Resonance frequency of microbubbles: Effect of viscosity

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Resonance frequency of microbubbles: Effect of viscosityDamir B. Khismatullina)

Department of Biomedical Engineering, Duke University, Durham, North Carolina 27708

(Dated: April 19, 2004)

The transmitted frequency at which a gas bubble of millimeter or submillimeter size oscillates reso-nantly in a low-viscosity liquid is approximately equal to the undamped natural frequency (referred toas the Minnaert frequency if surface tension effects are disregarded). Based on a theoretical analysisof bubble oscillation, this paper shows that such an approximation cannot be validated for microbub-bles used in contrast-enhanced ultrasound imaging. The contrast-agent microbubbles represent eitherencapsulated bubbles of size less than10 µm or free (nonencapsulated) bubbles of submicron size.The resonance frequency of the microbubbles deviates significantly from the undamped natural fre-quency over the whole range of microbubble sizes due to the increased viscous damping coefficient.The difference between these two frequencies is shown to have a tremendous impact on the resonantbackscatter by the microbubbles. In particular, the first and second harmonics of the backscatteredsignal from the microbubbles are characterized by their own resonance frequencies, equal to neitherthe microbubble resonance frequency nor the undamped natural frequency.

PACS numbers: 43.35.Ei, 43.80.Qf, 43.35.Bf

I. INTRODUCTION

Due to its safety and low cost, ultrasound scanning, orechosonography, is actively used in obstetrics and gynecol-ogy. However it loses other imaging modalities in regard tothe image quality. An excellent way to increase the qualityof sonograms is the use of micron-size bubbles as ultrasoundcontrast agents,1–3 i.e., as scatterers of ultrasound waves in-side the human body. When injected into the bloodstream,the microbubbles, less than 10µm in size, move with bloodcells toward the measuring site. Once reached the bloodvessel exposed to ultrasound, they break into oscillation andthus scatter the incoming ultrasound waves in all directions.Due to the large acoustic impedance difference at the inter-face between the gas and blood, scattering by microbubblesis of high intensity and results in a strong signal on the re-ceiver, i.e., it enhances the echo from blood. Moreover, be-cause microbubbles are nonlinear scatterers, the backscatteredsignal from them contains higher harmonics of the transmit-ted frequency. This has made possible the contrast harmonicimaging technique,4,5 which, in the Doppler mode, can imageblood flow in smallest blood capillaries, including myocardialperfusion.3

Although the overall effect of contrast agents on thebackscattered signal can be easily detected, it is very difficultto measure oscillations of an individual microbubble in thepopulation situated within the insonated blood vessel. The mi-crobubble behavior can be recorded by optical imaging with aframe rate of several MHz. Such fast-speed imaging of con-trast agent microbubbles has been reported by de Jong andco-workers.6 This group of scientists measured the radius-time curves for microbubbles but did not relate these results

a) Electronic address:[email protected]

to the contribution of each microbubble to the backscatteredsignal. In the absence of experimental data, we can only relyupon theory to realize how a given microbubble interacts withultrasound. Particularly, a theoretical analysis of microbub-ble dynamics can be used (a) to evaluate the transmitted fre-quency at which scattering by a microbubble of a given sizereaches the maximum and (b) to determine the intensity (scat-tering cross-section) and damping rate of the scattered waveproduced by the microbubble. The acoustic response of themicrobubble population located near the measuring site canbe calculated from these functions of bubble size, if the sizedistribution of microbubbles in the population is known or hasbeen measured.

All theoretical works concerning oscillations of ultrasoundcontrast agents are based on the Raleigh-Plesset equation.7

A number of corrections to this model have been made,including the effects of the encapsulating shell,8,9 bloodcompressibility10 and viscoelasticity,11 and interaction be-tween microbubbles.12 Hilgenfeldt et al.13 analyzed theoret-ically the effects of nonlinearity and polychromaticity (pulseddriving) on the resonant backscatter by free microbubbles. Ex-cept for Refs. 11 and 13, these works analyze backscatteringby contrast agents from the viewpoint of large bubbles. If abubble is covered with a high-viscosity shell and its size isless than 10µm, its resonance frequency differs significantlyfrom its natural frequency due to the increased viscous damp-ing coefficient.11 For a free bubble, the shift in resonance fre-quency due to damping effects were discussed by Leighton14

and Brennen.15 As shown here, the difference between thesetwo frequencies has a tremendous impact on backscatteringby ultrasound contrast agents. For example, it is believed1 thatcontrast-enhanced ultrasound provides images with the high-est quality if it operates at the frequency at which the majorpart of microbubble population resonates. To put it differ-ently, if the backscattered signal has a peak at the transmit-

J. Acoust. Soc. Am. D. B. Khismatullin: Resonance frequency of microbubbles 1

ted frequency or at multiple of this frequency, it comes frombubbles which resonate at this frequency. However, we willdemonstrate thatthe frequency at which the signal scatteredby a microbubble has a resonance peak is not equal to theresonance frequency of the microbubble.

The commercially available ultrasound contrast agents canbe divided into two different types: (i) suspensions of encap-sulated microbubbles and (ii) emulsions of liquid dodecafluo-ropentane (DDFP) and superheated drops. Encapsulated mi-crobubbles are coated by a polymer,1 protein,1 or lipid layer16

with the aim to increase stability of microbubbles in bloodflow. The thickness and mechanical properties (viscosity andelasticity) of the layer differ for different contrast agents. Anemulsion of liquid dodecafluoropentane17 boils at 28 C, i.e.,it becomes a suspension of free (non-encapsulated) microbub-bles after intravenous injection. A superheated emulsion18

also transforms itself into a suspension of free microbubblesbut under the action of ultrasound. In this paper, the reso-nant properties of both free and encapsulated microbubblesare studied theoretically.

II. ENCAPSULATED MICROBUBBLES

Free gas microbubbles dissolve away in blood very quickly.This is caused by the over-pressure in the gas inside the bubblecompared to the pressure in blood due to surface tension at thegas-blood interface.19 Encapsulation of the microbubble by a

biocompatible surface-active layer lowers the over-pressure,and, as a result, makes the microbubble more stable againstdissolution. The increased stability is the reason why the greatbulk of available contrast agents represent encapsulated mi-crobubbles. However, the encapsulating shell inevitably con-strains the bubble oscillation due to the shell viscosity, whichis much higher than the viscosity of blood plasma, and thus re-sults in decreasing the amplitude of the backscattered signal.The shell-induced damping of the oscillation becomes morepronounced with increasing the shell thickness.11

The encapsulating shell shows viscoelastic properties.Elasticity of the shell material arises from changes in config-urations of its macromolecules under flow.20 The viscoelas-tic solid model of the shell, in the form of the Kelvin-Voigtconstitutive equation, has been studied in Refs. 9 and 11. Ac-cording to this model, the shell elasticity plays a positive rolefor scattering. An increase in the shell elasticity leads to in-creasing the magnitude of the resonance peaks in the scatter-ing cross-section curves.11 In this section, we analyze the ef-fect of viscosity on the resonance frequency of and resonantbackscatter from microbubbles encapsulated by a viscoelasticsolid shell.

