Modeling effective dosages in hormetic dose-response studies

Abstract

Background: Two hormetic modifications of a monotonically decreasing log-logistic dose-response function are most often used to model stimulatory effects of low dosages of a toxicant in plant biology. As just one of these empirical models is yet properly parameterized to allow inference about quantities of interest, this study contributes the parameterized functions for the second hormetic model and compares the estimates of effective dosages between both models based on 23 hormetic data sets. Based on this, the impact on effective dosage estimations was evaluated, especially in case of a substantially inferior fit by one of the two models.

Methodology/principal findings: The data sets evaluated described the hormetic responses of four different test plant species exposed to 15 different chemical stressors in two different experimental dose-response test designs. Out of the 23 data sets, one could not be described by any of the two models, 14 could be better described by one of the two models, and eight could be equally described by both models. In cases of misspecification by any of the two models, the differences between effective dosages estimates (0-1768%) greatly exceeded the differences observed when both models provided a satisfactory fit (0-26%). This suggests that the conclusions drawn depending on the model used may diverge considerably when using an improper hormetic model especially regarding effective dosages quantifying hormesis.

Conclusions/significance: The study showed that hormetic dose responses can take on many shapes and that this diversity can not be captured by a single model without risking considerable misinterpretation. However, the two empirical models considered in this paper together provide a powerful means to model, prove, and now also to quantify a wide range of hormetic responses by reparameterization. Despite this, they should not be applied uncritically, but after statistical and graphical assessment of their adequacy.

Figure 1. Both logistic hormetic models unsuitable.

Dose-response relationship for the effect of 2-phenylethyl-isothiocyanate on root growth of Amaranthus hybridus and its description by the hormetic dose-response models after Brain and Cousens (grey curve) or Cedergreen et al. (black curve) (A). (B) Response modelling by the An-Johnson-Lovett Model II (black curve). Only the An-Johnson-Lovett Model II showed a significant hormetic effect. Error bars represent standard deviation.

Figure 2. Brain and Cousens model more suitable.

Dose-response relationships for effects of different phytotoxins and an aqueous leaf extract of Parthenium hysterophorus at the flowering stage on root growth of Lactuca sativa and their description by the hormetic dose-response models after Brain and Cousens (grey curve) or Cedergreen et al. (black curve). The Cedergreen et al. model did not detect significant hormetic responses (f<0) (A–E) or provided an inferior fit (F). Error bars represent standard deviation. DM = dry leaf mass.

Figure 3. Cedergreen et al. model more suitable.

Dose-response relationships for effects of different phytotoxins on root growth of Lactuca sativa (A–G) and for effects of root exudates of Hordeum vulgare on root growth of Sinapis alba (H) and their description by the hormetic dose-response models after Brain and Cousens (grey curve) or Cedergreen et al. (black curve). The Brain and Cousens model detected significant hormetic responses, but provided inferior fits. Error bars represent standard deviation.

Figure 4. Both logistic hormetic models equally suitable.

Dose-response relationships for effects of different phytotoxins on root growth of Amaranthus hybridus (A), Lactuca sativa (B–F), or Medicago sativa (G) and for effects of root exudates of Triticum aestivum on root growth of Sinapis alba (H) and their description by the hormetic dose-response models after Brain and Cousens (grey curve) or Cedergreen et al. (black curve). Both models provided an adequate, marginally varying fit. Error bars represent standard deviation.