Ultrasound attenuation by encapsulated microbubbles: time and pressure effects

Ultrasound attenuation by encapsulated microbubbles: time and pressure effects

Ultrasound in Med. & Biol., Vol. 30, No. 6, pp. 793– 802, 2004 Copyright © 2004 World Federation for Ultrasound in Medicine & Biology Printed in the U...

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Ultrasound in Med. & Biol., Vol. 30, No. 6, pp. 793– 802, 2004 Copyright © 2004 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved 0301-5629/04/$–see front matter

doi:10.1016/j.ultrasmedbio.2004.03.010

● Original Contribution ULTRASOUND ATTENUATION BY ENCAPSULATED MICROBUBBLES: TIME AND PRESSURE EFFECTS BORIS KRASOVITSKI, EITAN KIMMEL, MICHAL SAPUNAR and DAN ADAM The Department of Biomedical Engineering Technion, Haifa, Israel (Received 6 October 2003; revised 8 March 2004; in final form 18 March 2004)

Abstract— Ultrasound (US) contrast agents (UCA) consist of artificial encapsulated microbubbles filled with low-diffusivity gas. This study evaluated, both experimentally and theoretically, the behavior of a cloud of encapsulated microbubbles while the surrounding pressure was modified within the physiological range. The theoretical analysis included calculation of US attenuation caused by a bubble cloud. The radius and gas content of each bubble were determined from a solution of a diffusion problem. Shell permeability and rigidity were taken into account. Both experiments and theory demonstrated that, for fixed ambient pressures, higher pressures result in increased rate of attenuation decay. Pulsatile ambient pressure induces pulsations of attenuation of the same frequency. In general, theoretical predictions are in good agreement with experimental data. (E-mail: [email protected]) © 2004 World Federation for Ultrasound in Medicine & Biology. Key Words: Ultrasound attenuation, Ultrasound contrast agents, Microbubbles, Diffusion, Shell mechanics, Shell permeability.

INTRODUCTION

the surrounding liquid (Podell et al. 1999; Kabalnov et al. 1998b). The introduction of UCAs, with slowed dissolution and deterioration allowed transport of microbubbles by the vascular system into each perfused organ. This allowed better imaging of the various organs, but also opened up other objectives (e.g., the measurement of pressure). The use of UCAs for noninvasively measuring blood pressure in specified locations within the cardiovascular system is a promising clinical application (Shapiro et al. 1990; Brayman et al. 1996; Bouakaz et al, 1999). The first attempts to use US and air bubbles to measure blood pressure (Fairbank and Scully 1977; Hok 1981; Shankar et al. 1986) proved the concept, but it could not be implemented because the bubbles dissolved very fast. Some efforts were made to use the UCA only as a vehicle to transport the gas bubbles to the location of interest and, there, to burst the UCA and study the behavior of the free gas microbubbles, either their disappearance time (Bouakaz et al. 1999) or response properties (Frinking et al. 2001). Yet, free bubbles have a wide dispersion of sizes, which does not allow measuring their resonance frequency, as suggested by Fairbank and Scully (1977), and allows a very short measurement time due to their fast disappearance. Commercial UCAs consist of encapsulated gas microbubbles with diameters less than 10 ␮m to allow for passage through the lung capillaries. The large difference

Encapsulated microbubbles, known also as ultrasound (US) contrast agents (UCA), are used now in medicine for imaging of blood vessels. UCA are small gas bubbles with diameters of few ␮m, encapsulated (e.g., by a thin albumin coating) and filled with gas of high molecular weight and low diffusivity. The UCA are usually injected IV and traverse the lung capillaries before reaching the heart and the internal organs, where imaging is usually performed. The microbubbles undergo a rough environment of cyclic pressure changes, shear stresses due to the flow conditions and diffusive forces, which may destroy some microbubbles, damage others and affect their size (Van Liew and Raychaudhuri 1997; Bouakaz et al. 1998). When the UCA are exposed to a US field, even when low-energy nondestructive levels are used, the backscattering properties (as well as attenuation) deteriorate with time. The inability to obtain stationary US measures hampers efforts to obtain quantitative data when using UCA for clinical purposes. The deterioration is associated usually with shrinkage of the gas bubbles, caused mostly by diffusion of gas from the bubble into

