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Theoretical model for the threshold onset of contrast

microbubble oscillations (L)

Alexander A. Doinikova兲and Ayache Bouakaz

INSERM U930 CNRS ERL 3106, Université François Rabelais, CHU Bretonneau, 2 Boulevard Tonnellé,

37044 Tours Cedex 9, France

共Received 15 June 2009; revised 6 October 2009; accepted 6 December 2009兲

It has been reported recently through high speed optical observations of phospholipid-coated

contrast microbubbles that there is a threshold value for the acoustic pressure amplitude below

which the radial oscillation of the microbubbles does not occur. In this Letter, it is suggested that this

threshold behavior results from the fact that a phospholipid layer, as a physical material, has a

certain value of the limiting shear stress so that its deformation does not start until this limiting value

is exceeded. A theoretical model is proposed for the description of this phenomenon. The model

explains the experimentally observed dependence of the threshold onset of microbubble oscillation

on the initial bubble radius. © 2010 Acoustical Society of America. 关DOI: 10.1121/1.3278607兴

PACS number共s兲: 43.25.Yw, 43.35.Ei, 43.80.Qf 关AJS兴Pages: 649–651

I. INTRODUCTION

Emmer et al. 共2007a兲investigated experimentally the

onset of radial oscillation of phospholipid-coated contrast

microbubbles 共BR14兲with radii ranging from 1 to 5.5

m.

The microbubbles were insoniﬁed at a driving frequency of

1.7 MHz and acoustic pressure amplitudes ranging from 20

to 250 kPa. It has been found that for smaller microbubbles,

with resting radii R0⬍2.5

m, there is a threshold value for

the acoustic pressure amplitude below which little or no os-

cillation is observed. The microbubble oscillation was not

detected until the acoustic driving pressure reached the

threshold value, and as the resting radius of the microbubbles

decreased, the threshold pressure increased. In this study, it is

hypothesized that the observed threshold behavior results

from the fact that an encapsulating phospholipid layer, as a

physical material, has a certain value of the limiting shear

stress so that its deformation does not start until this limiting

value is exceeded. A theoretical model is proposed for the

description of this behavior. The model explains the experi-

mentally observed dependence of the threshold onset of mi-

crobubble oscillation on the initial bubble radius.

II. THEORETICAL MODEL

It is well known that many elastic solids and viscous

ﬂuids start moving 共deforming or ﬂowing, respectively兲only

if the applied stress exceeds a critical value, which is called

the limiting shear stress 共Reiner, 1958兲. Let us denote this

threshold by

el for an elastic solid and by

vis for a viscous

ﬂuid. If a medium possesses both elasticity and viscosity, it

will begin to move when the applied stress exceeds the

greater of the thresholds

el and

vis. Let us denote this

greater threshold by

0.

The commonly accepted opinion is that phospholipid

possesses both elasticity and viscosity. Therefore one can

assume that phospholipid, as a physical material, has a cer-

tain value of the threshold shear stress

0. We suggest that it

is just this general property inherent in most physical mate-

rials that is responsible for the threshold onset of radial os-

cillation of a phospholipid-coated microbubble. It should be

emphasized, however, that

0is treated here as the threshold

shear stress for phospholipid as a material, not as the thresh-

old value for a phospholipid spherical shell. The importance

of this distinction will become clear below.

The fact that the phospholipid shell of a contrast mi-

crobubble is spherical and ultrathin poses the following

question: How to relate the bulk physical properties of phos-

pholipid to the behavior of a phospholipid shell, which is

usually described by zero-thickness encapsulation models

such as modiﬁed Rayleigh–Plesset equations 共de Jong et al.,

1994;Morgan et al., 2000;Sarkar et al., 2005;Marmottant

et al., 2005兲? This problem can be solved using the results of

Church 共1995兲and Doinikov et al. 共2009兲.

Generalizing the Rayleigh–Plesset equation for a free

bubble 共Plesset, 1949兲to the case of a bubble enclosed in a

ﬁnite-thickness shell, Church 共1995兲derived the following

equation:

R1R

¨1

冋

1+

冉

L

S

−1

冊

R1

R2

册

+3

2R

˙1

2

冋

1+

冉

L

S

−1

冊

⫻

冉

4R2

3−R1

3

3R3

3

冊

R1

R2

册

=1

S

冋

Pg0

冉

R10

R1

冊

3

␥

−2

1

R1

−2

2

R2

−4

L

R1

2R

˙1

R2

3−P0−Pac共t兲−S

册

,共1兲

where R1共t兲and R2共t兲are the inner and the outer radii of the

encapsulating shell, respectively, the overdot denotes the

time derivative,

Land

Sare the equilibrium densities of the

surrounding liquid and the shell, respectively, Pg0is the equi-

librium gas pressure within the bubble,

␥

is the ratio of spe-

ciﬁc heats of the gas, R10 and R20 are the inner and the outer

radii of the shell at rest,

1and

2are the surface tension

a兲Author to whom correspondence should be addressed. Electronic mail:

