Oral drug kinetics under first-pass metabolism: A fractional modeling framework

K. B. Shadiya Nasrin; J. Kokila; M. Z. Youssef; K. B. Shadiya Nasrin; J. Kokila; M. Z. Youssef

doi:10.3934/math.2026883

AIMS Mathematics

2026, Volume 11, Issue 7: 21819-21845. doi: 10.3934/math.2026883

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Oral drug kinetics under first-pass metabolism: A fractional modeling framework

1.
Department of Statistics and Applied Mathematics, Central University of Tamil Nadu, Thiruvarur 610005, India
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia

Received: 18 March 2026 Revised: 01 July 2026 Accepted: 07 July 2026 Published: 22 July 2026
MSC : 33E12, 44A10, 60J70, 65L09, 92C45, 93A30, 93E24

This study proposes compartment models with the Caputo fractional derivative to capture drug dynamics across absorption, central, and peripheral compartments, and to account for memory effects and anomalous kinetics. The first part of the study covers mathematical modeling, which enables analysis of the amount of drug in each compartment using closed-form expressions. The second part is to estimate the pharmacokinetic parameters and compare models using the Akaike information criterion and Bayesian information criterion for the drugs pindolol, fenoprofen, and cyclosporin A. The regression analysis was performed by using the corresponding Mittag-Leffler regression model and the trust-region algorithm with the Adams predictor-corrector method.
- pharmacokinetics,
- absorption,
- mathematical modeling,
- Mittag-Leffler function,
- Laplace transform,
- inverse problems,
- nonlinear least squares method
Citation: K. B. Shadiya Nasrin, J. Kokila, M. Z. Youssef. Oral drug kinetics under first-pass metabolism: A fractional modeling framework[J]. AIMS Mathematics, 2026, 11(7): 21819-21845. doi: 10.3934/math.2026883

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Abstract

This study proposes compartment models with the Caputo fractional derivative to capture drug dynamics across absorption, central, and peripheral compartments, and to account for memory effects and anomalous kinetics. The first part of the study covers mathematical modeling, which enables analysis of the amount of drug in each compartment using closed-form expressions. The second part is to estimate the pharmacokinetic parameters and compare models using the Akaike information criterion and Bayesian information criterion for the drugs pindolol, fenoprofen, and cyclosporin A. The regression analysis was performed by using the corresponding Mittag-Leffler regression model and the trust-region algorithm with the Adams predictor-corrector method.

