Abstract
BackgroundThe 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.MethodsAn 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.ResultsThe 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.ConclusionThe 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.
Highlights
The classification of effects caused by mixtures of agents as synergistic, antagonistic or additive depends critically on the reference model of ‘null interaction’
Logistic functions and the Hill response surface Let A and B be active ingredients with dose-response curves a(x) and b(y) depending on the variables x and y and let U be a combination of A and B with the dose-response surface u(x, y)
The differential equations for the one- and two-dimensional cases, describing the variation of the effects a(x), b(y) and u(x, y) with the variation of x and y, are da(x) = αa(x) 1 − a(x) db(y) = βb(y) 1 − b(y) dx amax dy bmax ux + uy = γ (x, y)u(x, y) u(x, y) umax(x, y) where ux = ∂u(x, y)/∂x and uy = ∂u(x, y)/∂y denote the partial derivatives of u, and α, β and amax, bmax are constants
Summary
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. 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. The description and prediction of dose-response curves of pure compounds and estimating the combined effect of simultaneous administration of several active ingredients (a.i.s) is a field of active research in pharmacology, anesthesiology, toxicology, environmental science, and agrochemistry. The terms ‘agent’, ‘drug’, ‘chemical’, and ‘a.i.’ are used interchangeably
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