The physics of nanoshelled microbubbles

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The physics of nanoshelled microbubbles

Michiel Postema, Nico de Jong, Georg Schmitz

To cite this version:

Michiel Postema, Nico de Jong, Georg Schmitz. The physics of nanoshelled microbubbles.

Biomedi-zinische Technik. Ergänzungsband (Berlin. Zeitschrift), Fachverlag Schiele & Schön GmbH, 2005,

Medical Physics: Proceedings of the jointly held Congresses ICMP 2005 14th International

Confer-ence of Medical Physics of the International Organization for Medical Physics (IOMP), the

Euro-pean Federation of Organizations in Medical Physics (EFOMP) and the German Society of Medical

Physics (DGMP); BMT 2005 39th Annual Congress of the German Society for Biomedical Engineering

(DGBMT) within VDE 14th -17th September 2005, Nuremberg, Germany, 50 (Suppl. 1), pp.748-749.

�hal-03189339�

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The physics of nanoshelled microbubbles

Michiel Postema1_{, Nico de Jong}2_{, Georg Schmitz}1

1_{Institute for Medical Engineering, Ruhr-Universit¨at Bochum, Bochum, Germany} 2_{Department of Biomedical Engineering, Erasmus MC, Rotterdam, The Netherlands}

Abstract

Nanoshelled microbubbles are suitable markers for perfused areas in ultrasonic imaging, and have potential appli-cations in therapy. With radii up to 5 microns, their resonance frequencies are in the lower megahertz range. We explored the physics of nanoshelled microbubbles, with special attention to the influence of the nanoshell on the oscillation offset with respect to the driving phase. Microbubbles above resonance size oscillate π rad out of phase with respect to microbubbles under resonance size. As the damping becomes less, this transition in offset becomes more abrupt. Therefore, the damping due to the friction of the nanoshell can be derived from this abruptness. We support our results with some high-speed optical observations of oscillating microbubbles in an ultrasonic field.

1 Introduction

Ultrasound contrast agents consist of gas bubbles with equilibrium radii R0 up to 5 µm. These microbubbles

are small enough to be transported intravascularly and to pass through capillary vessels. Because their reso-nance frequencies coincide with those applied in ultra-sonic imaging, they are suitable markers for the detec-tion of perfused areas. The only ultrasound contrast agent currently allowed for clinical use in Europe is SonoVueTM_{. It is a second generation ultrasound}

con-trast agent, consisting of SF6 gas microbubbles

encap-sulated by an elastic lipid nanoshell. To enhance detec-tion techniques, predicting the dynamic behavior of ul-trasound insonified nanoshelled microbubbles has been of much clinical interest.

In this paper, we present a brief overview of the physics of ultrasound-insonified oscillating microbub-bles that are encapsulated by a nanoshell. More specif-ically, we investigate the influence of the nanoshell on the oscillation offset with respect to the driving phase. We support our results with some high-speed optical ob-servations of oscillating microbubbles in an ultrasonic field.

2 Theory

Let us consider a microbubble with an equilibrium ra-dius R0and a shell thickness a ¿ R0. In equilibrium,

the gas pressure inside the bubble pg0can be expressed

as:

pg0= p∞0 − pv+ 2σ

R0

. (1)

Here, p∞

0 is the static pressure of the liquid, pv is the

vapor pressure, and σ is the surface tension.

For an encapsulated gas bubble, the oscillating be-havior has been described by a modified RPNNP equa-tion, named after its developers Rayleigh, Plesset, Nolt-ingk, Neppiras, and Poritsky [1]:

ρ R ¨R +3 2ρ ˙R2= pg0 µ R0 R ¶3γ + pv− p∞0 −2σ R − 2Sp µ 1 R0 − 1 R ¶ − δ ω ρ R ˙R − pa(t) , (2) where pa(t) is the acoustic pressure in time, R is the

instantaneous microbubble radius, Spis a shell

elastic-ity parameter, δ is the total damping coefficient, γ is the polytropic exponent, ρ is the liquid density, and ω is the angular driving frequency.

