Laser-driven resonance of dye-doped oil-coated microbubbles: experimental study

Photoacoustic imaging oﬀers several attractive features as a biomedical imaging modality in-cluding excellent spatial resolution and functional information such as tissue oxygenation. A key limitation, however, is the contrast to noise ratio that can be obtained from tissue depths greater than 1-2 mm. Microbubbles coated with an optically absorbing shell have been proposed as a possible contrast agent for photoacoustic imaging, oﬀering greater signal ampliﬁcation and improved biocompatibility compared to metallic nanoparticles. We have previously developed a theoretical description of the dynamics of a coated microbubble subject to laser irradiation. The aim of this study was to test the predictions of the model. We fabricated two diﬀerent types oil-coated microbubbles and exposed them to both pulsed and continuous wave (CW) laser irradiation. Their response was characterized using ultra high-speed imaging. Although there was considerable vari-ability across the population, we found good agreement between the experimental results and theoretical predictions in terms of the frequency and amplitude of microbubble oscillation following pulsed excitation. Under CW irradiation, highly nonlinear behavior was observed which may be of considerable interest for developing new photoacoustic imaging techniques with greatly improved contrast enhancement.


I. INTRODUCTION
The details of the theoretical derivation are given in Appendix A. Here, we give the most 49 important results, e.g. the eigenfrequencies and the scaling behavior. Everywhere, the subscripts w and o refer to the water and the oil, respectively. The microbubble oscillations may be shown to obey a Rayleigh-Plesset type equation of the form: where P g is determined by the heat transfer into and out of the system, and assumes the 52 following expression for the pulsed laser case: where F a is the thermal energy per unit volume deposited by the laser pulse (in J/m 3 ). The more complex interactions relevant for the CW laser irradiation case are described by: R e,eq 1 − λ o λ w + R 2 e,eq 0.5 + where B a (t) is the heat density deposited by the laser (W/m 3 ). Inserting Eq. 3 into Eq. 1 54 results in: The expression of the coefficients follows directly from Eq. 1 and Eq. 3. Their detailed 56 expression is given in Appendix A18 and A19. The key features of the microbubble behavior 57 can be further described by linearizing Eq. 4 for small bubble oscillation amplitudes and by 58 Fourier transformation to the frequency domain: Here ω is the angular frequency of the laser excitation. The third-order transfer function 60 (Eq. 5) reduces to the product of a resonator transfer function (of order 2) and an integrator 61 1 jω . In the more simple case of pulsed illumination, the integrator disappears since the 62 driving consists in an quasi instantaneous energy deposition rather than a time modulated 63 power deposition. The gain of the transfer function of Eq. 5 is expressed as: and, similarly, for the case of pulsed irradiation. Here, R i is the internal bubble radius and V oil is the volume 66 of the optically absorbing liquid surrounding the microbubble. The only other requirement 67 on the encapsulating liquid is that it is immiscible in water. This liquid is hereafter referred 68 to as 'oil'. R 0 is the initial bubble gas core radius. Equations 6 and 7 show that the overall 69 gain of the transfer function is highest for low density and low heat capacity oils. The major 70 determinant of the oscillation amplitude at resonance, however, is the damping coefficient 71 (directly obtained from the canonical transfer function Eq. 5) that was found to depend 72 dramatically on the viscosity of the oil chosen: where x = R i /R e is the ratio of the inner to outer bubble radii. Thus the damping coefficient 74 decreases with increasing bubble size, with decreasing layer thickness (in competition with 75 the laser energy absorption), and, primarily, with decreasing oil viscosity. This relation 76 emphasizes the necessity for using low viscosity oils.

B. Undamped natural frequency
The undamped natural frequency of the system follows directly from the expression of 79 the linear resonance curve (Eq. 5) as follows: A crucial feature of this expression is the dominance of R i , the instantaneous or 'hot' 81 bubble radius, in determining the natural frequency. Also, the influence of the oil density, 82 which is present in the denominator, can shift significantly the natural frequency, whereas 83 the surface tension only has a secondary effect.

84
C. Scaling function 85 One challenge in comparing the results from the theoretical model to the experimental 86 data is the difficulty in fully characterizing the system. For example, the variations in 87 thickness of the oil layer within individual microbubbles is extremely difficult to measure.

