 ## Fractional kinetics in multi-compartmental systems

Aristides Dokoumetzidis, Richard Magin, Panos Macheras

University of Athens

Objectives: Fractional calculus, the branch of calculus dealing with derivatives of non-integer order (e.g., the half-derivative) allows the formulation of fractional differential equations (FDEs), which have recently been applied to pharmacokinetics (PK) . In this work we extend this theory to multi-compartmental models; we introduce a method to solve numerically such models and we also present applications in PK.

Methods: The solution of the fractional  “one-compartment” model with linear elimination is a Mittag-Leffler function (MLF), which is the fractional analogue of the exponential function . The MLF has good properties and behaves as a power law for long time scales while as an exponential for early times, hence it can describe kinetic data that follows power law terminal kinetics without exploding at t = 0. However, considering fractional multi-compartmental models is not as simple as changing the order of the ordinary derivatives on the left-hand side of the ODEs to fractional orders. We present a rationale of fractionalization of ODEs and a method of solving any linear system of FDEs based on a numerical inverse Laplace transform algorithm.

Results: Fractionalization of ODEs with different orders of fractional derivatives, when performed naively, may produce inconsistent systems, which violate mass balance. Our approach to fractionalization produces consistent systems and allows considering processes of different fractional orders to coexist in the same system. As an application, a two compartment model is considered, where elimination and transfer from compartment 1 to 2 are of the usual order 1, while transfer from compartment 2 to 1 is of fractional order α < 1, accounting for anomalous kinetics and deep tissue trapping. The system is solved using numerical inverse Laplace transform which produces the correct profile when α = 1 (classic 2-compartment model), hence verifying that the algorithm works. The system is fitted to PK data and parameters are estimated.

Conclusions: FDEs are a useful tool in pharmacokinetics, effectively modeling datasets that have power-law kinetics and accounting for anomalous diffusion and deep tissue trapping. Our approach allows the formulation of systems of FDEs, mixing different fractional orders, in a consistent manner and also provides for a numerical solution to these systems.

References:
 Dokoumetzidis, A., Macheras, P. (2009) Fractional kinetics in drug absorption and disposition processes. J. Pharmacokinet. Pharmacodyn. 36: 165-178.

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