bubbles pulsating radially. The fact that

the rate of clot dissolution has also been found to be

enhanced to a greater extent by stable rather than

inertial cavitation further supports the hypothesis

that microstreaming is a key mechanism underlying

sonothrombolysis [113].



4.4 Heating enhancement

The absorption of energy as ultrasound propagates

through a medium also produces a heating effect. In

most materials, including tissue, the rate of absorp-

tion increases with frequency [104], and thus the

presence of bubbles can enhance this heating effect

owing to their ability to generate higher harmonics

of the excitation frequency (Fig. 3). As above,

bubbles will also dissipate energy as heat, both as a

result of viscous friction in the surrounding liquid

and any coating material, and via conduction during

compression. The relative significance of each of

these dissipation mechanisms depends on the size of

the bubble, the physical properties of the surround-

ing liquid, and the frequency and intensity of the

ultrasound field [43, 115]. Similarly, the magnitude

of the temperature rise generated will also depend

on these parameters, as well as on the concentration

of bubbles present, the pulse repetition frequency/

duty cycle, and proximity to blood vessels [116, 117].

Large temperature rises are clearly beneficial for

tissue ablation, e.g. in HIFU surgery, and for certain

types of drug and gene delivery where thermal

activation is required [118, 119]. In diagnostic

imaging and where therapeutic agents are tempera-

ture sensitive, however, significant heating is nor-

mally undesirable, and the relevance of safety

indices (e.g. thermal index) for applications employ-

ing microbubbles require further investigation.

4.5 Chemical effects

The large and rapid reduction in volume experienced

by a bubble undergoing inertial collapse can pro-

duce a significant rise in pressure and temperature

(several thousand degrees centigrade or more) [100].

These extreme conditions are confined to the centre

of the bubble [95], but highly reactive chemical

species may be produced within this space [120]. Of

particular interest in the context of medical ultra-

sound is the potential for the formation of free

radicals and toxic chemicals such as hydrogen

peroxide (H2O2), which may be harmful to cells.

Riesz and Kondo have reported the production of

high concentrations of these species in the presence

of contrast agent microbubbles [121], but these

measurements were made at excitation frequencies

much lower than those used in ultrasound imaging

or HIFU (20–50 kHz) and at relatively high intensities

(although it should be noted that these conditions

are relevant for some biomedical applications, e.g.

transdermal drug delivery [122].) Juffermans et al.

[123], however, have shown that rat cardiomyoblast

cells experience a calcium ion (Ca2+) influx upon

exposure to low-intensity ultrasound with SonoVue,

causing localized hyperpolarization of the cell

membrane, which may promote molecular uptake,

and this may also be related to the generation of

H2O2 despite the much lower ultrasound intensities.

5 FUTURE DEVELOPMENTS

5.1 Safety considerations

Each of the effects described above can also be

regarded as potential damage mechanisms in

healthy tissue exposed to contrast agents or cavita-

tion activity. Clearly the effectiveness of microbub-

bles for enhancing procedures such as thrombolysis

and HIFU demonstrates their potential for causing

damage at high insonation pressures. To date,

however, the evidence for damage under diagnostic

conditions is controversial; the consensus among

clinicians is that the benefits offered by ultrasound

contrast agents outweigh the potential risks, partic-

ularly in comparison to other diagnostic techniques

[124, 125]. The action taken in 2007 by the US Food

and Drug Administration (FDA) in issuing a ‘black

box’ warning for ultrasound contrast agents stimu-

lated heated debate and, while this warning has now

been revised, there are increasing calls for further

investigation of the safety of microbubble-mediated

imaging and therapy [126]. (A black box warning is

the highest level warning required by the FDA and

must appear on the package insert for pharmaceu-

tical products, to indicate that the contents may

cause serious adverse effects; the name refers to the

thick black border surrounding the warning.) Further

discussion may be found in the article on safety and

bio-effects in this special issue.

