Mathematical model of glucose–insulin homeostasis in healthy rats

According to the World Health Organization there are over 220 million people in the world with diabetes and 3.4 million people died in 2004 as a consequence of this pathology. Development of an artificial pancreas would allow to restore control of blood glucose by coupling an infusion pump to a continuous glucose sensor in the blood. The design of such a device requires the development and application of mathematical models which represent the gluco-regulatory system. Models developed by other research groups describe very well the gluco-regulatory system but have a large number of mathematical equations and require complex methodologies for the estimation of its parameters. In this work we propose a mathematical model to study the homeostasis of glucose and insulin in healthy rats. The proposed model consists of three differential equations and 8 parameters that describe the variation of: blood glucose concentration, blood insulin concentration and amount of glucose in the intestine. All parameters were obtained by setting functions to the values of glucose and insulin in blood obtained after oral glucose administration. In vivo and in silico validations were performed. Additionally, a qualitative analysis has been done to verify the aforementioned model. We have shown that this model has a single, biologically consistent equilibrium point. This model is a first step in the development of a mathematical model for the type I diabetic rat.     

Hypoglycemia in childhood type 1 diabetes mellitus: Understanding and managing the dark side of intensive insulin therapy

Background: Despite the availability of advanced insulin delivery systems, blood glucose-monitoring equipment, and insulin analogue formulations, hypoglycemia remains a significant concern in the treatment of children and adolescents with type 1 diabetes mellitus (DM). Furthermore, patients who manage their blood glucose levels most effectively may also be the ones at greatest risk for hypoglycemia.Objective: The aim of this article was to review current issues surrounding the pathophysiology and frequency of hypoglycemia in children and adolescents with type 1 DM.Methods: Relevant articles for this review were identified through a search of MEDLINE (1992–2007; English-language articles only). The search terms used were children, adolescents, hypoglycemia, diabetes, insulin, and continuous subcutaneous insulin infusion.Results: The threat of severe hypoglycemia remains a major obstacle to the effective treatment of type 1 DM. Basalbolus therapy, using continuous subcutaneous insulin infusion or multiple daily injections, is the most effective and flexible method available for maintaining good glycemic control in children as well as in adults. Insulin analogues can be used effectively in these regimens and may be helpful toward addressing risks for hypoglycemia. Patient education should also be given a high priority in addressing the risk of hypoglycemia in children and adolescents with type 1 DM. The development of continuous glucose-monitoring systems offers the potential for an even brighter future for this group of patients.Conclusions: Recent advances in DM technology reduce but do not eliminate the risk of hypoglycemia in youth with type 1 DM. These observations underscore the need for a closed-loop insulin delivery system in which the rate of insulin infusion is regulated by real-time changes in glucose concentrations. (Insulin. 2007;2:157–165)Key words: type 1 diabetes mellitus; hypoglycemia; children; adolescents; insulin analogue; continuous subcutaneous insulin infusion; multiple daily injections; basal-bolus therapy.Accepted for publication 09052007     

Modeling the physiological glucose-insulin dynamic system on diabetics

A novel mathematical model is presented to describe the dynamic behavior of plasma glucose and insulin on diabetic subjects. Though various models have been proposed to simulate the short-term (a variety of intravenous glucose or insulin injection) glucose-insulin dynamics, it is intended to construct a modified delay differential equations (DDEs) model based on the human glucose-insulin metabolic system. Five specific adjustable parameters inside the model are defined as the factors of the major physiological functions. Then several clinical data sets (56 subjects) which includes the information of food ingestion and exogenous insulin injection are verified and the model could practically reflect the dynamics and oscillation behavior on diabetic subjects by varying the adjustable parameters. Moreover, the corresponding parameters are fairly helpful to identify the patient's conditions of major physiological functions. This generic glucose-insulin dynamic model can be expected to develop such advanced therapy strategies for diabetics in the future.     

