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

A mechanism for slow rhythms in coordinated pancreatic islet activity

Insulin levels in the blood oscillate with a variety of periods, including rapid (5-10 min), ultradian (50-120 min), and circadian (24 h). Oscillations of insulin are beneficial for lowering blood glucose and disrupted rhythms are found in people with type 2 diabetes and their close relatives. These in vivo secretion dynamics imply that the oscillatory activity of individual islets of Langerhans are synchronized, although the mechanism for this is not known. One mechanism by which islets may synchronize is negative feedback of insulin on whole-body glucose levels. In previous work, we demonstrated that a negative feedback loop with a small time delay, to account for the time required for islets to be exposed to a new glucose concentration in vivo, results in small 3-6 islet populations synchronizing to produce fast closed-loop oscillations. However, these same islet populations could also produce slow closed-loop oscillations with periods longer than the natural islet oscillation periods. Here, we investigate the origin of the slow oscillations and the bistability with the fast oscillations using larger islet populations (20-50 islets). In contrast to what was observed earlier, larger islet populations mainly synchronize to longer-period oscillations that are approximately twice the delay time used in the feedback loop. A mean-field model was also used as a proxy for a large islet population to uncover the underlying mechanism for the slow rhythm. The heterogeneous intrinsic oscillation periods of the islets interferes with this rhythm mechanism when islet populations are small, and is similar to adding noise to the mean-field model. Thus, the effect of a time delay in the glucose feedback mechanism is similar to other examples of time-delayed systems in biology and may be a viable mechanism for ultradian oscillations.

Computer modeling of digestive processes in the alimentary tract and their physiological regulation mechanisms: closing the gap between digestion models and in vivo behavior

Introduction

A model has been developed for in silico simulation of digestion and its physiological feedback mechanisms.

Methods

The model is based on known physiology described in the literature and is able to describe the complexity of many simultaneous processes related to food digestion.

Results

Despite the early stage of development of the model, it already encompasses a large number of processes that occur simultaneously, enabling the prediction of a large number of post-prandial physiological markers, which can be highly functional in combination with in vitro, organ-on-a-chip and digital twin models purposed to measure the physiological properties of organs and to predict the effect of adjusted food composition in normal and diseased states.

Discussion

Input from and collaboration between science fileds is needed to further develop and refine the model and to connect with in vitro, in vivo, and ex vivo (organ-on-a-chip) models.

Kinetic modeling of leucine-mediated signaling and protein metabolism in human skeletal muscle

Skeletal muscle protein levels are governed by the relative rates of muscle protein synthesis (MPS) and breakdown (MPB). The mechanisms controlling these rates are complex, and their integrated behaviors are challenging to study through experiments alone. The purpose of this study was to develop and analyze a kinetic model of leucine-mediated mTOR signaling and protein metabolism in the skeletal muscle of young adults. Our model amalgamates published cellular-level models of the IRS1-PI3K-Akt-mTORC1 signaling system and of skeletal-muscle leucine kinetics with physiological-level models of leucine digestion and transport and insulin dynamics. The model satisfactorily predicts experimental data from diverse leucine feeding protocols. Model analysis revealed that total levels of p70S6K are a primary determinant of MPS, insulin signaling substantially affects muscle net protein balance via its effects on MPB, and p70S6K-mediated feedback of mTORC1 signaling reduces MPS in a dose-dependent manner.

Dynamical modeling the effect of glucagon-like peptide on glucose-insulin regulatory system based on mice experimental observation

As an emerging global epidemic, type 2 diabetes mellitus (T2DM) represents one of the leading causes of morbidity and mortality worldwide. Existing evidences demonstrated that glucagon-like peptide-1 (GLP-1) modulate the glucose regulatory system by enhancing the β-cell function. However, the detailed process of GLP-1 in glycaemic regulator for T2DM remains to be clarified. Thus, in this study, we propose an Institute of Cancer Research (ICR) mice high fat and cholesterol dietary experimental data-driven mathematical model to investigate the secretory effect of GLP-1 on the dynamics of glucose-insulin regulatory system. Specifically, we develop a mathematical model of GLP-1 dynamics as part of the interaction model of β-cell, insulin, and glucose dynamics. The parameter estimation and data fitting are in agreement with the data in mice experiments In addition, uncertainty quantification is performed to explore the possible factors that influence the pathways leading to the pathological state. Model analyses reveal that the high fat or high cholesterol diet stimulated GLP-1 plays an important role in the dynamics of glucose, insulin and β cells in short-term. These results show that enhanced GLP-1 may mitigate the dysregulation of glucose-insulin regulatory system via promoting the β cells function and stimulating secretion of insulin, which offers an in-depth insights into the mechanistic of hyperglycemia from dynamical approach and provide the theoretical basis for GLP-1 served as a potential clinical targeted drug for treatment of T2DM.

