A dynamical systems analysis of the indirect response model with special emphasis on time to peak response

In this paper we present a mathematical analysis of the four classical indirect response models. We focus on characteristics such as the evolution of the response R(t) with time t, the time of maximal/minimal response T(max) and the area between the response and the baseline AUC(R), and the way these quantities depend on the drug dose, the dynamic parameters such as E(max) and EC50 and the ratio of the fractional turnover rate k(out) to the elimination rate constant k of drug in plasma. We find that depending on the model and on the drug mechanism function, T(max) may increase, decrease, decrease and then increase, or stay the same, as the drug dose is increased. This has important implications for using the shift in T(max) as a diagnostic tool in the selection of an appropriate model.