Post on 16-Mar-2023
Acoustic characterization of monodisperse lipid-coatedmicrobubbles: Relationship between size and shell viscoelasticproperties
Miguel A. Parrales,a) Juan M. Fernandez, and Miguel Perez-SaboridAerospace Engineering and Fluid Mechanics Department, University of Seville,Camino de los Descubrimientos s/n, 41092 Sevilla, Spain
Jonathan A. Kopechek and Tyrone M. PorterMechanical Engineering Department, Boston University, 110 Cummington Street, Boston,Massachusetts 02215
(Received 24 October 2013; revised 9 May 2014; accepted 7 July 2014)
The acoustic attenuation spectrum of lipid-coated microbubble suspensions was measured in order
to characterize the linear acoustic behavior of ultrasound contrast agents. For that purpose, micro-
bubbles samples were generated with a very narrow size distribution by using microfluidics techni-
ques. A performance as good as optical characterization techniques of single microbubbles was
achieved using this method. Compared to polydispersions (i.e., contrast agents used clinically),
monodisperse contrast agents have a narrower attenuation spectrum, which presents a maximum
peak at a frequency value corresponding to the average single bubble resonance frequency. The
low polydispersity index of the samples made the estimation of the lipid viscoelastic properties
more accurate since, as previously reported, the shell linear parameters may change with the equi-
librium bubble radius. The results showed the great advantage of dealing with monodisperse popu-
lations rather than polydisperse populations for the acoustic characterization of ultrasound contrast
agents. VC 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4890643]
PACS number(s): 43.35.Bf, 43.35.Yb, 43.80.Qf, 43.20.Fn [TGL] Pages: 1077–1084
I. INTRODUCTION
Ultrasound contrast agents are gas-filled microbubbles,
which are injected into the blood pool in order to increase
the blood echogenicity for ultrasound imaging applica-
tions.1–3 These contrast agents also have a great potential
use for drug and gene delivery for treating different dis-
eases.4–6 Due to their compressibility, the microbubbles per-
form volumetric oscillations, giving rise to a strong resonant
echo when driven by an ultrasound field with a specific fre-
quency. Consequently, they scatter more energy than rigid
particles of the same size, or even larger liquid-filled par-
ticles (red blood cells, for instance).7–9 Traditional methods
for contrast microbubbles generation (agitation, sonication)
produce very polydisperse suspensions, whose acoustic
response is difficult to optimize. Effectively, the size of each
microbubble plays a crucial role since resonance is only
reached when it precisely matches the operating frequency
of the ultrasound equipment. Nevertheless, these production
techniques are widely used in clinical applications due to the
simplicity and the high production ratio.10
Contrast microbubbles for medical applications are
commonly coated by phospholipid shells to prevent a rapid
dissolution of the gas core.5 This kind of shell also alters the
frequency response of the microbubble to acoustic excita-
tion. The stiffness of the shell leads to an increase in the
microbubble resonance frequency while the shell viscosity
increases attenuation, due to the phospholipid viscous
dissipation during the oscillatory motion of the coated bub-
ble.11,12 The performance of lipid-coated microbubbles as
ultrasound contrast agents strongly depends on the shell
viscoelastic behavior; consequently, considerable effort has
been invested in estimating these properties. Estimates of the
viscoelastic properties are based on two different techniques:
acoustic attenuation measurements of microbubbles suspen-
sions13,14 and optical measurements of the oscillatory motion
of single microbubbles.15,16 While optical methods are more
accurate and sophisticated, they need a considerable invest-
ment in instruments such as ultra-high-speed cameras or
lasers. In contrast, acoustic methods use basic ultrasonic
equipment and are able to provide a direct measurement of
the more relevant physical quantities.
