Review Mathematical modeling of the glucose–insulin system: A review Pasquale Palumbo a , Susanne Ditlevsen b,⇑ , Alessandro Bertuzzi c , Andrea De Gaetano a a BioMatLab, IASI ‘‘A. Ruberti’’, National Council of Researches, UCSC Largo A. Gemelli 8, 00168 Rome, Italy b Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark c IASI ‘‘A. Ruberti’’, National Council of Researches, Viale Manzoni 30, 00185 Rome, Italy article info Article history: Received 3 December 2012 Received in revised form 10 May 2013 Accepted 16 May 2013 Available online 1 June 2013 Keywords: Diabetes Minimal Model IVGTT and OGTT Insulin secretion and oscillations Euglycemic Hyperinsulinemic Clamp Long-term diabetes abstract Mathematical modeling of the glucose–insulin feedback system is necessary to the understanding of the homeostatic control, to analyze experimental data, to identify and quantify relevant biophysical param- eters, to design clinical trials and to evaluate diabetes prevention or disease modification therapies. Much work has been made over the last 30 years, and the time now seems ripe to provide a comprehensive review. The one here proposed is focused on the most important clinical/experimental tests performed to understand the mechanism of glucose homeostasis. The review proceeds from models of pancreatic insulin production, with a coarser/finer level of detail ranging over cellular and subcellular scales, to short-term organ/tissue models accounting for the intra-venous and the oral glucose tolerance tests as well as for the euglycemic hyperinsulinemic clamp, to total-body, long-term diabetes models aiming to represent disease progression in terms of b-cell population dynamics over a long period of years. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction The glucose–insulin system offers one of the clearest and sim- plest examples of homeostatic control in the organism. The level of glucose in blood needs to be kept within a narrow range. Since it represents the main metabolic substrate, or energy source, for brain tissue, abnormally low glucose concentrations give rise to anxiety, tremors, aggressiveness, obfuscation, coma and eventually death. On the other hand, excessive plasma glucose concentrations produce microvascular damages (notably in the retina and kidney) and neural damages, leading among others to blindness and chronic renal insufficiency. The way the body controls glycemia seems deceptively simple. Essentially a single hormone (insulin) is secreted by the b-cells of the pancreas in response to rising glu- cose concentrations (hyperglycemia). Insulin effects include increasing peripheral tissue glucose uptake (mainly by the muscle and fat tissues) and decreasing spontaneous glucose output by the liver. When insulin secretion by the pancreas is insufficient or ab- sent, due to (autoimmune) destruction of b-cells, the clinical pic- ture of Type 1 Diabetes Mellitus (T1DM) results; when insulin is secreted in normal, or supranormal amounts, but it is ineffective in lowering glycemia to normal levels, Type 2 Diabetes Mellitus (T2DM) is said to be present. A number of hormones contribute to rescuing the organism from hypoglycemia (adrenalin, glucagon, growth hormone, cortisol): however, since in clinical practice the situation of interest is normally inappropriately high glycemia, concentrating attention on the response to hyperglycemia by insu- lin seems justified, at least as a first modeling approach. We may therefore consider, as a first approximation, a simplified system in which a single metabolite (glucose) is controlled by a single hor- mone (insulin). This system will have to maintain glycemia in the absence of food intake, and will have to suppress hyperglycemia rapidly after meals, without incurring in dangerous hypoglyce- mias. We see therefore that the glucose–insulin system could be viewed, at least approximately, as a feedback control with a con- troller (the pancreas) and multiple effectors (muscle, liver, fat tis- sue), but where the only state variables of interest are glycemia and insulinemia. The present review has the goal of highlighting the biomedical problem of the glucose–insulin homeostasis from a physiological and clinical viewpoint, then describing the main combined exper- imental-modeling tools which are currently employed in investi- gating the behavior of the control system in individuals or populations. The review is structured as follows. The next section focuses on one specific biological function played by the pancreas, the organ responsible for glucose homeostasis by means of glu- cose-stimulated insulin production. In this case the range of the available models for insulin release spans a subcellular (models dealing with the molecular mechanisms leading to the ejection of the insulin packet from a beta-cell), and a cellular/organ level (models treating the pancreas as a population of secretory units). In these cases the loop is often (but not always) closed with the ac- tion of remote organs and tissues (explaining the occurrence of 0025-5564/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.mbs.2013.05.006 ⇑ Corresponding author. Tel.: +45 35320785. E-mail address: susanne@math.ku.dk (S. Ditlevsen). Mathematical Biosciences 244 (2013) 69–81 Contents lists available at SciVerse ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs