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The purpose of this paper is to present a fractional nonlinear mathematical model with beta-cell kinetics and glucose–insulin feedback in order to describe changes in plasma glucose levels and insulin levels over time that may be associated with changes in beta-cell kinetics. We discuss the solution to the problem with respect to its existence, uniqueness, non-negativity, and boundedness. Using three different fractional derivative operators, the proposed model is examined. To approximate fractional-order systems, we use an efficient numerical Euler method in Caputo, Caputo–Fabrizio, and Atangana–Baleanu sense. Several asymptomatic behaviors are observed in the proposed models based on these three operators. These behaviors do not appear in integer-order derivative models. These behaviors are essential for understanding fractional-order systems dynamics. Our results provide insight into fractional-order systems dynamics. These operators analyze local and global stability and Hyers–Ulam stability. Furthermore, the numerical solutions for the proposed model are simulated using the three methods. [ABSTRACT FROM AUTHOR]

Millions of people worldwide suffer from diabetes, a medical condition that is spreading at an accelerating pace. Numerous studies show that risk factors that may arise from diabetes can be avoided if the disease is detected early. The health-care monitoring system has benefited greatly from early diabetes prediction made possible by the integration of Deep Learning (DL) and Machine Learning (ML) algorithms. The objective of many early studies was to increase the prediction model accuracy. However, DL algorithms often cannot fully exploit the potential of the available datasets because they are too small. This study includes a very accurate DL model as well as a novel system that integrates cloud services and allows users to directly enhance an existing data set, which can increase the accuracy of DL techniques. Therefore, the Long Short-Term Memory (LSTM) model with controller is chosen for efficient type-1 diabetes prediction. Experimental validation of the proposed Nonlinear Model Predictive Control (NMPC)_LSTM algorithm method is compared with other conventional DL algorithms. The proposed controller method achieves excellent blood glucose set point tracking and the proposed algorithms achieves 98.95% accuracy for the obtained data. It outperforms other existing methods with an increase in percentage accuracy compared to the Benchmark Pima Indian Diabetes Datasets (PIDD).

This survey reviews glucose regulatory systems using mathematical methods. We discuss the utilization of various minimal and comprehensive models of glucose metabolism combined with model predictive control, data assimilation, and physics-informed neural networks to develop decision support systems. These systems will help doctors and nurses administrating insulin levels in patients with diabetes and those in intensive care units. [ABSTRACT FROM AUTHOR]

Background: Due to the increased prevalence of diabetes and the irreparable complications of this disease, it is important to measure and monitor the blood glucose levels of diabetic patients. The only way to treat type 1 diabetes is monitoring insulin, and in this type of diabetes, insulin should be injected into the body in order to reduce the patient’s blood glucose as prescribed by the physician at certain times. In addition, the only way to treat type 2 diabetes is through diet and exercise daily. Objective: We aim to use an ordinary differential equation model with two-delays to control the rate of changes in blood glucose levels throughout the day, based on the amount of food that the person consumes. Material and Methods: In this analytical study, we extended an ODE model which is parameterized by data collected in this study to capture dynamics of glucose and insulin. We used global sensitivity analysis method to assess model robustness with respect to parameter perturbations. Results: Our results have shown that utilizing the dynamics of changes in blood glucose levels throughout the day can be used to prevent hypoglycemia and hyperglycemic in the diabetic patients.Conclusion: Dynamic modeling can help us to prevent hypoglycemia and hyperglycemia in the diabetic patients.

