Mathematical modelling of the collapse time of an unfolding shelled microbubble

There is considerable interest at the moment on using shelled microbubbles as a transportation mechanism for localised drug delivery, specifically in the treatment of various cancers. In this report a theoretical model is proposed which predicts the collapse time of an unfolding shelled microbubble. A neo-Hookean, compressible strain energy density function is used to model the potential energy per unit volume of the shell. This is achieved by considering a reference configuration (stress free) consisting of a shelled microsphere with a hemispherical cap removed. This is then displaced angularly and radially by applying a stress load to the free edge of the shell. This forms a deformed open sphere possessing a stress. This is then used as an initial condition to model the unfolding of the shell back to its original stress free configuration. Asymptotic expansion along with the conservation of mass and energy are then used to determine the collapse times for the unfolding shell and how the material parameters influence this. The theoretical model is compared to published experimental results.