Date Published: March 20, 2019
Publisher: Public Library of Science
Author(s): Afroza Shirin, Fabio Della Rossa, Isaac Klickstein, John Russell, Francesco Sorrentino, Abhyudai Singh.
The Glucose-Insulin-Glucagon nonlinear model accurately describes how the body responds to exogenously supplied insulin and glucagon in patients affected by Type I diabetes. Based on this model, we design infusion rates of either insulin (monotherapy) or insulin and glucagon (dual therapy) that can optimally maintain the blood glucose level within desired limits after consumption of a meal and prevent the onset of both hypoglycemia and hyperglycemia. This problem is formulated as a nonlinear optimal control problem, which we solve using the numerical optimal control package PSOPT. Interestingly, in the case of monotherapy, we find the optimal solution is close to the standard method of insulin based glucose regulation, which is to assume a variable amount of insulin half an hour before each meal. We also find that the optimal dual therapy (that uses both insulin and glucagon) is better able to regulate glucose as compared to using insulin alone. We also propose an ad-hoc rule for both the dosage and the time of delivery of insulin and glucagon.
Insulin and glucagon are pancreatic hormones that help regulate the levels of glucose in the blood [1–4]. Insulin is produced by the beta-cells in the pancreas and carries glucose from the bloodstream to the cells throughout the body. Glucagon releases glucose from the liver into the bloodstream in order to prevent hypoglycemia. In people affected by diabetes insulin is either absent (type I diabetes) or not produced in the proper amount (type II diabetes). In type I diabetes the body’s immune system attacks and destroys the beta cells. As a result, insulin is not produced and glucose accumulates in the blood which may cause serious harm to several organs. Type II diabetes is a metabolic disorder in which the beta cells are unable to properly regulate the blood glucose within limits. Common therapies for diabetes involve the administration of exogenous insulin. Currently glucagon is not typically included in therapies because it does not preserve its chemical properties at room temperature and also because diabetic patients are still able to produce it.
We now describe in more detail the optimal control problem in Eqs (2)–(9) by setting the constraint and parameter values. In Fig 1A we plot the function BGI(G) versus the glucose G. The minimum BGI(G) occurs at G = Gd = 112.51 mg/dL, which corresponds to a clinical target set for the glucose level . Based on the data in , the average fasting plasma glucose level of patients with type I diabetes is Gb = 130 (mg/dL). Thus, we set the the basal glucose level Gb = 130 (mg/dL). The parameters ΘGb are set so that the steady state glucose is 130 (mg/dL) in the absence of a meal and of exogenously supplied insulin, i.e., we compute Θ130.
In this paper we have used the Glucose-Insulin-Glucagon mathematical model proposed in [2–4], which describes how the body responds to exogenously supplied insulin and glucagon in patients affected by Type I diabetes and designed an optimal dosing schedule of either insulin or insulin and glucagon together to regulate the blood glucose index (BGI), while limiting the total amount of insulin and glucagon administered. The numerical optimal control software PSOPT has been used to solve this optimal control problem. While the numerical solution requires knowledge of the set of model parameters, which are patient specific, the solutions we obtain provide insight into the best possible glucose regulation with insulin or with insulin and glucagon together. Our approach is in agreement with the results of references [60–62], in which simplified models are used to analytically establish general theoretical properties and control limitations for the glucose regulation problem.