It is important to note that the measured attenuation
spectrum is influenced heavily by the microbubble size dis-
tribution. In fact, it has been shown that larger microbubbles
tend to be the primary source of attenuation in polydisper-
sions, thus complicating the analysis of the measurements
and the estimation of the shell properties.17,18 An inability to
control the microbubble size distribution accurately can
result in inaccurate estimates of the shell viscoelastic param-
eters (elasticity and viscosity). Therefore, in order to get a
novel generation of monodisperse contrast agents, it is essen-
tial to achieve both ultrasound echo optimization in medical
applications as well as the simple and accurate determination
of the lipid-shell mechanical behavior with acoustic charac-
terization methods. To address these needs, researchers
began using microfluidic devices to produce monodisperse
lipid-coated microbubbles.19,20 Unlike polydisperse micro-
bubble suspensions, monodispersions allow one to measure
a)Author to whom correspondence should be addressed. Electronic mail:
mparrales@us.es
J. Acoust. Soc. Am. 136 (3), September 2014 VC 2014 Acoustical Society of America 10770001-4966/2014/136(3)/1077/8/$30.00
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attenuation as a function of mean microbubble diameter and
to more accurately estimate the shell viscoelastic
properties.21
In this article, we present an experimental procedure to
estimate the viscoelastic properties of lipid-coated microbub-
bles. The theoretical background for these estimations is
based on the linearized Rayleigh–Plesset equation modified
with the Marmottant model11 to account for the shell rheol-
ogy. Consequently, in order to assure accurate measurements,
we generated monodisperse samples of microbubbles and
acquired the acoustic attenuation for different equilibrium ra-
dius, thus avoiding the difficulties when using polydisperse
contrast agents. We found that the linear viscoelastic parame-
ters vary with the equilibrium radius, which is effectively
observed in single coated bubble characterization with optical
methods15 and light scattering.16 These results finally provide
us with confirmation of the accuracy and reliability of our ex-
perimental method, which may stimulate great interest for
broadening further investigations in the field of the rheology
of membranes.
II. THEORY
An isolated gas-filled microbubble with radius Ro at
rest, is immersed in an infinite liquid medium, with density
q1 and viscosity lL, under a hydrostatic pressure p1. An
acoustic perturbation pa with a characteristic wavelength
k� Ro, will make the bubble oscillate radially and symmet-
rically. The equations of motion describing the radial oscilla-
tions of a coated microbubble, and the acoustic scattering
and absorption properties of both a single bubble and a
microbubble suspension, is described in this section. We
applied the incompressible fluid motion equations over the
oscillating bubble, and balanced the inertial forces of the
moving boundary with the pressure difference in the sur-
rounding liquid medium, leading to the well-known
Rayleigh–Plesset equation,9,22
RR::
þ 3
2_R
2 ¼ 1
q1pL � pa � p1ð Þ; (1)
where pL stands for the liquid pressure just outside the bubble.
This term can be related with the gas pressure inside the bub-
ble pg by setting the normal stress balance across the bound-
ary, pL ¼ pg � 2r=R� 4lE_R=R, where pg ¼ pg0ðR=RoÞ�3j
,
where pg0 is the gas pressure at rest and r is the surface ten-
sion. In this last expression, a polytropic gas behavior, with a
coefficient j, has been assumed, and the total losses
have been modeled by using a linear effective viscosity
lE ¼ ðlL þ lac þ lthÞ, which takes into account the three
main damping mechanisms during the oscillations: viscous
dissipation, acoustic reradiation, and heat transfer.23–25
Thex acoustic and thermal viscosities have been defined as
lac ¼ q1x2oR3
o=4c1 and lth ¼ �pg0ImU=4xo, respectively,
where xo is the characteristic oscillation frequency and c1 is
the sound speed in the liquid. We found that both the poly-
tropic coefficient, defined as j ¼ ReU=3, and the thermal vis-
cosity were related to a complex function U, which can be
written as25
U ¼ 3c
1� 3 c� 1ð ÞiPe
ffiffiffiffiffiffiffiiPep
cotffiffiffiffiffiffiffiiPep
� 1� � ; (2)
where c is the adiabatic coefficient and Pe ¼ xoR2o=Dg is the
Peclet number, with Dg the gas thermal diffusivity.