Dynamical analysis of a glucose-insulin regulatory system with insulin-degrading enzyme and multiple delays

This paper investigates the dynamics of a glucose-insulin regulatory system model that incorporates: (1) insulin-degrading enzyme in the insulin equation; and (2) discrete time delays respectively in the insulin production term, hepatic glucose production term, and the insulin-degrading enzyme. We provide rigorous results of our model including the asymptotic stability of the equilibrium solution and the existence of Hopf bifurcation. We show that analytically and numerically at a certain value the time delays driven stability or instability occurs when the corresponding model has an interior equilibrium. Moreover, we illustrate the oscillatory regulation and insulin secretion via numerical simulations, which show that the model dynamics exhibit physiological observations and more information by allowing parameters to vary. Our results may provide useful biological insights into diabetes for the glucose-insulin regulatory system model.

Causal discovery approach with reinforcement learning for risk factors of type II diabetes mellitus

BACKGROUND

Statistical correlation analysis is currently the most typically used approach for investigating the risk factors of type 2 diabetes mellitus (T2DM). However, this approach does not readily reveal the causal relationships between risk factors and rarely describes the causal relationships visually.

RESULTS

Considering the superiority of reinforcement learning in prediction, a causal discovery approach with reinforcement learning for T2DM risk factors is proposed herein. First, a reinforcement learning model is constructed for T2DM risk factors. Second, the process involved in the causal discovery method for T2DM risk factors is detailed. Finally, several experiments are designed based on diabetes datasets and used to verify the proposed approach.

CONCLUSIONS

The experimental results show that the proposed approach improves the accuracy of causality mining between T2DM risk factors and provides new evidence to researchers engaged in T2DM prevention and treatment research.

Dynamics of a glucose-insulin model

Diabetes mellitus is a noncommunicable disease, which is a serious threat to human health around the world. In this paper, we propose a simple glucose-insulin model with Michaelis-Menten function as insulin degradation rate to mimic the pathogenic mechanism of diabetes. By theoretical analysis, a unique positive equilibrium of model exists and it is globally asymptotically stable. The four strategies are designed for diabetes patients based on the sensitivity of parameters, including insulin injection and medicine treatments. Numerical simulations are given to support the theoretical results.

Inferring Insulin Secretion Rate from Sparse Patient Glucose and Insulin Measures

The insulin secretion rate (ISR) contains information that can provide a personal, quantitative understanding of endocrine function. If the ISR can be reliably inferred from measurements, it could be used for understanding and clinically diagnosing problems with the glucose regulation system. Objective: This study aims to develop a model-based method for inferring a parametrization of the ISR and related physiological information among people with different glycemic conditions in a robust manner. The developed algorithm is applicable for both dense or sparsely sampled plasma glucose/insulin measurements, where sparseness is defined in terms of sampling time with respect to the fastest time scale of the dynamics. Methods: An algorithm for parametrizing and validating a functional form of the ISR for different compartmental models with unknown but estimable ISR function and absorption/decay rates describing the dynamics of insulin accumulation was developed. The method and modeling applies equally to c-peptide secretion rate (CSR) when c-peptide is measured. Accuracy of fit is reliant on reconstruction error of the measured trajectories, and when c-peptide is measured the relationship between CSR and ISR. The algorithm was applied to data from 17 subjects with normal glucose regulatory systems and 9 subjects with cystic fibrosis related diabetes (CFRD) in which glucose, insulin and c-peptide were measured in course of oral glucose tolerance tests (OGTT). Results: This model-based algorithm inferred parametrization of the ISR and CSR functional with relatively low reconstruction error for 12 of 17 control and 7 of 9 CFRD subjects. We demonstrate that when there are suspect measurements points, the validity of excluding them may be interrogated with this method. Significance: A new estimation method is available to infer the ISR and CSR functional profile along with plasma insulin and c-peptide absorption rates from sparse measurements of insulin, c-peptide, and plasma glucose concentrations. We propose a method to interrogate and exclude potentially erroneous OGTT measurement points based on reconstruction errors.

A methodical survey of mathematical model-based control techniques based on open and closed loop control approach for diabetes management

