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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.
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Theoretical model for the threshold onset of contrast
microbubble oscillations (L)
Alexander A. Doinikovaand Ayache Bouakaz
INSERM U930 CNRS ERL 3106, Université François Rabelais, CHU Bretonneau, 2 Boulevard Tonnellé,
37044 Tours Cedex 9, France
Received 15 June 2009; revised 6 October 2009; accepted 6 December 2009
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. DOI: 10.1121/1.3278607
PACS numbers: 43.25.Yw, 43.35.Ei, 43.80.Qf AJSPages: 649–651
Emmer et al. 2007ainvestigated experimentally the
onset of radial oscillation of phospholipid-coated contrast
microbubbles BR14with radii ranging from 1 to 5.5
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 R02.5
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.
It is well known that many elastic solids and viscous
fluids start moving deforming or flowing, respectivelyonly
if the applied stress exceeds a critical value, which is called
the limiting shear stress Reiner, 1958. 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
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 de Jong et al.,
1994;Morgan et al., 2000;Sarkar et al., 2005;Marmottant
et al., 2005? This problem can be solved using the results of
Church 1995and Doinikov et al. 2009.
Generalizing the Rayleigh–Plesset equation for a free
bubble Plesset, 1949to the case of a bubble enclosed in a
finite-thickness shell, Church 1995derived the following
where R1tand R2tare the inner and the outer radii of the
encapsulating shell, respectively, the overdot denotes the
time derivative,
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,
2are the surface tension
aAuthor to whom correspondence should be addressed. Electronic mail:
J. Acoust. Soc. Am. 127 2, February 2010 © 2010 Acoustical Society of America 6490001-4966/2010/1272/649/3/$25.00
Author's complimentary copy
coefficients for the gas-shell and the shell-liquid interfaces,
Lis the shear viscosity of the liquid, P0is the
hydrostatic pressure in the liquid, and Pactis the driving
acoustic pressure at the location of the bubble. The effect of
encapsulation is described by the term S, which is given by
where ris the radial coordinate of a spherical coordinate
system with the origin at the center of the bubble, and
rrr,tis the radial component of the stress deviator of the
shell. For thin-shelled bubbles, such as phospholipid-coated
microbubbles considered here, Eq. 1reduces to
where Rtis the radius of the microbubble, R0is the resting
value of Rt, and
is the surface tension at the gas-liquid
interface. Doinikov et al. 2009showed that in this limit the
term Sbecomes
where denotes the shell thickness. This equation allows
one to transform existing constitutive equations for the stress
ij, which are normally specified in the bulk form, into
a surface form that is required in Eq. 3. Equation 4can 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. 4it follows that if a phospholipid layer, as a
physical material, has the threshold shear stress equal to
then Sshould include a third part, S0, which can be written as
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. 5, the smaller the bubble radius, the greater the
value of S0, which is in agreement with Emmer et al.’s
2007aexperimental observations.
Using the measurements of Emmer et al. 2007a, 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.
2007a. The solid line shows a fit to the experimental points
by Eq. 5. The fitting was made using the program package
MATHEMATICA Wolfram Research, Inc., Champaign, IL.
The equation of the fitting curve is given by
cr = 0.12/R0.6
It follows that
00.04 N/m. Assuming that the thickness
of the phospholipid encapsulating layer is on the order of 1
nm Morgan et al., 2000, 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. 2005, the bulk elastic modulus ofa1nm
thick phospholipid layer is estimated to be equal to 333.3
MPa; see also Doinikov and Dayton 2006. Thus the value
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. 5, 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. Emmer et al., 2007b,2007c.In
this connection it should be noted that Eq. 5alone 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-
0f,Pa,N,.... 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.
1.5 2.0
3.0 3.5 4.0
FIG. 1. Threshold amplitude of the driving pressure as a function of the
resting bubble radius. Circles show experimental data from Emmer et al.
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
A.A.D. gratefully acknowledges the financial support
from the le STUDIUM®, Institute for Advanced Studies Or-
leans, France.
