Analytical Solutions for the Developed Five-Compartment Pharmacokinetic Model
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Abstract
This study presents analytical solutions for a developed compartment pharmacokinetic model consisting of five compartments to describe the dynamics of drug concentration in the body, namely the absorption compartment, the central compartment representing the blood circulation system, the rapidly equilibrating tissue compartment, the slowly equilibrating tissue compartment, and the effect-site compartment. The model is formulated as a system of ordinary differential equations describing drug transfer and elimination between compartments. The analytical solution process is divided into three stages: (1) deriving the solution for the absorption compartment using the method of separation of variables, (2) analyzing the system of equations for the central and tissue compartments using eigenvalue analysis combined with the Laplace transform, and (3) obtaining the solution for the effect-site compartment using the method of undetermined coefficients. The results show that, upon applying the initial conditions, closed-form analytical solutions for drug concentrations in all compartments can be obtained. Although some solutions involve complex eigenvalues, they effectively characterize the dynamic behavior of the system. Furthermore, a numerical
example is presented using rate constant parameters consistent with ranges reported in the pharmacokinetic literature, and the concentration-time profiles for all compartments are illustrated using the derived closed-form solutions, demonstrating drug behavior that is consistent with pharmacokinetic theory. These analytical solutions serve as a fundamental framework for validating numerical solutions and supporting future pharmacokinetic model development.
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References
Anton, H. and Rorres, C. (2013). Elementary Linear Algebra: Applications Version (11thed). Wiley.
Browning, A.P., Drovandi, C., Turner, I.W., Jenner, A.L. and Simpson, M.J. (2021). Exact Solutions and Equi-Dosing Regimen Regions for Multi-Dose Pharmacokinetics Models with Transit Compartments. Journal of Pharmacokinetics and Pharmacodynamics, 48, 99-131. https://doi.org/10.1007/s10928-020-09719-8
Cascone, S., Lamberti, G., Titomanlio, G. and Piazza, O. (2013). Pharmacokinetics of Remifentanil: A three-Compartmental Modeling Approach. Translational Medicine UniSa, 7, 18-22.
Gabrielsson, J. and Weiner, D. (2016). Pharmacokinetic and Pharmacodynamic Data Analysis: Concepts and Applications (5thed). CRC Press.
Jan, M., Kubra, K.T., Farooqi, S.H. and Asghar, A. (2025). A Mathematical Approach to Multi-Route Drug Distribution via Compartmental Models. Modeling Earth Systems and Environment, 11, 385. https://doi.org/10.1007/s40808-025-02571-0
Kreyszig, E. (2011). Advanced Engineering Mathematics (10thed). Wiley.
Nickalls, R.W.D. (1993). A New Approach to Solving the Cubic: Cardan's Solution Revealed. The Mathematical Gazette, 77(480), 354-359. https://doi.org/10.2307/3619777
Kunderu, L.N. and Reddy, S.R. (2018). Administration of Phenytoin in Normal and Uremic Subjects Studied by Two Compartment Mathematical Model. International Journal of Research, 7, 2657-2666. https://www.researchgate.net/publication/382493137_Administration_Of_Phenytoin_In_Normal_And_Uremic_Subjects_Studied_By_Two_Compartment_Mathematical_Model
Reddy, V.S.R.K. and Narayan, K.L. (2019). The Concentration of Digoxin After Intravenous and Oral Administration Studied by a Two-Compartment Model. Letters in Biomathematics, 6(2s), 1-15. https://lettersinbiomath.org/manuscript/index.php/lib/article/view/107/98
Rowland, M. and Tozer, T.N. (2011). Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications (4thed). Lippincott Williams & Wilkins.
Shargel, L., Wu-Pong, S. and Yu, A.B.C. (2012). Applied Biopharmaceutics and Pharmacokinetics (6thed). McGraw-Hill.
Sivasubramaniam, S. (2014). Review of Pharmacokinetic Models for Target-Controlled Infusions in Anes-Thesia. International Journal of Basic & Clinical Pharmacology, 3(3), 417-423. https://www.ijbcp.com/index.php/ijbcp/article/view/998/895
Strang, G. (2016). Introduction to Linear Algebra (5thed). Wellesley-Cambridge Press.
Świętaszczyk, C. and Jødal, L. (2024). First-Order Compartment Model Solutions: Exponential Sums and Beyond. Journal of Pharmacological and Toxicological Methods, 128. https://doi.org/10.1016/j.vascn.2024.107534
Uno, Y., Ogawa, E., Aiyoshi, E. and Arai, T. (2018). A Three-Compartment Pharmacokinetic Model to Predict the Interstitial Concentration of Talaporfin Sodium in the Myocardium for Photodynamic Therapy: A Method Combining Measured Fluorescence and Analysis of the Compartmental Origin of the Fluorescence. Bioengineering, 6(1). https://doi.org/10.3390/bioengineering6010001
Yakubu, G.D., Abdulhameed, M., Adamu, G.T., Kadas, S.A. and Tukur, S.A. (2023). Accurate Numerical Integration of Highly Stiff Pharmacokinetics Models Using Continuous Block Implicit Hybrid One-Step Collocation Methods. Journal of Algorithms & Computational Technology, 17. https://doi.org/10.1177/17483026231157197