In the human glucose-insulin regulatory system, diverse metabolic issues can arise, including diabetes type I and type II, hyperinsulinemia, hypoglycemia, etc. Therefore, the analysis and characterization of such a biological system is a must. It is well known that mathematical models are an excellent option to study and predict natural phenomena to some extent. On the other hand, fractional-order calculus provides a generalization of derivatives and integrals to arbitrary orders giving us a framework to add memory properties and an extra degree of freedom to the mathematical models to approximate real-world phenomena with higher accuracy. In this work, we introduce a fractional-order version of a mathematical model of the glucose-insulin regulatory system. Using the fractional-order Caputo derivative, we can investigate different concentration rates among insulin, glucose, and healthy beta cells. Additionally, the model incorporates two time-lags to represent the elapsed time in insulin secretion in response to blood glucose level and the delay in glucose drop due to increased insulin concentration. Analytical results of the equilibrium points and their corresponding stability are given. Numerical results, including phase portraits and bifurcation diagrams, reveal that the fractional-order increases the chaotic regions, leading to potential metabolic problems. Vice versa, the system seems to work correctly when the behavior evolves to periodic windows.

Background: Previous studies reported that proprotein convertase subtilisin/kexin type 9 (PCSK9) was a key player in the regulations of lipid metabolism and glucose homeostasis. The current study aimed to detect the expression of PCSK9 in pregnant women with gestational diabetes mellitus (GDM) and investigate the possible relationships between PCSK9 and related metabolic phenotypes in GDM. Methods: Circulating PCSK9 levels were determined by ELISA kit in a cohort of subjects with GDM (n = 170) and normal glucose tolerance (NGT; n = 130). We collected blood samples from all participants for the biochemical index determinations. Diagnosis of GDM was made according to the International Association of the Diabetes and Pregnancy Study Groups Consensus Panel. Correlation analysis and logistic regression analysis were used to study the potential associations between PCSK9 and GDM. Results: GDM women presented significantly higher circulating PCSK9 levels than those in NGT pregnant subjects (268.07 ± 77.17 vs. 254.24 ± 74.22 ng/ml, P < 0.05). In the GDM group, serum PCSK9 levels were positively correlated with fasting plasma glucose (FPG) (R = 0.251, P = 0.015), glycated hemoglobin (HbA1c) (R = 0.275, P = 0.009), total cholesterol (TC) (R = 0.273, P = 0.010), and low-density lipoprotein cholesterol (LDL-C) (R = 0.326, P = 0.002) after adjustment of age and gestational age. Logistic regression found that age [odds ratio (OR) = 5.412, P = 0.02] and serum PCSK9 levels (OR = 4.696, P = 0.03) were independently associated with GDM. Compared with the lowest serum PCSK9 level quartile group, the prevalence of GDM was significantly higher in the highest quartile group, the ORs of GDM were 3.485 (95% CI 1.408–8.627, P < 0.05 for the trend), after adjusting for potential confounders. Conclusions: Circulating PCSK9 levels were associated with dyslipidemia, pathoglycemia, and the risk of incident GDM, indicating a potential link between PCSK9 and GDM. [ABSTRACT FROM AUTHOR]

Background: Due to the increased prevalence of diabetes and the irreparable complications of this disease, it is important to measure and monitor the blood glucose levels of diabetic patients. The only way to treat type 1 diabetes is monitoring insulin, and in this type of diabetes, insulin should be injected into the body in order to reduce the patient's blood glucose as prescribed by the physician at certain times. In addition, the only way to treat type 2 diabetes is through diet and exercise daily. Objective: We aim to use an ordinary differential equation model with two-delays to control the rate of changes in blood glucose levels throughout the day, based on the amount of food that the person consumes. Material and Methods: In this analytical study, we extended an ODE model which is parameterized by data collected in this study to capture dynamics of glucose and insulin. We used global sensitivity analysis method to assess model robustness with respect to parameter perturbations. Results: Our results have shown that utilizing the dynamics of changes in blood glucose levels throughout the day can be used to prevent hypoglycemia and hyperglycemic in the diabetic patients. Conclusion: Dynamic modeling can help us to prevent hypoglycemia and hyperglycemia in the diabetic patients. [ABSTRACT FROM AUTHOR]

