Modeling non-monotonic dose–response relationships: Model evaluation and hormetic quantities exploration

Non-monotonic (biphasic) dose–response relationships, known as hormetic relationships, have been observed across multiple experimental systems. Several models were proposed to describe non-monotonic relationships. However, few studies provided comprehensive description of hermetic quantities and their potential application. In this study, five biphasic models were used to fit five hormetic datasets from three different experimental systems of our lab. The bisection algorithm based on individual monotone functions was proposed to calculate arbitrary hormetic quantities instead of traditional methods (e.g., model reparameterization) which need complex mathematical manipulation. Results showed that all the five biphasic models could describe those datasets fairly well with coefficient of determination (R2adjR2adj) greater than 0.95 and root mean square error (RMSE) smaller than 0.10. The best-fit model could be selected based on R2adjR2adj, RMSE, and a supplemental criterion of Akaike information criterion (AIC). Hormetic quantities that trigger 10% stimulation at the left (ECL10) and right (ECR10) side of stimulatory peak were calculated and emphasized for their implication in hormesis exploration for the first time. Furthermore, the ECL10, proposed as an alarm threshold for hormesis, was expected to be useful in risk assessment of environmental chemicals. This study lays a foundation in the quantitative description of the low dose hormetic effect and the investigation of hormesis in environmental risk assessment.

► We evaluated five biphasic models using five hormetic dose-response data.
► We calculated effect concentrations at hormetic range using bisection method.
► Concentration causes 10% stimulation was set as alarm threshold for hormetic effect.