How to avoid unbounded drug accumulation with fractional pharmacokinetics
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A number of studies have shown that certain drugs follow an anomalous kinetics that can hardly be represented by classical models. Instead, fractional-order pharmacokinetics models have proved to be better suited to represent the time course of these drugs in the body. Unlike classical models, fractional models can represent memory effects and a power-law terminal phase. They give rise to a more complex kinetics that better reflects the complexity of the human body. By doing so, they also spotlight potential issues that were ignored by classical models. Among those issues is the accumulation of drug that carries on indefinitely when the infusion rate is constant and the elimination flux is fractional. Such an unbounded accumulation could have important clinical implications and thus requires a solution to reach a steady state. We have considered a fractional one-compartment model with a continuous intravenous infusion and studied how the infusion rate influences the total amount of drug in the compartment. By taking an infusion rate that decays like a power law, we have been able to stabilize the amount of drug in the compartment. In the case of multiple dosing administration, we propose recurrence relations for the doses and the dosing times that also prevent drug accumulation. By introducing a numerical discretization of the model equations, we have been able to consider a more realistic two-compartment model with both continuous infusion and multiple dosing administration. That numerical model has been applied to amiodarone, a drug known to have an anomalous kinetics. Numerical results suggest that unbounded drug accumulation can again be prevented by using a drug input function that decays as a power law.
KeywordsFractional kinetics Compartmental models Drug accumulation Amiodarone
The implementation of the numerical examples presented in this paper have been made much easier thanks to the use of the chebfun library  and Igor Podlubny’s MATLAB routine for evaluating the Mittag-Leffler function.
- 1.Anderson J, Tomlinson RW (1966) The distribution of calcium-47 in the rat. J Physiol 182(3):664–670Google Scholar
- 3.Boyd JP (2001) Chebyshev and Fourier Spectral Methods (2nd edition). Dover PublicationsGoogle Scholar
- 4.Brockmann D (2009) Human mobility and spatial disease dynamics, vol 10. In: Schuster HG (ed) Reviews of nonlinear dynamics and complexity. Wiley-VCH Weinheim, pp 1–24Google Scholar
- 5.Carpinteri A, Mainardi F (1997) Fractals and fractional calculus in continuum mechanics. Springer-Verlag LondonGoogle Scholar
- 10.del Castillo Negrete D, Carreras BA, Lynch VE (2005) Nondiffusive transport in plasma turbulence: A fractional diffusion approach. Phys Rev Lett 94(065003)Google Scholar
- 15.Fuite J, Marsh R, Tuszynski J (2002) Fractal pharmacokinetics of the drug mibefradil in the liver. Phys Rev E 66(2):021904(1–11)Google Scholar
- 22.Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. North-Holland Mathematics Studies, Elsevier Science, Salt Lake CityGoogle Scholar
- 23.Magin R (2006) Fractional calculus in bioengineering. Begell House Publishers, New York Google Scholar
- 24.Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Rep 339:1–77Google Scholar
- 27.Podlubny I (1999) Fractional differential equations. Mathematics in science and engineering, Volume 198. Academic Press, San Diego Google Scholar
- 33.Trefethen LN et al (2011) Chebfun Version 4.0. The Chebfun Development Team, http://www.maths.ox.ac.uk/chebfun/. Accessed 5 Nov 2013