Statistical modeling of effective doses in hormetic dose-response relationships by reparameterization of a bilogistic model for inverted U-shaped curves

Abstract

Hormesis is a biphasic dose (concentration)-response phenomenon characterized by low dose stimulatory and high dose inhibitory effects exerted by stressors on living organisms. Recently, there has been increased interest in the phenomenon and statistical models for its exploration in toxicological studies. The bilogistic model of Beckon et al. is one of such models for modeling biphasic dose-response relationships in toxicological studies. However, there is no explicit formula for the estimation of effective doses (EDK) with the model. In this study, a simple general approach was suggested to reparameterize the model, leading to a range of mathematical models for determination of effective doses at both stimulatory and inhibitory ranges in inverted U-shaped hormetic dose-response relationships. The reparameterized models were tested on experimental data from three different in vitro experimental systems obtained from literature and our experiment. They were successfully applied to test for significance of hormesis and estimate effective doses and their statistical properties. In addition, reparameterization of the model for a particular effective dose (EDK) did not affect estimation of other parameters (such as x1, x2, β1, β2 and M). The reparameterized models provided useful tools for adequate exploration of the tested hormetic dose-response relationships. The extended models could hopefully be versatile in characterization of variable hormetic dose response relationships in many toxicological disciplines.