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DIFFERENTIAL EQUATIONS WITH FRACTIONAL DERIVATIVES

The fractional differential calculus is the branch of calculus dealing with derivatives, integrals and fractional differential equations, i.e. non-integer order e.g., the half-derivative is the derivative order of 1/2. The fractional derivatives were defined for the first time 300 years ago by Leibnitz, but the theoretical interest for them, revived a couple of decades earlier and the first implementations have appeared only recently.

The main implementation of the fractional differential calculus on natural sciences refers to phenomena in which the abnormal diffusion dominates, i.e. diffusion that deviates by Fick law.

The laboratory of Biopharmaceutics and Pharmacokinetics of UOA was the first that introduced the fractional differential calculus in the field of pharmacokinetics but also in pharmaceutics in general. Pharmacokinetics is the branch of pharmaceutics that quantitatively describes the course of the drug in the body and uses mathematical models that are usually expressed by differential equations.

Drug monitoring, from the time of intake until its excretion from the body

FRACTIONAL PHARMACOKINETICS

In the human body, the diffusion processes play a major role since the drug distribution is mainly carried out through diffusion. Despite the fact that the human body is a complex and heterogeneous structure, the diffusion processes often deviate from classical ideal rates.

The differential equations with fractional derivatives are suitable for describing the courses of some drugs in the body, where the classical expressions are a special case of the new wording with the fractional formalism. The laboratory of Biopharmaceutics and Pharmacokinetics, which introduced this concept in the field of pharmaceutics, presents a series of significant achievements that are a major progress in the area, for both the pharmaceutical and the mathematics community that deals with the applications of fractional differential equations.

The above picture shows a diagram of the fractional multi-compartmental pharmacokinetic model. The drug removal as well as its transfer from the central to the peripheral compartment are carried out with classical first-order kinetics, while the return of the drug from the peripheral to the central compartment is considered to follow slow fractional kinetics, due to entrapment of the drug inside the peripheral tissues.

Pharmaceutical Sciences World Congress, 2010

Professor Panos Macheras received the Research Achievement Award during the Pharmaceutical Sciences World Congress in 2010, for his contribution to the study of the heterogeneity of drug kinetics, part of which is the fractional pharmacokinetics.

The contribution of the laboratory on the development of fractional pharmacokinetics is summarized:
- to the introduction of concepts of fractional differential calculus and formulation of first pharmacokinetics models.
- to the expansion of fractional mathematical formalism to multi-compartmental models and evaluation of their parameters from pharmacokinetic data.
- to the development and application of numerical methods for solving differential equations with fractional derivatives.
- to the use of fractional kinetics for the description of the gastrointestinal drug absorption.

NATIONAL AND KAPODISTRIAN UNIVERSITY OF ATHENS

Dokoumetzidis Aristeidis, Lecturer the Pharmaceutical Technology Department, Faculty of Pharmacy.

Macheras Panos, Professor of Biopharmaceutics and Pharmacokinetics, Director of the Pharmaceutical Technology Department, Faculty of Pharmacy.

http://www.pharm.uoa.gr http://www.fip.org