Disturbance of blood sugar level is controlled through well-known biomechanical feedback loops: high levels of glucose in blood facilitate to release insulin from the pancreas which accelerates the absorption rate of cellular glucose. Low glucose levels encourage to release pancreatic glucagon which induces glycogen breakdown to glucose in the liver. These bio-control systems do not function properly in diabetic patients. Though the control of disease seems intuitively easy, in real life, due to many differences in structure by diet and fasting, exercise, medications, patient’s profile and other stressors, it is not that easy. The mathematical models of the glucose-insulin regulatory system follow the patient’s physiological conditions which make it difficult to identify and estimate all the model parameters. In this paper, we have given a systematic literature review on mathematical models of the diabetic patients, and various kinds of disease control techniques through the development of open and closed loop insulin deliver command system and optimization of exogenous insulin rate. It demonstrates the open and closed loop type model-based control strategies underlying the assumptions of the concerned models. The combination of mathematical model with control strategies such as genetic algorithm (GA), neural network (NN), sliding mode controller (SMC), model predictive controller (MPC), and fuzzy logic control (FLC) has been considered, which provides an overview of this area, highlighting the control profile over the diabetic model with promising clinical results, outlining key challenges, and identifying needs for the future research. Also, the significance of these control algorithms has been discussed in the presence of the noises, the controller’s robustness and various other disturbances. It provides substantial information on diabetes management through various control techniques.

An adaptive technique based blood glucose control in type-1 diabetes mellitus patients

This study proposes Grasshopper Optimization Algorithm (GOA) based type 1 diabetes mellitus system utilizing the nonlinear Bergman minimal model with proportional integral derivative (PID) controller. GOA is the optimization algorithm, which is utilized for selecting the optimized tuning parameters of the PID controller also solves the nonlinear system parameter identification problem. The novelty of the proposed study is to stabilize the glucose level in blood for type 1 diabetic patients by infusion of insulin in reduced time with optimal quantity. Without any intervention to the normal activities of patients, the supply of insulin injection and glucose monitoring is performed automatically for type 1 diabetic patients using this controller. In between the measured variable and set point, the difference is calculated by the PID controller to evaluate an error values. In realistic patient oriented conditions, the control performance evaluation, control optimization, and advanced patient modelling should be highly concentrated during the research/analysis on blood glucose control. Evaluation is performed to analyze control performances and implementation is done on Simulink/MATLAB environment. The performance analysis of the type 1 diabetes mellitus system with GOA technique is also discussed and to improve the control performance, to optimize the controller parameters. The simulation results have proved the substantial improvement in the performance of proposed algorithm with the better results achieved than the other conventional controllers such as PSO-PID and EHO-PID.

Design principles of the paradoxical feedback between pancreatic alpha and beta cells

Mammalian glucose homeostasis is controlled by the antagonistic hormones insulin and glucagon, secreted by pancreatic beta and alpha cells respectively. These two cell types are adjacently located in the islets of Langerhans and affect each others’ secretions in a paradoxical manner: while insulin inhibits glucagon secretion from alpha cells, glucagon seems to stimulate insulin secretion from beta cells. Here we ask what are the design principles of this negative feedback loop. We systematically simulate the dynamics of all possible islet inter-cellular connectivity patterns and analyze different performance criteria. We find that the observed circuit dampens overshoots of blood glucose levels after reversion of glucose drops. This feature is related to the temporal delay in the rise of insulin concentrations in peripheral tissues, compared to the immediate hormone action on the liver. In addition, we find that the circuit facilitates coordinate secretion of both hormones in response to protein meals. Our study highlights the advantages of a paradoxical paracrine feedback loop in maintaining metabolic homeostasis.

Quantitative approaches to energy and glucose homeostasis: machine learning and modelling for precision understanding and prediction

Obesity is a major global public health problem. Understanding how energy homeostasis is regulated, and can become dysregulated, is crucial for developing new treatments for obesity. Detailed recording of individual behaviour and new imaging modalities offer the prospect of medically relevant models of energy homeostasis that are both understandable and individually predictive. The profusion of data from these sources has led to an interest in applying machine learning techniques to gain insight from these large, relatively unstructured datasets. We review both physiological models and machine learning results across a diverse range of applications in energy homeostasis, and highlight how modelling and machine learning can work together to improve predictive ability. We collect quantitative details in a comprehensive mathematical supplement. We also discuss the prospects of forecasting homeostatic behaviour and stress the importance of characterizing stochasticity within and between individuals in order to provide practical, tailored forecasts and guidance to combat the spread of obesity.

Asymptotic tracking and disturbance rejection of the blood glucose regulation system

Type 1 diabetes patients need external insulin to maintain blood glucose within a narrow range from 65 to 108 mg/dl (3.6 to 6.0 mmol/l). A mathematical model for the blood glucose regulation is required for integrating a glucose monitoring system into insulin pump technology to form a closed-loop insulin delivery system on the feedback of the blood glucose, the so-called "artificial pancreas". The objective of this paper is to treat the exogenous glucose from food as a glucose disturbance and then develop a closed-loop feedback and feedforward control system for the blood glucose regulation system subject to the exogenous glucose disturbance. For this, a mathematical model for the glucose disturbance is proposed on the basis of experimental data, and then incorporated into an existing blood glucose regulation model. Because all the eigenvalues of the disturbance model have zero real parts, the center manifold theory is used to establish blood glucose regulator equations. We then use their solutions to synthesize a required feedback and feedforward controller to reject the disturbance and asymptotically track a constant glucose reference of 90  mg/dl. Since the regulator equations are nonlinear partial differential equations and usually impossible to solve analytically, a linear approximation solution is obtained. Our numerical simulations show that, under the linear approximate feedback and feedforward controller, the blood glucose asymptotically tracks its desired level of 90 mg/dl approximately.

