Compartmental Models with Application to Pharmacokinetics

Abstract

In this paper, a comprehensive study on different forms of a system of fractional differential equations (SFDE) on the compartmental system is investigated, distinguishing the characteristics surrounding these types: (1) commensurable SFDE; (2) non-commensurable SFDE; (3) implicit non-commensurable SFDE. With further classification, we illustrate the types and their characteristics by first modeling a classical derivative two-compartmental system with a single intravenous (I.V) dose through which the three models would be constructed. It is shown that the properties surrounding a non-commensurable SFDE type violate the idea of mass balance. It however, survives the theory of fractional calculus in modeling anomalous kinetics especially in an area where compartmental analysis is applicable such as Pharmacokinetics. We may affirm this by fitting the various forms of the proposed models to an experimental dataset with parameters determined by the least square method. In addition, the results show that the non-commensurable model predicts a good fit as compare to that of the other two models nevertheless, these fractional models explain the anomalous behavior better than the classical models. The fractional derivative used in this work is described in the Caputo sense. Since analytic solutions are sometimes difficult to be obtained, we implement the use of fractional finite difference method (FFDM) to simulate solutions to the fractional compartmental models. Numerical results also illustrate that the proposed numerical method is equally efficient to solve this and any complicated compartmental models since they perform well when the simulations are done for the classical case of the model.