A general approach to modeling biphasic relationships.

Biphasic relationships can be found throughout the sciences, especially in the dose-response relationships of pharmacology, toxicology, agriculture, and nutrition. Accurate modeling of biphasic dose-response is an essential step in establishing effective guidelines for the protection of human and ecosystem health, yet currently-used biphasic mathematical models lack biological rationale and fit only limited sets of biphasic data. To model biphasic relationships more closely over wider ranges of exposures, we suggest a simple, general, biologically reasonable modeling approach leading to a family of mathematical models that combine log-logistic functions: at least one for the upslope and one forthe downslope of the biphasic relationship. All parameters employed are meaningfully interpretable. These models can be used to test for the presence of biphasic effects, and they simplify to a standard log-logistic model in the special case where no biphasic effect can be detected. They offer the promise of improvement in assessment of the safety and efficacy of pharmaceuticals and nutrients as well as in determination of the toxicity of contaminants. Additionally, they may be useful in modeling nonmonotonic cause-effect relationships in other scientific disciplines.