Application of the Fractional Calculus in Pharmacokinetic Compartmental Modeling


  • Tahmineh Azizi Department of Mechanical Engineering, Florida State University, Tallahassee, Florida 32310, United States



Fractional calculus, NSFD method, Pharmacokinetic models, Grunwald-Letinkov method


In this study, we present the application of fractional calculus (FC) in biomedicine. We present three different integer order pharmacokinetic models which are widely used in cancer therapy with two and three compartments and we solve them numerically and analytically to demonstrate the absorption, distribution, metabolism, and excretion (ADME) of drug in different tissues. Since tumor cells interactions are systems with memory, the fractional-order framework is a better approach to model the cancer phenomena rather than ordinary and delay differential equations. Therefore, the nonstandard finite difference analysis or NSFD method following the Grunwald-Letinkov discretization may be applied to discretize the model and obtain the fractional-order form to describe the fractal processes of drug movement in body. It will be of great significance to implement a simple and efficient numerical method to solve these fractional-order models. Therefore, numerical methods using finite difference scheme has been carried out to derive the numerical solution of fractional-order two and tri-compartmental pharmacokinetic models for oral drug administration. This study shows that the fractional-order modeling extends the capabilities of the integer order model into the generalized domain of fractional calculus. In addition, the fractional-order modeling gives more power to control the dynamical behaviors of (ADME) process in different tissues because the order of fractional derivative may be used as a new control parameter to extract the variety of governing classes on the non local behaviors of a model, however, the integer order operator only deals with the local and integer order domain. As a matter of fact, NSFD may be used as an effective and very easy method to implement for this type application, and it provides a convenient framework for solving the proposed fractional-order models.


Lin, Z., Gehring, R., Mochel, J.P., Lave, T. and Riviere, J.E., Mathematical modeling and simulation in animal health-Part II: principles, methods, applications, and value of physiologically based pharmacokinetic modeling in veterinary medicine and food safety assessment, Journal of veterinary pharmacology and therapeutics, 39(5), pp. 421-438, 2016.

Brown, R.P., Delp, M.D., Lindstedt, S.L., Rhomberg, L.R. and Beliles, R.P., Physiological parameter values for physiologically based pharmacokinetic models, Toxicology and industrial health, 13(4), pp.407-484, 1997.

Azizi, T. and Mugabi, R., Global Sensitivity Analysis in Physiological Systems, Applied Mathematics, 11(3), pp.119-136, 2020.

Azizi, T, Mathematical modeling with applications in biological systems, physiology, and neuroscience, Kansas State University, 2021.

Pitchaimani, A., Nguyen, T.D.T., Marasini, R., Eliyapura, A., Azizi, T., Jaberi-Douraki, M. and Aryal, S., Biomimetic natural killer membrane camouflaged polymeric nanoparticle for targeted bioimaging, Advanced Functional Materials, 29(4), p.1806817, 2019.

Riviere, J.E., Jaberi-Douraki, M., Lillich, J., Azizi, T., Joo, H., Choi, K., Thakkar, R. and Monteiro-Riviere, N.A., Modeling gold nanoparticle biodistribution after arterial infusion into perfused tissue: Effects of surface coating, size and protein corona, Nanotoxicology, 12(10), pp.1093-1112, 2018.

Marino, S., Hogue, I.B., Ray, C.J. and Kirschner, D.E., A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of theoretical biology, 254(1), pp.178-196, 2008.

Blower, S.M. and Dowlatabadi, H., Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example, International Statistical Review/Revue Internationale de Statistique, pp.229-243, 1994.

Zi, Z., Sensitivity analysis approaches applied to systems biology models, IET systems biology, 5(6), pp. 336-346, 2011.

Dalberg, J., Gimenez, H., Keeley, A., Azizi, T., Xi, X. and Jaberi-Douraki, M., Local and Global Dynamics of Discrete Type 1 Diabetes Model, 2019.

Zhao, P., Zhang, L., Grillo, J.A., Liu, Q., Bullock, J.M., Moon, Y.J., Song, P., Brar, S.S., Madabushi, R., Wu, T.C. and Booth, B.P., Applications of physiologically based pharmacokinetic (PBPK) modeling and simulation during regulatory review, Clinical Pharmacology Therapeutics, 89(2), pp.259-267, 2011.

Barrett, J.S., Alberighi, O.D.C, Laer, S. and Meibohm, B., Physiologically based pharmacokinetic (PBPK) modeling in children, Clinical Pharmacology & Therapeutics, 92(1), pp. 40-49, 2012.

