Mathematical formalism for the properties of four basic models of indirect pharmacodynamic responses

Four basic models for characterizing indirect pharmacodynamic responses were proposed previously and applied using differential equations. These models consider inhibition or stimulated by drug of the production or loss of mediators or response variables. This report develops partially integrated solutions for these models which allow more detailed examination of the roles of model parameters and pharmacokinetic functions in affecting the time course of drug effects. Because of the nonlinear Hill function, the solutions are represented by means of definite integrals containing kinetic and dynamic functions. These solutions allow a qualitative examination, using calculus, of how response is controlled by Dose, IC₅₀ or SC₅₀, I(max) or S(max), and k(out) for drugs exhibiting monotonic or biphasic disposition. Characteristics of the response curves that were identified include shape, maximum or minimum, and changes with the above parameters and time. These relationships, together with simulation studies, provide a fundamental basis for understanding the temporal aspects of the basic indirect response models.