COEXISTENCE OF THREE OSCILLATORY MODES OF INSULIN SECRETION: MATHEMATICAL MODELING AND RELEVANCE TO GLUCOSE REGULATION

Insulin secretion in pancreatic [Formula: see text]-cells exhibits three oscillatory modes with distinct period ranges, called fast, slow, and ultradian modes. To unveil the mechanism underlying such oscillatory behaviors and their roles in blood glucose regulation, we propose a combined model for the glucose–insulin regulation system, incorporating both the cell-level insulin secretion mechanism and inter-organ interactions in the blood glucose regulation. Special emphasis is placed on the identification of the mechanism of the slow oscillation and its role associated with the whole-body glucose regulation. Via extensive numerical simulations, we obtain macroscopic behaviors of the three types of insulin/glucose oscillations in the whole-body as well as microscopic behaviors of the membrane potential and the calcium concentration in the [Formula: see text]-cell. Finally, optimal regulatory strategies for the blood glucose level are discussed on the basis of the quantitative information obtained from the mathematical modeling and numerical simulations.

The Pancreatic β-Cell: A Bioenergetic Perspective

The pancreatic β-cell secretes insulin in response to elevated plasma glucose. This review applies an external bioenergetic critique to the central processes of glucose-stimulated insulin secretion, including glycolytic and mitochondrial metabolism, the cytosolic adenine nucleotide pool, and its interaction with plasma membrane ion channels. The control mechanisms responsible for the unique responsiveness of the cell to glucose availability are discussed from bioenergetic and metabolic control standpoints. The concept of coupling factor facilitation of secretion is critiqued, and an attempt is made to unravel the bioenergetic basis of the oscillatory mechanisms controlling secretion. The need to consider the physiological constraints operating in the intact cell is emphasized throughout. The aim is to provide a coherent pathway through an extensive, complex, and sometimes bewildering literature, particularly for those unfamiliar with the field.

Mathematical investigation of diabetically impaired ultradian oscillations in the glucose-insulin regulation

We study the effect of diabetic deficiencies on the production of an oscillatory ultradian regime using a deterministic nonlinear model which incorporates two physiological delays. It is shown that insulin resistance impairs the production of oscillations by dampening the ultradian cycles. Four strategies for restoring healthy regulation are explored. Through the introduction of an instantaneous glucose-dependent insulin response, explicit conditions for the existence of periodic solutions in the linearised model are formulated, significantly reducing the complexity of identifying an oscillatory regime. The model is thus shown to be suitable for representing the effect of diabetes on the oscillatory regulation and for investigating pathways to reinstating a physiological healthy regime.

Simulation and qualitative analysis of glucose variability, mean glucose, and hypoglycemia after subcutaneous insulin therapy for stress hyperglycemia

Background

The critically ill can have persistent dysglycemia during the “subacute” recovery phase of their illness because of altered gene expression; it is also not uncommon for these patients to receive continuous enteral nutrition during this time. The optimal short-acting subcutaneous insulin therapy that should be used in this clinical scenario, however, is unknown. Our aim was to conduct a qualitative numerical study of the glucose-insulin dynamics within this patient population to answer the above question. This analysis may help clinicians design a relevant clinical trial.

Methods

Eight virtual patients with stress hyperglycemia were simulated by means of a mathematical model. Each virtual patient had a different combination of insulin resistance and insulin deficiency that defined their unique stress hyperglycemia state; the rate of gluconeogenesis was also doubled. The patients received 25 injections of subcutaneous regular or Lispro insulin (0-6 U) with 3 rates of continuous nutrition. The main outcome measurements were the change in mean glucose concentration, the change in glucose variability, and hypoglycemic episodes. These end points were interpreted by how the ultradian oscillations of glucose concentration were affected by each insulin preparation.

Results

Subcutaneous regular insulin lowered both mean glucose concentrations and glucose variability in a linear fashion. No hypoglycemic episodes were noted. Although subcutaneous Lispro insulin lowered mean glucose concentrations, glucose variability increased in a nonlinear fashion. In patients with high insulin resistance and nutrition at goal, “rebound hyperglycemia” was noted after the insulin analog was rapidly metabolized. When the nutritional source was removed, hypoglycemia tended to occur at higher Lispro insulin doses. Finally, patients with severe insulin resistance seemed the most sensitive to insulin concentration changes.