88
Some influences, however, such as differences in the absorption of individual microbubbles 89 (i.e. encapsulation efficiency) and potential inhomogeneities in the irradiating laser beam 90 due to its finite size can be compensated for by expressing the amplitude at resonance: and looking at its dependency on thermal dilatation, which can be measured experimentally 92 and individually for each bubble. This results in a scaling function that compensates for the 93 influence of variations in most of the parameters: where the variable y =

152
For the pulsed laser irradiation experiments, the laser was first fired only in the second 153 of the series of 6 movies (sequence of 128 images) recorded by the Brandaris camera, so that 154 the first movie would provide a reference. The pulsed laser was fired for a total of three 155 times at low energy setting (∼ 50 mJ/cm 2 ). The laser output before entering the microscope 156 was set close to the minimum (∼ 3 mJ, calibrated using a Coherent FieldMax II laser energy 157 meter) and further reduced using neutral density filters (ND 1.65).

158
The CW laser output was modulated in time at a chosen frequency by an acousto-optic 159 modulator (AA-optoelectronics, France). The CW laser was turned on 233 µs before the Here A is the amplitude of the response, z is the damping coefficient of the oscillator and

207
Finally, the conversion of the time to a bubble radius was performed with the help of a 208 finite differences simulation of the heat transfers using the method described previously [12].

209
During the CW laser irradiation, the bubble heats up and grows in time following a function 210 f (1/τ ) with τ = R 2 ieq /D w with τ the typical time scale, D w the thermal diffusivity of water 211 and R ieq the equilibrium radius of the bubble, which is also equal to the thickness of the 212 thermal boundary layer at equilibrium [12]. The function f is given by the finite differences  In order to better understand the behavior of the microbubble population as a whole, we 293 now look at the response of each bubble at a single time point given by the optical recording.

294
By doing so, we look at the global behavior of such laser-driven microbubbles. The influence of the variation in absorbance from one microbubble to another can be compensated for using 296 Eq. 11. Using the range of damping coefficients measured from the impulse response of the 297 microbubbles, a theoretical range for the microbubble responses can be calculated. This

315
From the results it appears that the harmonic generation mostly occurs around reso-316 nance (Fig. 7a, c and e) with an amplitude up to -10 dB as compared to the fundamental, 317 which makes them practically usable. Furthermore, only a few bubbles did not generate a 318 significant second harmonic. Interestingly, the generation of subharmonic components was 319 not highest from bubbles of resonant size but in the case of DCM-coated microbubbles for 320 radii ∼ 10% bigger for the DCM-coated microbubbles ( Fig. 7b and d). In some cases the 321 subharmonic was actually larger than that at the fundamental frequency, with a maximum these experiments to be between 50 and 500 kW/cm 2 which, integrated over the irradiation 357 time is too high for safe use. Furthermore, after 200 microsecond of laser irradiated, less 358 than ten percent of the input energy is stored in the bubble itself, which limits the efficacy 359 of the system. Nonetheless, achieving the measured signal amplitudes with these relatively 360 low powers during the concept phase is a good sign that this technology has the potential 361 to meet the biomedical requirements for the use of CW lasers.

362
As explained in the methods and results section, the coating is not accounted for in the 363 theory. Here again the model could be extended to include a more detailed treatment of 364 the water and water/air interface dynamics. It is however more relevant to develop this 365 modification for more common coatings, such as phospholipid, which does not apply for the 366 present test system. Also, the stabilization of these bubble was assured by the presence of 367 BSA. It is possible that heat induced a denaturation of the BSA while the oscillations would 368 prevent a stiff reticulation. This effect was not investigated and not accounted for since it 369 has a limited interest for this proof of concept. It would, however, become important in a 370 next step where the obtained damping coefficient is explained in details.

371
The damping of the oscillations sustained by these BSA-stabilized microbubbles is larger 372 than that predicted for a surfactant-free bubble. It is however interesting to note that the 373 presence of a low viscosity oil layer is sufficient, in principle, to significantly decrease viscous 374 damping. This conclusion also applies to the case of ultrasound-driven bubbles, which offers 375 interesting leads for the design of even more efficient acoustic (and photoacoustic) contrast 376 agents.

377
The CW laser in this study was modulated using a sine wave for practical simplicity. In 378 theory, a square wave is expected to induce a slightly higher response and to also generate 379 different harmonic response peaks. A more simple sine modulation containing a single har-380 monic frequency seemed therefore more suited for a proof of concept study where unexpected 381 effects have to be pointed out, but this should be explored further in future work.

