A mathematical approach to study combined effects of toxicants in vitro: Evaluation of the Bliss independence criterion and the Loewe additivity model

The study of interactions among toxicants is of fundamental interest and practical importance in toxicological sciences. However, a final agreement on the definition of agent interaction is lacking, and therefore, particular care should be adopted when using the terms additivity, synergism and antagonism, unless the exact toxicological pathways of the compounds studied are known. Two main different approaches, the Bliss independence criterion and the Loewe additivity model, have been generally used in co-exposure experiments. In some cases, they can present dramatically different results, depending on the slope of the pure dose-response curves of single substances. Here, we consider both models in in vitro experiments, where the dose-response curves can be extrapolated. Advantages and limitations of both approaches are discussed, using several mathematical simulations to better explain them, and applying the Hill function for the dose-response model curve. Overall we conclude that the Loewe additivity model is slightly preferable because of a general higher biological plausibility. However, which model to use must be determined case by case and the evaluation can be aided by experimental approaches, such as the study of multiple biomarkers and asynchronous exposures.