Conclusions

Subcutaneous regular insulin consistently lowered mean glucose concentrations and glucose variability; its linear dose-response curve rendered the preparation better suited for a sliding-scale protocol. The longer duration of action of subcutaneous regular insulin resulted in better glycemic-control metrics for patients who were continuously postprandial. Clinical trials are needed to examine whether these numerical results represent the glucose-insulin dynamics that occur in intensive care units; if present, their clinical effects should be evaluated.

Quasi-Steady-State Analysis based on Structural Modules and Timed Petri Net Predict System's Dynamics: The Life Cycle of the Insulin Receptor

The insulin-dependent activation and recycling of the insulin receptor play an essential role in the regulation of the energy metabolism, leading to a special interest for pharmaceutical applications. Thus, the recycling of the insulin receptor has been intensively investigated, experimentally as well as theoretically. We developed a time-resolved, discrete model to describe stochastic dynamics and study the approximation of non-linear dynamics in the context of timed Petri nets. Additionally, using a graph-theoretical approach, we analyzed the structure of the regulatory system and demonstrated the close interrelation of structural network properties with the kinetic behavior. The transition invariants decomposed the model into overlapping subnetworks of various sizes, which represent basic functional modules. Moreover, we computed the quasi-steady states of these subnetworks and demonstrated that they are fundamental to understand the dynamic behavior of the system. The Petri net approach confirms the experimental results of insulin-stimulated degradation of the insulin receptor, which represents a common feature of insulin-resistant, hyperinsulinaemic states.

MATHEMATICAL MODELING OF INTRA-VENOUS GLUCOSE TOLERANCE TEST MODEL WITH TWO DISCRETE DELAYS

A mathematical model for Intra-Venous Glucose Tolerance Test (IVGTT) with explicit glucose–insulin interaction is presented as a system of delay differential equation with discrete time delays and its important mathematical features are analyzed. This model includes the positivity and boundedness of the solution. An unique equilibrium point is found and its local stability is investigated. Using the Lyapunov functional approach, we show the global stability of the unique equilibrium point. The length of delay that preserves the stability is estimated. Sensitivity analysis is performed on a delay differential equation model for IVGTT that suggests the parameter value has a major impact on the model dynamics. Numerical calculations are performed to support and elaborate the analytical findings.

Homeostatic regulation in physiological systems: A versatile Ansatz

A generic modelling formalism is described for homeostatic dynamics in physiological systems. The method is particularly suited where the peripheral, physiological system itself is well-characterised, but the details of the central, regulatory component (the nervous and endocrine systems) have not necessarily been characterised in full detail. The method is applied to temperature regulation in Cardinalis cardinalis, C. sinuatus, Lepus alleni, and Passer domesticus, and furthermore to hydromineral regulation in Lymnaea stagnalis. These case studies demonstrate that the method allows a comprehensive analysis and integration of the available data and is capable of furnishing physiologically relevant predictions. We discuss the method in relation to optimal control theory as well as more conventional modelling approaches.

Comparison of Subcutaneous Regular Insulin and Lispro Insulin in Diabetics Receiving Continuous Nutrition: A Numerical Study

BACKGROUND

Optimal management of non-critically ill patients with diabetes maintained on continuous enteral feeding (CEN) is poorly defined. Subcutaneous (SQ) lispro and SQ regular insulin were compared in a simulated type 1 and type 2 diabetic patient receiving CEN.

METHOD

A glucose-insulin feedback mathematical model was employed to simulate type 1 and type 2 diabetic patients on CEN. Each patient received 25 SQ injections of regular insulin or insulin lispro, ranging from 0-6 U. Primary endpoints were the change in mean glucose concentration (MGC) and change in glucose variability (GV); hypoglycemic episodes were also reported. The model was first validated against patient data.

RESULTS

Both SQ insulin preparations linearly decreased MGC, however, SQ regular insulin decreased GV whereas SQ lispro tended to increase GV. Hourly glucose concentration measurements were needed to capture the increase in GV. In the type 2 diabetic patient, "rebound hyperglycemia" occurred after SQ lispro was rapidly metabolized. Although neither SQ insulin preparation caused hypoglycemia, SQ lispro significantly lowered MGC compared to SQ regular insulin. Thus, it may be more likely to cause hypoglycemia. Analyses of the detailed glucose concentration versus time data suggest that the inferior performance of lispro resulted from its shorter duration of action. Finally, the effects of both insulin preparations persisted beyond their duration of actions in the type 2 diabetic patient.

CONCLUSIONS

Subcutaneous regular insulin may be the short-acting insulin preparation of choice for this subset of diabetic patients. Clinical trial is required before a definitive recommendation can be made.

