Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems

Winner of the Pharmaceutics 2021 Best Paper Awards

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.

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Article information: Mircioiu, C.; Voicu, V.; Anuta, V.; Tudose, A.; Celia, C.; Paolino, D.; Fresta, M.; Sandulovici, R.; Mircioiu, I. Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems. Pharmaceutics 2019, 11, 140. https://doi.org/10.3390/pharmaceutics11030140

Conclusions
The predictability of release kinetics of active substances represents an essential characteristic applied to supramolecular carrier systems in order to be accepted as drugs. Both safety and efficacy depend on the rate and extent of availability of active substances at the place of absorption and at the site of action.
The measuring, the modeling, and the prediction of release kinetics represent research of high complexity, implying an in-depth understanding of physicochemical, physiological, and mathematical aspects. Unavoidably, almost all approaches start from one domain, and from one scientific language, while neglecting the other domains. Although many papers, many books, and many reviews were written, all of these satisfied only specific cases from one or two marginal sub-domains. Consequently, all future papers and reviews are welcome, but they surely cannot overcome some of these irreducible difficulties.
On the other hand, it is continually emphasized that the more complex the model is, the more data are needed in order to validate it. Uncertainty, lack of uniqueness, and robustness increase with the number of parameters.
Furthermore, since the complexity and diversity of mechanistic models is huge, a clustering of these models as a function of boundary conditions, as tried in this paper, would probably allow a better understanding of the phenomena, and more efficient research for new models.