Identification of combined action types in experiments with two toxicants: a response surface linear model with a cross term

Abstract

Within the framework of the response surface linear model with a cross term, i.e. a model of the type Y(x₁, x₂) = b₀ + b₁x₁ + b₂x₂ + b₁₂x₁x₂ (hyperbolic paraboloid), a complete solution of identification of combined action types of two toxicants x₁ and x₂ is presented. It is shown that the type of combined effect in this model is determined by two factors: the direction in which the toxicants act (unidirectional or oppositely directed), and the position of the saddle point S of a hyperbolic paraboloid. For unidirectional actions of toxicants, already-known ways to identify the type of combined effect (including a shape of the isobole: concave-up or concave-down) provided identical and unambiguous answers regarding the type of combined effect (antagonism or synergism). For oppositely directed actions of toxicants, the shape of the isobole (concave-up or concave-down) did not allow us to determine the type of combined action type unambiguously. We show that in both cases (unidirectional or oppositely directed actions of toxicants) the signs of the model coefficients b₁, b₂ and b_12, in conjunction with the coordinates of the saddle point S help unambiguously identify the type of combined action by comparing the observed effect with the zero interaction response surface. An atlas of all possibly combined action types for two toxicants for the hyperbolic paraboloid model was created. Applications of the developed formalism to experimental data are provided.

• Combined action of toxicants
• isobole approach
• linear model with cross term
• mathematical modeling
• response surface methodology
• toxicants that produce opposite effects

Declaration of interest

The authors report no declarations of interest.