References

[1]	S. S. Jambhekar, P. J. Breen, Basic pharmacokinetics, London: Pharmaceutical Press, 2009.
[2]	H. Zhou, Pharmacokinetic strategies in deciphering atypical drug absorption profiles, J. Clin. Pharmacol., 43 (2003), 211–227. https://doi.org/10.1177/0091270002250613 doi: 10.1177/0091270002250613
[3]	A. Y. Abuhelwa, D. B. Williams, R. N. Upton, D. J. Foster, Food, gastrointestinal pH, and models of oral drug absorption, Eur. J. Pharm. Biopharm, 112 (2017), 234–248. https://doi.org/10.1016/j.ejpb.2016.11.034 doi: 10.1016/j.ejpb.2016.11.034
[4]	L. Cheng, H. Wong, Food effects on oral drug absorption: Application of physiologically-based pharmacokinetic modeling as a predictive tool, Pharmaceutics, 12 (2020), 672. https://doi.org/10.3390/pharmaceutics12070672 doi: 10.3390/pharmaceutics12070672
[5]	D. Fleisher, C. Li, Y. Zhou, L. H. Pao, A. Karim, Drug, meal and formulation interactions influencing drug absorption after oral administration: clinical implications, Clin. Pharmacokinet., 36 (1999), 233–254. https://doi.org/10.2165/00003088-199936030-00004 doi: 10.2165/00003088-199936030-00004
[6]	G. Lamberti, S. Cascone, F. Marra, G. Titomanlio, M. d'Amore, A. A. Barba, Gastrointestinal behavior and ADME phenomena: Ⅱ. In silico simulation, J. Drug Deliv. Sci. Tec., 35 (2016), 165–171. https://doi.org/10.1016/j.jddst.2016.06.014 doi: 10.1016/j.jddst.2016.06.014
[7]	Y. K. Tam, Individual variation in first-pass metabolism, Clin. Pharmacokinet., 25 (1993), 300–328. https://doi.org/10.2165/00003088-199325040-00005 doi: 10.2165/00003088-199325040-00005
[8]	D. J. Back, S. M. Rogers, First‐pass metabolism by the gastrointestinal mucosa, AP & T, 1 (1987), 339–357. https://doi.org/10.1111/j.1365-2036.1987.tb00634.x doi: 10.1111/j.1365-2036.1987.tb00634.x
[9]	M. V. Varma, R. S. Obach, C. Rotter, H. R. Miller, G. Chang, S. J. Steyn, et al., Physicochemical space for optimum oral bioavailability: contribution of human intestinal absorption and first-pass elimination, J. Med. Chem., 53 (2010), 1098–1108. https://doi.org/10.1021/jm901371v doi: 10.1021/jm901371v
[10]	J. Kokila, M. Vellappandi, D. Meghana, V. Govindaraj, Optimal control study on Michaelis-Menten kinetics-A fractional version, Math. Comput. Simul., 210 (2023), 571–592. https://doi.org/10.1016/j.matcom.2023.03.033 doi: 10.1016/j.matcom.2023.03.033
[11]	X. Wu, W. Zhang, X. S. Wang, J. Li, A mechanistic model-to-model approach for solving a nonlinear pharmacokinetic model, Bull. Malays. Math. Sci. Soc., 48 (2025), 53. https://doi.org/10.1007/s40840-025-01827-3 doi: 10.1007/s40840-025-01827-3
[12]	B. Shiri, Well-posedness of the mild solutions for incommensurate systems of delay fractional differential equations, Fractal Fract., 9 (2025), 60. https://doi.org/10.3390/fractalfract9020060 doi: 10.3390/fractalfract9020060
[13]	M. A. Zaky, A new endpoint-adaptive spectral method for nonlinear tempered fractional terminal value problems with non-smooth solutions, Appl. Numer. Math., 225 (2026), 76–86. https://doi.org/10.1016/j.apnum.2026.02.003 doi: 10.1016/j.apnum.2026.02.003
[14]	M. A. Zaky, A linearized two-dimensional Galerkin-L1 spectral method with diagonalization for time-fractional diffusion equations with delay, Appl. Numer. Math., 220 (2026), 1–12. https://doi.org/10.1016/j.apnum.2025.09.008 doi: 10.1016/j.apnum.2025.09.008
[15]	S. Kumawat, S. Bhatter, B. Bhatia, S. D. Purohit, H. M. Baskonus, D. L. Suthar, Novel application of q-HAGTM to analyze Hilfer fractional differential equations in diabetic dynamics, Math. Comput. Simul., 238 (2025), 136–149. https://doi.org/10.1016/j.matcom.2025.04.029 doi: 10.1016/j.matcom.2025.04.029
[16]	S. Kumawat, S. Bhatter, S. D. Purohit, D. L. Suthar, Mathematical modeling of tuberculosis using Caputo fractional derivative: A comparative analysis with real data, Sci. Rep., 15 (2025), 12672. https://doi.org/10.1038/s41598-025-97502-5 doi: 10.1038/s41598-025-97502-5
[17]	A. D'Ambrosio, F. Itaj, F. Cacace, V. Piemonte, Mathematical modeling of the gastrointestinal system for preliminary drug absorption assessment, Bioengineering, 11 (2024), 813. https://doi.org/10.3390/bioengineering11080813 doi: 10.3390/bioengineering11080813
[18]	M. Pompa, M. Capocelli, V. Piemonte, A new gastro-intestinal mathematical model to study drug bioavailability, Med. Eng. Phys., 74 (2019), 106–114. https://doi.org/10.1016/j.medengphy.2019.09.015 doi: 10.1016/j.medengphy.2019.09.015
[19]	E. Sjögren, J. Westergren, I. Grant, G. Hanisch, L. Lindfors, H. Lennernäs, et al., In silico predictions of gastrointestinal drug absorption in pharmaceutical product development: Application of the mechanistic absorption model GI-Sim, Eur. J. Pharm. Sci., 49 (2013), 679–698. https://doi.org/10.1016/j.ejps.2013.05.019 doi: 10.1016/j.ejps.2013.05.019
[20]	S. J. Achar, T. Sathiyaraj, S. Harshavarthini, Fractional order modelling of water scarcity and its social effects, Model. Earth Syst. Environ., 11 (2025), 335. https://doi.org/10.1007/s40808-025-02510-z doi: 10.1007/s40808-025-02510-z