Address correspondence to: Dr. Boris Krasovitski, The Department of Biomedical Engineering, Technion, Haifa 32000, Israel. E-mail: [email protected] 793

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of acoustic impedance between the gas bubble and that of the surrounding medium produces large backscattered US signals and attenuation of the pass-through energy. The compressibility of the microbubbles, which is very different from that of blood, allows microbubbles to change considerably in size in response to changes in hydrostatic pressure and, thus, even demonstrate cyclic changes in video density (Shapiro et al. 1990). Variation in the size, in turn, should affect the acoustic characteristics of microbubbles, such as their resonance frequency (Fairbank and Scully 1977), the reflectivity or attenuation of the US waves. The value of the surrounding pressure could be derived from the acoustic characteristics of UCA, even though their compressibility is lower than that of free gas bubbles, because their size is known and nearly uniform and their deterioration in the blood is much slower. Few attempts have been made to study the problem of estimating the dynamic changes of pressure from the response of encapsulated microbubbles to US. Because the frequency of the pressure fluctuations is much smaller than the natural frequencies of the microbubbles, the behavior of the microbubbles can be considered to be “quasisteady” (Ran and Katz 1991). This assumption may be hardly accepted for UCAs containing air. An accurate model of the acoustic response of microbubbles containing inert gas should, therefore, incorporate the US insonation and the diffusion processes, which are affected by cyclic changes in ambient pressure. Previous studies that were concerned with the effects of timevarying pressure on the intensity of the US image or its brightness found only low correlation (Ran and Katz 1991; Brayman et al. 1996; Padial et al. 1995; Greim et al. 2000). Yet these studies dealt with air-filled UCA. When the UCA contain high molecular weight gas, of less solubility and diffusivity, the behavior of the microbubbles is very different when exposed to changes in pressure and to US insonation (Podell et al. 1999). Timedependent processes take place from the moment the UCA is injected into the blood, or even into saline. The study of these processes that are related to the usage of UCA is important for both quantitative imaging (e.g., of perfusion) and blood-pressure estimation. Dissolution of gas bubbles in liquid with steady ambient pressure was first considered by Epstein and Plesset (1950). Plesset and Zwick (1952) extended the solution to a gas bubble in an ultrasonic field. Both studies were concerned with a free gas bubble and used a quasistationary solution for the concentration field. Kabalnov et al. (1998b) and Chen et al. (2002) studied the process of diffusion from an encapsulating single bubble. They also used a quasisteady solution of the diffusion problem and did not take into account the permeability and mechanical properties of the encapsu-

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Fig. 1. Schematic of the diffusion problem. pg ⫽ pressure inside the bubble, R ⫽ external radius of the shell. Curved arrows mean possible directions of the diffusion.

lating shell. As a result, the theoretical analysis of Kabalnov et al. (1998a) predicted lifetime of the gas bubble as much shorter than the observed values. In the present study, the unsteady problem of an encapsulated gas bubble in liquid subjected to varying ambient pressures is solved. The solution provides the time-dependant backscattering and attenuation of a single bubble and a group of bubbles with a given size distribution. It is assumed that the gas concentrations at the bubble boundary may vary with time and both the mechanical and permeability properties of the shell are taken into account. MATERIALS AND METHODS The diffusion problem This part deals with the diffusion process, which starts after the UCA bubbles are submerged into a liquid containing dissolved air (Fig. 1). The UCA bubbles are usually subjected to pulsed US sonication, where the sonication periods are of a very small fraction of the process period and, therefore, the effects of rectified diffusion during sonication may be ignored here. In addition, it is assumed that the mechanical properties of the shell are not influenced by sonication of low-intensity US (i.e., the mechanical index, MI, is smaller than 0.5, Mayer and Grayborn 2001). The bubbles initially contain osmotic gas. The diffusion of both air and the osmotic gas may be described using the Fourier equations: ⭸C m ⫽ D mⵜ 2C m, ⭸t

m ⫽ 1, 2

(1)

where Cm is the mole concentration of the gases in the surrounding liquid and Dm is the diffusivity of the gases

Attenuation of microbubbles ● B. KRASOVITSKI et al.

in the liquid. The assumption of spherical symmetry is based on the fact that the size of the bubbles is much smaller than the typical distance between the bubbles. Therefore, the influence of neighboring bubbles may be ignored. Note that m ⫽ 1 is designated for air and m ⫽ 2, for the osmotic agent. The initial and boundary conditions for eqn (1) are: C m共r,0兲 ⫽ C im; C m共R,t兲 ⫽ C sm;

r ⬎ R; t ⬎ 0;

m ⫽ 1, 2 m ⫽ 1, 2.

