Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems

Abstract

Embedding of active substances in supramolecular systems has as the main goal to ensure the controlled release of the active ingredients. Whatever the final architecture or entrapment mechanism, modeling of release is challenging due to the moving boundary conditions and complex initial conditions. Despite huge diversity of formulations, diffusion phenomena are involved in practically all release processes. The approach in this paper starts, therefore, from mathematical methods for solving the diffusion equation in initial and boundary conditions, which are further connected with phenomenological conditions, simplified and idealized in order to lead to problems which can be analytically solved. Consequently, the release models are classified starting from the geometry of diffusion domain, initial conditions, and conditions on frontiers. Taking into account that practically all solutions of the models use the separation of variables method and integral transformation method, two specific applications of these methods are included. This paper suggests that "good modeling practice" of release kinetics consists essentially of identifying the most appropriate mathematical conditions corresponding to implied physicochemical phenomena. However, in most of the cases, models can be written but analytical solutions for these models cannot be obtained. Consequently, empiric models remain the first choice, and they receive an important place in the review.

Keywords: boundary conditions; diffusion equation; drug carriers; release kinetics.

Figure 1

Spatial distribution of the active substance at different (t_n1, …, t_nk) time points.

Figure 2

Release kinetics of a drug from microemulsions in an experiment using a dialysis membrane.

Figure 3

Spatial distribution of the concentrations of active substance at different time intervals: (a) case $c_1<c_0$, transfer into the membrane; (b) case $c_1>c_0$, transfer out of the membrane.

Figure 4

Distribution of concentration in a membrane separating two domains with constant concentrations.

Figure 5

Radial transfer across a hollow sphere in a release medium where the concentration of drug has a constant value $c_1$.

Figure 6

Release of active substance embedded in liposomes.

Figure 7

Higuchi’s moving boundary model inside solid and semisolid formulations.

Figure 8

Higuchi’s model for release from a spherical tablet of radius R, in the condition of a moving solvent front.

Figure 9

Swelling of a spherical polymer particle following the intrusion of solvent across the outer surface.

Figure 10

Marginal-type erosion models.

Figure 11

Black-box model of transfer (weighting) function, defined in the space of image functions obtained after the application of an integral transformation.