Insulin injections and exercise scheduling for individuals with diabetes: An optimal control model

Abstract

The aim of this paper is to promote the development of new biological models and the application of optimization methods to these models. Fast and cheap computing power allows for the ready implementation of increasingly complex dynamic models in biology. However, these models are normally developed in isolation, and their highly coupled nature can make it difficult to incorporate features from one model into another. In addition, there are many recent advances in numerical optimal control, which have not yet been applied to biological models. In this paper, we illustrate how an existing biological model can be extended to incorporate features from other models, and we demonstrate that numerical optimal control techniques can readily determine optimal strategies for managing the resulting system. In particular, we develop a new composite dynamic model for the blood glucose regulatory system by incorporating the effects of exercise and insulin injections into an existing model with 8 state variables. We formulate an optimal control problem, in which the aim is to determine optimal injection times, optimal injection volumes, and an optimal exercise regime to regulate the blood glucose level. A numerical approach, based on the concept of control parameterization and a time-scaling transformation, is then developed for solving the optimal control problem. Numerical results for 5 scenarios show that optimal treatment regimes can be readily determined via the proposed approach.