Let us consider a spherical gas bubble covered by a vis-coelastic layer of finite thickness (Fig. 1). The bubble residesin an unbounded Newtonian liquid (blood plasma) and oscil-lates under the action of ultrasound. The radial oscillation ofan encapsulated bubble is described by the equation11:

ρs0

(1+

∆ρa

R− F

Cl

da

dt

)ad2a

dt2+

[32

+∆ρa

R

(2− a3

2R3

)− 1

2Cl

d

dt(aF )

](da

dt

)2

= pa−pI +ρl0

ρs0Cl

d

dt

[a(pa − pI)1+∆ρa/R

], (1a)

pa = pg0

(a

a0

)−3κ

− 2(σ1

a+

σ2

R

)+ 3

[∫ R

a

τ(s)rr

rdr +

∫ ∞

R

τ(l)rr

rdr

]. (1b)

Heret is time,a the inner radius of the bubble,R the outerradius of the bubble (d = R − a is the shell thickness),ρs0

the shell density,ρl0 the initial density of the liquid (bloodplasma),pg0 = p0 the initial gas pressure,pI the incidentpressure in the liquid, andCl the speed of sound in the liq-uid. Surface tensions at the inner (gas-shell) and outer (shell-liquid) interfaces are denoted byσ1 andσ2. The parameters∆ρ andF are defined as

∆ρ =ρl0 − ρs0

ρs0, F =

ρl0

ρs0

1 + ∆ρ(a/R)4

1 + ∆ρa/R. (2)

A polytropic exponentκ is different from γg (ratio ofconstant-pressure to constant-volume specific heats for thegas) because of heat transfer through the bubble walls. It takesthe value between1 (isothermal behavior) andγg (adiabaticbehavior). However, we disregard the energy dissipation due

to thermal effects because the thermal damping coefficient foran encapsulated microbubble is three orders of magnitude lessthan the viscous damping coefficient if the ultrasound scanneroperates at frequenciesf = 1 - 10 MHz.11 In the case of freemicrobubbles, thermal damping is two orders of magnitudeless than viscous damping.11

The integrals in (1b) represent the contributions of the shelland liquid viscoelasticities to the bubble oscillation. The ra-dial component of the shear stress tensor (radial stress) is de-noted byτrr. In the shell domain,τrr = τ

(s)rr . In the liquid

domain,τrr = τ(l)rr . The liquid elasticity weakly affects the

bubble oscillation as compared to the shell elasticity.11 There-fore, we assume that the liquid is Newtonian, i.e.,τ

(l)rr depends

linearly on the radial component of the rate-of-strain tensor:

τ (l)rr = 2µl

.γrr= 2µl

∂vr

∂r. (3)

2 J. Acoust. Soc. Am. D. B. Khismatullin: Resonance frequency of microbubbles

outerinterface

Shell

d a

R

Gas

Liquid

innerinterface

FIG. 1. Schematic sketch of an encapsulated microbubble of outerradiusR. The microbubble resides in an unbounded Newtonian liq-uid. It is covered with a shell (viscoelastic layer of finite thicknessd),i.e., it is characterized by two interfaces: inner (gas-shell) and outer(shell-liquid) interfaces.

(µl is the liquid viscosity). Near the outer interface of theoscillating bubble, the radial velocity of the liquid is

vr =a2

r2

da

dt=

R2

r2

dR

dt. (4)

Providedτ(l)rr

∣∣∣r→∞

= 0, substitution of Eqs. (3) and (4) into

the second integral of Eq. (1b) gives

3∫ ∞

R

τ(l)rr

rdr = −4µla

2

R3

da

dt= −4µl

R

dR

dt. (5)

We restrict our attention to the case of small-amplitude os-cillation of the microbubble in the sinusoidal pressure field

pI = p0 − PA sin(Ωt) = p0 +PA

2[i exp(iΩt) + c.c.] . (6)

Here p0 is the undisturbed liquid pressure in the region farfrom the microbubble (blood pressure),PA the acoustic pres-sure amplitude,Ω the angular transmitted frequency [thetransmitted frequencyf = Ω/(2π)]. The symbolc.c. de-notes complex conjugate. The acoustic pressure amplitude isfar less thanp0, i.e., there exists a small parameterεp suchthat PA = εpp0P andP is a real number of the order of 1.Then, the solution to Eq. (1) is expanded in powers ofεp as11

a = a0(1 + x), (7a)

R = R0

[1+

a30

R30

x +a30

R30

(1− a3

0

R30

)x2+O(x3)

], (7b)

x = x(t; εp) = εpx1(t) + ε2px2(t) + O(ε3

p) (7c)

(a0 andR0 are the equilibrium values of the inner and outerradii of the microbubble). The linear and second-order non-linear acoustic responses of the microbubble (the first and sec-ond harmonics of microbubble oscillation) are specified by thefunctionsX1(t) = εpa0x1(t) andX2(t) = ε2

pa0x2(t). Theymay be expressed as

X1(t) =A1r(Ω)

2PA exp(iΩt), (8a)

X2(t) = A0r(Ω)P 2A +

A2r(Ω)2

P 2A exp(2iΩt), (8b)

whereA0r(Ω), A1r(Ω), andA2r(Ω) are the complex-valuedfunctions of the transmitted frequency. The magnitudesof these functions represent the zeroth-, first-, and second-harmonic responses of the microbubble to the ultrasound field.

