Vaporization dynamics of volatile perfluorocarbon droplets: a theoretical model and in vitro validation

Abstract

Purpose: Perfluorocarbon (PFC) microdroplets, called phase-change contrast agents (PCCAs), are a promising tool in ultrasound imaging and therapy. Interest in PCCAs is motivated by the fact that they can be triggered to transition from the liquid state to the gas state by an externally applied acoustic pulse. This property opens up new approaches to applications in ultrasound medicine. Insight into the physics of vaporization of PFC droplets is vital for effective use of PCCAs and for anticipating bioeffects. PCCAs composed of volatile PFCs (with low boiling point) exhibit complex dynamic behavior: after vaporization by a short acoustic pulse, a PFC droplet turns into a vapor bubble which undergoes overexpansion and damped radial oscillation until settling to a final diameter. This behavior has not been well described theoretically so far. The purpose of our study is to develop an improved theoretical model that describes the vaporization dynamics of volatile PFC droplets and to validate this model by comparison with in vitro experimental data.

Methods: The derivation of the model is based on applying the mathematical methods of fluid dynamics and thermodynamics to the process of the acoustic vaporization of PFC droplets. The used approach corrects shortcomings of the existing models. The validation of the model is carried out by comparing simulated results with in vitro experimental data acquired by ultrahigh speed video microscopy for octafluoropropane (OFP) and decafluorobutane (DFB) microdroplets of different sizes.

Results: The developed theory allows one to simulate the growth of a vapor bubble inside a PFC droplet until the liquid PFC is completely converted into vapor, and the subsequent overexpansion and damped oscillations of the vapor bubble, including the influence of an externally applied acoustic pulse. To evaluate quantitatively the difference between simulated and experimental results, the L2-norm errors were calculated for all cases where the simulated and experimental results are compared. These errors were found to be in the ranges of 0.043-0.067 and 0.037-0.088 for OFP and DFB droplets, respectively. These values allow one to consider agreement between the simulated and experimental results as good. This agreement is attained by varying only 2 of 16 model parameters which describe the material properties of gaseous and liquid PFCs and the liquid surrounding the PFC droplet. The fitting parameters are the viscosity and the surface tension of the surrounding liquid. All other model parameters are kept invariable.

Conclusions: The good agreement between the theoretical and experimental results suggests that the developed model is able to correctly describe the key physical processes underlying the vaporization dynamics of volatile PFC droplets. The necessity of varying the parameters of the surrounding liquid for fitting the experimental curves can be explained by the fact that the parts of the initial phospholipid shell of PFC droplets remain on the surface of vapor bubbles at the oscillatory stage and their presence affects the bubble dynamics.

FIG. 1.

Schematic of the model.

FIG. 2.

Examples of experimental measurements. (A) A DFB droplet near 2.7 μm in diameter vaporizes and expands to a maximum near 15.5 μm in diameter and eventually settles to a smaller resting diameter. (B) An OFP droplet near 2 μm in diameter expands to a maximum near 14.6 μm in diameter and settles to a smaller resting diameter. Scale bar represents 5 μm. Reprinted with permission from P. S. Sheeran, T. O. Matsunaga, and P. A. Dayton, Phys. Med. Biol. 59, 379–402 (2014). Copyright 2014, Institute of Physics and Engineering in Medicine.

FIG. 3.

Comparison of simulated and experimental results for OFP droplets. The best-fit values of η₃ and σ₁₃ are (a) η₃ = 0.006 Pa s, σ₁₃ = 0 N/m; (b) η₃ = 0.0065 Pa s, σ₁₃ = 0 N/m; (c) η₃ = 0.009 Pa s, σ₁₃ = 0 N/m; (d) η₃ = 0.02 Pa s, σ₁₃ = 0.25 N/m. The cross shows the moment when the droplet completely turns into vapor.

FIG. 4.

Comparison of simulated and experimental results for DFB droplets. The best-fit values of η₃ and σ₁₃ are (a) η₃ = 0.004 Pa s, σ₁₃ = 0 N/m; (b) η₃ = 0.0105 Pa s, σ₁₃ = 0.065 N/m; (c) η₃ = 0.007 Pa s, σ₁₃ = 0 N/m; (d) η₃ = 0.025 Pa s, σ₁₃ = 0 N/m. The cross shows the moment when the droplet completely turns into vapor.

FIG. 5.

Normalized mass flux through the surface of the vapor bubble for (a) the OFP droplet shown in Fig. 3(c) and (b) the DFB droplet shown in Fig. 4(c) at three values of the amplitude of the ultrasound pulse: 1, 2, and 4 MPa.

FIG. 6.

Comparison of simulated (solid line) and experimental (circles) results obtained for a 5.4 μm radius DDFP droplet activated by a six-cycle, 3.5 MHz, 4.5 MPa pulse at 15 °C superheating.

FIG. 7.

Bubble evolution inside droplets composed of different PFCs. The cross shows the moment when the droplet is completely converted into vapor.