Mathematical modeling of drug delivery systems with diffusion and absorption equations

Kumar, Vipan; Bhardwaj, Uma; Sood, Yamini; Bhanuse, Vijaykumar; Musale, Vinayak; Kumar, M. Ramesh

doi:10.47974/JIM-2369

Open Access Research Article

Mathematical modeling of drug delivery systems with diffusion and absorption equations

Vipan Kumarer.vipingupta14@gmail.comDepartment of Electronics and Communication EngineeringSri Sai College of Engineering and TechnologyBadhani, Punjab, 145002, IndiaView full profile → , Uma Bhardwajvc@niu.edu.inDepartment of Biotechnology and MicrobiologyNoida International UniversityNoida, Uttar Pradesh, 203201, IndiaView full profile → , Yamini Soodyaminisood1987@gmail.comDepartment of Computer Science and EngineeringSri Sai UniversityPalampur, Himachal Pradesh, IndiaView full profile → , ^*Vijaykumar BhanuseCorresponding authorvijaykumar.bhanuse@vit.eduDepartment of Instrumentation and Control EngineeringVishwakarma Institute of TechnologyPune, Maharashtra, 411037, IndiaView full profile → , Vinayak Musalevinayak.musale@gmail.comDepartment of Computer Engineering and TechnologyDr. Vishwanath Karad MIT World Peace UniversityPune, Maharashtra, 411038, IndiaView full profile → , M. Ramesh Kumarmrkpdy@gmail.comDepartment of MathematicsIFET Engineering CollegeVillupuram, Tamil Nadu, 605108, IndiaView full profile →

* Corresponding author · click or hover a name for details

pp. 2273–2280Vol. 28Issue 6September 2025DOI: 10.47974/JIM-2369XML

Received:: 10 Dec 2024
Published Online:: 30 Sep 2025
Article type:: Research Article
Language:: EN
Article no.:: JIM-2369
Pages:: 2273–2280

Abstract

Mathematical modelling plays a fundamental role in designing drug delivery systems, where the goal is to release an active molecule at a controlled rate and in a specific target region. Diffusion and absorption processes govern how drugs migrate through polymers and tissues and how they are taken up by the body. Challenges arise from spatially varying material properties, nonlinear boundary conditions and the coupling between diffusion within the device and absorption by biological tissues. This paper formulates a one dimensional diffusion–absorption model for a controlled‐release reservoir coupled to a sink that mimics systemic uptake. Starting from Fick’s laws, we derive the governing partial differential equation with a first order absorption term and solve it using separation of variables and Laplace transforms. The model predicts concentration profiles inside the device and the fraction of drug released as functions of time. Results show that higher diffusion coefficients accelerate release, whereas stronger absorption reduces internal concentrations but increases the rate of systemic delivery. The proposed framework provides insights into the interplay between device parameters and pharmacokinetic outcomes and offers guidelines for optimizing controlled release formulations.

Keywords

Drug delivery Diffusion equation Absorption kinetics Fick’s law Controlled release Mathematical modelling

Subject Classifications

65L60

References

[1] S. Jafari, M. Soleimani, and M. Badinezhad, “Application of different mathematical models for further investigation of in vitro drug release mechanisms based on magnetic nano-composite,” Polym. Bull., vol. 79, no. 2, pp. 1021–1038 (2022).
[2] X. Hu, C. Zhang, Y. Xiong, S. Ma, C. Sun, and W. Xu, “A review of recent advances in drug loading, mathematical modeling and applications of hydrogel drug delivery systems,” J. Mater. Sci., vol. 59, no. 32, pp. 15077–15116 (2024).
[3] F. Bisotti, F. Pizzetti, G. Storti, and F. Rossi, “Mathematical modelling of cross-linked polyacrylic-based hydrogels: physical properties and drug delivery,” Drug Deliv. Transl. Res., vol. 12, no. 8, pp. 1928–1942 (2022).
[4] N. Sahai, M. Gogoi, and N. Ahmad, “Mathematical modeling and simulations for developing nanoparticle-based cancer drug delivery systems: A review,” Curr. Pathobiol. Rep., vol. 9, no. 1, pp. 1–8 (2021).
[5] V. Miskovic-Stankovic and T. M. Atanackovic, “On a system of equations with general fractional derivatives arising in diffusion theory,” Fractal and Fractional, vol. 7, no. 7 (2023).
[6] M. Yang, J. Meng, L. Han, X. Yu, Z. Fan, and Y. Yuan, “Pharmacokinetic study of triptolide nanocarrier in transdermal drug delivery system—Combination of experiment and mathematical modeling,” Molecules, vol. 28, no. 2 (2023).
[7] M. Askarizadeh, N. Esfandiari, B. Honarvar, S. A. Sajadian, and A. Azdarpour, “Kinetic modeling to explain the release of medicine from drug delivery systems,” ChemBioEng Rev., vol. 10, no. 6, pp. 1006–1049 (Dec. 2023).
[8] P. Trucillo, “Drug carriers: A review on the most used mathematical models for drug release,” Processes, vol. 10, no. 6 (2022).
[9] P. R. Yadav, L. J. Dobson, S. K. Pattanayek, and D. B. Das, “Swellable microneedles based transdermal drug delivery: Mathematical model development and numerical experiments,” Chem. Eng. Sci., vol. 247, pp. 117005 (2022).
[10] M. K. Singh, “Mapping the Indian landscape of nanodrug delivery for cancer treatment: A bibliometric analysis,” COLLNET Journal of Scientometrics and Information Management, vol. 18, no. 2, pp. 155–178 (2024).
[11] A. Mazakova, S. Jomartova, T. Mazakov, G. Ziyatbekova, and A. Tursynbai, “Application of mathematical methods in pharmacology,” Journal of Interdisciplinary Mathematics, vol. 27, no. 6, pp. 1421–1435 (2024).

Views: 291Downloads: 168Citations: 0