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

The direct pharmacological effect E is described by the E_max model relating E to the drug plasma concentration C_p. The area under the effect vs. time curve (AUC_E) 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 C_p for large doses are derived and used as conditions which have to be satisfied by functions for which the asymptotics of the integral defining AUC_E are derived. The AUC_E is proportional to the time T_C>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 T_C>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.