Statistical modeling of the hormetic dose zone and the toxic potency completes the quantitative description of hormetic dose responses.

Quantifying the characteristics of hormesis provides valuable insights into this low-dose phenomenon and helps to display and capture its variability. A prerequisite to do so is a statistical procedure allowing quantification of general hormetic features, namely the maximum stimulatory response, the dose range of hormesis, and the distance from the maximum stimulation to the dose where hormesis disappears. Applying extensions of a hormetic dose-response model that is well-established in plant biology provides a direct estimation of several quantities, except the hormetic dose range. Another dose range that is difficult to model directly is the distance between the dose where hormesis disappears and the dose giving 50% inhibition, known as toxic potency. The present study presents 2 further model extensions allowing for a direct quantification of the hormetic dose range and the toxic potency. Based on this, a 4-step mathematical modeling approach is demonstrated to quantify various dose-response quantities, to compare these quantities among treatments, and to interrelate hormesis features. Practical challenges are exemplified, and possible remedies are identified. The software code to perform the analysis is provided as Supplemental Data to simplify adoption of the modeling procedure. Because numerous patterns of hormesis are observed in various sciences, it is clear that the proposed approach cannot cope with all patterns; however, it should be possible to analyze a great range of hormesis patterns.