Enhanced modelling of the glucose–insulin system and its applications in insulin therapies

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.

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.

Bifurcation analysis in a delay model of IVGTT glucose-insulin interaction

In this paper, a delayed differential model based on the intravenous glucose tolerance test is considered. The conditions to determine stability or instability of the model's steady state are obtained. We obtain the necessary conditions for the appearance of a bifurcation, and we investigate the direction and stability of the local bifurcation. For this purpose, the normal form theory is used. In addition, the numerical diagrams in the direction of theoretical results are drawn.

Low Molecular Weight Chitosan-Insulin Complexes Solubilized in a Mixture of Self-Assembled Labrosol and Plurol Oleaque and Their Glucose Reduction Activity in Rats

Oral insulin delivery that better mimics physiological pathways is a necessity as it ensures patient comfort and compliance. A system which is based on a vehicle of nano order where positively charged chitosan interacts with negatively charged insulin and forms a polyelectrolyte complex (PEC) solubilizate, which is then solubilized into an oily phase of oleic acid, labrasol, and plurol oleaque-protects insulin against enzymatic gastrointestinal reduction. The use of an anionic fatty acid in the oily phase, such as oleic acid, is thought to allow an interaction with cationic chitosan, hence reducing particle size. Formulations were assessed based on their hypoglycaemic capacities in diabetic rats as compared to conventional subcutaneous dosage forms. 50 IU/kg oral insulin strength could only induce blood glucose reduction equivalent to that of 5 IU/kg (1 International unit = 0.0347 mg of human insulin). Parameters that influence the pharmacological availability were evaluated. A preliminary investigation of the mechanism of absorption suggests the involvement of the lymphatic route.

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.

Predictive Control of the Blood Glucose Level in Type I Diabetic Patient Using Delay Differential Equation Wang Model

Because of increasing risk of diabetes, the measurement along with control of blood sugar has been of great importance in recent decades. In type I diabetes, because of the lack of insulin secretion, the cells cannot absorb glucose leading to low level of glucose. To control blood glucose (BG), the insulin must be injected to the body. This paper proposes a method for BG level regulation in type I diabetes. The control strategy is based on nonlinear model predictive control. The aim of the proposed controller optimized with genetics algorithms is to measure BG level each time and predict it for the next time interval. This merit causes a less amount of control effort, which is the rate of insulin delivered to the patient body. Consequently, this method can decrease the risk of hypoglycemia, a lethal phenomenon in regulating BG level in diabetes caused by a low BG level. Two delay differential equation models, namely Wang model and Enhanced Wang model, are applied as controller model and plant, respectively. The simulation results exhibit an acceptable performance of the proposed controller in meal disturbance rejection and robustness against parameter changes. As a result, if the nutrition of the person decreases instantly, the hypoglycemia will not happen. Furthermore, comparing this method with other works, it was shown that the new method outperforms previous studies.

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.

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.

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.

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.