Diabetes mellitus modeling and short-term prediction based on blood glucose measurements F. Ståhl, R. Johansson * Department of Automatic Control, Lund University, P.O. Box 118, SE22100 Lund, Sweden article info Article history: Received 21 January 2008 Received in revised form 24 September 2008 Accepted 6 October 2008 Available online 30 October 2008 Keywords: Diabetes Mathematical model System identification Predictor Glucose dynamics Insulin dynamics abstract Insulin-Dependent Diabetes Mellitus (IDDM) is a chronic disease characterized by the inability of the pancreas to produce sufficient amounts of insulin. Daily compensation of the deficiency requires 4–6 insulin injections to be taken daily, the aim of this insulin therapy being to maintain normoglycemia – i.e., a blood glucose level between 4 and 7 mmol/l. To determine the quantity and timing of these injec- tions, various different approaches are used. Currently, mostly qualitative and semi-quantitative models and reasoning are used to design such a therapy. Here, an attempt is made to show how system identi- fication and control may be used to estimate predictive quantitative models to be used in design of opti- mal insulin regimens. The system was divided into three subsystems, the insulin subsystem, the glucose subsystem and the insulin–glucose interaction. The insulin subsystem aims to describe the absorption of injected insulin from the subcutaneous depots and the glucose subsystem the absorption of glucose from the gut follow- ing a meal. These subsystems were modeled using compartment models and proposed models found in the literature. Several black-box models and grey-box models describing the insulin/glucose interaction were developed and analyzed. These models were fitted to real data monitored by an IDDM patient. Many difficulties were encountered, typical of biomedical systems: Non-uniform and scarce sampling, time- varying dynamics and severe nonlinearities were some of the difficulties encountered during the model- ing. None of the proposed models were able to describe the system accurately in all aspects during all conditions. However, all the linear models shared some dynamics. Based on the estimated models, short-term blood glucose predictors for up to two-hour-ahead blood glucose prediction were designed. Furthermore, we explored the issues that arise when applying prediction theory and control to short- term blood glucose prediction. Ó 2008 Elsevier Inc. All rights reserved. 1. Preface In January 2002, the first author was diagnosed with Diabetes Type 1 and soon realized the difficulty of maintaining normoglyce- mia. The question was raised whether control theory could be ap- plied to the problem. To analyze the system using methods of control theory, a model of the system is essential. This paper is an attempt to estimate such a model based on home-monitored data, typically found in a diabetes diary. The models used were pri- marily linear and were found to be partly insufficient to describe the system. 2. Introduction Diabetes Mellitus is a disease characterized by the inability of the pancreas to produce sufficient amounts of insulin. To cover the deficiency 4–6 insulin injections have to be taken daily, the aim being to keep the blood glucose level as constant as possible. To determine the amount and timing of these injections different approaches are used. Mostly qualitative and semi-quantitative models and reasoning are used to design such a therapy. Most pa- tients monitor their blood glucose using personal glucose meters, and determine their own insulin injections based on these results. Poorly controlled blood glucose levels may result in severe compli- cations. Hypoglycemia – i.e., low blood glucose levels – may lead to brain damage [2], coma and eventually death. On the other hand, hyperglycemia – i.e., high blood glucose levels – can result in chronic damages such as retinopathy, kidney failure and amputa- tion due to angiopathy. All patients set their own insulin regime with aid from their physician based on HbA 1c , personal observa- tions and a qualitative estimate of the glucose data. These regimes are to their nature rigid and non-flexible. They form the basis for the therapy and patients often have to – and are encouraged to – alter their injection doses when their behavior deviate from the routine the regime was based on. In these situations, the patients 0025-5564/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2008.10.008 * Corresponding author. Tel.: +4646 222 8791; fax: +4646 138118. E-mail address: Rolf.Johansson@control.lth.se (R. Johansson). Mathematical Biosciences 217 (2009) 101–117 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs