Optimization Problem in a Class of Linear System. Application to Multiple Drug Administration

2000 Mathematics Subject Classification: 62H15, 62P10.

The optimal control settings are useful tools in studying the behavior of the plasma concentration of a drug (medicine) after its application. Linear compartmental systems are widely used as mathematical models for studying this behavior. In this paper we develop a strategy for minimizing the total amount of applied drug under the restriction: the mean plasma concentrations belongs to prescribed therapeutic interval. The aim of the article is to give to the therapeutist an simple and applicable formula for the optimal input sequence of multiple doses as a function of the time and rate constant of drug elimination. The desired formula is given as a result of the theorems for optimal input function which we proved. The optimal input is important specially for patients, which needs of treatment with antibiotics but they have kidney shortage function. Because of this the relationship between the optimal dose and the time intervals of administration to maintain a effective drug concentration in plasma is considered too. The results are applied to the two compartment stochastic model using experimental data of plasma concentration after single administration of antibiotic Tobramicin to a patient with kidney shortage function.