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An Application of the Grünwald-Letinkov Fractional Derivative to a Study of Drug Diffusion in Pharmacokinetic Compartmental Models

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Mathematical Models and Computer Simulations for Biomedical Applications (MCHBS 2021)

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 33))

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Abstract

In this study, we present the application of fractional calculus (FC) in biomedicine. We present three different integer order pharmacokinetics 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 or nanoparticles (NPs) 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 Grünwald-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 pharmacokinetics models for oral drug administration. This study shows that the fractional order modeling extends the capabilities of 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.

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Acknowledgements

The author would like to thank the reviewers, whose comments really helped to improve the manuscript.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Florida State University, Tallahassee, FL, USA

    Tahmineh Azizi

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  1. Tahmineh Azizi
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Correspondence to Tahmineh Azizi .

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Editors and Affiliations

  1. Istituto per le Applicazioni del Calcolo (IAC) “M. Picone”, Consiglio Nazionale delle Ricerche (CNR - National Research Council), Roma, Italy

    Gabriella Bretti

  2. Istituto per le Applicazioni del Calcolo (IAC) “M. Picone”, Consiglio Nazionale delle Ricerche (CNR - National Research Council), Roma, Italy

    Roberto Natalini

  3. Department of Biotechnology and Biosciences, University of Milano-Bicocca, Milano, Italy

    Pasquale Palumbo

  4. Department of Mathematical Sciences, Politecnico di Torino, Torino, Italy

    Luigi Preziosi

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Azizi, T. (2023). An Application of the Grünwald-Letinkov Fractional Derivative to a Study of Drug Diffusion in Pharmacokinetic Compartmental Models. In: Bretti, G., Natalini, R., Palumbo, P., Preziosi, L. (eds) Mathematical Models and Computer Simulations for Biomedical Applications. MCHBS 2021. SEMA SIMAI Springer Series, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-031-35715-2_1

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