The effects of the coating on the microbubble dynamics
can be modeled according to Marmottant et al.11 formula-
tion. Assuming that the shell is represented by a thin lipid
monolayer with viscoelastic properties, the pressure differ-
ence across the bubble wall can now be written as
pL ¼ pg � 2rðRÞ=R� 4lE_R=Rþ 4js
_R=R2; (3)
where js is the shell surface viscosity and rðRÞ represents the
radius-dependent surface tension coefficient, which can
be linearized by rðRÞ ’ 2vðR=Ro � 1Þ in the elastic
regime, where v is the shell elastic modulus.11,12 The
Rayleigh–Plesset equation can be linearized assuming small
oscillations around the equilibrium radius, Ro. Therefore, con-
sidering R ’ Roð1þ XÞ with jXj � 1, the linear microbubble
radial dynamics equation yields24,25
X::
þ 2d _X þ x2nX ¼ � pa
q1R2o
; (4)
where d ¼ ð2lE=q1R2o þ 2js=q1R3
oÞ is the damping factor
and xn is the bubble natural frequency, defined by
x2n ¼
3jp1q1R2
o
þ 4vq1R3
o
: (5)
The differential equation for the radial dynamics shows that
the coated microbubble behaves as a damped harmonic os-
cillator, which reaches the maximum amplitude response
when driven at xn. If we compare Eq. (4) with the equation
of motion for a free microbubble, we observe that the lipid
coating adds an additional viscous damping term.
Additionally, lipid-coated microbubbles have a higher natu-
ral frequency due to the stiffness of the lipid shell.12
The oscillating bubble behaves as a monopolar acoustic
source and reradiates acoustic energy from the excitation
pulse.9 The radiated acoustic pressure at a distance rfrom the linearly oscillating bubble can be written as
ps ¼ q1R3oX::
ðsÞ=r, where s ¼ ðt� r=c1Þ is the retarded
time. Assuming time-harmonic acoustic excitations and oscil-
lations of the form X ¼ X̂ expð�ixotÞ, we can finally write
p̂s ¼p̂aRo
xn=xoð Þ2 � 1� iC
eikr
r¼ p̂afs
eikr
r; (6)
where k ¼ xo=c1 is the wavenumber, fs is the omnidirec-
tional scattering function, and C ¼ 2d=xo stands for the total
damping dimensionless coefficient. Therefore, in relation
with the oscillatory response, the sound scattered by the bub-
ble reaches a relative maximum at the natural frequency.9,13
The ratio of the total acoustic power scattered to the
external acoustic intensity is given by the scattering cross
section
1078 J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014 Parrales et al.: Viscoelastic properties of coated bubbles
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rs ¼ð
Xjfsj2dX ¼ 4pjfsj2; (7)
where dX is a solid angle element. This parameter measures
the efficiency of the microbubble as a sound scatterer.9 The
absorption cross section ra is defined as the ratio of the
absorbed acoustic power (due to viscous and thermal losses) to
the incident intensity,13 and it is given by ra ¼ ðC=Cac � 1Þrs,
where Cac ¼ 4lac=q1xoR2o. Finally, the total acoustic power
removed by the bubble is obtained by using the extinction cross
section, defined as
re ¼ rs þ ra ¼ ðC=CacÞrs: (8)
This last parameter is related to the attenuation properties of
a bubbly medium as we will show.13
Let us now assume a dilute microbubble homogeneous
suspension at a relatively low concentration n (bubbles per
cubic meter), so that multiple scattering can be ignored. The
size distribution of the bubble population is given by the
probability density function f ðRoÞ. The reduction in the
acoustic intensity over a distance dz through the suspension
is given by dIðzÞ ¼ �aIðzÞdz, where a is the attenuation
coefficient.13 By linear superposition, the attenuation coeffi-
cient is written in terms of the extinction cross section as
a ¼ð1
0
nref ðRoÞdRo: (9)
Note that, if the microbubble suspension were monodisperse,
this parameter would be just a ¼ nre. In this case, the maxi-
mum value of the attenuation is reached at the natural fre-
quency of the bubbles xn. Experimentally, the attenuation
coefficient of each sample is obtained by comparison with a
reference acoustic intensity measurement Iref associated to a
bubble-free medium.14,21,26 More details about the experi-
mental characterization and attenuation measurements are
given in Sec. III.
