Asymptotics of the total net direct pharmacological effect for large drug doses.

The direct pharmacological effect E is described by the Emax model relating E to the drug plasma concentration Cp. The area under the effect vs. time curve (AUCE) is used as the measurement of the total net pharmacological effect. The drug plasma concentrations are solutions of compartmental systems of ordinary differential equations with the input terminated after a finite time and controlled in a proportional manner by a single dose-like parameter. The asymptotics of the time derivative of Cp for large doses are derived and used as conditions which have to be satisfied by functions for which the asymptotics of the integral defining AUCE are derived. The AUCE is proportional to the time TC > EC50 for which the drug concentration stays above the threshold level EC50. The threshold EC50 denotes the drug plasma concentration which elicits 50% of the maximum effect. The parameter TC > EC50 is proportional to the logarithm of drug dose for large doses and its asymptotics is calculated up to the order o(1) as dose increases to infinity. The results are applied to basic pharmacokinetic systems.