Optimal drug dosage subject to constraints (Q1842438)

In various areas of practical medicine we often need to create a specified time dependence of the concentration of a drug in the blood stream. Repeated or intermittent administration of drugs is a basic treatment technique, and we accordingly use it in our formulation of the optimal dosage problem. In the broad sense, intermittent administration gives rise to several difficult problems: a) the number of injections is fixed in advance; b) the dose at each instant is fixed, but the injection time has to be chosen; c) the injection time is fixed, and the dose has to be chosen; d) the time, the dose, and the number of doses all have to be chosen. We consider the following problem: provide a mathematical method for finding the doses \(\{d_ i \}_{i=1}^ n\) that must be injected at fixed, not necessarily equally spaced times \(\{ \tau_ i \}_{i=1}^ n\) in order to sustain a concentration of the active substance in the blood which provides the best (in some sense) approximation of a given (nonlinear) ``concentration vs. time'' dependence on the interval \([0,T]\) subject to a constraint on the total injected dose \(D\). We thus analyze problem c) above with an additional constraint on the total administered dose.