III. EXPERIMENTS
A. Monodisperse microbubble generation
The production of monodisperse contrast microbubbles
can be achieved by using microfluidics techniques. In our case,
we have manufactured two different polydimethylsiloxane-
(PDMS-) based microdevices via soft-lithography methods.
The microchannel geometry of each microdevice was designed
to operate them under two different microbubble generation
regimes, flow-focusing27–29 and co-flow.30 Once all the micro-
channels were fabricated in the PDMS device, it was then oxy-
gen plasma treated and bonded to a glass slide. An inverted
microscope was used to monitor microbubble production
(Fig. 1).
For microbubble production via flow-focusing, microflui-
dic devices were fabricated based upon the design published
by Hettiarachchi et al.19 To generate contrast microbubbles, a
lipid solution and octafluoropropane were pushed through their
respective microchannels. The lipid solution consisted of a
mixture of dipalmitoylphosphatidylcholine, dipalmitoylphos-
phatidic acid, and N–(carbonyl-methoxypolyethyleneglycol
2000)–1,2-dipalmitoyl-sn-glycero-3-phosphoethanolamine in a
81:8:10 molar ratio. The flow rate (�0:3 ml/h) of the lipid
solution was controlled with a syringe pump (KDS100,
Fisher Scientific), while the octafluoropropane gas flow was
controlled with a pressure regulator (�1:4 bar). By maintain-
ing the liquid flow rate and gas pressure constant, we pro-
duced microbubble suspensions with a very narrow size
distribution.21 For microbubble production via co-flow,
microfluidic devices were fabricated based upon the design
published by Castro-Hernandez et al.30 In this case, we used
the same lipid solution and air instead of octafluoropropane.
The gas and liquid flow rates were controlled via two inde-
pendent pressure regulators, so that the resulting operating
pressures were between 2.30 and 2.50 bar. Compared with
the flow-focusing devices, co-flow devices required a higher
gas pressure for microbubble production but had a signifi-
cantly higher rate of production (see Fig. 2). In order to col-
lect the resulting coated microbubbles for size measurements
and acoustic characterization, a pipette tip was inserted into
the outlet hole of the microdevice. The resulting suspension
was then removed from the tip successively by a syringe with
a hypodermic needle. An inverted Nikon microscope was
used both to monitor the microbubble production and capture
micrographs of the suspension. The size distribution of each
batch was finally measured using a Coulter counter (Z2,
Beckman Coulter, Inc.) or laser diffraction (LA-950, Horiba
Scientific). In order to compare the experimental acoustic
FIG. 1. (Color online) Microfluidic ex-
perimental setup for generating mono-
disperse lipid-coated microbubbles.
J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014 Parrales et al.: Viscoelastic properties of coated bubbles 1079
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characterization with the theoretical prediction for the
attenuation in a contrast microbubble suspension, the meas-
ured size distribution was fitted by a Weibull density function
(Fig. 3),
f Roð Þ ¼lK�Ro
lRo= �Ro
� �K�1exp � lRo= �Ro
� �Kh i
; (10)
where �Ro is the microbubble mean radius and l is a shape
parameter defined by l ¼ ½ðK � 1Þ=K�1=K.
We controlled the mean size of contrast microbubbles
precisely by adjusting the gas pressure and liquid flow rate.