⭸C m ⭸r



r⫽R

795

⫽ ␣ m共C m ⫺ C sm兲 r⫽R;

Here,

(2)

Here R is the instant external radius of the encapsulating shell, Cim is the initial homogeneous gas concentration outside the bubble in the surrounding liquid, and Csm is the gas concentration at the external surface of the shell. Following Eller and Flynn (1965), we assume that Csm equals the saturation concentration, which is related to the partial pressure by Henry’s law:

m ⫽ 1, 2. (7)

␣m ⫽

(3)

t ⬎ 0;

␣ pm␳ mk m . D m␮ m␦

(8)

Mass balance determines the mole contents (nm) of each gas inside the bubble: dn m ␳ mQ m ⭸C m ⫽ 4 ␲ R 2D m ⫽ ⫺ dt ␮m ⭸r



;

m ⫽ 1, 2.

r⫽R

(9) The pressure inside the bubble, pg, equals the sum of the partial pressures of the gases in the bubble:

冘p . 2

p me C sm ⫽ ; km

m ⫽ 1, 2.

␦Qm ; 4 ␲ R 2␣ pm

m ⫽ 1, 2.

(5)

Here, pm is the partial pressures of the gases in the bubble; ␦ is the shell thickness; ␣pm is the gas permeability through the shell. The difference pm ⫺ pme is influenced by surface tensions on the interfaces liquidshell and shell-gas and the mechanical response of the shell. The influence in explicit form will be shown below, eqn (12). The volume rate of the gas leaving the bubble, Qm, may be expressed through the concentration gradient at the bubble wall by:

␮ m ⭸C m Q m ⫽ ⫺ 4 ␲ R 2D m ␳ m ⭸r



;

(10)

Pressure-volume relationship for the gases in the bubble may be assumed to be isothermal for bubbles of small diameters (typical of UCA) when the applied frequency is in the MHz range (Prosperetti 1977): pm ⫽

3n mR gT ; 4␲R3

m ⫽ 1, 2

m ⫽ 1, 2.

(11)

Here, Rg is the universal gas constant, T is the absolute temperature, which does not change during the process, and nm is the mole content of the gas. A force equilibrium equation for the bubble shell may be written in the following form: p g ⫽ P ⬁ ⫹ P st ⫹ P s.

(12)

Here, the first term on the right hand side is the ambient pressure (P⬁); the second term P st ⫽

r⫽R

m

m⫽1

Here, pme is the partial pressure of the gas at the external surface of the shell and km is the corresponding Henry’s constant. Using the concept of permeability (Hennessy et al. 1967), the partial pressure inside the bubble and at the external surface of the shell may be related in the following way:

p me ⫽ p m ⫺

pg ⫽

(4)

2共 ␥ 1 ⫹ ␥ 2兲 R

(13)

(6)

where ␮m and ␳m are the molecular weight and density of the gases, respectively. Substituting eqn (6) into eqn (5) converts the boundary condition eqn (3), into a boundary condition of the third kind:

accounts for the surface tension force at the external and internal shell surfaces, where ␥1 is the surface tension at the shell-air boundary and ␥2 is the surface tension at the shell-liquid boundary; the third term (Ps) accounts for the contribution of stress within the shell. Simulations that deal with an encapsulated bubble subjected to ultrasonic

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Fig. 2. Components of the equilibrium eqn (12). Rcr ⫽ radius of the bubble corresponding to loss of stability, R* ⫽ the radius corresponding to zero internal pressure in elastic statement, pg ⫽ pressure inside the bubble, P⬁ ⫽ ambient pressure, Pst ⫽ the pressure accounting for surface tension forces on external and internal shell surfaces, Ps ⫽ the pressure accounting for contribution of stress in the shell, Pcol ⫽ the shell contribution after loss of stability; subscript “e” denotes corresponding parameters in elastic statement.