Grey-box pharmacokinetic/pharmacodynamic modelling of a euglycaemic clamp study

Grey-box pharmacokinetic/pharmacodynamic (PK/PD) modelling is presented as a promising way of modelling the pharmacokinetics and pharmacodynamics of the in vivo system of insulin and glucose and to estimate model and derived PK/PD parameters. The concept behind grey-box modelling consists in using a priori physiological knowledge along with information from data in the estimation of model parameters. The PK/PD properties of two types of insulin are investigated in a euglycaemic clamp study where a single bolus of insulin is injected subcutaneously. The effect of insulin on the glucose disappearance is investigated by artificially maintaining a blood glucose concentration close to the normal fasting level. The infused glucose needed to maintain the clamped blood glucose concentration can therefore be used as a measure for the glucose utilization. The PK and PD parameters are successfully estimated simultaneously thereby describing the uptake, distribution, and effect of the two different types of insulin.     

The role of glucagon in the regulation of blood glucose: Model studies

Mathematical models afford a procedure of unifying concepts and hypotheses by expressing quantitative relationships between observables. The model presented indicates the roles of both insulin and glucagon as regulators of blood glucose, albeit in different ranges of the blood glucose concentrations. Insulin secretion is induced during hyperglycemia, while glucagon secretion results during hypoglycemia. These are demonstrated by simulations of a mathematical model conformed to data from the oral glucose tolerance test and the insulin infusion test in normal control subjects and stable and unstable diabetic patients. The model studies suggest the parameters could prove of value in quantifying the diabetic condition by indicating the degree of instability. Presented at the Society for Mathematical Biology Meeting, University of Pennsylvania, Philadelphia, August 19–21, 1976.     

Glucose Homeostatic Law: Insulin Clearance Predicts the Progression of Glucose Intolerance in Humans

Homeostatic control of blood glucose is regulated by a complex feedback loop between glucose and insulin, of which failure leads to diabetes mellitus. However, physiological and pathological nature of the feedback loop is not fully understood. We made a mathematical model of the feedback loop between glucose and insulin using time course of blood glucose and insulin during consecutive hyperglycemic and hyperinsulinemic-euglycemic clamps in 113 subjects with variety of glucose tolerance including normal glucose tolerance (NGT), impaired glucose tolerance (IGT) and type 2 diabetes mellitus (T2DM). We analyzed the correlation of the parameters in the model with the progression of glucose intolerance and the conserved relationship between parameters. The model parameters of insulin sensitivity and insulin secretion significantly declined from NGT to IGT, and from IGT to T2DM, respectively, consistent with previous clinical observations. Importantly, insulin clearance, an insulin degradation rate, significantly declined from NGT, IGT to T2DM along the progression of glucose intolerance in the mathematical model. Insulin clearance was positively correlated with a product of insulin sensitivity and secretion assessed by the clamp analysis or determined with the mathematical model. Insulin clearance was correlated negatively with postprandial glucose at 2h after oral glucose tolerance test. We also inferred a square-law between the rate constant of insulin clearance and a product of rate constants of insulin sensitivity and secretion in the model, which is also conserved among NGT, IGT and T2DM subjects. Insulin clearance shows a conserved relationship with the capacity of glucose disposal among the NGT, IGT and T2DM subjects. The decrease of insulin clearance predicts the progression of glucose intolerance.     

A mathematical model and computer simulation study of insulin sensitive glucose transporter regulation

A mathematical model of insulin sensitive glucose transporter regulation is developed. Model structure is based on experimental evidence from adipocytes and myocytes. Model parameters correspond with known cellular processes. As an example, computer simulation results are compared with data from rat adipocytes. Cellular processes explicitly represented in the model include state-dependent glucose transporter synthesis and degradation rates, insulin sensitive glucose transporter translocation rates, and a glucose transporter endocytosis rate. Most of these processes are represented as first-order events. Using more complex representations of the model structure (e.g. higher order rate constants or saturable pathways) or alternative structures did not result in qualitatively better results. The model is able to accurately simulate the insulin sensitive, insulin concentration dependent, reversible translocation of glucose transporters observed in normal adipocytes. The model is also able to accurately simulate the changes in regulation of glucose transporter translocation observed with increases in cell surface area. Finally, the model can simulate pathogenic states which induce impairment of glucose transporter regulation (e.g. altered glucose transporter regulation in adipocytes from rats on high fat diets, rats with streptozotocin induced diabetes, and fasted rats). Since the structure of our model is sufficient to explain glucose transporter regulation in both normal and pathological states, it may aid in understanding the post-receptor components of insulin resistance (decreased sensitivity or responsiveness to insulin) seen in pathological states such as obesity and diabetes mellitus.     

