Pharmacodynamic models: parameterizing the hill equation, Michaelis-Menten, the logistic curve, and relationships among these models.

The Hill equation is often used in dose-response or exposure-response modeling. Aliases for the Hill model include the Emax model, and the Michaelis-Menten model. There is confusion about the appropriate parameterization, how to interpret the parameters, what the meaning is of the various parameterizations found in the literature, and which parameterization best approximates the statistical inferences produced when fitting the Hill equation to data. In this paper, we present several equivalent versions of the Hill model; show that they are equivalent in terms of yielding the same prediction for a given dose, and are equivalent to the four-parameter logistic model in this same sense; and deduce which parameterization is optimal in the sense of having the least statistical curvature and preferable multicollinearity.