Fractional kinetics in multi-compartmental systems

Abstract

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) for one-compartment models. In this work we extend that theory to multi-compartmental models. Unlike systems defined by a single ordinary differential equation (ODE), considering fractional multi-compartmental models is not as simple as changing the order of the ordinary derivatives of the left-hand side of the ODEs to fractional orders. The latter may produce inconsistent systems which violate mass balance. We present a rationale for fractionalization of ODEs, which produces consistent systems and allows processes of different fractional orders in the same system. We also apply a method of solving such systems based on a numerical inverse Laplace transform algorithm, which we demonstrate that is consistent with analytical solutions when these are available. As examples of our approach, we consider two cases of a basic two-compartment PK model with a single IV dose and multiple oral dosing, where the transfer from the peripheral to the central compartment is of fractional order α < 1, accounting for anomalous kinetics and deep tissue trapping, while all other processes are of the usual order 1. Simulations with the studied systems are performed using the numerical inverse Laplace transform method. It is shown that the presence of a transfer rate of fractional order produces a non-exponential terminal phase, while multiple dose and constant infusion systems never reach steady state and drug accumulation carries on indefinitely. The IV fractional system is also fitted to PK data and parameter values are estimated. In conclusion, our approach allows the formulation of systems of FDEs, mixing different fractional orders, in a consistent manner and also provides a method for the numerical solution of these systems.

Robust model predictive control for discrete-time fractional-order systems

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Critically Safe General Anaesthesia in Closed Loop: Availability and Challenges

“…The emerging non-classical concepts of fractional derivatives and fractional functions (in time domain) and fractional order impedance models (in frequency domain) have made the core of the original research performed in the last decade by few research groups (Ionescu 2012;Dokomuetzidis et al, 2010;Popovic et al, 2010Popovic et al, , 2011. However, research is still needed to prove that these kind of models can improve the perception of today's patient response to drug interactions.…”

Controlled Drug Administration by a Fractional PID

“…Fractional dynamics can be cast as Physiologically Based Pharmacokinetic Models (PBPK) with fractional-order derivatives (see Dokoumetzidis et al [2010a]) where the mass balance equations are rewritten using fractionalorder derivatives. This offers a mechanistic understanding of the interplay among the main factors of drug distribution, allows us to draw individualized concentrationtime profiles and study drug-drug interactions using the fractional calculus approach.…”

“…Amiodarone is well-known for its highly nonlinear non-exponential dynamics and singular long-term accumulation pattern. Recently, Dokoumetzidis et al [2010a] modelled the pharmacokinetic distribution of Amiodarone with a fractional compartmental model following a single i.v. and a single oral dose.…”