How to avoid unbounded drug accumulation with fractional pharmacokinetics

A number of studies have shown that certain drugs follow an anomalous kinetics that can hardly be represented by classical models. Instead, fractional-order pharmacokinetics models have proved to be better suited to represent the time course of these drugs in the body. Unlike classical models, fractional models can represent memory effects and a power-law terminal phase. They give rise to a more complex kinetics that better reflects the complexity of the human body. By doing so, they also spotlight potential issues that were ignored by classical models. Among those issues is the accumulation of drug that carries on indefinitely when the infusion rate is constant and the elimination flux is fractional. Such an unbounded accumulation could have important clinical implications and thus requires a solution to reach a steady state. We have considered a fractional one-compartment model with a continuous intravenous infusion and studied how the infusion rate influences the total amount of drug in the compartment. By taking an infusion rate that decays like a power law, we have been able to stabilize the amount of drug in the compartment. In the case of multiple dosing administration, we propose recurrence relations for the doses and the dosing times that also prevent drug accumulation. By introducing a numerical discretization of the model equations, we have been able to consider a more realistic two-compartment model with both continuous infusion and multiple dosing administration. That numerical model has been applied to amiodarone, a drug known to have an anomalous kinetics. Numerical results suggest that unbounded drug accumulation can again be prevented by using a drug input function that decays as a power law.