The total damping coefficient is a summation of the damping coefficient due to radiation δr, the damping

co-efficient due to the friction of the shell δs, the damping

coefficient due to heat conduction δt, and the damping

coefficient due to the viscosity of the surrounding liquid

δv[1]: δ = δr+ δs+ δt+ δv, δr=ω R c , δs= Sf mω, δv= 4η ρ ω R2, (3) δt= sinh z+sin z cosh z−cos z−2z z

3(γ−1)+cosh z−cos zsinh z−sin z

,

where c is the speed of sound in the liquid, m is the replacive mass of the microbubble [2], Sf is the shell

friction, η is the viscosity of the liquid, and z = R0/lD,

in which lDis the thermal boundary layer thickness:

lD=

s

Kg

2 ω ρgCp. (4)

Here, Cp is specific heat of the gas, Kg is the thermal

(3)

For a ¿ R0, δ may be assumed constant over an

oscil-lation.

3 Materials and methods

We simulated the oscillating behavior of free microbub-bles with various sizes in a harmonic acoustic field

pa(t) = p+ sin ωt, where p+is the pressure amplitude.

Equation (2) was computed with a MATLAB° (TheR MathWorks, Inc., Natick, MA) program. The following fixed parameters were used: c=1480 m s−1_{, C}

p=1000

J kg−1_K−1_{, K}

g=0.0234 J m−1s−1K−1, p∞0 =1 atm,

pv=2.33 Pa, Sp=0 kg s−2, γ=1.4, η=0.001 Pa s, ρ=998

kg m−3_{, ρ=1 kg m}−3_{, σ=0.072 kg s}−2_{, and ω/2π=0.5}

MHz. For microbubble radii R0 <10 µm, the offsets

φ between paand R were computed by comparing the

maximal microbubble radii R(ωt − φ) = Rmax with

pa(ωt) = p+.

4 Results and Discussion

An example of simulated oscillating behavior of a mi-crobubble is shown in the figure below.

0 0.5 1 1.5 2 2.5 3 3.5 4 99 100 101 R /R 0 [%] _R 0 = 0.5 Rr 0 0.5 1 1.5 2 2.5 3 3.5 4 95 100 105 R /R0 [%] _R 0 = 1 Rr 0 0.5 1 1.5 2 2.5 3 3.5 4 99.8 100.0 100.2 R /R 0 [%] _R 0 = 2 Rr ω t [π rad] 0 0.5 1 1.5 2 2.5 3 3.5 4 -5 0 5 pa [kPa]

Figure 1: Simulations of the relative bubble surface ex-cursion as a function of the driving phase for under-damped (δ = 0.2) microbubbles with equilibrium radii

R0 = 1₂Rr, 1Rr, 2Rr, where Rris the resonant radius,

during insonification at ω/2π=0.5 MHz and p+_{=5 kPa.}

The microbubble with half resonance size oscillates ex-actly φ = π rad out of phase with the driving pressure. At resonance, φ = 3

2π rad, and at double resonance

φ = 2π rad. As the damping becomes less, the

transi-tion in offset becomes more abrupt, as Figure 2 demon-strates. 0 1 2 3 4 5 6 7 8 9 10 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 R0 [µm] φ [ π rad ] S_f = 4.0×10-6 S f = 0.27×10 -6_{kg s}-1 S_f = 0 kg s -1 kg s -1 1.0

Figure 2: φ as a function of R0, for typical values

of the shell friction: Sf=0 kg s−1 (unencapsulated),

Sf=0.27×10−6 kg s−1 (SonoVueTM), and Sf=4×10−6 kg s−1_(Albunex°).R

1

5

2

6

3

7

4

8

Figure 3: High-speed movie of experimental nanoshelled microbubbles (Bracco Research SA, Gen`eve, Switzerland). Frame 1 has been captured prior to ultrasound arrival. The other seven frames cover one full ultrasonic cycle. Each frame corresponds to a 40×40 µm2 _{area. The central bubble oscillates} 1

3π

out of phase with the upper left microbubble. Clearly, the damping due to the shell friction is minimal in this example.

References

[1] N. de Jong, Acoustic properties of ultrasound contrast agents. PhD thesis: Erasmus University Rotterdam, 1993. [2] H. Medwin, “Counting bubbles acoustically: a review,”

Ultrasonics, vol. 15, pp. 7–13, 1977. Correspondence: michiel.postema@rub.de