Wagner, C., Zhao, P., Pan, Y., Hsu, V., Grillo, J., Huang, S.M. and Sinha, V., Application of physiologically based pharmacokinetic (PBPK) modeling to support dose selection: report of an FDA public workshop on PBPK, CPT: pharmacometrics & systems pharmacology, 4(4), pp.226-230, 2015.

Sopasakis, P., Sarimveis, H., Macheras, P. and Dokoumetzidis, A., Fractional calculus in pharmacokinetics, Journal of pharmacokinetics and pharmacodynamics, 45(1), pp.107-125, 2018.

Chen, B., Abuassba, Adnan O.M.A., Compartmental Models with Application to Pharmacokinetics, Procedia Computer Science, 187, pp. 60-70, 2021.

Atici, F.M., Atici, M., Nguyen, N., Zhoroev, T. and Koch, G., A study on discrete and discrete fractional pharmacokineticspharmacodynamics models for tumor growth and anti-cancer effects, Computational and Mathematical Biophysics, 7(1), pp. 10-24, 2019.

Gomez-Aguilar, J. F., Lopez-Lopez, M. G., Alvarado-Martinez, V.M., Baleanu, D. and Khan, H., Chaos in a cancer model via fractional derivatives with exponential decay and Mittag-Leffler law, Entropy, 19(12), p. 681, 2017.

Rihan, F. A., Baleanu, D., Lakshmanan, S. and Rakkiyappan, R., On fractional SIRC model with salmonella bacterial infection, Abstract and Applied Analysis, Hindawi, 2014.

Rihan, F.A., Lakshmanan, S., Hashish, A.H., Rakkiyappan, R. and Ahmed, E., Fractional-order delayed predator-prey systems with Holling type-II functional response, Nonlinear Dynamics, 80(1), pp.777-789, 2015.

Rihan, F.A., Hashish, A., Al-Maskari, F., Hussein, M.S., Ahmed, E., Riaz, M.B. and Yafia, R., Dynamics of tumor-immune system with fractional-order, Journal of Tumor Research, 2(1), pp. 109-115, 2016.

Hilfer, R., Applications of fractional calculus in physics, World scientific, pp. 497-528, 2000.

Zeinadini, M. and Namjoo, M., Approximation of fractional-order Chemostat model with nonstandard finite diff erence scheme, Hacettepe Journal of Mathematics and Statistics, 46(3), pp. 469-482, 2017.

Gorenflo, R and Mainardi, F., Fractional calculus, Fractals and fractional calculus in continuum mechanics, pp. 223-276, 1997.

Mainardi, F., Fractional calculus, Fractals and fractional calculus in continuum mechanics, Springer, pp. 291-348, 1997.

Machado, J, T., Kiryakova, V. and Mainardi, F., Recent history of fractional calculus, Communications in nonlinear science and numerical simulation, 16(3), pp. 1140-1153, 2011.

Carpinteri, A. and Mainardi, F., Fractals and fractional calculus in continuum mechanics, Springer, 2014.

Liouville, J., Memoire sur quelques questiona de geometrie et de mechanique, et sur un nouveau genre de calcul pour resoudre ces questions, Journal de l ecole polytechniqu, 13, pp. 16-18, 1831.

Oldham, K. and Spanier, J., The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.

Caputo, M. and Fabrizio, M., A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation & Applications, 1(2), pp. 73-85, 2015.

Atangana, A. and Baleanu, D., Application of fixed point theorem for stability analysis of a nonlinear Schrodinger with Caputo-Liouville derivative, Filomat, 31(8), pp. 2243-2248, 2017.

Miller, K. S., Ross, B., An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.

Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.

Mickens, R, E., Nonstandard finite difference models of differential equations, world scientific, 1994.

Mickens, R, E., Nonstandard finite difference schemes for reaction-diffusion equations, Numerical Methods for Partial Differential Equations: An International Journal, 15(2), pp. 201-214, 1999.

Mickens, R, E., A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion, Computers & Mathematics with Applications, 45(1-3), pp. 429-436, 2003.

Lee, H.A., Imran, M., Monteiro-Riviere, N.A., Colvin, V.L., Yu, W.W. and Riviere, J.E., Biodistribution of quantum dot nanoparticles in perfused skin: evidence of coating dependency and periodicity in arterial extraction, Nano Letters, 7(9), pp.2865-2870, 2007.