Fractional kinetics in drug absorption and disposition processes
We explore the use of fractional order differential equations for the analysis of datasets of various drug processes that present anomalous kinetics, i.e. kinetics that are non-exponential and are typically described by power-laws. A fractional differential equation corresponds to a differential equation with a derivative of fractional order. The fractional equivalents of the “zero-” and “first-order” processes are derived. The fractional zero-order process is a power-law while the fractional first-order process is a Mittag–Leffler function. The latter behaves as a stretched exponential for early times and as a power-law for later times. Applications of these two basic results for drug dissolution/release and drug disposition are presented. The fractional model of dissolution is fitted successfully to datasets taken from literature of in vivo dissolution curves. Also, the proposed pharmacokinetic model is fitted to a dataset which exhibits power-law terminal phase. The Mittag–Leffler function describes well the data for small and large time scales and presents an advantage over empirical power-laws which go to infinity as time approaches zero. The proposed approach is compared conceptually with fractal kinetics, an alternative approach to describe datasets with non exponential kinetics. Fractional kinetics offers an elegant description of anomalous kinetics, with a valid scientific basis, since it has already been applied in problems of diffusion in other fields, and describes well the data.
KeywordsFractional kinetics Anomalous kinetics Power-law
- 11.Phan G, Le Gall B, Deverre JR, Fattal E, Bénech H (2006) Predicting plutonium decorporation efficacy after intravenous administration of DTPA formulations: Study of pharmacokinetic-pharmacodynamic relationships in rats. Pharm Res 23:2030–2035. doi: 10.1007/s11095-006-9046-x PubMedCrossRefGoogle Scholar
- 15.Podlubny I (1999) Fractional differential equations. Academic Press, San DiegoGoogle Scholar
- 26.Podlubny I, Kacenak M (2005) Matlab implementation of the Mittag-Leffler function. MATLAB Central, Available online: http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=8738