A mathematical model of the human metabolic system and metabolic flexibility

In healthy subjects some tissues in the human body display metabolic flexibility, by this we mean the ability for the tissue to switch its fuel source between predominantly carbohydrates in the postprandial state and predominantly fats in the fasted state. Many of the pathways involved with human metabolism are controlled by insulin and insulin-resistant states such as obesity and type-2 diabetes are characterised by a loss or impairment of metabolic flexibility. In this paper we derive a system of 12 first-order coupled differential equations that describe the transport between and storage in different tissues of the human body. We find steady state solutions to these equations and use these results to nondimensionalise the model. We then solve the model numerically to simulate a healthy balanced meal and a high fat meal and we discuss and compare these results. Our numerical results show good agreement with experimental data where we have data available to us and the results show behaviour that agrees with intuition where we currently have no data with which to compare.

Modeling and qualitative analysis of diabetes therapies with state feedback control

For the therapies of diabetes mellitus, a novel mathematical model with two state impulses: impulsive injection of insulin and impulsive injection of glucagon, is proposed. To avoid hypoglycemia and hyperglycemia, the injections of insulin and glucagon are determined by closely monitoring the plasma glucose level of the patients. By using differential equation geometry theory, the existence of periodic solution and the attraction region of the system have been obtained, which ensures that injections in such an automated way can keep the blood glucose concentration under control. The simulation results verify that the better insulin injection strategy in closed-loop control is a larger dose but longer interval rather than a smaller dose but shorter interval. Besides, our numerical analysis reveals that medicine studies and practice that slow down the insulin degradation are helpful for the plasma glucose control. Our findings can provide significant guidance in both design of artificial pancreas and clinical treatment.

Determination of personalized diabetes treatment plans using a two-delay model

Diabetes cases worldwide have risen steadily over the past few decades, lending urgency to the search for more efficient, effective, and personalized ways to treat the disease. Current treatment strategies, however, may fail to maintain oscillations in blood glucose concentration that naturally occur multiple times per day, an important element of normal human physiology. Building upon recent successes in mathematical modeling of the human glucose-insulin system, we show that both food intake and insulin therapy likely demand increasingly precise control over insulin sensitivity if oscillations at a healthy average glucose concentration are to be maintained. We then model and describe personalized treatment options for patients with diabetes that maintain these oscillations. We predict that for a person with type II diabetes, both blood glucose levels can be controlled and healthy oscillations maintained when the patient gets an hour of daily exercise and is placed on a combination of Metformin and sulfonylurea drugs. We note that insulin therapy and an additional hour of exercise will reduce the patient׳s need for sulfonylureas. Results of a modeling analysis suggest that, with constant nutrition and controlled exercise, the blood glucose levels of a person with type I diabetes can be properly controlled with insulin infusion between 0.45 and 0.7μU/mlmin. Lastly, we note that all suggested strategies rely on existing clinical techniques and established treatment measures, and so could potentially be of immediate use in the design of an artificial pancreas.

Qualitative analysis of subcutaneous Lispro and regular insulin injections for stress hyperglycemia: a pilot numerical study

Increased glucose variability (GV) is an independent risk factor for mortality in the critically ill; unfortunately, the optimal insulin therapy that minimizes GV is not known. We simulate the glucose-insulin feedback system to study how stress hyperglycemia (SH) states, taken to be a non-uniform group of physiologic disorders with varying insulin resistance (IR) and similar levels of hyperglycemia, respond to the type and dose of subcutaneous (SQ) insulin. Two groups of 100 virtual patients are studied: those receiving and those not receiving continuous enteral feeds. Stress hyperglycemia was facilitated by doubling the gluconeogenesis rate and IR was stepwise varied from a borderline to a high value. Lispro and regular insulin were simulated with dosages that ranged from 0 to 6 units; the resulting GV was analyzed after each insulin injection. The numerical model used consists of a set of non-linear differential equations with two time delays and five adjustable parameters. The results show that regular insulin decreased GV in both patient groups and rarely caused hypoglycemia. With continuous enteral feeds and borderline to mild IR, Lispro showed minimal effect on GV; however, rebound hyperglycemia that increased GV occurred when the IR was moderate to high. Without a nutritional source, Lispro worsened GV through frequent hypoglycemia episodes as the injection dose increased. The inferior performance of Lispro is a result of its rapid absorption profile; half of its duration of action is similar to the glucose ultradian period. Clinical trials are needed to examine whether these numerical results represent the glucose-insulin dynamics that occur in intensive care units, and if such dynamics are present, their clinical effects should be evaluated.

A novel mathematical model detecting early individual changes of insulin resistance

BACKGROUND

Insulin resistance (IR) and hyperinsulinemia as well as obesity play a key role in the metabolic syndrome (MetS), type 2 diabetes (T2D), and associated cardiovascular disease. Unfortunately, IR and hyperinsulinemia are often diagnosed late (i.e., when the MetS is already clinically evident). An earlier diagnosis of IR would be desirable to reduce its clinical consequences, in particular in view of the increasing prevalence of obesity and diabetes conditions. For this purpose, we developed a mathematical model capable of detecting early onset of IR through small variations of insulin sensitivity, glucose effectiveness, and first- or second-phase responses.