irradiation commonly express the latter term as (Church 1995; Hoff et al. 2000):

冋 冉





12 R 20 ␦ 0 Re U␮s P s ⫽ P se ⫽ Gs 1 ⫺ ⫹ . R3 R R

(14)

Here, Gs is the shell rigidity modulus, Re is the bubble radius in unstrained equilibrium position, R0 is the initial radius, ␦0 is the initial thickness of the shell, ␮s is the shell viscosity and U is the radial velocity of the shell surface. Note that eqn (14) corresponds to the Kelvin model for viscoelastic material. In this study, only the diffusion processes determine the movement of the bubble surface and, therefore, U is low and the viscous effects may be ignored. It is not possible to introduce an elastic model [e.g., eqn (14)] into the Ps term in eqn (12), because it leads to a nonphysical situation of negative pressure inside the bubble for bubble radius smaller than R* (Fig. 2). The effect of the elastic model is shown in Fig. 2 by the dashed curve pge. This happens because, when R approaches zero, the surface tension term behaves like 1/R, and the more dominant term of the shell response Pse is negative and behaves like ⫺1/R4. Here, Pse denotes the shell response in elastic statement. For example, for a bubble with initial radius 2 ␮m and actual radius 1.66 ␮m, shell thickness 15 nm and the shell physical properties given in the Results section, the elastic shell response Pse ⫽ ⫺0.143 MPa, surface tension pressure Pst ⫽ 0.043 MPa and ambient pressure P⬁ ⫽ 0.1 MPa. Corresponding internal gas pressure pg ⫽ 0 and decreases while the actual bubble radius decreases. Because pg should be positive, the bubble radius cannot

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become less than R*, implying that the bubble will not completely disappear. However, this situation contradicts the observations of complete disappearance after several minutes of UCA bubbles in gas-deficient liquid (Kabalnov et al. 1998a). To overcome this contradiction, a different stress-strain relationship for the shell is suggested here. The shell is assumed to be elastic as long as the radius is greater than some bubble radius Rcr (Rcr ⬍ Re), which corresponds to the radius at which the spherical shell under compression loses stability. Similarly, a punched ball, losing air, at a certain moment loses its spherical form. Until this moment, the ball shell may be considered as an elastic material and, after this moment, its behavior deviates from the elastic model. Following a technique developed in (Vol’mir 1963; Volmir 1957), value of Rcr is the root of the transcendental equation:

冉 冊 R cr R0

3



冉 冊

R e R cr R0 R0

2





␦0 1 ⫹ ␯ ⫽ 0. 3 冑3 R 0 1 ⫺ ␯ 1

(15)

As for the mechanical response of the shell, the following expression is here suggested: P se ⫽





12G sR 20 ␦ 0 Re 1⫺ R3 R

P s ⫽ P col共R, q兲

for R ⬎ R cr;

(16)

for R ⬍ R cr.

The function Pcol (R, q), which represents the shell response after the loss of stability, is discussed in detail in Appendix 1. The parameter q is determined by best fitting to experimental data. The diffusion problem, eqns (1), (2) and (7), is solved numerically using a finite-difference scheme (Samarski 1963). The main difficulty in solving such a problem is the moving boundary of the half-infinite domain. To circumvent this difficulty, a modified transformation is used, first suggested by Plesset and Zwick (1952). The transformation converts the original domain into a fixed-boundaries finite domain. The algorithm for solving the problem consists of calculating, at each time-step, the instantaneous distribution of concentration of both gases and the concentration gradients at the surface of the bubble. Then, these values are introduced into eqn (9) to yield the instantaneous gas mole contents inside the bubble. Finally, by solving eqns (10) and (12), the instantaneous value of the bubble radius is found and the procedure is repeated for the next time step. Scattering and attenuation It is common to describe the acoustic properties of bubbles by two measures (scattering and extinction

Attenuation of microbubbles ● B. KRASOVITSKI et al.

cross-section), where both depend on bubble size, resonant frequency and the damping coefficient. The total scattering cross-section of a single bubble may be expressed as follows (Medwin 1977; Hoff et al. 2000):

␴s ⫽ 4␲R2

⍀4 . 关⍀ 2 ⫺ 1兴 2 ⫹ ⍀ 2␦ 2d

(17)