Church, C. C. 1995. “The effect of an elastic solid surface layer on the
radial pulsations of gas bubbles,” J. Acoust. Soc. Am. 97, 1510–1521.
de Jong, N., Cornet, R., and Lancée, C. T. 1994. “Higher harmonics of
vibrating gas-filled microspheres. Part one: Simulations,” Ultrasonics 32,
Doinikov, A. A., and Dayton, P. A. 2006. “Spatio-temporal dynamics of an
encapsulated gas bubble in an ultrasound field,” J. Acoust. Soc. Am. 120,
Doinikov, A. A., Haac, J. F., and Dayton, P. A. 2009. “Modeling of non-
linear viscous stress in encapsulating shells of lipid-coated contrast agent
microbubbles,” Ultrasonics 49, 269–275.
Emmer, M., Matte, G., van Neer, P., van Wamel, A., and de Jong, N.
2007c. “Improved ultrasound contrast agent detection in a clinical set-
ting,” in Proceedings of the IEEE Ultrasonics Symposium, New York, pp.
Emmer, M., van Wamel, A., Goertz, D. E., and de Jong, N. 2007a. “The
onset of microbubble vibration,” Ultrasound Med. Biol. 33, 941–949.
Emmer, M., Vos, H. J., van Wamel, A., Goertz, D. E., Versluis, M., and de
Jong, N. 2007b. “Vibrating microbubbles at low acoustic pressures,” in
The Abstract Book of the 12th European Symposium on Ultrasound Con-
trast Imaging, Rotterdam, The Netherlands, p. 84.
Marmottant, P., van der Meer, S., Emmer, M., Versluis, M., de Jong, N.,
Hilgenfeldt, S., and Lohse, D. 2005. “A model for large amplitude os-
cillations of coated bubbles accounting for buckling and rupture,” J.
Acoust. Soc. Am. 11 8, 3499–3505.
Morgan, K. E., Allen, J. S., Dayton, P. A., Chomas, J. E., Klibanov, A. L.,
and Ferrara, K. W. 2000. “Experimental and theoretical evaluation of
microbubble behavior: Effect of transmitted phase and bubble size,” IEEE
Trans. Ultrason. Ferroelectr. Freq. Control 47, 1494–1509.
Plesset, M. S. 1949. “The dynamics of cavitation bubbles,” ASME J. Appl.
Mech. 16, 277–282.
Reiner, M. 1958.Rheology Springer-Verlag, Berlin.
Sarkar, K., Shi, W. T., Chatterjee, D., and Forsberg, F. 2005. “Character-
ization of ultrasound contrast microbubbles using in vitro experiments and
viscous and viscoelastic interface models for encapsulation,” J. Acoust.
Soc. Am. 11 8, 539–550.
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
... We propose a correction to the acoustic contrast factor to account for secondary Bjerknes forces and summarise our findings in a dynamical measurement of the compressibility of coated microbubbles: a key parameter for the uptake of microbubble-based therapies [8] and sensing applications [15,16]. Thanks to a direct estimation of key shell parameters-obtained by milling and compressing a selection of bubbles under a Focussed Ion Beam Scanning Electron Microscope (FIB-SEM)-we discuss our results in terms of the onset of volume oscillations [25,30] and of buckling [26]. ...
... In the case of phospholipid-coated bubbles, Emmer et al. [25] showed that the linear oscillator model behind Equation (1) is only valid above a threshold pressure, which they related to the onset of volume oscillations, highlighting how smaller bubbles have a larger threshold. According to Doinikov and Bouakaz [30], this behaviour would be due to the shear stress τ 0 of the phospholipid as a material: before oscillations start, the acoustic pressure would need to overcome the value: ...
... In this way, we account for the dependence on concentration by noting that secondary Bjerknes forces modify the acoustic pressure locally acting on the bubbles: compressibility should be related not to p a , but to the effective pressure p meas from Figure 2a. Figure 7, which presents compressibility κ p as a function of effective pressure, allows the threshold to be estimated between 1.7 ± 0.5 kPa and 2.0 ± 0.3 kPa (164.33 kHz). The observed (quadratic) trend with increasing pressure is not compatible with the onset of oscillations, as described by Emmer et al. [25] and Doinikov and Bouakaz [30], but instead agrees with the models of shell compressibility based on Hooke's law [41,49], which account for buckling. Paul et al. [59], in particular, proposes near the transition to buckling a quadratic dependence of the radial compression rate R/R 0 , where R 0 is the equilibrium radius of the bubble. ...
