Modeling synergistic effects by using general Hill-type response surfaces describing drug interactions.

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 to describe co-operative effects are currently in use, the Additive Dose (ADM) or concentration addition (CA) and the Multiplicative Survival (MSM) or independent action (IA) models. Recently we proposed an approach which describes 'zero-interaction' surfaces based on the only requirement that simultaneous administration of different drugs leads to Hill-type response surfaces, which are solutions of the underlying logistic differential equations. No further assumptions, neither on mechanisms of action nor on limitations of parameter combinations are required. This defines-and limits-the application range of our approach. Resting on the same principle, we extend this ansatz in the present paper in order to describe deviations from the reference surface by generalized Hill-type functions. To this end we introduce two types of parameters, perturbations of the pure drug Hill-parameters and interaction parameters that account for n-tuple interactions between all components of a mixture. The resulting 'full-interaction' response surface is a valid solution of the basic partial differential equation (PDE), satisfying appropriate boundary conditions. This is true irrespective of its actual functional form, as within our framework the number of parameters is not fixed. We start by fitting the experimental data to the 'full-interaction' model with the maximum possible number of parameters. Guided by the fit-statistics, we then gradually remove insignificant parameters until the optimum response surface model is obtained. The 'full-interaction' Hill response surface ansatz can be applied to mixtures of n compounds with arbitrary Hill parameters including those describing baseline effects. Synergy surfaces, i.e., differences between full- and null-interaction models, are used to identify dose-combinations showing peak synergies. We apply our approach to binary and ternary examples from the literature, which range from mixtures behaving according to the null-interaction model to those showing strong synergistic or antagonistic effects. By comparing 'null-' and 'full-response' surfaces we identify those dose-combinations that lead to maximum synergistic or antagonistic effects. In one example we identify both synergistic and antagonistic effects simlutaneously, depending on the dose-ratio of the components. In addition we show that often the number of parameters necessary to describe the response can be reduced without significantly affecting the accuracy. This facilitates an analysis of the synergistic effects by focussing on the main factors causing the deviations from 'null-interaction'.