The expected effect of a combination of agents: the general solution

Interactions between agents (drugs, carcinogens, physiological stimuli, environmental pollutants, etc.) in producing their effects are of fundamental interest and practical importance in virtually every branch of biology and medicine. A combination of agents is said to show interaction when the magnitude of its effect is greater or smaller than expected, expectation being based on the dose-effect relations of the individual agents in the combination. The crux of the matter is to decide what is expected, and various rules have been proposed to this end (for example, that the expected effect is the sum of the effects of the individual constituents of the combination, or that it is the product of these effects, or that it may be calculated from the law of mass action). These rules are valid for combinations of agents with particular and rather restricted types of dose-effect relations, but they have no general validity. A general solution to this problem is given here, that enables the effects of non-interactive combinations to be calculated directly from the dose-effect relations of the individual agents (whether expressed algebraically or numerically), regardless of the particular types of dose-effect relations involved. This solution is based on the fact that, when an effect of particular magnitude is produced by a combination of n agents which do not interact to produce that effect, the point representing the combination in the n-dimensional space spanned by the dose-axes of the individual agents lies in the same (n-1)-dimensional hyperplane as those representing other combinations iso-effective with it and iso-effective amounts of the individual agents. Methods for calculating the effect of a non-interactive combination as the sum or product of the effects of its constituents, or from the law of mass action, each of which is correct in appropriate cases, may be deduced (without invoking mechanisms of action) by applying this general principle to particular types of dose-effect relations.