Blood glucose level neural model for type 1 diabetes mellitus patients

This paper deals with the blood glucose level modeling for Type 1 Diabetes Mellitus (T1DM) patients. The model is developed using a recurrent neural network trained with an extended Kalman filter based algorithm in order to develop an affine model, which captures the nonlinear behavior of the blood glucose metabolism. The goal is to derive a dynamical mathematical model for the T1DM as the response of a patient to meal and subcutaneous insulin infusion. Experimental data given by continuous glucose monitoring system is utilized for identification and for testing the applicability of the proposed scheme to T1DM subjects.     

Checking of some hypotheses of the pathogenesis of diabetes mellitus by mathematical modeling

A mathematical model of normal regulation of carbohydrate metabolism by the pancreas endocrine apparatus is presented. In a numerical experiment the model imitated changed levels of sucrose, insulin glucagon and gastrointestinal hormones in the blood in response to the ingested 50 g of glucose. The model of normal regulation was damaged in the way which theoretically should result in diabetes development. Then an estimation was made to what extent the disturbances of carbohydrate metabolism characteristic of diabetes were reproduced by the changed model. It has been shown that disturbances specific for diabetes appear when the sensitivity of beta-cells to glucose stimulus or hyperproduction of glucagon decreased. No changes in the behaviour of blood glucose typical of diabetes were obtained in the model when a decrease of the sensitivity of insulin receptors due to hyperinsulinemia in insulin-dependent tissues was imitated, as well as an increased activity of liver insulinase or hyposecretion of gastrointestinal hormones. These results point to the necessity of further development of these hypotheses.     

Mathematical simulation of the body fluid regulating system in dog

A mathematical model of body fluid volume and osmolality regulation was developed which incorporated the major nonlinearities of fluid assimilation, exchange, distribution and excretion. The non-linear differential equations define compartmental material balances for water, urea, sodium, protein and antidiuretic hormone (ADH). The parameters of these equations were calculated using analytical solutions and available steady-state experimental data. The model was used to simulate the renal response to five input forcings: (1) intraesophageal water infusion; (2) water ingestion; (3) intravenous ADH injection; (4) intravenous water infusion; and (5) intermittent water loading. The model yielded continuous simulation curves which agreed reasonably well with the available transient and steady-state experimental data. The model predicted that stimulating volume receptors via changes in left atrial pressure accounts for only 15–20% of changes in ADH secretion rate, whereas stimulation of the osmotic receptors via changes in plasma osmolality accounts for the remaining 80–85% of changes. Thus, it appears that regulation of ADH secretion is largely dependent upon plasma osmolality during forcings which do not appreciably alter the cardiovascular blood volume.     

Multi-compartmental modelling and experimental design for glucose transport studies in insulin-resistant rats

Many pathologies are associated with abnormalities of glucose metabolism or with perturbations of its transport (type 2 diabetes or insulin-resistance). The pre-diabetic state is characterised by a state of insulin-resistance, in others words a defect of glucose transport in insulin-sensible tissues, such as muscles and adipose tissues. The mathematical modelling of experimental data can be an excellent method to explore the mechanisms implied in the studied biological phenomenon. Thus, starting from a symbolic formulation like the compartmental modelling, it can be possible to develop a theoretical basis for the observation and to consider the best-adapted experiments for the study. We showed with mathematical models that [123I]-6-deoxy-6-iodo-D-glucose (6-DIG), shown as a tracer of glucose transport in vitro, could point out this transport abnormality. To quantify the insulin resistance, we estimated the fractional transfer coefficients of 6-DIG from the blood to the organs. We realised many studies to lead to a satisfying model; special attention has been paid to the precision of the parameter to select the best model. The results showed that by associating experimental data obtained with 6-DIG activities and an adapted mathematical model, discriminating parameters (in and out fractional transfer coefficients) between the two groups (control and insulin-resistant rats) could be pointed out.     

