A mathematical investigation on the active substance pulsatory release from a solution-charged liposome

We adopt a model in which the time evolution of a solution-charged liposome with selective permeability is seen as a sequence of two-stage cycles. In the first stage the lipid vesicle with a certain input concentration of the active substance is swelling up in an osmotic process until it reaches a size that one pore is formed on its surface. The second stage follows, where the vesicle content is partially eliminated through the pore to induce a decrease of the vesicle volume up to its original size. A new cycle is then initiated with the remaining concentration of the active substance for input, and so on. The process is described mathematically by an ordinary differential equation for the first stage and by a system of three such equations for the second but the numerical treatment is quite demanding for the system because this is stiff. We use a recently proposed Runge-Kutta method of a special form which, in contrast to the standard versions, shares the salient advantages of being both explicit and A-stable. The results on a case with realistic biophysical parameters are given for the swelling time in each cycle, total number of cycles, and the active substance amount delivered during each cycle. The model can be successfully used in a large variety of situations of direct practical interest, in particular when the active substance is pharmacological.