To calculate the contribution of the microbubble to thebackscattered signal, the scattering cross-sections of the mi-crobubble at the transmitted frequency and at twice the trans-mitted frequency are considered. The scattering cross-sectionat the transmitted frequency, or the first-harmonic scatteringcross-section, is denoted byσs1. It is related to the ratio ofthe total acoustic powerW1 scattered by the microbubble atthe first harmonic to the intensityI0 of the incident acousticfield9:

σs1 =W1

I0, W1 =

4πr2|Ps1|22ρlCl

, I0 =P 2

A

2ρlCl. (9)

The symbolσs2 denotes the scattering cross-section at twicethe transmitted frequency, or the second-harmonic scatteringcross-section:

σs2 =W2

I0, W2 =

4πr2|Ps2|22ρlCl

. (10)

(W2 is the total acoustic power scattered by the microbubbleat the second harmonic). Herer is the radial coordinate,Ps1

andPs2 are the first- and second-harmonic amplitudes of thepressure wave scattered by the microbubble. Based on the re-lationship between the scattered pressure field and the bubbleradius11,21–23

ps(t, r) =ρl0

3r

d2

dξ2a3(ξ), ξ = t− r/Cl, (11)

the expressions forPs1 andPs2 can be found [see Eqs. (86)and (87) in Ref. 11]. Then, the first- and second-harmonicscattering cross-sections depend onA1r(Ω) and A2r(Ω) asfollows:

σs1 = 4πρ2l0a

40Ω

4|A1r(Ω)|2, (12a)

σs2 = 64πρ2l0a

40P

2AΩ4

∣∣∣∣A2r(Ω) +A2

1r(Ω)2a0

∣∣∣∣2

. (12b)

It is important to say that the expression for the second-harmonic scattering cross-section given by Church [Eq. (26b)in Ref. 9] is in error because it has been derived without con-sidering (a) the nonlinear relationship between the scatteredpressure field and bubble oscillation and (b) the dependence


of the scattered pressure on the second time derivative of thebubble radius. Due to the same reasons, it is also inappropriateto use the well-known formula for the scattered pressure24

ps = iΩρl0

(a20

da

dt

)exp[−i(Ω/Cl)r]

r(13)

for evaluating nonlinear scattering by microbubbles.If the shell behaves as a viscoelastic solid, its behavior can

be described by the Kelvin-Voigt model. Because we ignorethe shape deformation of the microbubble, this model is re-duced to the following relation11:

τ (s)rr = −4a2

r3

[Gs(a− ae) + µs

da

dt

](14)

whereGs andµs are the elasticity modulus and viscosity ofthe shell,ae is the unstrained inner radius of the bubble, whichis different from the equilibrium radiusa0 due to radial pre-stress in the shell.11 Under the assumption that the pre-stressis counterbalanced by the over-pressure, the unstrained radiusdepends on the shell elasticity and surface tensions9:

ae = a0(1 + Z), Z =1

2Gs

(σ1

a0+

σ2

R0

)R3

0

R30 − a3

0

. (15)

In view of Eq. (15), substitution of Eq. (14) into the first inte-gral of Eq. (1b) gives

3∫ R

a

τ(s)rr

rdr = −4(R3

0 − a30)

R3

[Gs

(1− a0

a

)+

µs

a

da

dt

]

+a0R

30

aR3

(2σ1

a0+

2σ2

R0

), (16)

BecauseR3 = a3 + R30 − a3

0 due to the shell incompress-ibility, equations (1), (5) and (16) can be combined into asingle ODE in the inner radiusa (or in the outer radiusR).The resulting differential equation represents the mathemati-cal model for radial oscillation of an encapsulated microbub-ble in a compressible Newtonian liquid provided the encap-sulating shell behaves as a viscoelastic solid. By applyingthe perturbation technique [see Eqs (6)-(8)] to this model andusing Eqs (12) , we obtain the following expressions for thefirst- and second-harmonic scattering cross-sections of the mi-crobubble:

σs1 =4π(1 + ∆ρa0/R0)−2ρ2

l0a20Ω

4

ρ2s0

[Ω2

0 − Ω2 + Sac(Ω)]2 + 4Ω2β2(Ω) , (17a)

σs2 =16π(1 + ∆ρa0/R0)−4ρ2

l0P2AΩ4Γ(Ω)

ρ4s0a

20

[Ω2

0−Ω2+Sac(Ω)]2+4Ω2β2(Ω)2 . (17b)

HereΩ0 = 2πf0 andf0 is the undamped natural frequency,i.e., the natural frequency of microbubble oscillation upon ig-noring viscous damping and liquid compressibility:

f0 =1

2πa0 [ρs0(1+∆ρa0/R0)]1/2

3κp0− 2σ1

a0− 2σ2a

30

R40

+4Gs

(1− a3

0

R30

)[1+

(1+

3a30

R30

)Z

]1/2

. (18)

The functionSac(Ω) is the contribution of acoustic radiationto the bubble stiffness:

Sac(Ω)=T 2

acΩ4

1 + T 2acΩ2

, Tac =ρl0a0

ρs0Cl

(1+

∆ρa0

R0

)−1

. (19)

If thermal damping is ignored, the total linear damping coef-ficientβ(Ω) = βvis + βac(Ω), where

βvis =2

ρs0a20

(1+

∆ρa0

R0

)−1[µs

(1− a3

0

R30

)+µl

a30

R30

], (20)

βac(Ω) =TacΩ2

2(1 + T 2acΩ2)

(21)

are the viscous damping and acoustic radiation damping coef-ficients. The functionΓ(Ω) can be written as

Γ(Ω) =

[Ω2

n − αΩ2 + Sn(Ω)]2 + 16Ω2β2

n(Ω)

[Ω20 − 4Ω2 + Sac(2Ω)]2 + 16Ω2β2(2Ω)

. (22)

Here

α =3[1 + ∆ρ(a0/R0)4]2(1 + ∆ρa0/R0)

, (23)

Sn(Ω) = Sac(2Ω)− 4α

3T 2

acΩ4(4 + T 2

acΩ2)

(1+4T 2acΩ2)(1+T 2

acΩ2), (24)

Ωn =1

a0 [ρs0(1+∆ρa0/R0)]1/2

9κ(κ + 1)

2p0

−4σ1

a0− 4σ2a

60

R70

+ 4Gs

(1− a3

0

R30

)

×[2+

3a30

R30

+(

2+3a3

0

R30

+9a6

0

R60

)Z

]1/2

, (25)

βn(Ω) =3

ρs0a20

(1+

∆ρa0

R0

)−1[µs

(1− a6

0

R60

)+µl

a60

R60

]

+βac(2Ω)− α

2TacΩ2(1− T 2

acΩ2)

(1+4T 2acΩ2)(1+T 2

acΩ2), (26)

F andTac are given in Eqs (2) and (19), respectively.By definition,14 the linear resonance frequency of bubble

oscillation fres is the frequency at which the bubble first-harmonic response (linear amplitude-frequency response) hasa local maximum. If the bubble is covered with a shell, itsfirst-harmonic response is of the form:

|A1r(Ω)|= ρ−1s0 a−1

0 (1 + ∆ρa0/R0)−1

[Ω2

0−Ω2+Sac(Ω)]2+4Ω2β2(Ω)1/2

, (27)

Even if the surrounding liquid is assumed to be incompress-ible, the total damping coefficient is sufficient large to influ-ence the resonance frequency of microbubbles encapsulatedby the high-viscosity shell. Damping has a negligible effectonfres if

β(Ω) ¿ Ω0/2 (28)


This condition is satisfied if the Reynolds numbers for theshell Res = a0ρs0U/µs and for the liquid Rel = a0ρl0U/µl

are much more than unity (U =√

p0/ρl0 is the characteristicvelocity). It is clear that the difference between the resonanceand undamped natural frequencies increases with decreasingthe bubble size and increasing viscosity. For microbubblesof radius1 µm suspended in water and encapsulated by theshell, which density is equal to the water density,Res < 1if µs > 0.01 Pa·s = 10 cP. Meanwhile, the shell viscosity ofcontrast microbubbles is much higher than10 cP.8,25 There-fore, using the term ”resonance frequency” as a synonym of”natural frequency” is validated only for large bubbles.

