DOI 10.15507/2079-6900.24.202202.200-214

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 35Q92, 35K05, 35R30, 65K05

Mathematical study of medicine diffusion from swelling chitosan film

A. O. Syromyasov1, A. S. Shurshina2, D. V. Galkin3

1National Research Mordovia State University (Saransk, Russian Federation)

2Bashkir State University (Ufa, Russian Federation)

3«Coder» LLC (Saransk, Russian Federation)

Abstract. One of modern dosage forms is a medicine-saturated organic film: after putting this film onto a skin the medicine releases thus providing healing effect. Present article concerns films based on chitosan and containing amikacinum or cefazolinum. The most important characteristic of such film is rate of medicine release described by diffusion coefficient. To find it the film is placed in water and the average medicine concentration in the film is measured at different time moments. Two problems arise here. First, the film properties change because of its swelling. Second, diffusion is not the only process that takes place inside the film. To deal with these effects, authors suppose diffusion coefficient to be time-variable and complete the mathematical model with ODE describing detachment of medicine molecules from high-molecular matrix. All the equations in the model are solved analytically, so average medicine concentration in the film is known function of time. Thus, to solve stated inverse problem it is sufficient to find unknown scalar parameters of known functions using least-squares framework. Expressions arising in the solution are complicated so non-gradient methods are preferrable for optimization. Applying described procedure to experimental data leads to a good accuracy and the results may be explained from physicochemical point of view. In particular, the film swelling doesn’t influence release rate. In fact, the diffusion rate during first hours of experiment is large, and the main part of the medicine is released before swelling starts to play important role.

Key Words: diffusion equation, swelling chitosan film, inverse problem, numerical optimization, Nelder–Mead method, least squares method, high-molecular compound

For citation: A. O. Syromyasov, A. S. Shurshina, D. V. Galkin. Mathematical study of medicine diffusion from swelling chitosan film. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 24:2(2022), 200–214. DOI:

Submitted: 25.02.2022; Revised: 01.04.2022; Accepted: 25.05.2022

Information about the authors:

Alexey O. Syromyasov, Associate Professor, Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), Ph.D. (Phys.-Math.), ORCID:,

Anzhela S. Shurshina, Associate Professor, Department of High Molecular Compounds and General Chemical Technology, Bashkir State University (32 Zaki Validi St., Ufa 450076, Russia), ORCID:, Ph.D. (Chemistry),

Dmitry V. Galkin, Senior Programmer, «Coder» LLC (84V Sovetskaya St., Saransk 430005, Russia), ORCID:,

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.