Robust statistical methods are important to the evaluation of toxicological interactions (i.e., departures from additivity) among chemicals in a mixture. However, different concepts of joint toxic action as applied to the statistical analysis of chemical mixture toxicology data or as used in environmental risk assessment often appear to conflict with one another. A unifying approach for application of statistical methodology in chemical mixture toxicology research is based on consideration of change(s) in slope. If the slope of the dose-response curve of one chemical does not change in the presence of other chemicals, then there is no interaction between the first chemical and the others. Conversely, if the rate of change in the response with respect to dose of the first chemical changes in the presence of the other chemicals, then an interaction is said to exist. This concept of zero interaction is equivalent to the usual approach taken in additivity models in the statistical literature. In these additivity models, the rate of change in the response as a function of the i(th) chemical does not change in the presence of other chemicals in a mixture. It is important to note that Berenbaum's (1985, J. Theor. Biol. 114, 413-431) general and fundamental definition of additivity does not require the chemicals in the mixture to have a common toxic mode of action nor to have similarly shaped dose response curves. We show an algebraic equivalence between these statistical additivity models and the definition of additivity given by Berenbaum.

Environmental exposures generally involve chemical mixtures instead of single chemicals. Statistical models such as the fixed-ratio ray design, wherein the mixing ratio (proportions) of the chemicals is fixed across increasing mixture doses, allows for the detection and characterization of interactions among the chemicals. In this study, we tested for interaction(s) in a mixture of five organophosphorus (OP) pesticides (chlorpyrifos, diazinon, dimethoate, acephate, and malathion). The ratio of the five pesticides (full ray) reflected the relative dietary exposure estimates of the general population as projected by the US EPA Dietary Exposure Evaluation Model (DEEM). A second mixture was tested using the same dose levels of all pesticides, but excluding malathion (reduced ray). The experimental approach first required characterization of dose-response curves for the individual OPs to build a dose-additivity model. A series of behavioral measures were evaluated in adult male Long-Evans rats at the time of peak effect following a single oral dose, and then tissues were collected for measurement of cholinesterase (ChE) activity. Neurochemical (blood and brain cholinesterase [ChE] activity) and behavioral (motor activity, gait score, tail-pinch response score) endpoints were evaluated statistically for evidence of additivity. The additivity model constructed from the single chemical data was used to predict the effects of the pesticide mixture along the full ray (10-450 mg/kg) and the reduced ray (1.75-78.8 mg/kg). The experimental mixture data were also modeled and statistically compared to the additivity models. Analysis of the 5-OP mixture (the full ray) revealed significant deviation from additivity for all endpoints except tail-pinch response. Greater-than-additive responses (synergism) were observed at the lower doses of the 5-OP mixture, which contained non-effective dose levels of each of the components. The predicted effective doses (ED20, ED50) were about half that predicted by additivity, and for brain ChE and motor activity, there was a threshold shift in the dose-response curves. For the brain ChE and motor activity, there was no difference between the full (5-OP mixture) and reduced (4-OP mixture) rays, indicating that malathion did not influence the non-additivity. While the reduced ray for blood ChE showed greater deviation from additivity without malathion in the mixture, the non-additivity observed for the gait score was reversed when malathion was removed. Thus, greater-than-additive interactions were detected for both the full and reduced ray mixtures, and the role of malathion in the interactions varied depending on the endpoint. In all cases, the deviations from additivity occurred at the lower end of the dose-response curves.