NEFA-glucose comodulation model of beta-cell insulin secretion in 24-h multiple-meal test

There is experimental evidence that a source of fatty acids (FAs) that is either exogenous or endogenous is necessary to support normal insulin secretion. Therefore, FAs comodulate the glucose-induced pancreatic insulin secretion. To assess the role of FAs, 16 morbidly obese nondiabetic patients and 6 healthy volunteers were studied. The controls and the obese subjects, before and after diet-induced weight loss, spent 24 h in a calorimetric chamber, where they consumed standardized meals. Hourly blood samples were drawn from a central venous catheter for the measurement of glucose, C-peptide, and nonesterified fatty acid (NEFA) concentrations. Insulin sensitivity was measured (as the M value) by euglycemic hyperinsulinemic clamp. In the present study, we propose a mathematical model in which insulin secretion rate (ISR) is expressed as a function of both plasma glucose and NEFA concentrations. Model parameters, obtained by fitting the individual experimental data of plasma C-peptide concentration, gave an estimated ISR comparable with that obtained by the deconvolution method. To evaluate the performance of the model in an experimental condition in which incretin effect was minimized, previous data on insulin secretion following a butter load and subsequent hyperglycemic clamp were reanalyzed. This model of nutrient-stimulated insulin secretion is the first attempt to represent in a simple way the interaction between glucose and NEFA in the regulation of insulin secretion in the beta-cell and explains, at least in part, the "potentiation factor" used in previous models to account for other control factors different from glucose after either an intravenous infusion of glucose or a mixed meal.     

Continuous modeling of metabolic networks with gene regulation in yeast and in vivo determination of rate parameters

A continuous model of a metabolic network including gene regulation to simulate metabolic fluxes during batch cultivation of yeast Saccharomyces cerevisiae was developed. The metabolic network includes reactions of glycolysis, gluconeogenesis, glycerol and ethanol synthesis and consumption, the tricarboxylic acid cycle, and protein synthesis. Carbon sources considered were glucose and then ethanol synthesized during growth on glucose. The metabolic network has 39 fluxes, which represent the action of 50 enzymes and 64 genes and it is coupled with a gene regulation network which defines enzyme synthesis (activities) and incorporates regulation by glucose (enzyme induction and repression), modeled using ordinary differential equations. The model includes enzyme kinetics, equations that follow both mass-action law and transport as well as inducible, repressible, and constitutive enzymes of metabolism. The model was able to simulate a fermentation of S. cerevisiae during the exponential growth phase on glucose and the exponential growth phase on ethanol using only one set of kinetic parameters. All fluxes in the continuous model followed the behavior shown by the metabolic flux analysis (MFA) obtained from experimental results. The differences obtained between the fluxes given by the model and the fluxes determined by the MFA do not exceed 25% in 75% of the cases during exponential growth on glucose, and 20% in 90% of the cases during exponential growth on ethanol. Furthermore, the adjustment of the fermentation profiles of biomass, glucose, and ethanol were 95%, 95%, and 79%, respectively. With these results the simulation was considered successful. A comparison between the simulation of the continuous model and the experimental data of the diauxic yeast fermentation for glucose, biomass, and ethanol, shows an extremely good match using the parameters found. The small discrepancies between the fluxes obtained through MFA and those predicted by the differential equations, as well as the good match between the profiles of glucose, biomass, and ethanol, and our simulation, show that this simple model, that does not rely on complex kinetic expressions, is able to capture the global behavior of the experimental data. Also, the determination of parameters using a straightforward minimization technique using data at only two points in time was sufficient to produce a relatively accurate model. Thus, even with a small amount of experimental data (rates and not concentrations) it was possible to estimate the parameters minimizing a simple objective function. The method proposed allows the obtention of reasonable parameters and concentrations in a system with a much larger number of unknowns than equations. Hence a contribution of this study is to present a convenient way to find in vivo rate parameters to model metabolic and genetic networks under different conditions.     

Model-Based Quantification of the Systemic Interplay between Glucose and Fatty Acids in the Postprandial State

In metabolic diseases such as Type 2 Diabetes and Non-Alcoholic Fatty Liver Disease, the systemic regulation of postprandial metabolite concentrations is disturbed. To understand this dysregulation, a quantitative and temporal understanding of systemic postprandial metabolite handling is needed. Of particular interest is the intertwined regulation of glucose and non-esterified fatty acids (NEFA), due to the association between disturbed NEFA metabolism and insulin resistance. However, postprandial glucose metabolism is characterized by a dynamic interplay of simultaneously responding regulatory mechanisms, which have proven difficult to measure directly. Therefore, we propose a mathematical modelling approach to untangle the systemic interplay between glucose and NEFA in the postprandial period. The developed model integrates data of both the perturbation of glucose metabolism by NEFA as measured under clamp conditions, and postprandial time-series of glucose, insulin, and NEFA. The model can describe independent data not used for fitting, and perturbations of NEFA metabolism result in an increased insulin, but not glucose, response, demonstrating that glucose homeostasis is maintained. Finally, the model is used to show that NEFA may mediate up to 30–45% of the postprandial increase in insulin-dependent glucose uptake at two hours after a glucose meal. In conclusion, the presented model can quantify the systemic interactions of glucose and NEFA in the postprandial state, and may therefore provide a new method to evaluate the disturbance of this interplay in metabolic disease.     

A mathematical model of metabolic insulin signaling pathways

We develop a mathematical model that explicitly represents many of the known signaling components mediating translocation of the insulin-responsive glucose transporter GLUT4 to gain insight into the complexities of metabolic insulin signaling pathways. A novel mechanistic model of postreceptor events including phosphorylation of insulin receptor substrate-1, activation of phosphatidylinositol 3-kinase, and subsequent activation of downstream kinases Akt and protein kinase C-zeta is coupled with previously validated subsystem models of insulin receptor binding, receptor recycling, and GLUT4 translocation. A system of differential equations is defined by the structure of the model. Rate constants and model parameters are constrained by published experimental data. Model simulations of insulin dose-response experiments agree with published experimental data and also generate expected qualitative behaviors such as sequential signal amplification and increased sensitivity of downstream components. We examined the consequences of incorporating feedback pathways as well as representing pathological conditions, such as increased levels of protein tyrosine phosphatases, to illustrate the utility of our model for exploring molecular mechanisms. We conclude that mathematical modeling of signal transduction pathways is a useful approach for gaining insight into the complexities of metabolic insulin signaling.     

An extension to the compartmental model of type 1 diabetic patients to reproduce exercise periods with glycogen depletion and replenishment

The purpose of this work is to present the main interactions promoted by exercise and synthesize them into mathematical equations. It is intended to extend the ability of the compartmental glucose-insulin model introduced by Sorensen [1985. A physiologic model of glucose metabolism in man and its use to design and assess improved insulin therapies for diabetes. Ph.D. Dissertation, Chemical Engineering Department, MIT, Cambridge] to reproduce variations in the blood glucose concentration induced by exercise in diabetic patients and to complement the previous work by Lenart and Parker [2002. Modeling exercise effects in type I diabetic patients. In: Proceedings of the 15th Triennial World Congress, Barcelona, Spain] and Lenart, DiMascio and Parker [2002. Modeling glycogen-exercise interactions in type I diabetic patients. In: Proceedings of the A.I.Ch.E. Annual Meeting, Indianapolis, IN]. The immediate consequences of exercise are incorporated in this research: redistribution of blood flows, increments in peripheral glucose and insulin uptakes, and increment in hepatic glucose production. The extended model was verified with experimental data for light and moderate intensity exercise. In addition, data extrapolation was introduced to simulate heavy intensity exercise. The hepatic glycogen reservoir limits the peripheral glucose uptake for prolonged exercise. Therefore, the depletion and replenishment of hepatic glycogen were modeled, looking to reproduce the blood glucose levels for a type 1 diabetic patient during a normal day, with meal intakes, insulin infusions and/or boluses, and a predefined exercise regime. From the extensive simulation evaluation, it is found that the new exercise model provides a good approximation to the available experimental data from literature.     