doinikov@bsu.by

J. Acoust. Soc. Am. 127 共2兲, February 2010 © 2010 Acoustical Society of America 6490001-4966/2010/127共2兲/649/3/$25.00

Author's complimentary copy

coefﬁcients for the gas-shell and the shell-liquid interfaces,

respectively,

Lis the shear viscosity of the liquid, P0is the

hydrostatic pressure in the liquid, and Pac共t兲is the driving

acoustic pressure at the location of the bubble. The effect of

encapsulation is described by the term S, which is given by

S=−3

冕

R1

R2

rr共r,t兲

rdr,共2兲

where ris the radial coordinate of a spherical coordinate

system with the origin at the center of the bubble, and

rr共r,t兲is the radial component of the stress deviator of the

shell. For thin-shelled bubbles, such as phospholipid-coated

microbubbles considered here, Eq. 共1兲reduces to

RR

¨+3

2R

˙2=1

L

冋

Pg0

冉

R0

R

冊

3

␥

−2

R−4

L

R

˙

R−P0

−Pac共t兲−S

册

,共3兲

where R共t兲is the radius of the microbubble, R0is the resting

value of R共t兲, and

is the surface tension at the gas-liquid

interface. Doinikov et al. 共2009兲showed that in this limit the

term Sbecomes

S=−共3/R兲

rr共r,t兲兩r=R,共4兲

where denotes the shell thickness. This equation allows

one to transform existing constitutive equations for the stress

tensor

ij, which are normally speciﬁed in the bulk form, into

a surface form that is required in Eq. 共3兲. Equation 共4兲can be

used to establish a criterion for the onset of oscillation of

phospholipid-coated microbubbles.

Normally, it is assumed that Sincludes two parts, Sel and

Svis, which describe the shell elasticity and the shell viscos-

ity. From Eq. 共4兲it follows that if a phospholipid layer, as a

physical material, has the threshold shear stress equal to

0,

then Sshould include a third part, S0, which can be written as

S0=3

0/R0.共5兲

The proposed criterion is that the microbubble oscillation

starts when the acoustic pressure amplitude Paexceeds the

magnitude equal to S0; i.e., the critical value of the acoustic

pressure amplitude, Pa

cr, is equal to S0. Note that, as follows

from Eq. 共5兲, the smaller the bubble radius, the greater the

value of S0, which is in agreement with Emmer et al.’s

共2007a兲experimental observations.

III. RESULTS AND DISCUSSION

Using the measurements of Emmer et al. 共2007a兲, one

can estimate the magnitude of

0and thereby check the pro-

posed theoretical model by comparing

0with the value of

the bulk elastic modulus reported in literature for lipid. Fig-

ure 1displays the acoustic pressure threshold as a function of

microbubble initial radius. The circles in Fig. 1correspond to

experimental data presented in Fig. 4 in Emmer et al.

共2007a兲. The solid line shows a ﬁt to the experimental points

by Eq. 共5兲. The ﬁtting was made using the program package

MATHEMATICA 共Wolfram Research, Inc., Champaign, IL兲.

The equation of the ﬁtting curve is given by

Pa

cr = 0.12/R0.共6兲

It follows that

0⬇0.04 N/m. Assuming that the thickness

of the phospholipid encapsulating layer is on the order of 1

nm 共Morgan et al., 2000兲, the magnitude of the critical shear

stress for the phospholipid layer as a material should be of

the order of 40 MPa. According to the experimental data of

Marmottant et al. 共2005兲, the bulk elastic modulus ofa1nm

thick phospholipid layer is estimated to be equal to 333.3

MPa; see also Doinikov and Dayton 共2006兲. Thus the value

of

0is found to be smaller by almost one order of magnitude

than the bulk elastic modulus. This is a reasonable difference

for such material parameters, and hence this result conﬁrms

the plausibility of the proposed theory.

To sum up, the assumption that the threshold onset of

radial oscillation of phospholipid-coated microbubbles is a

consequence of the rheological properties of a phospholipid

layer allows one to suggest a simple physical explanation for

this experimental observation. This explanation is that phos-

pholipid, as a physical material, has the limiting shear stress,

which in turn results in the threshold onset of microbubble

oscillation. The derived expression for the critical acoustic

pressure, Eq. 共5兲, reveals that the observed dependence of the

threshold pressure on the resting bubble radius is a geometri-

cal factor that results from the spherical shape of the phos-

pholipid coating.

In conclusion, we should mention that experiments re-

veal that the threshold behavior of phospholipid-coated mi-

crobubbles is dependent on the ultrasound parameters of the

excitation pulse, such as the driving frequency f, the acoustic

pressure amplitude Pa, the number of pulses in the imposed

ultrasound signal N, etc. 共Emmer et al., 2007b,2007c兲.In

this connection it should be noted that Eq. 共5兲alone cannot

explain such effects since they result from the rheological

properties of a phospholipid layer as a material. That is, they

are accounted for in the quantity

0, which accordingly

should be considered as a function of the parameters of ex-

citation,

0=

0共f,Pa,N,...兲. Further experiments are thus re-

quired in order to understand the speciﬁc rheological laws

that can be applied to describe the dependence of

0on the

excitation parameters.

20

40

60

80

100

1.5 2.0

2.5

3.0 3.5 4.0

R

0

[m]

FIG. 1. Threshold amplitude of the driving pressure as a function of the

resting bubble radius. Circles show experimental data from Emmer et al.

共2007a兲.

650 J. Acoust. Soc. Am., Vol. 127, No. 2, February 2010 A. A. Doinikov and A. Bouakaz: Letters to the Editor

Author's complimentary copy

ACKNOWLEDGMENT

A.A.D. gratefully acknowledges the ﬁnancial support

from the le STUDIUM®, Institute for Advanced Studies 共Or-

leans, France兲.

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J. Acoust. Soc. Am., Vol. 127, No. 2, February 2010 A. A. Doinikov and A. Bouakaz: Letters to the Editor 651

Author's complimentary copy