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The merits of ultrasound contrast agents (UCAs) were already known in the 1960s. It was, however, not until the 1990s that UCAs were clinically approved and marketed. In these years, it was realized that the UCAs are not just efficient ultrasound scatterers, but that their main constituent, the coated gas microbubble, acts as a nonlinear resonator and, as such, is capable of generating harmonic energy. Subharmonic, ultraharmonic, and higher harmonic frequencies of the transmitted ultrasound frequency have been reported. This opened up new prospects for their use and several detection strategies have been developed to exploit this harmonic energy to discriminate the contrast bubbles from surrounding tissue. This insight created a need for tools to study coated bubble behavior in an ultrasound field and the first models were developed. Since then, 20 years have elapsed, in which a broad range of UCAs and UCA models have been developed. Although the models have helped in understanding the responses of coated bubbles, the influence of the coating has not been fully elucidated to date and UCA models are still being improved. The aim of this review paper is to offer an overview in these developments and indicate future directions for research.
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We present a model applicable to ultrasound contrast agent bubbles that takes into account the physical properties of a lipid monolayer coating on a gas microbubble. Three parameters describe the properties of the shell: a buckling radius, the compressibility of the shell, and a break-up shell tension. The model presents an original non-linear behavior at large amplitude oscillations, termed compression-only, induced by the buckling of the lipid monolayer. This prediction is validated by experimental recordings with the high-speed camera Brandaris 128, operated at several millions of frames per second. The effect of aging, or the resultant of repeated acoustic pressure pulses on bubbles, is predicted by the model. It corrects a flaw in the shell elasticity term previously used in the dynamical equation for coated bubbles. The break-up is modeled by a critical shell tension above which gas is directly exposed to water.
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Coupled equations describing the radial and translational dynamics of an encapsulated gas bubble in an ultrasound field are derived by using the Lagrangian formalism. The equations generalize Church's theory by allowing for the translation motion of the bubble and radiation losses due to the compressibility of the surrounding liquid. The expression given by Church for the inner bubble radius corresponding to the unstrained state of the bubble shell is also refined, assuming that the shell can be of arbitrary thickness and impermeable to gas. Comparative linear analysis of the radial equation is carried out relative to Church's theory. It is shown that there are substantial departures from predictions of Church's theory. The proposed model is applied to evaluate radiation forces exerted on encapsulated bubbles and their translational displacements. It is shown that in the range of relatively high frequencies encapsulated bubbles are able to translate more efficiently than free bubbles of the equivalent size.
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Zero-thickness interface models are developed to describe the encapsulation of microbubble contrast agents. Two different rheological models of the interface, Newtonian (viscous) and viscoelastic, with rheological parameters such as surface tension, surface dilatational viscosity, and surface dilatational elasticity are presented to characterize the encapsulation. The models are applied to characterize a widely used microbubble based ultrasound contrast agent. Attenuation of ultrasound passing through a solution of contrast agent is measured. The model parameters for the contrast agent are determined by matching the linearized model dynamics with measured attenuation data. The models are investigated for its ability to match with other experiments. Specifically, model predictions are compared with scattered fundamental and subharmonic responses. Experiments and model prediction results are discussed along with those obtained using an existing model [Church, J. Acoust. Soc. Am. 97, 1510 (1995) and Hoff et al., J. Acoust. Soc. Am. 107, 2272 (2000)] of contrast agents.
Most previous theoretical investigations of gas bubble dynamics have assumed an uncontaminated gas-liquid interface. Recently, however, the potential importance of layers of surface active agents on bubble dynamics has been increasingly recognized. In this work it is assumed that a continuous layer of incompressible, solid elastic material separates the gas from the bulk Newtonian liquid. Elasticity is modeled to include viscous damping. A Rayleigh-Plesset-like equation describing the dynamics of such surface-contaminated gas bubbles is derived. The equation predicts that the surface layer supports a strain that counters the Laplace pressure and thereby stabilizes the bubble against dissolution. An analytical solution to this equation which includes both the fundamental and second-harmonic response is presented. The dispersion relation describing the propagation of linear pressure waves in liquids containing suspensions of these bubbles also is presented. It is found that (1) the resonance frequencies of individual bubbles tend to increase as the modulus of rigidity increases; (2) the damping provided by the viscosity of the shell dominates thermal effects for bubble radii less than ~10 μm; (3) the attenuation coefficient in a bubbly liquid decreases as either the rigidity or the viscosity of the surface layer increases; (4) encapsulated bubbles with shell rigidity greater than ~85 MPa provide a greater total scattering cross section per unit attenuation in the lower biomedical frequency range than do free bubbles of the equivalent size.