Upon neglecting the liquid compressibility, the simple ex-pression for the resonance frequency is obtained from themaximization of|A1r(Ω)|:

fres = fres0 =

√f20 −

β2vis

2π2. (29)

As follows from Eq. (27), microbubbles resonate in a com-pressible viscous liquid at the frequency

fres =1

2πTac

−1

+[1 +

4T 2ac(2π2f2

0 − β2vis)

1+ 4Tacβvis(1+Tacβvis)

]1/21/2

. (30)

Note that the acoustic contribution to the resonance frequencyof free bubbles was discussed by Prosperetti.26

Figure 2 compares the resonance frequency with the un-damped natural frequency for air microbubbles encapsulatedby the shell of viscosityµs = 0.45 Pa·s and elasticityGs =12 MPa. The shell thicknessd is 5% of the initial outer ra-dius R0. As estimated by Hoffet al.,25 these values of pa-rameters are typical of ”polymeric microbubbles” preparedby Nycomed.27 We assume that the microbubbles reside inblood plasma (viscosityµl = 0.0012 Pa·s, densityρl0 = 1027kg·m−3, and speed of soundCl = 1543 m·s−1).28 The shelldensity is supposed to be slightly higher than the plasma den-sity: ρs0 = 1100 kg·m−3. The initial pressure in the liquidp0 = 0.1 MPa. The surface tensionsσ1 = 0.04 N·m−1 andσ2 = 0.005 N·m−1 (see Ref. 9). In the case of encapsulatedmicrobubbles, the best matching between the numerical solu-tion, which accounts for heat conduction through the bubblewall, and the analytical formula is observed atκ = 1.1.11

This value of the polytropic exponent is used in all calcula-tions here.

As illustrated in Fig. 2, there is a notable difference be-tween the resonance and undamped natural frequencies overthe whole range of microbubble sizes. In the case of Nycomedsuspension, this difference varies from -0.39 MHz (15.9%) atR0 = 5 µm to -3.30 MHz (141.8%) atR0 = 2.5 µm. In addi-tion, encapsulated microbubbles are characterized by a criticalsize below which they never resonate. The critical size is inthe size range of contrast agents. The critical radius of Ny-comed microbubbles is 2.24µm. From both Eqs. (29) and(30) it follows that microbubbles cease to resonate if

βvis ≥ π√

2f0. (31)

Bubble radius, ( m)

Fre

quen

cy(M

Hz)

1 4 7 100

1

2

3

4

5

6

fres

fres0

f0

Nycomed microbubbles

µR0

2.24

FIG. 2. Resonance frequency (solid) of an air microbubble encap-sulated by a polymeric shell25,27 as a function of the outer radiusof the microbubble, compared with the undamped natural frequency(dotted). The microbubble resides in human blood plasma. Thedashed line is the resonance frequency under the assumption thatblood plasma is incompressible [Eq. (29)]. The solid and dotted linesare calculated from Eqs. (30) and (18), respectively. The encapsu-lating shell is characterized by viscosity of 0.45 Pa·s and elasticityof 12 MPa. Its thickness is 5% of the bubble outer radius. Otherparameters:µl = 0.0012 Pa·s, Cl = 1543 m·s−1, ρl0 = 1027kg·m−3, ρs0 = 1100 kg·m−3, p0 = 0.1 MPa,σ1 = 0.04 N·m−1,σ2 = 0.005 N·m−1, andκ = 1.1.

The more the shell and liquid viscosities, the more the criticalradius (and hence the difference between the resonance andundamped natural frequencies). Although the critical radiusdecreases with increasing the shell elasticity, the shell viscos-ity is a more important parameter: a ten-fold decrease in shellviscosity has approximately the same effect on the critical ra-dius of Nycomed microbubbles as a 100-fold increase in shellelasticity.

Figure 2 also shows that ignoring the liquid compressibil-ity leads to overestimation of the resonance frequency: themicrobubble situated in a less compressible liquid resonatesat a higher frequency. This happens due to acoustic radia-tion damping. The difference betweenfres0 andfres reaches0.165 MHz (≈ 5.6%) for Nycomed bubbles of radius 3.1µm.Another result of Fig. 2 is that there exists an upper limit to theresonance frequency of microbubbles: the Nycomed suspen-sion does not resonate if the transmitted frequency exceeds 3MHz. This result is, however, not applicable to the backscat-tered signal produced by the suspension, as shown later.

Even in the case of Albunexr, which is a suspension ofprotein-encapsulated microbubbles, there is a significant de-viation of the resonance frequency from the undamped natu-ral frequency for all possible sizes of microbubbles.11 As es-


Bubble radius, ( m)

Res

onan

cefr

eque

ncy

(MH

z)

0 2 4 6 8 100

1

2

3

4

5

6

NycomedAlbunex

µR0

2.97 MHz

2.241.46

5.47 MHz

FIG. 3. Resonance frequencies of Nycomed and Albunexr mi-crobubbles (solid and dashed) as functions of the outer radius [basedon Eq. (30)]. The Albunexr microbubble is encapsulated by a pro-tein shell of viscosityµs = 1.77 Pa·s, elasticityGs = 88.8 MPa,and constant thicknessd = 15 nm (see Ref.9). Both microbubblesreside in human blood plasma. The parameters of the Nycomed mi-crobubble and blood plasma are as in Fig. 2.