MATERIALS AND METHODS

Murine models provide controlled conditions to study various stages of IR. Various degrees of hypercholesterolemia, obesity, IR, and atherosclerosis were induced in low-density lipoprotein receptor-deficient mice by feeding them cholesterol- or sucrose-rich diets. IR was assessed by oral glucose tolerance tests. Controls included animals fed exclusively, or switched back to, regular chow. A nonlinear mathematical model of the order of 5 was developed by refining Bergman's "Minimal Model" and then applied to experimental data.

RESULTS

Different metabolic constellations consistently corresponded to specific and close-meshed changes in model parameters. Reduced second-phase glucose sensitivity characterized an early impaired glucose tolerance. Later stages showed an increased first-phase glucose sensitivity compensating for decreased insulin sensitivity. Finally, T2D was associated with both first- and second-phase sensitivities close to zero.

CONCLUSIONS

The new mathematical model detected various insulin-sensitive or -resistant metabolic stages of IR. It can therefore be implemented for quantitative metabolic risk assessment and may be of therapeutic value by anticipating the start of therapeutic interventions.

A Generic Integrated Physiologically based Whole-body Model of the Glucose-Insulin-Glucagon Regulatory System

Models of glucose metabolism are a valuable tool for fundamental and applied medical research in diabetes. Use cases range from pharmaceutical target selection to automatic blood glucose control. Standard compartmental models represent little biological detail, which hampers the integration of multiscale data and confines predictive capabilities. We developed a detailed, generic physiologically based whole-body model of the glucose-insulin-glucagon regulatory system, reflecting detailed physiological properties of healthy populations and type 1 diabetes individuals expressed in the respective parameterizations. The model features a detailed representation of absorption models for oral glucose, subcutaneous insulin and glucagon, and an insulin receptor model relating pharmacokinetic properties to pharmacodynamic effects. Model development and validation is based on literature data. The quality of predictions is high and captures relevant observed inter- and intra-individual variability. In the generic form, the model can be applied to the development and validation of novel diabetes treatment strategies.

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.

Glucose-insulin regulation model with subcutaneous insulin injection and evaluation using diabetic inpatients data

Closed-loop insulin delivery systems often implement glucose measurement and insulin administration in the subcutis. However some existing models for glucose-insulin system ignored the dynamics of subcutaneous glucose and subcutaneously-injected insulin. This paper reports a two-compartment model that includes glucose and insulin dynamics in subcutis, and its evaluation using patient data. Clinical information such as glucose level, insulin dosage, insulin injection time and meals of anonymous diabetes inpatients was collected. Measured glucose level of the diabetic inpatients agrees with that of computer simulation. Due to the lack of glucose-insulin model with subcutaneously-injected insulin for type 2 diabetic patients, our model was compared with existing model for type 1 subjects. The new glucose-insulin model can mimic dynamics of glucose and insulin under the disturbance of insulin injections and meals. Model parameters were estimated using nonlinear least square method and their effect on pathology and physiology of diabetes were analyzed.

The impact of mathematical modeling on the understanding of diabetes and related complications

Diabetes is a chronic and complex multifactorial disease caused by persistent hyperglycemia and for which underlying pathogenesis is still not completely understood. The mathematical modeling of glucose homeostasis, diabetic condition, and its associated complications is rapidly growing and provides new insights into the underlying mechanisms involved. Here, we discuss contributions to the diabetes modeling field over the past five decades, highlighting the areas where more focused research is required.

Dynamic processes in regulation and some implications for biofeedback and biobehavioral interventions

Systems theory has long been used in psychology, biology, and sociology. This paper applies newer methods of control systems modeling for assessing system stability in health and disease. Control systems can be characterized as open or closed systems with feedback loops. Feedback produces oscillatory activity, and the complexity of naturally occurring oscillatory patterns reflects the multiplicity of feedback mechanisms, such that many mechanisms operate simultaneously to control the system. Unstable systems, often associated with poor health, are characterized by absence of oscillation, random noise, or a very simple pattern of oscillation. This modeling approach can be applied to a diverse range of phenomena, including cardiovascular and brain activity, mood and thermal regulation, and social system stability. External system stressors such as disease, psychological stress, injury, or interpersonal conflict may perturb a system, yet simultaneously stimulate oscillatory processes and exercise control mechanisms. Resonance can occur in systems with negative feedback loops, causing high-amplitude oscillations at a single frequency. Resonance effects can be used to strengthen modulatory oscillations, but may obscure other information and control mechanisms, and weaken system stability. Positive as well as negative feedback loops are important for system function and stability. Examples are presented of oscillatory processes in heart rate variability, and regulation of autonomic, thermal, pancreatic and central nervous system processes, as well as in social/organizational systems such as marriages and business organizations. Resonance in negative feedback loops can help stimulate oscillations and exercise control reflexes, but also can deprive the system of important information. Empirical hypotheses derived from this approach are presented, including that moderate stress may enhance health and functioning.

