A unifying concept for assessing toxicological interactions: changes in slope

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.