[21]	M. Jan, K. T. Kubra, S. H. Farooqi, A. Asghar, A mathematical approach to multi-route drug distribution via compartmental models, Model. Earth Syst. Environ., 11 (2025), 385. https://doi.org/10.1007/s40808-025-02571-0 doi: 10.1007/s40808-025-02571-0
[22]	V. Singh, P. Thakur, S. K. Srivastava, A nonlinear time-fractional-order epidemic model for tobacco smoking with sensitivity analysis, Model. Earth Syst. Environ., 11 (2025), 318. https://doi.org/10.1007/s40808-025-02468-y doi: 10.1007/s40808-025-02468-y
[23]	M. Vellappandi, S. Lee, Role of fractional derivatives in pharmacokinetic/pharmacodynamic anesthesia model using BIS data, Comput. Biol. Med., 187 (2025), 109783. https://doi.org/10.1016/j.compbiomed.2025.109783 doi: 10.1016/j.compbiomed.2025.109783
[24]	S. Bhatter, K. Jangid, S. D. Purohit, Fractionalized mathematical models for drug diffusion, Chaos Soliton. Fract., 165 (2022), 112810. https://doi.org/10.1016/j.chaos.2022.112810 doi: 10.1016/j.chaos.2022.112810
[25]	A. Dokoumetzidis, R. Magin, P. Macheras, Fractional kinetics in multi-compartmental systems, J. Pharmacokinet. Pharmacodyn, 37 (2010), 507–524. https://doi.org/10.1007/s10928-010-9170-4 doi: 10.1007/s10928-010-9170-4
[26]	V. F. Morales-Delgado, M. A. Taneco-Hernández, C. Vargas-De-León, J. F. Gómez-Aguilar, Exact solutions to fractional pharmacokinetic models using multivariate Mittag-Leffler functions, Chaos Soliton. Fract., 168 (2023), 113164. https://doi.org/10.1016/j.chaos.2023.113164 doi: 10.1016/j.chaos.2023.113164
[27]	D. Verotta, Fractional compartmental models and multi-term Mittag–Leffler response functions, J. Pharmacokinet. Pharmacodyn., 37 (2010), 209–215. https://doi.org/10.1007/s10928-010-9155-3 doi: 10.1007/s10928-010-9155-3
[28]	M. C. Wenlock, Profiling the estimated plasma concentrations of 215 marketed oral drugs, Med. Chem. Commun., 7 (2016), 706–719. https://doi.org/10.1039/C5MD00583C doi: 10.1039/C5MD00583C
[29]	M. C. Wenlock, Oral drug suitability parameters, Med. Chem. Commun., 9 (2018), 460–470. https://doi.org/10.1039/C7MD00586E doi: 10.1039/C7MD00586E
[30]	R. Gugler, W. Herold, H. J. Dengler, Pharmacokinetics of pindolol in man, Eur. J. Clin. Pharmacol., 7 (1974), 17–24. https://doi.org/10.1007/bf00614385 doi: 10.1007/bf00614385
[31]	J. F. Nash, L. D. Bechtol, R. J. Bopp, K. Z. Farid, C. T. Spradlin, C. A. Bunde, Linear pharmacokinetics of orally administered fenoprofen calcium, J. Pharm. Sci., 68 (1979), 1087–1090. https://doi.org/10.1002/jps.2600680907 doi: 10.1002/jps.2600680907
[32]	J. Grevel, E. Nüesch, E. Abisch, K. Kutz, Pharmacokinetics of oral cyclosporin A (Sandimmun) in healthy subjects, Eur J. Clin. Pharmacol., 31 (1986), 211–216. https://doi.org/10.1007/bf00606661 doi: 10.1007/bf00606661
[33]	L. Debnath, D. Bhatta, Integral transforms and their applications, New York: Chapman and Hall/CRC, 2016. https://doi.org/10.1201/9781420010916
[34]	K. Diethelm, The analysis of fractional differential equations: An application oriented exposition using differential operators of Caputo type, Berlin: Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2
[35]	I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[36]	B. Shiri, Y. G. Shi, D. Baleanu, The well-posedness of incommensurate FDEs in the space of continuous functions, Symmetry, 16 (2024), 105. https://doi.org/10.3390/sym16081058 doi: 10.3390/sym16081058
[37]	D. Baleanu, J. J. Nieto, B. Shiri, Gronwall inequality for incommensurate systems of weakly singular integral equations and its application to FDEs, Differ. Integral Equ., 39 (2026), 1–14. https://doi.org/10.57262/die039-0102-1 doi: 10.57262/die039-0102-1
[38]	W. H. Aellig, Clinical pharmacology of pindolol, Am. Heart J., 104 (1982), 346–356. https://doi.org/10.1016/0002-8703(82)90125-9 doi: 10.1016/0002-8703(82)90125-9
[39]	B. J. Clark, K. Menninger, A. Bertholet, Pindolol‐the pharmacology of a partial agonist, Brit. J. Clin. Pharmaco., 13 (1982), 149S–158S. https://doi.org/10.1111/j.1365-2125.1982.tb01904.x doi: 10.1111/j.1365-2125.1982.tb01904.x
[40]	D. C. Montgomery, E. A. Peck, G. G. Vining, Introduction to linear regression analysis, John Wiley & Sons, 2021.
[41]	C. K. Ward, R. E. Schirmer, Fenoprofen calcium, Anal. Profiles Drug Substances, 6 (1977), 161–182. https://doi.org/10.1016/S0099-5428(08)60344-5 doi: 10.1016/S0099-5428(08)60344-5
[42]	A. Fahr, Cyclosporin clinical pharmacokinetics, Clin. Pharmacokinet., 24 (1993), 472–495. https://doi.org/10.2165/00003088-199324060-00004 doi: 10.2165/00003088-199324060-00004
[43]	G. Ismailos, C. Reppas, J. B. Dressman, P. Macheras, Unusual solubility behaviour of cyclosporin A in aqueous media, J. Pharm. Pharmacol., 43 (1991), 287–289. https://doi.org/10.1111/j.2042-7158.1991.tb06688.x doi: 10.1111/j.2042-7158.1991.tb06688.x

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