timated by Church,9 the protein (albumin) shell of these mi-crobubbles is thin (d = 15 nm), very viscous (µs = 1.77 Pa·s)and elastic (Gs = 88.8 MPa). Figure 3 shows that the criti-cal radius of Albunexr bubblesRc = 1.46 µm. This valueis less than the critical radius of Nycomed bubbles mainlydue to a difference in shell thickness between these two con-trast agents. The upper limit to the resonance frequency ishigher for Albunexr (fmax

res = 5.47 MHz) than for Nycomed(fmax

res = 2.97 MHz). This can be explained only by a higherelasticity of the albumin shell. (A decrease in shell thicknessleads to a decrease in the critical radius but has no effect onthe upper limit to the resonance frequency.)11

The widely used assumption that the resonances of thebackscattered signal (resonance peaks in the output-level ver-sus frequency curve) are produced by microbubbles resonat-ing at the transmitted frequency may lead to misinterpreta-tion of experimental data. The first-harmonic resonance ofthe signal scattered by an encapsulated microbubble occurs atthe frequencyf1, which is different from the resonance fre-quency of the microbubblefres. If the surrounding liquid isconsidered to be incompressible, the maximization of the first-harmonic scattering cross-sectionσs1 [see Eq. (17a)] gives thefollowing formula forf1:

f1 =f20

fres0=

f20√

f20 − β2

vis/(2π2). (32)

In the case of large bubbles, i.e., when the difference between

the resonance and undamped natural frequencies of bubbleoscillation is negligible small, it is correctly reasoned thatf1 = fres = f0. In such a situation, resonating bubblesare responsible for the resonance peaks in the output-levelversus frequency curve. In the case of small bubbles, theviscous damping coefficient is large and, therefore, nonres-onating bubbles make the major contribution to the resonantbackscatter.

Considering the liquid compressibility, the first-harmonicresonance frequency of the backscattered signalf1 =

√X.

The quantityX, as is evident from the maximization ofσs1,is one of the positive real roots of the cubic equation

X3 + c2X2 + c1X + c0 = 0 (33)

with

c0 =f40

2π2T 2ac(1 + 2Tacβvis)2

, (34a)

c1 =8π2T 2

acf40 + β2

vis/(2π2)− f20

2π2T 2ac(1 + 2Tacβvis)2

, (34b)

c2 =2(4π2T 2

acf40 + β2

vis/π2 − 2f20 )

(1 + 2Tacβvis)2. (34c)

Figure 4(a) plots the resonance and undamped natural fre-quencies of the Nycomed microbubble (dash-dot and dashedlines) situated in blood plasma as well as the first-harmonicresonance frequency of the signal scattered by the microbub-ble (solid line) as functions of the outer radiusR0. The first-harmonic resonance frequency is calculated from the solutionof Eq. (33) (see p. 17 in Ref. 29). Like the undamped naturalfrequencyf0, the first-harmonic resonance frequency of thebackscattered signalf1 decreases monotonically with increas-ing the bubble radius. However,f0 tends to infinity when theinner radiusa0 approaches zero, whilef1 reaches the high-est value (several tens of MHz) ata0 > 0. If we continueto use the assumptionf1 = fres = f0 for bubbles of arbi-trary small size, we will be led to the conclusion that the bub-ble resonance and the resonance peak in the first-harmonicscattering cross-section always exist, no matter how bubblesare small. We will, then, argue that very small bubbles res-onate at very high frequencies. Figure 4 demonstrates thatthis conclusion is wrong. Bubbles are characterized by thecritical radius. Not only bubbles of subcritical size cease toresonate, but they also never produce the resonant backscatterat the transmitted frequency. (As will be shown later, bub-bles of subcritical size can produce the resonant backscatterat twice the transmitted frequency.) The difference betweenthe first-harmonic resonance frequency of the backscatteredsignal and the microbubble resonance frequency always ex-ceeds the difference between the undamped natural and reso-nance frequencies of the microbubble. For Nycomed bubblesof outer radiusR0 = 5 µm, f1 − fres ≈ 0.73 MHz (30.0%),while f0 − fres ≈ 0.38 MHz (15.9%). If R0 = 2.54 µm,f1− fres ≈ 11.89 MHz (487.2%) andf0− fres ≈ 3.09 MHz(126.8%).

As indicated in Fig. 4(b), a restriction on the first-harmonicresonance frequency is solely the effect of the liquid com-pressibility. If blood plasma is treated as an incompressible


liquid, f1 is calculated from Eq. (32). According to this equa-tion, f1 tends to infinity when the bubble radius approachesthe critical value (Rcrit = 2.24 µm for Nycomed microbub-bles). However, Nycomed microbubbles oscillating in bloodplasma are not able to produce the resonance peak in the first-harmonic scattering cross-section if the transmitted frequencyexceeds 14.33 MHz. The existence of the upper limit tof1

results in another value of the critical bubble radius, specificfor f1. In particular, the resonant backscatter from Nycomedbubbles of radius less thanRcrit1 = 2.54 µm seems not toexist.

In the most experimental works on contrast agents, theproperties of microbubbles, such as the shell viscosity andelasticity,25,30 are evaluated through frequency analysis ofthe backscattered echo, i.e., from the output-level versus fre-quency curves. It is generally believed1,31 that the highestpeaks in these curves are associated with the resonance fre-quency of microbubbles. In reality, the highest peaks are as-sociated with the first-harmonic resonance frequency of thebackscattered signalf1, which is not coincident with the mi-crobubble resonance frequencyfres. Moreover, it is believedthat the frequency at which the highest peak is observed de-pends on the microbubble parameters according to Eq. (18).In reality, equation (18) describes the undamped natural fre-quencyf0, which is different from the first-harmonic reso-nance frequency of the backscattered signal. The undampednatural frequency can be used instead off1 and fres if themicrobubble radius is more than5 µm, i.e., beyond the sizerange of contrast agents. Figure 5 shows how the differencebetweenf1 (measured in experiments) andf0 (used insteadof f1) depends on the microbubble radius for two types ofcontrast agents: Nycomed25,27and Albunexr.9 Although thisdifference is relatively small for bubbles of radiusR0 = 5µm, in particular,f1 − f0 = 0.34 MHz for Nycomed bubblesand0.025 MHz for Albunexr bubbles, it becomes very largewhen the bubble radius approaches the critical valueRcrit1.Note that in the case of Albunexr bubbles the critical ra-dius for f1 is again more than the critical radius forfres:Rcrit1 = 1.61 µm > Rcrit = 1.46 µm. The maximal dif-ference betweenf1 andf0 is 8.8 MHz for Nycomed bubblesand14.1 MHz for Albunexr bubbles. Therefore, the use ofEq. (18) for the interpretation of experimental data on the res-onant backscatter from microbubbles cannot be tolerated.

