Kinetic interpretation of log-logistic dose-time response curves.

Walter W Focke,Ndeke Musee,Isbe Van Der Westhuizen,Mattheüs Theodor Loots

doi:10.1038/s41598-017-02474-w

Abstract

A Hill-type time-response curve was derived using a single-step chemical kinetics approximation. The rate expression for the transformation is a differential equation that provides an interpolation formula between the logistic growth curve and second order kinetics. The solution is equivalent to the log-logistic cumulative distribution function with the time constant expressed in terms of a kinetic rate constant. This expression was extended to a full dose-time-response equation by postulating a concentration dependence for the rate constant. This was achieved by invoking a modified form of Haber’s law that connects an observed toxic effect with the concentration of the active agent and the elapsed exposure time. Analysis showed that the concept of Concentration Addition corresponds to a special case where the rate constant for the overall transformation rate is proportional to the sum of the rate constants that apply when the agents act individually. Biodiesel “survival” curves were measured and used to test the applicability of the empirical model to describe the effects of inhibitor dosage and binary inhibitor mixtures. Positive results suggest that the proposed dose-response relationship for the toxicity of agents to organisms can be extended to inanimate systems especially in cases where accurate mechanistic models are lacking.

Highlights

A Hill-type time-response curve was derived using a single-step chemical kinetics approximation
The property of central interest is the time tm where the transformation equals the median effect value Pm = (P∞ − Po)/2. (b) Dose response curve for a fixed elapsed treatment time derived from the set of timeresponse curves
A Hill-type time-response function follows from the assumption that the transformation of a substrate, under the influence of an effector agent, is described by an autocatalytic rate expression

Summary

Introduction

A Hill-type time-response curve was derived using a single-step chemical kinetics approximation. The solution is equivalent to the log-logistic cumulative distribution function with the time constant expressed in terms of a kinetic rate constant This expression was extended to a full dose-time-response equation by postulating a concentration dependence for the rate constant. Dose-response models are used to describe the effect of the amount of a toxin or a therapeutic drug on survival in biological communities; the combined effect of inhibitors on enzymes[10]; the relationships among exposure time, concentration, and toxicity of insecticides[11]; the quantity of applied fertiliser on agricultural yield[12]; the relationship between herbicide dose and plant response[13]; the decomposition kinetics[14] or the polymorphic transformations of solids[15], and the concentration of antioxidants on the oxidative stability of organic materials, etc. The median response is attained at a dosage equivalent to LC50

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