A combination of doses d1, d2, dn of n agents shows zero interaction when d1/D1 + d2/D2. + dn/Dn = 1, where D1,D2, Dn are the doses of the individual agents isoeffective with the combination. In synergy, the sum in this equation is less than 1, and in antagonism, it exceeds 1. This equation may be used to calculate the expected (zero interactive) effect of any combination, irrespective of the shapes of the dose-response curves of the agents and of whether they are linear or nonlinear, similar or dissimilar. For a given set of agents, finding the combination that has the maximum therapeutic (or toxic) effect may be logistically a huge problem because of the large number of variables (e.g., dose, dose interval, number of doses) that are generally involved. This problem may be tackled by (1) response surface methods, in which an equation (usually a low-order polynomial) is fitted to the observed effects of a number of different combinations, and the maximum on this response surface is found mathematically, or by (2) direct search methods, in which the response surface is explored one combination at a time without preconceived ideas about its form. For problems with many variables, direct search methods are more economic.
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