Nonstandard finite difference approach for solving 3-compartment pharmacokinetic models

Abstract

Complex nature of the analytical solutions to 3-compartment pharmacokinetic models leads to the discrete approximation of the continuous differential equation been mostly used. In this paper, we applied nonstandard finite difference method to 3-compartment pharmacokinetic models. This method was introduced to compensate for the weaknesses of methods such as the standard finite difference methods, numerical instabilities being a prime example of such. Three-compartment pharmacokinetic model with 2 different routes of administration (IV bolus injection and IV bolus infusion) is considered for simulations. For the case when the system is homogeneous (models arising from IV bolus injection mode of administration), "exact" finite difference scheme is obtained for any step size while in the case of nonhomogeneous (models arising from IV bolus infusion route of administration), scheme that has the same qualitative behavior as the analytical solution for all step sizes is obtained. It was shown computationally that the nonstandard finite difference scheme for 3-compartment pharmacokinetic model with IV bolus mode of administration is exact while 3-compartment pharmacokinetic model with IV infusion is dynamically consistent with the continuous model for all step sizes.

Keywords: exact finite difference scheme; intravenous infusion; intravenous injection; nonstandard finite difference; pharmacokinetic model; solve.