Analytical Solution of a Model for Shrinking Drug-loaded Microspheres

The dynamics of shrinking drug-loaded microspheres were studied using a diffusion equation in spherical coordinates and with a radially modulated diffusivity. A movable boundary condition that represents the shrinking was incorporated using an approximation based on the Laplace transform. The resulting diffusive problem with radially modulated diffusivity was solved using Laplace transform techniques with the Bromwich integral, the residue theorem and special functions. Analytical solutions in the form of infinity series of special functions were derived for the general case of shrinking microspheres and for the particular case with exponential shrinking. All computations were made using computer algebra, specifically Maple. Some numerical simulations were made in the case of microspheres with exponential shrinking. The analytical results were used to derive the effective constant time for the shrinking microsphere. As future line of investigation, it is proposed the analysis of models with boundary condition that shows the memory effect. It is expected that the obtained analytical results could be very important in pharmaceutical engineering.