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From a theoretical point of view, it has often been argued that the model of independent action (IA) is the most correct reference model to use for predicting the joint effect of mixtures of chemicals with different molecular target sites. The theory of IA, however, relies on a number of assumptions that are rarely fulfilled in practice. It has even been argued that, theoretically, the concentration addition (CA) model could be just as correct. In the present study, we tested the accuracy of both IA and CA in describing binary dose-response surfaces of chemicals with different molecular targets using statistical software. We compared the two models to determine which best describes data for 158 data sets. The data sets represented 98 different mixtures of, primarily, pesticides and pharmaceuticals tested on one or several of seven test systems containing one of the following: Vibrio fischeri, activated sludge microorganisms, Daphnia magna, Pseudokirchneriella subcapitata, Lemna minor, Tripleurospermum inodorum, or Stellaria media. The analyses showed that approximately 20% of the mixtures were adequately predicted only by IA, 10% were adequately predicted only by CA, and both models could predict the outcome of another 20% of the experiment. Half of the experiments could not be correctly described with either of the two models. When quantifying the maximal difference between modeled synergy or antagonism and the reference model predictions at a 50% effect concentration, neither of the models proved significantly better than the other. Thus, neither model can be selected over the other on the basis of accuracy alone.

Binary-mixture studies often are conducted with the aim of elucidating the effect of one specific chemical on the biological action of another. The results can be interpreted in relation to reference models by the use of response-surface analyses and isobolograms. The amount of data needed for these analyses is, however, extensive, and the experiments therefore rarely are repeated. In the present study, we investigate the reproducibility of isobole shapes of binary-mixture toxicity experiments in terms of deviation from the reference model of concentration addition (CA), dose-level dependence, and isobole asymmetry. We use data from four herbicide mixtures tested in three to five independent experiments on the aquatic test plant Lemna minor and the terrestrial plant Tripleurospermum inodorum. The results showed that the variation both within and among experiments was approximately half the size for the aquatic test system compared to the terrestrial system. As a consequence, a consistent deviation from CA could be obtained in three of four herbicide mixtures for L. minor, whereas this was only the case for one or two of the herbicide mixtures tested on T. inodorum. For one mixture on T. inodorum, both CA synergism and antagonism were detected. Dose-dependent effects could not be repeated consistently, just as the asymmetry found in some isoboles could not. The study emphasizes the importance of repeating mixture toxicity experiments, especially for test systems with large variability, and using caution when drawing biological conclusions from the test results.

During the past two decades, the phenomenon of hormesis has gained increased recognition. To promote research in hormesis, a sound statistical quantification of important parameters, such as the level and significance of the increase in response and the range of concentration where it occurs, is strongly needed. Here, we present an improved statistical model to describe hormetic dose-response curves and test for the presence of hormesis. Using the delta method and freely available software, any percentage effect dose or concentration can be derived with its associated standard errors. Likewise, the maximal response can be extracted and the growth stimulation calculated. The new model was tested on macrophyte data from multiple-species experiments and on laboratory data of Lemna minor. For the 51 curves tested, significant hormesis was detected in 18 curves, and for another 17 curves, the hormesis model described that data better than the logistic model did. The increase in response ranged from 5 to 109%. The growth stimulation occurred at an average dose somewhere between zero and concentrations corresponding to approximately 20 to 25% of the median effective concentration (EC50). Testing the same data with the hormesis model proposed by Brain and Cousens in 1989, we found no significant hormesis. Consequently, the new model is shown to be far more robust than previous models, both in terms of variation in data and in terms of describing hormetic effects ranging from small effects of a 10% increase in response up to effects of an almost 100% increase in response.

Theoretically, the effect of two independently acting compounds in a mixture will depend on the slope of the dose-response curves of the individual compounds if evaluated in relation to the model of concentration addition (CA). In the present study, we explored development of the shape of the dose-response relationship for four different recommended endpoints (surface area, frond number, fresh weight-specific, and dry weight-specific relative growth rates [RGRA, RGRF, RGRFW, and RGRDW, respectively]) and for two differently acting herbicides (metsulfuron-methyl and terbuthylazine) over time (3-15 d) on the standard test plant Lemna minor to identify endpoints and experiment times for which predictions of independent action (IA) would depart the most from those of CA. After a test time of 6 d, predictions of IA based on RGRA and RGRFW showed antagonism in relation to CA. Based on RGRDW, synergy was predicted, whereas IA based on RGRF was indistinguishable from CA. To test the prediction of choice of endpoint giving different results in mixture-toxicity experiments, three endpoints and six combinations of independently acting herbicides were evaluated using isobolograms. The experiments showed that in four of six herbicide combinations, different conclusions were reached depending on endpoint. The contradictory isoboles did not follow the theory of IA and, therefore, are more likely to be related to differences in susceptibility of the physiological processes affecting each endpoint than to the shape of the dose-response curve.