Here, ⍀ ⫽ f/fr, fr is the resonant frequency of the bubble with an actual radius R, f is the applied frequency and ␦d is the damping coefficient. The expression for the resonance frequency of an encapsulated bubble is based on Rayleigh–Plesset-like equation of capsulated bubble dynamics (Church 1995; Hoff et al. 2000), 3G s␦ 2 f r2 ⫽ f rg ⫹ 2 3 ; ␲ R ␳l

冑冋

(18)



1 1 2共 ␥ 1 ⫹ ␥ 2兲 f rg ⫽ 3␬pg ⫺ . 2␲R ␳l R

G sf ⫽

P sR 4 12 R 20 ␦ 0共R ⫺ R e兲

(19)

which reflects “softening” of the shell when it collapses. The function Ps was defined earlier by eqn (16). Obviously, Gsf ⫽ Gs for the elastic regime, R ⬎ Rcr, eqn (16), and, for R ⬍ Rcr, the value of Gsf decreases with decreasing R and approaches zero at a rate which is a function of R3. The damping constant of the encapsulated bubble,

␦ d ⫽ ␦ r ⫹ ␦ ␯ ⫹ ␦ th

(20)

accounts for the reradiation

␦r ⫽

2 ␲ f 2R , f rC

the viscosity of liquid and the shell:

␦␯ ⫽

(21)

2共 ␮ lR ⫹ 3 ␮ s␦ 0兲 ␲␳ l f rR 3

(22)

and the thermal conductivity (␦th) (Hoff et al. 2000; Church 1995). The latter may be ignored by assuming an isothermal process (see above). The extinction cross-section of a bubble cloud per unit volume (Se) can be described by:

Se ⫽





␴ en共a兲da

(23)

0

where n is the size distribution function of the bubble cloud and ␴e, is the extinction cross-section of a single bubble as given by (Hoff et al. 2000; Church 1995):

␴e ⫽

It includes the resonance frequency of free bubble (frg) and a term that accounts for the contribution of the shell. Here, the polytropic constant of the gas (␬) equals unity, because the process is assumed to be isothermal (see above). Equation (18) was derived assuming an elastic behavior of the shell. For the case presented here, the assumption is valid only for as long as the shell does not lose stability (R ⬎ Rcr). For R ⬍ Rcr, one cannot consider the shell to be elastic anymore. An alternative is suggested here, to replace the rigidity modulus Gs in eqn (18) by some effective rigidity modulus (Gsf):

797

⍀4 2 Rc s␦ d , 2 f r 关⍀ ⫺ 1兴 2 ⫹ ⍀ 2␦ d2

(24)

cs is the sound velocity in the liquid. The attenuation rate of a bubble cloud is:

␣ b ⫽ 4.34 S e

(25)

where ␣b is in dB/m and Se is in m⫺1 (Medwin 1977). The bubble cloud consists of bubbles with a continuous size distribution. For the numerical solution of the problem, the whole range of bubble sizes was divided into 50 groups, represented by their mean initial radii. For each group size, the diffusion problem described above was solved to yield the changes in time, the bubble radius, mole content, resonant frequency and cross-section. The integrated parameters of the bubble cloud were then calculated. The experimental setup The experimental setup included a 13 ⫻ 13 ⫻ 17 cm Perspex pressure container filled with 0.5 L of saline solution, which included two holes for placing the US transducers/receivers, with proper sealing, and an inlet and an outlet allowing injection of UCA microbubble suspensions and for applying and controlling external hydrostatic pressures. The pressure was adjusted by a precision regulator IR1000-01B-R (SMC Pneumatics, Inc.) and monitored by a digital pressure switch ISE4LB01-65 (SMC Pneumatics, Indianapolis, Indiana) with accuracy 1 kPa. Pressure gauge (SMC, ZSE40F) allowed feedback control of the pressure changes, run by a LabView program.

