Influence of Dose Range on Degree of Nonlinearity Detected in Dose-Proportionality Studies for Drugs with Saturable Elimination: Single-Dose and Steady-State Studies

Abstract

Deviation from proportionality occurs when the ratio of area under the curve (AUC) values is not equal to the ratio of administered doses. The degree of nonlinearity (f_NL) can be quantitated as the ratio of AUCs divided by the ratio of doses. We explore positive deviation from proportionality (f_NL > 1) using the classical Michaelis–Menten model of nonlinear elimination after a single dose (n = 1) or at steady state (ss). The degree of nonlinearity is related to the ratio of the highest dose to the lowest dose (Rd = D_H/D_L): f_NLⁿ⁼¹ = (2 + Rd · ε)/(2 + ε), f_NL^ss= (Rd · Ω −1) /(Rd · Ω −Rd), where ε is the ratio of the initial concentration after the lowest dose to the K_m (ε = D_L/K_m · V) and Ω is the ratio of the V_max to the average rate of input for the highest dose (Ω = V_maxτ/F · D_H). From these relationships, we find that (1) for single-dose studies, K_m is the important Michaelis–Menten parameter, while V_max is important at steady state; (2) the degree of nonlinearity cannot exceed the ratio of doses in single-dose studies, and when doses in extreme excess of K_m· V are chosen, the degree of nonlinearity is equal to the dose range; and (3) at steady state, the degree of nonlinearity can exceed the ratio of doses and approaches infinity as the average input rate approaches V_max. Literature data (phenytoin and ethanol) support these findings. We conclude that the degree of nonlinearity is not a useful measure of nonlinearity in and of itself and propose percentage saturation as being more informative.