Single toxin dose-response models revisited.

The goal of this paper is to offer a rigorous analysis of the sigmoid shape single toxin dose-response relationship. The toxin efficacy function is introduced and four special points, including maximum toxin efficacy and inflection points, on the dose-response curve are defined. The special points define three phases of the toxin effect on mortality: (1) toxin concentrations smaller than the first inflection point or (2) larger then the second inflection point imply low mortality rate, and (3) concentrations between the first and the second inflection points imply high mortality rate. Probabilistic interpretation and mathematical analysis for each of the four models, Hill, logit, probit, and Weibull is provided. Two general model extensions are introduced: (1) the multi-target hit model that accounts for the existence of several vital receptors affected by the toxin, and (2) model with a nonzero mortality at zero concentration to account for natural mortality. Special attention is given to statistical estimation in the framework of the generalized linear model with the binomial dependent variable as the mortality count in each experiment, contrary to the widespread nonlinear regression treating the mortality rate as continuous variable. The models are illustrated using standard EPA Daphnia acute (48h) toxicity tests with mortality as a function of NiCl or CuSO4 toxin.