Growth and dissolution of an encapsulated contrast microbubble: effects of encapsulation permeability

Gas diffusion from an encapsulated microbubble is modeled using an explicit linear relation for gas permeation through the encapsulation. Both the cases of single gas (air) and multiple gases (perfluorocarbon inside the bubble and air dissolved in surrounding liquid) are considered. An analytical expression for the dissolution time for an encapsulated air bubble is obtained; it showed that for small permeability the dissolution time increases linearly with decreasing permeability. A perfluorocarbon-filled contrast microbubble such as Definity was predicted to experience a transient growth because of air infusion before it dissolves in conformity with previous experimental findings. The growth phase occurs only for bubbles with a critical value of initial mole fraction of perfluorocarbon relative to air. With empirically obtained property values, the dissolution time of a 2.5-micron diameter (same as that of Definity), lipid-coated octafluoropropane bubble, with surface tension 25 mN/m, is predicted to be 42 min in an air-saturated medium. The properties such as shell permeability, surface tension and relative mole fraction of octafluoropropane are varied to investigate their effects on the time scales of bubble growth and dissolution, including their asymptotic scalings where appropriate. The dissolution dynamics scales with permeability, in that when the time is nondimensioanlized with permeability, curves for different permeabilities collapse on a single curve. Investigation of bubbles filled with other gases (nonoctafluoropropane perfluorocarbon and sulfur hexafluoride) indicates longer dissolution time because of lower solubility and lower diffusivity for larger gas molecules. For such micron-size encapsulated bubbles, lifetime of hours is possible only at extremely low surface tension (<1 mN/m) or at extreme oversaturation.