5.2 New agents

Advances in both diagnostic and therapeutic appli-

cations of microbubble agents have generated

demand for preparation techniques that provide a

much higher degree of control over microbubble

characteristics. Several techniques for generating

near-monodisperse coated microbubbles have been

reported, e.g. see reference [127], and are reviewed

in reference [128]. By controlling the particle size

distribution, these methods offer the ability both to

predetermine the acoustic response of a microbub-

ble suspension and their destruction threshold, as

well as to control the loading of therapeutic com-

ponents – which is critical in order to ensure



accurate dosing. There is also considerable interest

in ‘engineering’ of new agents, whereby the compo-

sition and structure of the agent’s constituent

particles (microbubbles, droplets, liposomes, etc.)

are deliberately tailored, for instance, in order to

improve their acoustic response, e.g. see reference

[129]; to optimize them for targeted therapy [7] and

molecular imaging [87, 130]; to increase their

capacity for drug delivery and gene therapy [131,

132]; or to enable penetration of tumour vasculature

[64].

5.3 Diagnostic and therapeutic applications

More sensitive methods of microbubble detection

and improved characterization of individual micro-

bubbles’ [133] propagation through suspensions

[134] are enabling the development of more accu-

rate schemes for quantitative imaging [135–137],

which will improve both diagnostic capability and

treatment monitoring as described in section 3.3.

Another application undergoing exciting develop-

ments is sonothrombolysis, where microbubbles

have been shown to increase markedly the effective-

ness of tissue plasminogen activator (t-PA) and the

rate of clot lysis in vitro and in vivo [138–140]. There

is, moreover, considerable evidence that microbub-

bles increase the permeability of not only individual

cell membranes but also the endothelium [103, 141]

including temporary opening of the blood–brain

barrier [142].

As above, there is still considerable uncertainty

regarding the mechanisms underlying the well-

documented but poorly understood observations of

enhanced cell uptake in the presence of microbub-

bles exposed to ultrasound. As already mentioned,

there are a number of very recent studies which

indicate that more subtle effects on the scale of the

cell membrane may be involved than was previously

thought [123, 103]. This is an area which has only

recently started to be investigated in detail, and

future developments will be of great importance in

optimizing the use of microbubbles in therapeutic

procedures and assessment of their safety for both

imaging and therapy.

ACKNOWLEDGEMENTS

The authors would like to thank Dr Sergey Martynovfor his help in preparing the figures.