Moreover, the periodic and stable generation of lipid-coated
microbubbles with these two microfluidic devices avoids the
non-uniform distribution of lipids covering the bubbles, as
reported in previous studies using agents generated via agita-
tion/sonication.31,32
B. Acoustic characterization
The experimental setup for measuring frequency-
dependent attenuation for suspended monodisperse lipid-
coated microbubbles is shown in Fig. 4. Monodispersions
were dispensed into a sample holder made of polymethylme-
thacrylate plastic. The sample holder had acoustic windows
consisting of a 4 mm depth chamber, which were covered by
two Mylar (polyethylene terephthalate plastic) sheets of
12 lm thickness each. The sample holder was submerged in a
tank of deionized water and positioned just in front of a stain-
less steel reflector, which was located at the transducer focal
region. Attenuation measurements were made with a
2.25 MHz transducer (Panametrics, USA) and a 1 MHz trans-
ducer (Met-flow, Switzerland), operating independent of each
other. The 2.25 MHz transducer was excited by a pulser/
receiver (5072PR, Panametrics, USA), emitting an acoustic
pulse that traveled through the sample chamber, reflected off
the steel surface, and returned to the transducer active surface.
The received reflections were amplified by the pulser/receiver
and digitized with a digital oscilloscope (Wavesurfer 64XS,
LeCroy, USA) before being saved on a desktop computer for
frequency analysis using MATLAB software (The MathWorks,
Inc., MA, USA).21,26 The 1 MHz transducer was excited by a
one-cycle square wave (65% duty cycle) generated by an arbi-
trary waveform generator (3390, Keithley Instruments). In
this case, we removed the steel reflector and placed a needle
hydrophone (100-100-1, M€uller) behind the sample holder in
order to receive the transmitted acoustic signal after propagat-
ing through the microbubble suspension. The signal was
amplified (MVA-10, M€uller), digitalized, and saved on the
desktop computer for analysis.
As a preliminary stage, the transducer reference signal
needed to be acquired and saved. The reference acoustic
transmission spectrum Iref , measured using a free-bubble
sample, is shown in the Fig. 5 for the 1 MHz transducer.
FIG. 2. Generation of contrast micro-
bubbles. (a) Flow-focusing microde-
vice (�103 lbubbles/s). (b) Co-flow
microdevice (�105 lbubbles/s). (c)
Micrograph of a monodisperse sample
with a mean bubble size of approxi-
mately 12 lm.
FIG. 3. (Color online) Microbubble size distribution measured with a
Coulter counter, fitted by a Weibull distribution with K ¼ 6:8 and �Ro
’ 2:85 lm.
FIG. 4. (Color online) Acoustic experimental setup for measuring the acous-
tic attenuation of contrast microbubble suspensions.
1080 J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014 Parrales et al.: Viscoelastic properties of coated bubbles
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Effectively, we observe that the maximum intensity gain
was reached at the transducer center frequency. When a con-
trast microbubble sample was used instead, the measured
transmission Isample changed due to the frequency dependent
properties of bubble scattering and absorption. As we can
check in Fig. 5, a remarkable narrow transmission band-gap
appears in consequence. Thus, by using dI ¼ �aIðzÞdz, we
finally write the attenuation coefficient of each sample as
a ¼ 1
lln
Iref
Isample
� �; (11)
where l is the total path traveled by the acoustic wave
through the suspension.
