Dose–response regressions for algal growth and similar continuous endpoints: Calculation of effective concentrations

Abstract

We derive equations for the effective concentration giving 10% inhibition (EC10) with 95% confidence limits for probit (log-normal), Weibull, and logistic dose-response models on the basis of experimentally derived median effective concentrations (EC50s) and the curve slope at the central point (50% inhibition). For illustration, data from closed, freshwater algal assays are analyzed using the green alga Pseudokirchneriella subcapitata with growth rate as the response parameter. Dose-response regressions for four test chemicals (tetraethylammonium bromide, musculamine, benzonitrile, and 4-4-(trifluoromethyl)phenoxy-phenol) with ranges of representative slopes at 50% response (0.54-2.62) and EC50s (2.20-357 mg/L) were selected. Reference EC50s and EC10s with 95% confidence limits using probit or Weibull models are calculated by nonlinear regression on the whole dataset using a dose-response regression program with variance weighting and proper inverse estimation. The Weibull model provides the best fit to the data for all four chemicals. Predicted EC10s (95% confidence limits) from our derived equations are quite accurate; for example, with 4-4-(trifluoromethyl)phenoxy-phenol and the probit model, we obtain 1.40 (1.22-1.61) mg/L versus 1.40 (1.20- 1.64) mg/L obtained from the nonlinear regression program. The main advantage of the approach is that EC10 or ECx (where x = 1-99) can be predicted from well-determined responses around EC20 to EC80 without experimental data in the low- or high-response range. Problems with the estimation of confidence interval for EClow,x (concentration predicted to cause x% inhibition) from algal growth inhibition also are addressed. Large confidence intervals may be the result of experimental error and lack of a well-defined reference response value.