The study of combined action of agents using differential geometry of dose-effect surfaces

Although graphic surfaces have been used routinely in the study of combined action of agents, they are mainly used for display purposes. In this paper, it is shown that useful mechanistic information can be obtained from an analytical study of these surfaces using the tools of differential geometry. From the analysis of some simple dose-effect surfaces, it is proposed that the intrinsic curvature, referred to in differential geometry as the Gaussian curvature, of a dose-effect surface can be used as a general criterion for the classification of interaction between different agents. This is analogous to the interpretation of the line curvature of a dose-effect curve as an indication of self-interaction between doses for an agent. In this framework, the dose-effect surface would have basic uniform fabric with zero curvature in the absence of interaction, tentatively referred to as null-interaction. Pictorially speaking, this fabric is distorted locally or globally like the stretching and shrinking of a rubber sheet by the presence of interaction mechanisms between different agents. Since self-interaction with dilution dummies does not generate intrinsic curvature, this criterion of null-interaction would describe the interaction between two truly different agents. It is shown that many of the published interaction mechanisms give rise to dose-effect surfaces with characteristic curvatures. This possible correlation between the intrinsic geometric curvature of dose-effect surfaces and the biophysical mechanism of interaction presents an interesting philosophical viewpoint for the study of combined action of agents.