Conference Paper
Optical studies have shown threshold behaviour of phospholipid-coated contrast agent microbubbles. Below the acoustic pressure threshold, phospholipid-coated microbubbles oscillate significantly less than above the threshold. For microbubbles smaller than 3.0 mum diameter, pressure-dependent scattering was measured, which is believed to be the result of threshold behaviour. The aim of this study is to investigate if threshold behaviour is useful to enhance the contrast in power modulation images. For levovist and BR14 suspensions (filtered and native), a programmable ultrasound system recorded power modulation images at 2 MHz and acoustic pressures between 25 and 250 kPa. Results were compared to intensities recorded with a commercial ultrasound system. An inverse relationship between the pressure-dependency of the scattering and microbubble size was observed. Threshold behaviour enhances the contrast in power modulation images. Using a suspension with microbubbles smaller than 2.0 mum, at 2 MHz transmit frequency and an acoustic pressure of 250 kPa, the CTR value was 33 dB, which is 13 dB higher compared to a native BR14 suspension.
The dynamics of an adiabatic gas–filled bubble in a viscous liquid is studied when subjected to a tension wave produced by a shock wave reflected off the water surface. Below a critical bubble size the water does not cavitate. Numerical solutions show that cavitation causes severe prolongation and attenuation of the original tension wave.
The acoustic behaviour of an ideal gas bubble in water is considered and the equation of motion is extended to model an Albunex microsphere. Calculations reveal large differences in non-linear behaviour between ideal gas bubbles and Albunex microspheres, due to the additional restoring force of, and friction inside, the shell that surrounds the Albunex microsphere. Simulations with the Albunex contrast agent further reveal that the optimal driving frequency is 1 MHz, resulting in a second harmonic that is 20 dB below the first harmonic at an acoustic pressure of 50 kPa. The difference increases to 25 dB for a driving frequency of 2 MHz.
A general theoretical approach to the development of zero-thickness encapsulation models for contrast microbubbles is proposed. The approach describes a procedure that allows one to recast available rheological laws from the bulk form to a surface form which is used in a modified Rayleigh-Plesset equation governing the radial dynamics of a contrast microbubble. By the use of the proposed procedure, the testing of different rheological laws for encapsulation can be carried out. Challenges of existing shell models for lipid-encapsulated microbubbles, such as the dependence of shell parameters on the initial bubble radius and the "compression-only" behavior, are discussed. Analysis of the rheological behavior of lipid encapsulation is made by using experimental radius-time curves for lipid-coated microbubbles with radii in the range 1.2-2.5 microm. The curves were acquired for a research phospholipid-coated contrast agent insonified with a 20 cycle, 3.0 MHz, 100 kPa acoustic pulse. The fitting of the experimental data by a model which treats the shell as a viscoelastic solid gives the values of the shell surface viscosity increasing from 0.30 x 10(-8) kg/s to 2.63 x 10(-8) kg/s for the range of bubble radii, indicated above. The shell surface elastic modulus increases from 0.054 N/m to 0.37 N/m. It is proposed that this increase may be a result of the lipid coating possessing the properties of both a shear-thinning and a strain-softening material. We hypothesize that these complicated rheological properties do not allow the existing shell models to satisfactorily describe the dynamics of lipid encapsulation. In the existing shell models, the viscous and the elastic shell terms have the linear form which assumes that the viscous and the elastic stresses acting inside the lipid shell are proportional to the shell shear rate and the shell strain, respectively, with constant coefficients of proportionality. The analysis performed in the present paper suggests that a more general, nonlinear theory may be more appropriate. It is shown that the use of the nonlinear theory for shell viscosity allows one to model the "compression-only" behavior. As an example, the results of the simulation for a 2.03 microm radius bubble insonified with a 6 cycle, 1.8 MHz, 100 kPa acoustic pulse are given. These parameters correspond to the acoustic conditions under which the "compression-only" behavior was observed by de Jong et al. [Ultrasound Med. Biol. 33 (2007) 653-656]. It is also shown that the use of the Cross law for the modeling of the shear-thinning behavior of shell viscosity reduces the variance of experimentally estimated values of the shell viscosity and its dependence on the initial bubble radius.