Statistical modeling of the effects of drug combinations

A method is described for identifying and quantitating departures from additivity (i.e., synergism and antagonism) when drugs having like effects are given in combination. It is applicable for both graded and quantal (e.g., after probit or logit transformation) responses. Log(dose)-response curves of both drugs should be linear but need not be parallel. The following model is fitted to dose-response data for both the individual drugs and combinations of drugs: Y = β_o + β₁log(iA + p.b + β₄(A.P.B)^{$\text{1}\text{2}$} where Y is the response, A is the amount of drug A, B is the amount of drug B, and P is a relative potency of the drugs given by log(P) = β₂ + β₃log(B'), in which B' is the solution to B' — B — $\text{A}\text{P}$ = 0. If log(dose)-response curves of the two drugs are parallel, β₃ = 0, and P becomes a constant parameter to be estimated. A positive value of β₄ corresponds to synergism and a negative value to antagonism. Hypothesis tests may be carried out to determine whether β₄ is significantly different from zero.