Theory of synergistic effects: Hill-type response surfaces as 'null-interaction' models for mixtures

The classification of effects caused by mixtures of agents as synergistic, antagonistic or additive depends critically on the reference model of 'null interaction'. Two main approaches are currently in use, the Additive Dose (ADM) or concentration addition (CA) and the Multiplicative Survival (MSM) or independent action (IA) models. We compare several response surface models to a newly developed Hill response surface, obtained by solving a logistic partial differential equation (PDE). Assuming that a mixture of chemicals with individual Hill-type dose-response curves can be described by an n-dimensional logistic function, Hill's differential equation for pure agents is replaced by a PDE for mixtures whose solution provides Hill surfaces as 'null-interaction' models and relies neither on Bliss independence or Loewe additivity nor uses Chou's unified general theory.

An n-dimensional logistic PDE decribing the Hill-type response of n-component mixtures is solved. Appropriate boundary conditions ensure the correct asymptotic behaviour. Mathematica 11 (Wolfram, Mathematica Version 11.0, 2016) is used for the mathematics and graphics presented in this article.

The Hill response surface ansatz can be applied to mixtures of compounds with arbitrary Hill parameters. Restrictions which are required when deriving analytical expressions for response surfaces from other principles, are unnecessary. Many approaches based on Loewe additivity turn out be special cases of the Hill approach whose increased flexibility permits a better description of 'null-effect' responses. Missing sham-compliance of Bliss IA, known as Colby's model in agrochemistry, leads to incompatibility with the Hill surface ansatz. Examples of binary and ternary mixtures illustrate the differences between the approaches. For Hill-slopes close to one and doses below the half-maximum effect doses MSM (Colby, Bliss, Finney, Abbott) predicts synergistic effects where the Hill model indicates 'null-interaction'. These differences increase considerably with increasing steepness of the individual dose-response curves.

The Hill response surface ansatz contains the Loewe additivity concept as a special case and is incompatible with Bliss independent action. Hence, when synergistic effects are claimed, those dose combinations deserve special attention where the differences between independent action approaches and Hill estimations are large.