A monodisperse suspension of encapsulated microbubblescannot produce the resonant backscatter at both the transmit-ted frequency and twice the transmitted frequency. It is notquite correct to argue that if bubbles of one radius resonate,all the harmonics, subharmonics, and ultraharmonics of thebackscattered signal they produce will be resonant. This state-ment is validated only for large bubbles. In particular, the sec-ond harmonic of the backscattered signal from encapsulatedmicrobubbles of radiusR0 has a resonance peak at the fre-quencyf2 (second-harmonic resonance frequency), which isequal to neither the first-harmonic resonance frequencyf1 northe undamped natural frequencyf0. This is apparent from themaximization of the second-harmonic scattering cross-sectionσs2. It should be noted thatf2 is defined as thetransmitted fre-quencyat which the second harmonic of the backscattered sig-

Bubble radius, ( m)

Fre

quen

cy(M

Hz)

0 1 2 3 4 50

4

8

12

16

f1

fres

f0


µR0

(a)

2.24

Bubble radius, ( m)

Firs

t-ha

rmon

icre

sona

nce

frequ

ency

(MH

z)

2 2.25 2.5 2.75 30

10

20

30

40

µR0

(b)

2.54

incompressible liquid

compressibleliquid

2.24

14.33 MHz

FIG. 4. (a) Resonance and undamped natural frequencies of Ny-comed microbubbles (dash-dot and dashed) and first-harmonic reso-nance frequency of the backscattered signal from the microbubbles(solid line) as functions of the bubble outer radius. The liquid (bloodplasma) compressibility is taken into account. Open circle denotesthe highest frequency at which the resonant peak in the first-harmonicscattering cross-section can be observed. (b) The first-harmonic res-onance frequency of the backscattered signal produced by Nycomedmicrobubbles surrounded by compressible and incompressible liq-uids (thick and thin solid lines) as a function of the bubble outerradius.


Bubble radius, ( m)

f 1-

f 0(M

Hz)

1 2 3 4 50

4

8

12

16

NycomedAlbunex

µR0

2.54

8.8

1.61

14.1

FIG. 5. The difference between the first-harmonic resonance fre-quencyf1 and the undamped natural frequencyf0 for Nycomed andAlbunexr microbubbles (solid and dashed) as functions of the bub-ble outer radius. Nycomed microbubbles of radiusR0 < 2.54 µmand Albunexr microbubbles of radiusR0 < 1.61 µm do not pro-duce resonant scattering at the transmitted frequency. The parame-ters are listed in Figs. 2 and 3.

nal is resonant, i.e., in the output-level versus frequency curve,the second-harmonic resonance peak will be at2f2. Theσs2 isa very complicated function of the angular frequencyΩ [seeEq. (17b)]. It is impossible to obtain the analytical formulafor f2 even in the incompressible case. (IfTac = 0, the max-imization ofσs2 results in a fifth-order algebraic equation inΩ2.) To calculate numerically the second-harmonic resonancefrequency as a function of bubble radius, we use MapleTM

(Maplesoft, a division of Waterloo Maple Inc.). The results ofthe numerical calculation are presented in Figs. 6 and 7.

Figure 6 compares the second-harmonic resonance fre-quencyf2 (solid line), the first-harmonic resonance frequencyf1 (dash-dot line), and the undamped natural frequencyf0

(dashed line) for Nycomed microbubbles of radiusR0 ≤ 5µm. While f1 is always higher thanf0, f2 is always lessthan f0. Hence, the difference betweenf1 and f2 is morethan betweenf1 andf0. It increases with decreasing the bub-ble radius. For instance, when the outer radius of Nycomedmicrobubbles is changed from5 µm to 2.54 µm, f2 − f1 in-creases from0.9 MHz to 10.6 MHz. Also, there is no criticalradius for the second-harmonic resonance. Even submicronbubbles may have a resonance peak at twice the transmittedfrequency. Hence, ultrasound-based methods for detection ofvery small bubbles inside materials should exploit the secondharmonic, not the first harmonic.

As illustrated in Fig. 7, overestimation of the optimal trans-

Bubble radius, ( m)

Fre

quen

cy(M

Hz)

0 1 2 3 4 50

10

20

30

40

f2

f1

f0

µR0


FIG. 6. First- and second-harmonic resonance frequencies of thebackscattered signal (dash-dot and solid) and undamped natural fre-quency of Nycomed microbubbles (dashed) as functions of the outerradius. The liquid compressibility is taken into account. The second-harmonic resonance frequency is defined as the transmitted fre-quency at which the second harmonic of the backscattered signal isresonant. Open circle denotes the highest frequency at which theresonant peak in the first-harmonic scattering cross-section can beobserved.PA = 30 kPa. The remaining parameters are listed inFig. 2.

mitted frequency for second-harmonic imaging by usingf0

reaches several tens of MHz for microbubbles of radius lessthan1 µm. In this figure, the difference betweenf2 andf0

is plotted as a function of the outer radius for Nycomed andAlbunexr microbubbles (solid and dashed lines). This differ-ence remains relatively small (but cannot be ignored) for mi-crobubbles of radius between 3µm and 5µm. In this range,f2 − f0 varies from−0.55 to−1.43 MHz for Nycomed bub-bles and from−0.063 to−0.485 for Albunexr bubbles. Thesecond-harmonic resonance frequency deviates significantlyfrom the undamped natural frequency if the bubble radius isless than3 µm (or its size is less than 6µm).

The microbubble population is highly polydisperse. Thepolydisperse suspension is less sensitive to the transmitted fre-quency than the monodisperse suspension. If the ultrasoundtransducer operates at the frequency of several MHz (from2 to14 MHz), there always exist bubbles of one size, sayR1, thatproduce the resonant backscatter at the transmitted frequencyand bubbles of another size, sayR2, that produce the resonantbackscatter at twice the transmitted frequency. The backscat-tered signal from this suspension, therefore, has peaks at thetransmitted frequency and multiple of this frequency. It is im-portant to say thatR1 is not equal toR2 in the case of encap-


Bubble radius, ( m)

f 2-

f 0(M

Hz)

0 1 2 3 4 5-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

NycomedAlbunex

µR0

FIG. 7. The difference between the second-harmonic resonance fre-quencyf2 and the undamped natural frequencyf0 for Nycomed andAlbunexrmicrobubbles (solid and dashed) as functions of the outerradius.PA = 30 kPa. The remaining parameters are listed in Figs. 2and 3.

sulated microbubbles.

