Generalized concentration addition: a method for examining mixtures containing partial agonists

Environmentally relevant toxic exposures often consist of simultaneous exposure to multiple agents. Methods to predict the expected outcome of such combinations are critical both to risk assessment and to an accurate judgment of whether combinations are synergistic or antagonistic. Concentration addition (CA) has commonly been used to assess the presence of synergy or antagonism in combinations of similarly acting chemicals, and to predict effects of combinations of such agents. CA has the advantage of clear graphical interpretation: Curves of constant joint effect (isoboles) must be negatively sloped straight lines if the mixture is concentration additive. However, CA cannot be directly used to assess combinations that include partial agonists, although such agents are of considerable interest. Here, we propose a natural extension of CA to a functional form that may be applied to mixtures including full agonists and partial agonists. This extended definition, for which we suggest the term "generalized concentration addition," encompasses linear isoboles with slopes of any sign. We apply this approach to the simple example of agents with dose-response relationships described by Hill functions with slope parameter n=1. The resulting isoboles are in all cases linear, with negative, zero and positive slopes. Using simple mechanistic models of ligand-receptor systems, we show that the same isobole pattern and joint effects are generated by modeled combinations of full and partial agonists. Special cases include combinations of two full agonists and a full agonist plus a competitive antagonist.