IV. RESULTS AND DISCUSSION
Using the flow-focusing microfluidic device, we have
generated different batches of narrowly distributed lipid-
coated microbubbles, with a mean diameter of 5:960:2 lm,
as shown in Fig. 3. In Fig. 6, we show the attenuation
coefficient measurements for the monodisperse suspension
while excited by the 2.25 MHz transducer. The attenuation
spectrum had a maximum peak for a frequency value corre-
sponding to the average single bubble natural frequency,
Eq. (5). The measured spectrum was remarkably narrow
around the resonance peak, due to the quasi-monodisperse
size distribution. In contrast, when a polydisperse sample
(generated by agitation techniques and using the same gas
and lipid solution) was used instead, as shown in Fig. 7, the
measured attenuation had a much broader spectrum. These
results were in good agreement with previously published
measurements.18,21
As observed, the frequency-dependent attenuation meas-
ured for monodispersions was proportional to the microbubble
concentration n. This scaling is only valid for low concentra-
tion suspensions where multiple scattering between the bub-
bles can be neglected.14 In the concentration range used for
our study, we used the single scattering theoretical approach
(Sec. II) to get the total acoustic energy removed from the
propagating wave by the microbubbles: the attenuation coeffi-
cient within the sample is proportional to the sum of the
extinction cross section for all the bubbles. As we have
shown, the extinction depends on the microbubble radius, ex-
citation frequency, and, finally, on the viscoelastic properties
of the lipid shell, i.e., reðRo; xo; v; jsÞ. It is very difficult to
measure the shell elasticity and viscosity directly; alterna-
tively, these properties are estimated by fitting the attenuation
measurements using a single scattering approach.6–8,13,33
Knowing the size distribution of our monodisperse samples,
we successfully fitted the attenuation theoretical curves to the
measured attenuation spectra by setting the shell elastic mod-
ulus to v ¼ 0:28 N/m and the shell surface viscosity to js
¼ 3� 10�8 kg/s, as shown in Fig. 6. These values were in a
very good agreement with those obtained by previous experi-
mental studies for lipid-coated microbubbles,15,16,34,35 thus
showing the consistency and reliability of the method that we
propose. As expected, for the polydisperse samples, the diffi-
culty in identifying both the main resonance frequency peak
FIG. 5. Acoustic transmission power spectrum for the reference signal, and
the measured one for the microbubble monodisperse sample.
FIG. 6. (Color online) Attenuation coefficient in a monodisperse sample at
different concentrations n (�103 microbubbles/ml). �: n ¼ 1:4, �: n ¼ 3:5,
�: n ¼ 5:3, and þ: n ¼ 6:5. Solid lines represent the theoretical fitted
curves. The resulting phospholipid shell parameters were: elastic modulus
v ¼ 0:28 N/m and surface viscosity js ¼ 3� 10�8 kg/s. The inset corre-
sponds to a micrograph of the monodisperse coated microbubble
suspension.
FIG. 7. Attenuation coefficient in a polydisperse sample at different concen-
trations n (�105 microbubbles/ml). �: n ¼ 1:2, �: n ¼ 2:2, �: n ¼ 3:1, and
þ: n ¼ 4:5. The inset corresponds to a micrograph of the polydisperse
coated microbubble suspension.
J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014 Parrales et al.: Viscoelastic properties of coated bubbles 1081
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and the bandwidth associated with damping mechanisms
(Fig. 7) lead to a very high uncertainty range for the shell
parameters.21 In contrast, great accuracy can be achieved
when dealing with monodisperse samples.
It has been hypothesized recently that the viscoelasticity
of the lipid shell may not be linear, which may be reflected
by a dependence of the shell viscoelastic properties on the
microbubble size.36 To test this hypothesis with our experi-
mental method, we measured the acoustic attenuation of
monodisperse samples with different mean radius, generated
via the co-flow configuration.30 For that purpose, we excited
the samples with the 1 MHz transducer and acquired the
transmitted signal directly using the needle hydrophone. The
fitted results for the different suspensions are shown in
Fig. 8. As observed, the main resonance peak identified in
the attenuation spectrum was inversely related with micro-
bubble size. The viscoelastic properties estimated for each
sample as a function of microbubble radius are reported in
Table I. We observe in Fig. 9 that the elastic modulus and
the surface viscosity increase linearly as the equilibrium size
of the coated microbubble becomes bigger. This result is
consistent with the experimental trend found in previous
studies via optical characterization.15,16,36 These measure-
ments also confirmed that the lipid-shell behaves
non-linearly except when the bubbles oscillate with small
amplitudes. Effectively, for �Ro ¼ 4 lm, an oscillation ampli-
tude of jXj � 1% only implies Dv=vo � 0:2% and Djs=jso
� 0:1%. Our observations emphasize the need to develop
new constitutive relations to take into account this non-
linear behavior. Moreover, the non-linear correction for the
shell viscoelastic model explains theoretically the compres-sion-only behavior observed when lipid-coated microbub-
bles oscillate with large amplitudes.11,36
FIG. 8. (Color online) Attenuation coefficient for different monodisperse samples with increasing values of the mean microbubble radius. (a) �Ro ¼ 3:7 lm,
(b) �Ro ¼ 4:8 lm, (c) �Ro ¼ 5:5 lm, (d) �Ro ¼ 6:3 lm.
