Monotonicity of Time to Peak Response with Respect to Drug Dose for Turnover Models

In this paper we analyze the monotonicity of the time to peak response Tmax with respect to the drug dose D for the four different turnover models I IV, as introduced by Dayneka et al. [2]. We do this for the situation when the drug is supplied through an initial bolus, and eliminated according to a single exponential function and stimulation or inhibition takes place through a Hill function. We show that in Models I and III, in which the drug impacts the production term, the function Tmax(D) is increasing for all values of the systemand the drug parameters. For Model II (inhibition of the loss term) the situation is more delicate. Here we prove monotonicity of Tmax(D) for a substantial range of values of the rateand drug constants, but leave the question of monotonicity open for some values. Finally, in Model IV (stimulation of the loss term) the function Tmax(D) is known not to be monotone for some values of the rate constants and Imax [12].