To increase the quality of contrast-enhanced ultrasoundimaging, one needs (a) to create a suspension of microbub-bles with a mean radius equal to eitherR1 (for fundamen-tal mode imaging) orR2 (for second-harmonic mode imag-ing) or (b) to operate the transducer at the frequencyf1 inthe fundamental (first-harmonic) mode or atf2 in the second-harmonic mode, wheref1 andf2 are taken at the mean ra-dius of microbubble population. The former way should beused if the transducer is characterized by a narrow range ofoperating frequencies. Unfortunately, neither Albunexr norNycomed (and other polymeric contrast agents) are good can-didates for the fundamental mode imaging. The mean radiusof Albunexr bubbles is about1.7 µm,9 which is very closeto the critical radius forf1. As shown in Fig. 5,Rcrit1 = 1.61µm for Albunexr. (The critical radius will be even higher ifthermal damping is taken into account.) Polymeric microbub-bles are worse than Albunexr because of the thick shell. Themean radius of Nycomed microbubbles is between2.0 and2.5 µm,27 less than the critical radius (Rcrit1 = 2.54 µm forNycomed). These contrast agents can, however, be used forsecond-harmonic imaging. Figure 8(a) shows that if the trans-ducer operates at frequencyf = 5.0 MHz (this frequency wasused in experiments with Nycomed bubbles by Hoffet al.),25

a Nycomed suspension should be of mean radiusR1 = 4.36µm in order to provide optimal imaging in the fundamental(first-harmonic) mode. Such a high value ofR1 signifies thatthe major contribution to the first harmonic of the backscat-tered signal is from 4-5µm bubbles, i.e., from the largest mi-

crobubbles of the suspension. Smaller microbubbles plays aninsignificant role for first-harmonic imaging. Optimal second-harmonic imaging is possible if the mean radius of the Ny-comed suspension isR2 = 2.77 µm. This value is sufficientlyclose to the actual mean radius of the Nycomed suspension(between2.0 and2.5 µm), i.e., Nycomed microbubbles willscatter ultrasound more or less efficiently at twice the trans-mitted frequency. Nevertheless, one can increase the qualityof second-harmonic imaging from Nycomed microbubbles bya slight decrease in the transmitted frequency. As illustratedin Fig. 8(b), the optimal transmitted frequency for second-harmonic imaging isf2 = 4.61 MHz, if the mean radius ofNycomed microbubbles is2.0 µm. Figure 8(b) also showsthat there is no optimal frequency for first-harmonic imag-ing. This is because2.0 µm is less than the critical radiusRcrit1 = 2.54 µm. It is necessary to say that the present anal-ysis is expected to hold quantitatively for experiments on scat-tering by microbubbles under low-intensity continuous ultra-sound. Finite-amplitude effects and polychromaticity of theultrasound signal can lead to additional shifts in resonancefrequencies.13

Because a lipid monolayer is very thin as compared to apolymeric or protein layer, the viscoelastic solid shell modeldoes not work well for lipid-coated microbubbles. In thiscase, a Newtonian interface model suggested by Chatterjeeand Sarkar32 is more appropriate. This model introduces theinterfacial dilatational viscosityκs instead of the shell viscos-ity µs. (In this model, an encapsulated shell has zero thick-ness.) Therefore lipid-coated microbubbles can be consideredas free microbubbles with the decreased surface tension coef-ficient and the liquid viscosity replaced by

µlb = µl +κs

a0(35)

[see Eqs. (16) and (17) in Ref. 32]. Hence the viscousdamping coefficient for lipid-coated microbubbles is calcu-lated from Eq. (37), given below, whereµlb is used insteadof µl. The viscosity-induced shift in resonance frequency willclearly exist for these microbubbles because the second termin the right-hand side of Eq. (35) becomes very large whenbubble sizes are in micron or submicron range.

III. FREE MICROBUBBLES

Another type of the commercially available ultrasound con-trast agents represents emulsions that transform itself into sus-pensions of free gas microbubbles inside the human body.1 Inparticular, Echogen (Sonus Pharmaceuticals, Bothell, Wash-ington) is an emulsion of liquid dodecafluoropentane (DDFP)which boils at about28C.1 In the liquid phase, such an emul-sion is stable against dissolution. In the gas phase, it is moreechogenic than a suspension of encapsulated microbubbles.

To evaluate backscattering by Echogen microbubbles at thetransmitted frequency and twice the transmitted frequency, wecan still use Eqs. (17) in whichR0 = a0, σ1 = σ2 = σ,Z = 0, pg0 = p0 + 2σ/a0 (without the stabilizing layer, there

is the over-pressure in the gas inside a microbubble),τ(s)rr = 0,


Bubble radius, R0 ( m)

Sca

tterin

gcr

oss

-sec

tion

/(4

)

0 1 2 3 4 510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Second harmonicFirst harmonic

πa02

Nycomed

f = 5.0 MHz R1

=4.

36m

µµ

R2

=2.

77m

µ

(a)

Transmiited frequency (MHz)

Sca

tterin

gcr

oss

-sec

tion

/(4

)

0 2 4 6 8 1010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Second harmonicFirst harmonic

πa02

R0 = 2.0 m

f2 = 4.61 MHz

(b)

µ

FIG. 8. First- and second-harmonic scattering cross-sections normal-ized by4πa2

0 (dashed and solid) as functions of (a) bubble radiusR0

at the transmitted frequencyf = 5.0 MHz and (b) transmitted fre-quency atR0 = 2.0 µm. The parameters correspond to Nycomedmicrobubbles situated in blood plasma.

ρs0 = ρl0, and∆ρ = 0. In this case, the undamped naturalfrequencyf0 and the characteristic timeTac are given by

f0 =1

2πa0

√3κp0

ρl0+ (3κ−1)

2σ

ρl0a0, Tac =

a0

Cl. (36)

The viscous damping coefficient

βvis =2µl

ρl0a20

. (37)

The expressions forSac(Ω) and βac(Ω) remain unchanged[see Eqs. (19) and (21)].

