A commentary on fractionalization of multi-compartmental models

Fractional calculus, the branch of calculus that deals with derivatives of non-integer order, e.g., a half derivative, allows the formulation of fractional differential equations (FDEs), capable of describing a range of phenomena, most of them related in one way or another to anomalous diffusion processes [1, 2]. FDEs have recently found application in the field of pharmacokinetics (PK), since the presence of non-classical, anomalous kinetics has been established years ago and many articles have appeared in the literature trying to quantify these processes by the use of either empirical power-laws or fractal kinetics [3, 4]. Fractional pharmacokinetics (fPK) was first described by Dokoumetzidis and Macheras in [5] where the concept was introduced for a simple ‘‘one-compartment’’ model that gave rise to a Mittag-Leffler function (MLF). The MLF has very nice properties since it behaves as a power law for large time scales but as an exponential for small times, hence the MLF can describe kinetic data that follows power law terminal kinetics without presenting problems for t = 0. However, if we want to write models in more physiological terms then eventually we will need a formulation with more than one compartments. The first attempt to write multicompartmental fPK models was done by Popovic et al. [6], who basically fractionalized the classic compartmental pharmacokinetics in a straightforward manner, i.e., they generalized the classic first order derivatives found on the left hand side of ordinary differential equations (ODEs) by replacing them with fractional derivatives. But is this change in the order of the derivatives all that is needed to establish correct and consistent fPK models? In this note we will demonstrate that it is not.