TABLE I. Results obtained for the acoustic characterization and lipid-shell
properties estimations using monodisperse contrast agents.
�Ro (lm) fn (MHz) v (N/m) js (�10�8 kg/s)
2.9 1.37 0.28 3.0
3.7 1.14 0.35 3.6
4.8 0.84 0.50 4.2
5.5 0.78 0.76 5.4
6.3 0.70 0.85 6.0
1082 J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014 Parrales et al.: Viscoelastic properties of coated bubbles
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V. CONCLUSIONS
In this study, we have measured the attenuation in
monodisperse microbubble suspensions over a frequency
range that includes the resonance frequency of the bubbles.
The results reported in this experimental work show the
great advantage of using monodisperse lipid-coated micro-
bubbles rather than polydisperse ones for the acoustic char-
acterization of ultrasound contrast agents. We show that the
acoustic attenuation for lipid-coated monodisperse samples
is characterized by a very narrowband spectrum around the
natural frequency of the bubbles. Furthermore, the attenua-
tion measurements of such a suspension provide an accurate
estimation of the lipid-shell viscoelastic parameters by fitting
the theoretical model to the experimental curves.
First methods for obtaining the shell properties were based
on acoustic attenuation measurements of polydisperse sam-
ples,7,13,14 which are characterized by a very simple instrumen-
tation. More recently, the coating parameters have usually
been estimated from optical characterization of single bub-
bles,15,16,35,36 which avoid the uncertainty related to polydis-
persity. Here, we take advantage of previous methods by
combining the simplicity of acoustic attenuation measurements
and the accuracy of working with monodisperse suspensions.
The values obtained for the elasticity and the viscosity
of the shell were in good agreement with the reported values
in the literature.15,16,36 This confirms the reliability of the
present experimental methodology, which is able to get as
much accuracy as the optical characterization techniques.
The dependence that we observe for the viscoelastic proper-
ties with the equilibrium radius of the bubbles emphasize
that the coating shell behaves non-linearly for moderated
oscillations. This result may lead to the development of new
constitutive laws for lipid-membrane rheology.
ACKNOWLEDGMENTS
M.A.P. acknowledges the NanoMedAl group at Boston
University for the 4 months hosting, and Francisco del
Campo and Lucia Martin-Banderas (Dpto. Farmacia y
Tecnologia Farmaceutica) at University of Seville for the
experimental support. Also, he wishes to thanks D. Lohse
from the Physics of Fluids group at University of Twente for
the training on ultrasound contrast agent experimental
techniques. This work was funded by the Ministry of
Economy of Spain through the project No. DPI2011-28356-
C03-01 and funded by the National Science Foundation
(CBET 1134420).
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FIG. 9. (Color online) Viscoelastic parameters experimental estimation in
function of the mean equilibrium microbubble radius from different mono-
disperse samples. (a) Elastic modulus linear regression: v ¼ 1:8� 105 �Ro
� 0:28 N/m. (b) Surface viscosity linear regression: js ¼ 9� 10�3 �Ro
þ 2:7� 10�9 kg/s.
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