383
In this study, we have fabricated two types of oil-coated microbubbles, one with dichloro-384 methane and a second type with toluene. In both cases, the oil entraps a dye that absorbs 385 the laser light. The bubbles were first irradiated with a pulsed laser and good agreement 386 between the measured natural frequency and the theoretical predictions was found. The 387 non-dimensional damping coefficient was also measured and shown to be higher than that 388 expected for an oil-coated microbubble. There was also considerable variation from one 389 bubble to another. The microbubbles could successfully be driven by a modulated CW laser  1957-1962 (2003).  1915-1920 (1994). The Navier-Stokes equation for an incompressible, Newtonian fluid is as follows Body forces will be negligible and a spherical symmetry case is investigated leading to In the simulation, the bubbles will have an oscillation amplitude in the order of micrometers and the frequency will be in the order of MHz. Speeds will therefore be approximately 1m.s −1 and thus much lower than the speed of sound. For this reason, incompressibility of the liquid is assumed leading to: WhereṘ is dR dt . With this we find This equation can be written for both the oil layer and the water. When integrating from r = A to r = B the term µ∇ 2 v drops out and this gives Taking the inner bubble radius R i for R and integrating over the water domain: Integrating over the oil domain: rewritten as:

540
Over the oil gas interface Where σ o is the strain tensor and − → e r denotes that it is in the r direction. δP 1 is the difference 541 in pressure over the oil-gas interface.
Where u is the velocity derivative to the radius. σ is the surface tension. Knowing Over the oil water interface Resulting in 3. Combining

546
We know that P g − P ∞ = P (R − i ) − P ∞ because the pressure at the inside of the inner 547 radius of the bubble is by definition in the gas and therefore P g = P (R − i ). We can rewrite 548 by adding and subtracting similar terms Part 1 of (A6) is defined in (A4), part 2 is defined in (A3), part 3 is defined in (A5) and 550 part 4 is defined in (A1). Thus, the complete equation is: Rewriting gives And rewriting further To reach the modified Rayleigh-Plesset equation: Where the viscosity of water µ w is temperature dependent following the following relation µ w = 2.414 · 10 −5 · 10 247.8/(T −140) Where T is the temperature of the water at the water-oil interface. This new RP equation we make the assumption a priori (verified a posteriori) that the microbubble resonance 561 frequency is in the same range as its acoustic resonance frequency and using a Minnaert 562 approximation, the temperature in the gas can be considered constant for bubbles smaller 563 than: A diffusion distance estimation can be done for the water to find app. 0.1µm for 1 M Hz 565 frequency.

566
The change in temperature over time can be described as follows: .
with B amp being the amplitude of the oscillations in the power and ω = 2πf . Using the initial equilibrium solution: I is the average laser power. Filling in C 1o and C 2o and rewriting gives We know that Where T g can be filled in and P is the total pressure as used in the modified Rayleigh-Plesset equation (eq A10) : Where P g is the found P and P ∞ is the pressure at infinity. Expressing this in the 572 new variables P 0 as the pressure at the beginning of the static solution and P g as the total 573 pressure in the gas, and filling in T g and the found pressure, this gives: Organizing forṘ,Ṙ 2 andR and, as an approximation, taking all R i and R e to be R i,eq 575 and R e,eq in case they are multiplied byṘ,Ṙ 2 orR.
In order to find an equation that does not contain an integral, everything is derived to In which caseṘ e is assumed to be approximatelyṘ i and T gas,eq + T room is now called T g2 .

579
The term 2γR iṘi is of higher order and is therefore neglected. R i is expected to act as an 580 harmonic oscillator and will therefore have the shape of Which has the shape of a transfer function with G being the gain, O the output, I the input, z the damping and ω 0 the angular eigen 588 frequency. One thing that can be noted here is that this transfer function is of third order 589 where a standard RP equation would be of second order. The expected phase difference in 590 our case is therefore π at resonance instead of π/2 such as in the normal RP equation.

591
→ From (A15) we can find Therefore ζ can be simplified Where the equilibrium pressure P g,eq is the atmospheric pressure plus the Laplace pressure 594 jump over both interfaces: which altogether is an expression for the angular eigenfrequency as a function of R i,eq and 600 R e,eq . This shows the eigenfrequency is inversely related to the bubble size but also shows 601 that the oil layer thickness plays a role. The denominator under the square root shows 602 an inertial shift of the resonance curve: Because oil and water have different densities, the 603 thickness of the oil layer influences the mass to be displaced and therefore the resonance 604 frequency.

605
Now to find an expression for the damping. According to A27: