Quantitative analysis of dose-effect relationships: the combined effects of multiple drugs or enzyme inhibitors

A generalized method for analyzing the effects of multiple drugs and for determining summation, synergism and antagonism has been proposed. The derived, generalized equations are based on kinetic principles. The method is relatively simple and is not limited by whether the dose-effect relationships are hyperbolic or sigmoidal, whether the effects of the drugs are mutually exclusive or nonexclusive, whether the ligand interactions are competitive, noncompetitive or uncompetitive, whether the drugs are agonists or antagonists, or the number of drugs involved. The equations for the two most widely used methods for analyzing synergism, antagonism and summation of effects of multiple drugs, the isobologram and fractional product concepts, have been derived and been shown to have limitations in their applications. These two methods cannot be used indiscriminately. The equations underlying these two methods can be derived from a more generalized equation previously developed by us (59). It can be shown that the isobologram is valid only for drugs whose effects are mutually exclusive, whereas the fractional product method is valid only for mutually nonexclusive drugs which have hyperbolic dose-effect curves. Furthermore, in the isobol method, it is laborious to find proper combinations of drugs that would produce an iso-effective curve, and the fractional product method tends to give indication of synergism, since it underestimates the summation of the effect of mutually nonexclusive drugs that have sigmoidal dose-effect curves. The method described herein is devoid of these deficiencies and limitations. The simplified experimental design proposed for multiple drug-effect analysis has the following advantages: It provides a simple diagnostic plot (i.e., the median-effect plot) for evaluating the applicability of the data, and provides parameters that can be directly used to obtain a general equation for the dose-effect relation; the analysis which involves logarithmic conversion and linear regression can be readily carried out with a simple programmable electronic calculator and does not require special graph paper or tables; and the simplicity of the equation allows flexibility of application and the use of a minimum number of data points. This method has been used to analyze experimental data obtained from enzymatic, cellular and animal systems.