Optimal homeostasis necessitates bistable control

Bistability is a fundamental phenomenon in nature. In biology, a number of fine properties of bistability have been identified. However, these properties are only consequences of bistability at the physiological level, which do not explain why it had to emerge during evolution. Using optimal homeostasis as the first principle, I find that bistability emerges as an indispensable control mechanism. It is the only solution to a dilemma in glucose homeostasis: high insulin efficiency is required to confer rapidness in plasma glucose clearance, whereas an insulin sparing state is required to guarantee the brain's safety during fasting. The optimality consideration renders a clear correspondence between the molecular and physiological levels. This new perspective can illuminate studies on the twin epidemics of obesity and diabetes and the corresponding intervening strategies. For example, overnutrition and sedentary lifestyle may represent sudden environmental changes that cause the lose of optimality, which may contribute to the marked rise of obesity and diabetes in our generation. Because this bistability result is independent of the parameters of the mathematical model (for which the result is quite general), some other biological systems may also use bistability to control homeostasis.

Mathematical model for glucose regulation in the whole-body system

The human body needs continuous and stable glucose supply for maintaining its biological functions. Stable glucose supply comes from the homeostatic regulation of the blood glucose level, which is controlled by various glucose consuming or producing organs. Therefore, it is important to understand the whole-body glucose regulation mechanism. In this article, we describe various mathematical models proposed for glucose regulation in the human body, and discuss the difficulty and limitation in reproducing real processes of glucose regulation.

The range of time delay and the global stability of the equilibrium for an IVGTT model

Diabetes mellitus has become a prevalent disease in the world. Diagnostic protocol for the onset of diabetes mellitus is the initial step in the treatments. The intravenous glucose tolerance test (IVGTT) has been considered as the most accurate method to determine the insulin sensitivity and glucose effectiveness. It is well known that there exists a time delay in insulin secretion stimulated by the elevated glucose concentration level. However, the range of the length of the delay in the existing IVGTT models are not fully discussed and thus in many cases the time delay may be assigned to a value out of its reasonable range. In addition, several attempts had been made to determine when the unique equilibrium point is globally asymptotically stable. However, all these conditions are delay-independent. In this paper, we discuss the range of the time delay and provide easy-to-check delay-dependent conditions for the global asymptotic stability of the equilibrium point for a recent IVGTT model through Liapunov function approach. Estimates of the upper bound of the delay for global stability are given in corollaries. In addition, the numerical simulation in this paper is fully incorporated with functional initial conditions, which is natural and more appropriate in delay differential equation systems.

Interaction of fast and slow dynamics in endocrine control systems with an application to β-cell dynamics

Endocrine dynamics spans a wide range of time scales, from rapid responses to physiological challenges to with slow responses that adapt the system to the demands placed on it. We outline a non-linear averaging procedure to extract the slower dynamics in a way that accounts properly for the non-linear dynamics of the faster time scale and is applicable to a hierarchy of more than two time scales, although we restrict our discussion to two scales for the sake of clarity. The procedure is exact if the slow time scale is infinitely slow (the dimensionless ε-quantity is the period of the fast time scale fluctuation times an upper bound to the slow time scale rate of change). However, even for an imperfect separation of time scales we find that this construction provides an excellent approximation for the slow-time dynamics at considerably reduced computational cost. Besides the computation advantage, the averaged equation provided a qualitative insight into the interaction of the time scales. We demonstrate the procedure and its advantages by applying the theory to the model described by Tolić et al. [I.M. Tolić, E. Mosekilde, J. Sturis, Modeling the insulin-glucose feedback system: the significance of pulsatile insulin secretion, J. Theor. Biol. 207 (2000) 361-375.] for ultradian dynamics of the glucose-insulin homeostasis feedback system, extended to include β-cell dynamics. We find that the dynamics of the β-cell mass are dependent not only on the glycemic load (amount of glucose administered to the system), but also on the way this load is applied (i.e. three meals daily versus constant infusion), effects that are lost in the inappropriate methods used by the earlier authors. Furthermore, we find that the loss of the protection against apoptosis conferred by insulin that occurs at elevated levels of insulin has a functional role in keeping the β-cell mass in check without compromising regulatory function. We also find that replenishment of β-cells from a rapidly proliferating pool of cells, as opposed to the slow turn-over which characterises fully differentiated β-cells, is essential to the prevention of type 1 diabetes.