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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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Author's complimentary copy
