Numerical solution of nonlinear fractional compartmental systems
A. Dokoumetzidis, P Macheras
School of Pharmacy, University of Athens, Greece
Objectives: We present a method to formulate and solve numerically, nonlinear pharmacokinetic systems which include fractional rates. As an example we consider the fractional Michaelis - Menten (MM) kinetics.
Methods: In  a method of formulating compartmental systems with linear fractional differential equations (FDE) was introduced with applications in pharmacokinetics. Here we extent this formulation to nonlinear FDEs. We further report a method to solve numerically such systems by finite differences (FD) based on  but with modifications to account for the special form of the equations that we use. A simple linear FDE of a one-compartment PK model with fractional elimination and constant infusion which has an analytical solution was solved with the FD algorithm and the numerical solution was compared to the analytical one. Further, a nonlinear model of a two compartment model with fractional MM elimination of order α was solved with the FD method. The special case when the order was set to α=1 was also compared to the output of a Runge - Kutta algorithm, MATLAB ode45.
Results: Since the formulation of our FDEs includes derivatives on the left and the right hand side, the original algorithm of  had to be modified. The finite differences scheme became implicit instead of the explicit of  and therefore an additional step of numerically solving an algebraic equation was introduced at each step of the integrator. The FD numerical algorithm gave identical results to the analytical solution of a linear FDE, proving that it works well, for linear systems. The FD algorithm provided a solution for a nonlinear system of FDEs. In the case where the order of the FDEs was set to 1, the FD algorithm provided the same result as a common Runge - Kutta routine. This algorithm can also be used for linear systems as an alternative to the Numerical Inverse Laplace Transform method that we proposed in  since it may be faster and more stable.
Conclusion: An algorithm to solve numerically nonlinear systems of FDEs was shown to perform well. This algorithm can be considered as general purpose and may be used for linear systems too.
 Dokoumetzidis A., Magin R., Macheras P. Fractional kinetics in multi-compartmental systems. J. Pharmacokinet Pharmacodyn. 37: 508-524 (2010).
 Petras I. Fractional-Order Nonlinear Systems. Springer 2011.