Mathematical Decision of Optimal Drug Concentration with State-Space Computational Approach

Abstract

Mathematical modelling of drug concentration is pivotal in treating a disease. However, applying unsuitable drug concentrations to treatment will not cure patients and may delay their recovery. This paper describes a drug concentration problem through a mathematical model that is a system of nonlinear ordinary differential equations. In solving this model, a simplified model is proposed to characterize the actual drug concentration problem. Then, a loss function, which measures the differences between the simplified model and the real problem, is defined. A state space representation showing the linear relation between drug circulation and metabolism is expressed. Using a gradient method, parameters in the simplified model are updated iteratively until convergence is achieved. With these optimal parameters, the analytical solution of the simplified model, which approximates the result of the actual drug concentration, is obtained. In addition, by adding a control input to the simplified model, the optimal decision of drug concentration is suggested to speed up the process of drug circulation and metabolism. Hence, the time taken decreased for the drug to beat its target site and fasten the recovery. In conclusion, the efficiency of the simplified model with control input for handling the drug concentration problem is highly demonstrated.