F Authors 2010

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APPENDIX 1

Notation

a linear attenuation coefficient

b linear scattering coefficient

bb damping coefficients for the linear

equation of motion

B universal gas constant

c speed of sound in the surrounding

liquid

ci initial dissolved gas concentration in

the liquid surrounding the bubble

cs dissolved gas concentration at the

bubble surface

C thermal conductivity

D effective diffusivity of the bubble

surface

fI inertia factor

fL term describing resistance to motion

of the bubble provided by the

surrounding liquid

fP pressure factor

fRad correction factor describing the effect

of energy dissipation due to acoustic

reradiation

fS term describing resistance to motion

of the bubble provided by the bubble

surface

fTh correction factor describing the effect

of energy dissipation due to thermal

conduction

h scattering function for an individual

bubble

Iinc intensity of the incident field

m mass of gas contained in the bubble

mb mass coefficient for the linear

equation of motion

M molar mass of the gas

n size distribution of the population

N equivalent number density for a

monodisperse suspension containing

the same gas volume fraction

p pressure

pG pressure of the gas inside the bubble

pL pressure in the liquid

po hydrostatic pressure in the liquid

pscat pressure scattered or reradiated by a

bubble

pv vapour pressure inside the bubble

p‘ far field pressure in the liquid

Pabs power absorbed by the bubble

Pscat power scattered by the bubble

Q constant characterizing the

relationship between surface

tension and adsorbed molecular

concentration

r radial coordinate measured from the

bubble centre

R instantaneous bubble radius

Ro initial bubble radius

Rx radius at which buckling of the

surface will occur_RR rate of change of bubble radius



€RR acceleration of the bubble surface

sabs absorption cross-section

sext extinction cross-section

sscat scattering cross-section

Srr, L radial stress in the liquid�SS stress tensor

t time

T absolute temperature

u radial velocity

V volume

x constant characterizing the

relationship between surface tension

and adsorbed molecular

concentration

y spatial coordinate

z small quantity used for linearization

of the equation of motion

Z power law exponent characterizing

the relationship between surface

viscosity and adsorbed molecular

concentration

b time-dependent bubble volume

concentration

c ratio of specific heats

Co initial molecular concentration at the

bubble surface

dd effective damping coefficient

gso effective surface viscosity

k polytropic constant

mL dynamic viscosity of the liquid

mTh effective thermal viscosity

r density

rL density of the surrounding liquid

s interfacial/surface tension

so interfacial tension in the absence of

any surface contamination

v excitation frequency

vo undamped natural frequency

vR linear resonance frequency

Parameters used in the simulations

po 5 1.006105 Pa

Q 5 2.50610218 N m

Ro 5 2.0061026 m

Rx 5 1.661026 m

x 5 0

Z 5 0

Co 5 2.0061017/m2 (0 for uncoated bubbles)

gso 5 2.0061028 N s/m (0 for uncoated bubbles)

k 5 1

mL 5 0.0015 Pa s

rL 5 1000 kg/m3

so 5 0.05 N/m

APPENDIX 2

Derivation of linearized quantities

In order to derive suitable expressions for the linear

scattering and extinction cross-sections for an

individual microbubble, equations (8) and (20) must

be linearized. To do this, it is assumed that the gas

will follow a polytropic relation

pG~ poz2so

Ro

� �Ro

R

� �3k

and that the time-dependent radius R is replaced by

Ro[1 + z(t)], where z is a small quantity (% 1) [30] so

that equation (8) becomes

rL Ro 1zzð ÞRo€zzz3

2R2

o _zz2

� �zpo{pA tð Þ

{ poz2so

Ro

� �Ro 1{zð Þ

Ro

� �3k

z4mLRo _zz 1{zð Þ

Ro

~{2 1{zð Þ

Rosoz

QCxz1o

xz1ð Þ 1{Ro 1{zð Þ

Ro

� �2 xz1ð Þ( ) !

{4Ro _zz 1{2zð Þ

R2o

gso eZR2x= R2

o 1zzð Þ2{R2x½ � ð29Þ

Discarding all higher-order terms in z and taking

x 5 Z 5 0 for linear behaviour gives

rLR2o€zzz 4mLz

4gso

Ro

� �_zz

z 3k poz2so

Ro

� �{

2so

Roz

4QCo

Ro

� �z~pA tð Þ

ð30Þ

which can be written in the form

mb€zzzbb _zzzkbz~pA tð Þ ð31Þ

where

mb~rLR2o, bb~ 4mLz

4gso

Ro

� �

kb~ 3k poz2so

Ro

� �{

2so

Roz

4QCo

Ro

� �



For the purposes of determining the validity of eq-

uation (28) dd 5 bb/2mb and

vo~

ffiffiffiffiffiffiffikb

mb

s

For harmonic excitation, the solution will be of the

form

z~pAj j eivt

kb{mbv2ð Þzibbv½ �~pAj j ei vtzQð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kb{mbv2ð Þ2zb2bv2

h ir

ð32Þ

where

Q~tan{1 {bbv

kb{mbv2

� �

The scattered pressure will be

pscat r, tð Þ~ {rLR3ov2

r

� �pAj j eivt

kb{mbv2ð Þzibbv½ � ð33Þ

The scattering function, h, is defined as

h~pscat r, tð Þ

pAj j eivtr~

{rLR3ov2

kb{mbv2ð Þzibbv½ � ð34Þ

The scattering cross-section is defined as

sscat~Pscat

Iinc~

pscatj j2

2rLcL4pr2 2rLcL

pAj j2

~4pr2

LR6ov4

kb{mbv2ð Þ2zb2bv2

h i ð35Þ

and the viscous absorption cross-section

sabs~Pabs

Iinc~

4pbbR3ov2 pAj j2

kb{mbv2ð Þ2zb2bv2

h i 2rLcL

pAj j2

~8pbbR3

ov2rLcL

kb{mbv2ð Þ2zb2bv2

h i ð36Þ

The extinction cross-section may be approximated

as

sext~sscatzsabs ð37Þ

although, as mentioned in section 2, it should be

noted that: the total extinction coefficient should

also contain contributions from acoustic and ther-

mal damping; the expression for sscat does not take

into account attenuation in the surrounding liquid;

and the expression for sabs may differ depending on

whether the instantaneous or average dissipated

power is considered (cf. reference [143]).



special issue paper 171 cavitation and contrast: the...

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