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A RITEC RAM 5000 transmitter/receiver system, which included a high-power gated RF amplifier, an internal pulse generator and a broadband receiver, produced the US pulses. An acoustic single-element focused transmitter/receiver transducer (either V310, with a central frequency f0 ⫽ 5.5 MHz and 111.6% bandwidth, or focused V303, f0 ⫽ 0.95 MHz and 61.54% bandwidth, Panametrics, Waltham, MA), was placed in a hole in the container’s wall (“Transducer”). A hydrophone model PVDF-GL-0400IA (Specialty Engineering Associates, Soquel, CA), supplied with a 17-dB amplifier, was placed in the hole in the container wall precisely opposite the transducer (“Hydrophone”) for the attenuation measurements. The measured signals were connected to a sampling system, which consisted of the CompuScope CS14100 (Gage, Tektronix Technology Company, Lashine, Quebec, Canada), characterized by 14-bit resolution and 50-MHz sampling rate, connected to a Pentium III PC (“Digitizer”). The signals were processed off-line and their power spectra estimated. Total power of the passedthrough signals and the amplitude of echoes at 1/2, ⫻1, ⫻2 the imposed frequency, were measured and presented as a function of time vs. the ambient pressure changes. The UCA OptisonR (Mallinckrodt Medical GmbH, Hennef, Germany) was used in the current experiments, consisting of a sterile suspension of human serum albumin-coated microspheres filled with octafluoropropane (C3F8) with a concentration of 5.0 – 8.0 ⫻ 108 bubbles/ mL. The microbubbles have a mean diameter in the range of 2.0 – 4.5 ␮m, with 93% of the microbubbles being smaller than 10 ␮m. The approximate amount of octafluoropropane gas in each mL of OptisonR is 0.22 mg. OptisonR was injected into saline solution using a 20-gauge syringe, at a maximum injection rate of less than 1.0 mL/s. To limit excessive acoustic attenuation and multiple scattering, a diluted solution of OptisonR in saline with a concentration of 0.1 ␮L/mL was used. Saline solution (0.9% NaCl) was utilized as the medium. A slowly rotating magnetic stirrer kept it in circulation. All experiments were performed in buffer solution at 25 ⫾ 2°C. PO2 was measured to be 7.5 (⫹3/⫺2) mg/L in saline (exposed to air for 24 h) and ⬍ 1 mg/L in 24 h degassed saline. The experimental protocol The “RITEC” transmitter/receiver produced pulses of nine cycles in length, at 4 MHz, 200 kPa, with PRF ⫽ 10 Hz. This frequency corresponds to twice the resonance frequency of OptisonR (Shi and Forsberg 2000), so as to decrease the possibility of microbubble destruction (Chen and Shung 1998). In some experiments, the pulses were of 16 cycles in length, at 2 MHz, 50 kPa, with PRF ⫽ 5 Hz. All acoustic measurements were carried out within 11 min after OptisonR injection. During the 30 s

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after injection, the container was sealed and hydrostatic pressure was established. The acoustic measurements were started 30 s after setting the hydrostatic pressure (or 1 min after UCA injection) and conducted during the following 10 min every 2 s. The series of measurements were repeated 5 times, for every value of ambient pressure. The first series of experiments were performed at hydrostatic pressures of 0, 10, 20 kPa above room pressure. A second series of experiments was performed with cyclic-varying ambient pressures, modulated with amplitudes of 20 kPa at a repetition frequency of 0.05 Hz and modulated by 5 kPa at a repetition frequency of 1 Hz (the latter mimicking left ventricle pressure changes). The US insonation started 60 s after injection of OptisonR, to ensure proper UCA mixing. The measurements were carried out over 6 min. A total of 7 experiments were conducted for each condition of pressure change. RESULTS AND DISCUSSION In this study, the change with time of the acoustical properties of OptisonR were calculated and compared with our experimental data. The simulation parameters The simulation parameters were chosen as follows: the bubble size distribution was based on the measurements of de Jong et al. (1992). The thickness of bubble shell (␦0) is assumed to be of 20 nm, which is within the range given by various sources, from 15 nm (Christiansen et al. 1994) up to 30 –50 nm (Barnhart et al. 1990). The diffusion coefficient of air in water is D1 ⫽ 2 · 10⫺9 m2/s (Lide 1991) and of octafluoropropane is D2 ⫽ 7.45 · 10⫺10 m2/s (Chen et al. 2002). Shell permeability to octafluoropropane (␣p2 ⫽ 2.88 · 10⫺ 21 m2/s/Pa) and to air (␣p1 ⫽ 7.73 · 10⫺21 m2/s/Pa) were chosen to fit the experimental data for the attenuation changes in the air-saturated water, under the assumption that the permeability ratio to the two gases equals to the ratio of the corresponding diffusion coefficients. It is worth noting that the value of ␣p1 is close to the permeability of polymer films to air (e.g., permeability of polyvinydene chloride to air equals 10.8 · 10⫺21 m2/s/Pa) (Hennessy et al. 1967). The surface tension of shell-air (␥1 ⫽ 0.005 N/m) and shell-water (␥2 ⫽ 0.04 N/m) were chosen following Church (1995). The selection of the rigidity modulus of the shell (Gs) is based on the frequency of the peak of the attenuation curve (see eqn (18)). It was found that a value of Gs ⫽ 45 MPa corresponded to the peak frequency of 2 MHz that was observed in our experiments. Changes in bubble size Typical results for the change in time of the dimensionless radii of different bubbles (R/R0) are presented in