The viscous damping coefficient of free microbubbles issmaller than that of encapsulated microbubbles. For exam-ple, a free bubble of radius1 µm situated in blood plasma(ρl0 = 1027 kg·m−3, µl = 0.0012 Pa·s) is characterized byβvis ≈ 2.337×106 s−1, while the viscous damping coefficientof an Albunexr microbubble of the same (outer) radius is ap-proximately1.596× 108 s−1. Therefore, the deviation of theresonance frequency from the undamped natural frequency isexpected to be less for Echogen than for Albunexr. How-ever, it is not negligible. Like encapsulated bubbles, free bub-bles are characterized by a critical radius for bubble resonanceRcrit. From (31) it follows that

Rcrit =(

3κ−13κ

)σ

p0

[−1 +

√1 +

24κµ2l p0

(3κ−1)2σ2ρl0

](38)

for free bubbles. Ifσ = 0, the critical radius varies directlywith the liquid viscosityµl:

Rcrit =2√

2 µl√3κp0ρl0

. (39)

The liquid viscosity has a more effect on the critical radiusthan the surface tension. According to Eq. (38), if the sur-face tension increases from0 to 0.04 N·m−1, the critical ra-dius of bubbles oscillating almost isothermally (κ = 1.1) inblood plasma decreases from184 to 56 nm. An increase inthe liquid viscosity from0.0012 to 0.004 Pa·s (high-shear-rate viscosity of blood) atσ = 0.04 N·m−1 leads to increas-ing Rcrit from 56 to 396 nm. Because the mean radius ofEchogen microbubbles is about75 nm,17 a slight increase inviscosity and/or a decrease in surface tension, induced by sol-uble polymers and another surface-active molecules in blood,will make this suspension nonresonant. Experiments17 showno first-harmonic resonance of the backscattered signal fromthese microbubbles. Note that the presence of a thin surfactantlayer on the bubble surface, which is realized in lipid-shelledcontrast agents and leads to a decrease in surface tension, willfurther magnify the critical radius and hence the differencebetweenf0 andfres.

As illustrated in Fig. 9, the undamped natural frequencyf0

(dotted line) should not be used as the microbubble resonancefrequencyfres (solid line) for free bubbles in blood plasma ifthe bubble radius is less than 0.5µm. Also, in this size range,both the first- and second-harmonic resonance frequencies ofthe backscattered signalf1 andf2 (dash-dot and dashed lines)deviate strongly from the undamped natural frequency. In par-ticular, f1 − f0 andf0 − f2 are equal to0.28 MHz and0.61MHz at a0 = 0.5 µm but become32.3 MHz and24.8 MHzat a0 = 0.1 µm. The critical radius forf1 at which the firstresonance frequency reaches the upper limit (357 MHz) is66nm.

IV. CONCLUSION

The transmitted frequency at which the resonant excitationof microbubble oscillation occurs is taken to be equal to the


Bubble radius ( m)

Fre

quen

cy(M

Hz)

0 0.1 0.2 0.3 0.4 0.50

25

50

75

100

fres

f2

f1

f0

µ

Free microbubbles

56 nm

FIG. 9. Resonance and undamped natural frequencies of free mi-crobubbles in blood plasma (solid and dotted) and first- and second-harmonic resonance frequencies of the backscattered signal fromthe microbubbles (dash-dot and dashed) as functions of the bub-ble radius. The blood plasma compressibility is taken into account(Cl = 1543 m·s−1 andρl0 = 1027 kg·m−3). The pressure waveamplitudePA = 30 kPa. Other parameters:σ = 0.04 N·m−1,µl = 0.0012 Pa·s, p0 = 0.1 MPa, andκ = 1.1. Free bubbles donot resonate if their size is less thanRcrit = 56 nm. The upperlimit to the first-harmonic resonance frequencyf1 is 357 MHz (notshown here). The critical radius for the resonant backscatter at thefirst harmonic isRcrit1 = 66 nm.

undamped natural frequency in most experimental and theo-retical works on scattering by contrast agents. We have shownthat this assumption does not work for both free and encap-sulated microbubbles used in contrast-enhanced ultrasoundimaging. The resonance frequency of microbubbles deviatesfrom the undamped natural frequency due to the increasedviscous damping coefficient. The difference between thesetwo frequencies rises as the bubble size decreases becausethe viscous damping coefficient is inversely proportional tothe square of the bubble radius. The linear resonance of mi-crobubble oscillation is characterized by a critical radius be-low which it cannot be observed. The critical radius of encap-sulated microbubbles is typically between1 and3 µm and in-creases with increasing the shell thickness. The critical radiusof free bubbles depends significantly on the liquid viscosityand surface tension. In blood plasma, free bubbles cease toresonate if their radius is below56 nm.

The deviation of the resonance frequency from the un-damped natural frequency results in the resonant backscat-ter by nonresonating microbubbles. The resonances of thebackscattered signal (resonance peaks in the output-level ver-sus frequency curves) are characterized by the frequencies dif-ferent from both the microbubble resonance frequency and the

undamped natural frequency. Every harmonic of the backscat-tered signal has its own resonance frequency. In particu-lar, a monodisperse suspension of microbubbles cannot pro-duce the resonant backscatter at both the trasmitted frequencyand twice the transmitted frequency. (Because a microbub-ble suspension is polydisperse, there exist bubbles of one sizethat produce the resonant backscatter at the transmitted fre-quency and bubbles of another size that produce the resonantbackscatter at twice the transmitted frequency.) The differ-ence between the first- and second-harmonic resonance fre-quencies of the backscattered signalf1 andf2 reaches sev-eral MHz in the case of encapsulated microbubbles of radiusless than3 µm. To increase the quality of contrast-enhancedultrasound imaging, the transducer should operate at the fre-quencyf1 in the fundamental mode and at the frequencyf2 inthe second-harmonic mode, wheref1 andf2 are taken at themean radius of the ultrasound contrast agent.

Also, there exists a critical radius for the first-harmonic res-onance of the backscattered signal, larger than the critical ra-dius for the bubble resonance. The microbubbles of this radiusproduce the resonant backscatter at the transmitted frequencyequal to the upper limit to the first-harmonic resonance fre-quency. There is no critical radius for the second harmonicresonance of the backscattered signal.

The resonance peak in the first harmonic of the backscat-tered signal is not observed in many experiments with ultra-sound contrast agents. It is worth to mention Refs. 33 [Figs.4, 5, and 7], 34 [Figs. 4(a) and 9(b)], and 25 [Fig. 5, seealso Fig. 9]. The indirect evidence for the disappearance ofthe bubble resonance (and hence of the first-harmonic reso-nance of the backscattered signal) is presented in experimentsby Chenet al..35 As noted by Hoffet al.,25 ”the resonance fre-quency increases and the resonance peak broadens and almostdisappears due to the viscoelastic polymeric shell”.

Our next concerns will incorporating bubble polydispersityand pulsed driving conditions (with different pulse shapes andlengths) into the theoretical model and performing a directcomparison between theoretical predictions and experimentaldata on the resonance frequencies of the backscattered signalfrom ultrasound contrast agents.

ACKNOWLEDGMENTS

The author wishes to thank Ronald A. Roy and Ali Nadimfor fruitful discussions. This work was partially supported bythe North Atlantic Treaty Organization under Grant No. DGE-0000779. Any opinions, findings, conclusions or recommen-dations expressed in this publication are those of the authorand do not necessarily reflect the view of NATO.

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Resonance frequency of microbubbles: Effect of viscosity

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