**Multiple dosing for linear fractional pharmacokinetic systems**

A. Dokoumetzidis (1), R. Magin (2), P. Macheras (1)

(1) School of Pharmacy, University of Athens, Greece; (2) Department of Bioengineering, University of Illinois at Chicago, Chicago, IL, USA

**Objectives:** We investigate the implementation of multiple dosing in pharmacokinetic systems including fractional rates.

**Methods:** One of the problems of fractional calculus is the initialisation of fractional differential equations because of the time memory effects, which may have consequences in the implementation of multiple dosing systems. In this poster we investigate the implementation of a multiple dosing scheme in a one compartment model (i.e. a Mittag-Leffler (ML) function), with two methods, which both work in the classic, non-fractional case: (i) Considering each additional dose by reinitialization i.e. with ML functions where the initial value of each is the final value of the previous one; and (ii) implementing the multiple dosing by adding up several single dose profiles, using the superposition principle. We assess whether these techniques work for fractional systems by comparing them to the limit case of a one-compartment fractional model with constant infusion which has an analytical solution.

**Results:** We derive the analytical solution of a PK model with fractional elimination and constant infusion which involves a Mittag-Leffler function. A multiple dose system is implemented by reinitialization and by superposition. The two methods give the same profiles only when the order α=1. For a fractional α=0.5 the 2 methods give different profiles and only the superposition method gives profiles which follow the constant infusion model in the limit when the dose and the dosing interval become very small. The reinitialization method fails to do that. Both the constant infusion and the multiple dose system demonstrate the lack of a steady state and the ever ending accumulation of drug as a result of the presence of fractional kinetics. In the case of the constant infusion where there is analytical solution we prove mathematically that for infinite time the solution goes to infinity and not to a steady state. This is an important clinical implication of the presence of fractional kinetics.

**Conclusion:** Multiple dosing in linear pharmacokinetic systems with fractional rates can be implemented using the superposition principle exactly the same way as in ordinary PK systems, however the reinitialization method fails. The important implication of lack of a steady state for constant rate multiple dosing (or infusion) is also pointed out.