Attenuation of microbubbles ● B. KRASOVITSKI et al.

799

Fig. 5. Change of the attenuation coefficients with time for different ambient pressures; experimental results. Fig. 3. Actual dimensionless bubble radius vs. initial bubble radius for different time moments. f ⫽ 4 MHz; P⬁ ⫽ 0.12 MPa. See explanations in the text.

Fig. 3 for ambient pressure 0.12 MPa. For example, the bubble with initial radius of 4 ␮m initially expanded and reached in 10 s a radius of ⬃4.12 ␮m. Then it started shrinking and, in 120 s, it returned to its initial size, in 240 s, its radius equalled 3.8 ␮m, in 360 s, 2.6 ␮m and, in 600 s, 0.12 ␮m. As can be seen, the bubbles that initially were of smaller radius shrank more rapidly than those of larger initial radius. This known effect is explained by the higher surface-to-volume ratio of the smaller bubbles and the resulting stronger diffusion effects. Also one can see, there was an increase of the bubble size for small values of time (R/R0 ⬎ 1), because of inward flux of air. Attenuation by the bubble cloud The change with time of the attenuation coefficient caused by the bubble cloud (Fig. 4), was calculated for an ultrasonic frequency of 4 MHz, ambient pressure 0.1 (atmospheric pressure), 0.11 and 0.12 MPa and air-saturated water. When comparing the simulation results with experimental data (Fig. 5) measured under the same

Fig. 4. Change of the attenuation coefficients with time for different ambient pressures. (––––) Calculation for constant rigidity modulus of the shell; ambient pressure 0.12 MPa (- - - - -) simulation results.

conditions, high qualitative and quantitative agreement may be seen. In both simulations and experiments, atmospheric ambient pressure was associated with gradual decrease of attenuation with time. Increasing the ambient pressure accelerated the rate of decrease of the attenuation. For instance, although, after 12 min at atmospheric pressure, the attenuation decreased slightly by about 15%, at elevated ambient pressure, the attenuation practically disappeared at 12 min. This may be explained by increased diffusion rate, which was caused by the elevated pressure. In addition, elevated pressure was associated with a temporary rise of the attenuation, several min after the onset of the sonication. This effect was observed at elevated pressures in all our experiments. Interestingly, this effect appeared in the simulations only when the concept of variable rigidity modulus (Gs) was used; see eqn (19). For comparison, for the case of fixed Gs, this effect disappeared and the attenuation monotonously decreased with time for both atmospheric and elevated pressure (Fig. 4, dashed line for ambient pressure of 0.12 MPa). Pulsatile ambient pressure In the cardiovascular system, especially in the arteries and in the heart ventricles, UCA are subjected to pulsating pressures with a frequency dictated by the heartbeat. Thereafter, experimental and theoretical investigations were conducted to study the effects of the pulsating ambient pressure on UCA behavior in time. The attenuation coefficients of a bubble cloud were calculated for pressures pulsating between 0 and 20 kPa at a repetition frequency of 0.05 Hz and, for pressures pulsating between 15 and 20 kPa, at a repetition frequency of 1 Hz (Figs. 6a, 7a). The results of the measurements, at these conditions, are depicted in Figs 6b and 7b. Note that, because of technical reasons, the 0.05-Hz repetition frequency experiments and calculations were conducted with a decreased OptisonR concentration. As shown in these graphs, the attenuation pul-