Is Dynamic Autocrine Insulin Signaling Possible? A Mathematical Model Predicts Picomolar Concentrations of Extracellular Monomeric Insulin within Human Pancreatic Islets

Insulin signaling is essential for -cell survival and proliferation in vivo. Insulin also has potent mitogenic and anti-apoptotic actions on cultured -cells, with maximum effect in the high picomolar range and diminishing effect at high nanomolar doses. In order to understand whether these effects of insulin are constitutive or can be subjected to physiological modulation, it is essential to estimate the extracellular concentration of monomeric insulin within an intact islet. Unfortunately, the in vivo concentration of insulin monomers within the islet cannot be measured directly with current technology. Here, we present the first mathematical model designed to estimate the levels of monomeric insulin within the islet extracellular space. Insulin is released as insoluble crystals that exhibit a delayed dissociation into hexamers, dimers, and eventually monomers, which only then can act as signaling ligands. The rates at which different forms of insulin dissolve in vivo have been estimated from studies of peripheral insulin injection sites. We used this and other information to formulate a mathematical model to estimate the local insulin concentration within a single islet as a function of glucose. Model parameters were estimated from existing literature. Components of the model were validated using experimental data, if available. Model analysis predicted that the majority of monomeric insulin in the islet is that which has been returned from the periphery, and the concentration of intra-islet monomeric insulin varies from 50–300 pM when glucose is in the physiological range. Thus, our results suggest that the local concentration of monomeric insulin within the islet is in the picomolar ‘sweet spot’ range of insulin doses that activate the insulin receptor and have the most potent effects on -cells in vitro. Together with experimental data, these estimations support the concept that autocrine/paracrine insulin signalling within the islet is dynamic, rather than constitutive and saturated.     

Modeling a simplified regulatory system of blood glucose at molecular levels

In this paper, we propose a new mathematical control system for a simplified regulatory system of blood glucose by taking into account the dynamics of glucose and glycogen in liver and the dynamics of insulin and glucagon receptors at the molecular level. Numerical simulations show that the proposed feedback control system agrees approximately with published experimental data. Sensitivity analysis predicts that feedback control gains of insulin receptors and glucagon receptors are robust. Using the model, we develop a new formula to compute the insulin sensitivity. The formula shows that the insulin sensitivity depends on various parameters that determine the insulin influence on insulin-dependent glucose utilization and reflect the efficiency of binding of insulin to its receptors. Using Lyapunov indirect method, we prove that the new control system is input-output stable. The stability result provides theoretical evidence for the phenomenon that the blood glucose fluctuates within a narrow range in response to the exogenous glucose input from food. We also show that the regulatory system is controllable and observable. These structural system properties could explain why the glucose level can be regulated.     

Modeling the insulin-glucose feedback system: the significance of pulsatile insulin secretion

A mathematical model of the insulin-glucose feedback regulation in man is used to examine the effects of an oscillatory supply of insulin compared to a constant supply at the same average rate. We show that interactions between the oscillatory insulin supply and the receptor dynamics can be of minute significance only. It is possible, however, to interpret seemingly conflicting results of clinical studies in terms of their different experimental conditions with respect to the hepatic glucose release. If this release is operating near an upper limit, an oscillatory insulin supply will be more efficient in lowering the blood glucose level than a constant supply. If the insulin level is high enough for the hepatic release of glucose to nearly vanish, the opposite effect is observed. For insulin concentrations close to the point of inflection of the insulin-glucose dose-response curve an oscillatory and a constant insulin infusion produce similar effects.     

Decreased glucose tolerance but normal blood glucose levels in the field in the caviomorph rodent Ctenomys talarum: the role of stress and physical activity

Hystricomorph rodents have a divergent insulin molecule with only 1-10% of the biological activity in comparison to other mammalian species. In this study, we used the subterranean rodent Ctenomys talarum as a model and performed blood glucose tolerance tests (GTTs) with trained and untrained individuals to evaluate blood glucose regulation and the possible role of physical activity as a compensatory mechanism. Additionally, we evaluated the variations in blood glucose during acute and chronic stress and gathered data in the field to evaluate natural-occurring variations in blood glucose levels. The GTTs showed that C. talarum have a diminished capacity of regulating blood glucose levels in comparison to other mammals and suggest that unexplored differences in the compensatory mechanisms, insulin structure and/or glucose transporters exist within species of hystricomorph rodents. However, blood glucose levels in the field stayed within the normal mammalian range. Physical activity did not prove to be a compensatory mechanism for blood glucose regulation. The individuals did not display important increases in blood glucose after acute stressors and managed to adequately regulate blood glucose during chronic stress. We suggest that the species may not face a selective pressure favoring a more tightly, mammalian like, capacity of regulating blood glucose levels.