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Fig. 6. (a) Change of the attenuation coefficients with time for pulsatile ambient pressures, pulsation period 20 s; simulation results. (b) Change of the attenuation coefficients with time for pulsatile ambient pressures, pulsation period 20 s; experimental results.

sated with a frequency equal to that of the imposed frequency of the pulsating ambient pressure. According to the simulated results, during the initial period, the pulsations were rather weak and both pulses were in phase (first stage). Later (second stage), the attenuation pulse and the ambient pressure pulse were in antiphase, namely, when ambient pressure rose, the attenuation decreased and vice versa (see Figs. 6a, 7a). This behavior is in accord with the results shown in Fig. 4 for fixed different pressures. As shown in Figs. 6a and 7a for the second stage, the pulsation amplitude and the absolute value of the attenuation gradually decreased with time until they diminished completely after some 10 min. It is worth noting that the peak attenuation values during the second stage (see Fig. 6a), which correspond to the atmospheric pressure, decreased much faster than the attenuation values for fixed atmospheric pressure, shown in Fig 4. This may be explained by facilitated, irreversible loss of the gas in the bubble during the elevated pressure parts of the pulse. The behavior of the simulated attenuation pulse during the second stage (Figs. 6a, 7a)

Fig. 7. (a) Change of the attenuation coefficients with time for pulsatile ambient pressures, pulsation period 1 s. Simulation results. Complete results (top); zoom on 10 s (bottom). (b) Change of the attenuation coefficients with time for pulsatile ambient pressures, pulsation period 1 s. Experimental results; complete results (top); zoom on 10 s. (bottom).

Attenuation of microbubbles ● B. KRASOVITSKI et al.

was similar to the experimental results presented in Figs. 6b and 7b. In all experiments, the attenuation pulse was in antiphase to the ambient pressure pulse and both absolute value and amplitude of attenuation gradually decreased with time. The reason for the discrepancy between simulated results and the experimental results at the first stage is not yet clear. CONCLUSIONS

APPENDIX 1 There is no technique, to the best of our knowledge, for calculating the mechanical response of a shell after loss of stability. To reach this objective, a relationship is here derived between the pressure difference (Pcol) across the shell after collapse [see eqn (16)] and the actual radius of the bubble (R). The relationship is based on the following conditions: 1. Pcol is negative for 0 ⬍ R ⬍ Rcr because the shell response at this stage resists the action of the external pressure P⬁ and the surface tension force Pst which are positive [see eqn (12)]; 2. the function Ps should be smooth and continuous at R ⫽ Rcr; 3. the function pg [see eqn (12)] should be positive and finite for 0 ⬍ R ⬍ Rcr; and 4. Pcol should contain free parameter to allow best fit to the experimental data. After fulfilling the four conditions above, the function Pcol takes the following form:

冘a R ; 2

i

i

determined from the above conditions for the pressure function: P col共R cr, q兲 ⫽ P se共R cr兲;

dP col dR

R cr ⬎ R ⬎ R n

(A1)

i⫽0

P col共R, q兲 ⫽ P in共R n, q兲 ⫹ 2共 ␥ 1 ⫹ ␥ 2兲





1 1 ⫺ ; Rn R

Rn ⬎ R ⬎ 0

(A2)

The role of the intermediate function Pin, eqn (A1), is to provide a smooth transition from the pressure function Pse, eqn (16), and the function in eqn (A2). The R⫺1 dependency of eqn (A2) is the form that assures finite value of the function pg, as R approaches 0. The coefficients ai, i ⫽ 0, 1, 2 in the polynomial, eqn (A1), are



R⫽Rcr



dP se dR



; R⫽Rcr

(A3)



Pin共Rex, q兲 ⫽ ⫺ 2q共␥1 ⫹ ␥2兲

The main features of the attenuation-time curves that were observed in experiments with fixed ambient pressures are the gradual decay of attenuation with time, accelerated decay rate for increasing pressure and temporary rise of attenuation a few min after the beginning of the experiments. Our model predicts all these features. As for the pulsatile ambient pressure, some of the predicted features coincide with the experimental observations and some do not. The reason of this partial discrepancy is a topic of future investigations.

P in共R, q兲 ⬅

801

冊 冏

1 1 dPin ⫺ ; Rcr R dR

⫽ 0.

R⫽Rex

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