BackgroundPrevious studies reported that proprotein convertase subtilisin/kexin type 9 (PCSK9) was a key player in the regulations of lipid metabolism and glucose homeostasis. The current study aimed to detect the expression of PCSK9 in pregnant women with gestational diabetes mellitus (GDM) and investigate the possible relationships between PCSK9 and related metabolic phenotypes in GDM.MethodsCirculating PCSK9 levels were determined by ELISA kit in a cohort of subjects with GDM (n = 170) and normal glucose tolerance (NGT; n = 130). We collected blood samples from all participants for the biochemical index determinations. Diagnosis of GDM was made according to the International Association of the Diabetes and Pregnancy Study Groups Consensus Panel. Correlation analysis and logistic regression analysis were used to study the potential associations between PCSK9 and GDM.ResultsGDM women presented significantly higher circulating PCSK9 levels than those in NGT pregnant subjects (268.07 ± 77.17 vs. 254.24 ± 74.22 ng/ml, P < 0.05). In the GDM group, serum PCSK9 levels were positively correlated with fasting plasma glucose (FPG) (R = 0.251, P = 0.015), glycated hemoglobin (HbA1c) (R = 0.275, P = 0.009), total cholesterol (TC) (R = 0.273, P = 0.010), and low-density lipoprotein cholesterol (LDL-C) (R = 0.326, P = 0.002) after adjustment of age and gestational age. Logistic regression found that age [odds ratio (OR) = 5.412, P = 0.02] and serum PCSK9 levels (OR = 4.696, P = 0.03) were independently associated with GDM. Compared with the lowest serum PCSK9 level quartile group, the prevalence of GDM was significantly higher in the highest quartile group, the ORs of GDM were 3.485 (95% CI 1.408–8.627, P < 0.05 for the trend), after adjusting for potential confounders.ConclusionsCirculating PCSK9 levels were associated with dyslipidemia, pathoglycemia, and the risk of incident GDM, indicating a potential link between PCSK9 and GDM.

Abstract This paper proposes a delayed fractional-order model of glucose–insulin interaction in the sense of the Caputo fractional derivative with incommensurate orders. This fractional-order model is developed from the first-order model of glucose–insulin interaction. Firstly, we investigate the non-negativity and the boundedness of solutions of the fractional-order model. Secondly, the stability and the bifurcation of the model are studied by separating the associated characteristic equation of the model into its real and imaginary parts and taking a time delay as the bifurcation parameter. The asymptotic stability and the Hopf bifurcation are discussed via the condition of creation of the bifurcation. Furthermore, it is shown that the onset of the bifurcation is related to the fractional orders of the model. Finally, some numerical simulations of the model using the Adam–Bashforth–Moulton predictor corrector scheme are demonstrated to support our obtained theoretical results.

Glucose effectiveness, defined as the ability of glucose itself to increase glucose utilization and inhibit hepatic glucose production, is an important mechanism maintaining normoglycemia. We conducted a minimal modeling analysis of glucose effectiveness at zero insulin (GEZI) using intravenous glucose tolerance test data from subjects with type 2 diabetes (T2D, n=154) and non-diabetic (ND) subjects (n=343). A hierarchical statistical analysis was performed, which provided a formal mechanism for pooling the data from all study subjects, to yield a single composite population model that quantifies the role of subject specific characteristics such as weight, height, age, sex, and glucose tolerance. Based on the resulting composite population model, GEZI was reduced from 0.021 min–1 (standard error – 0.00078 min–1) in the ND population to 0.011 min–1 (standard error – 0.00045 min–1) in T2D. The resulting model was also employed to calculate the proportion of the non–insulin-dependent net glucose uptake in each subject receiving an intravenous glucose load. Based on individual parameter estimates, the fraction of total glucose disposal independent of insulin was 72.8% ± 12.0% in the 238 ND subjects over the course of the experiment, indicating the major contribution to the whole-body glucose clearance under non-diabetic conditions. This fraction was significantly reduced to 48.8% ± 16.9% in the 30 T2D subjects, although still accounting for approximately half of the total in the T2D population based on our modeling analysis. Given the potential application of glucose effectiveness as a predictor of glucose intolerance and as a potential therapeutic target for treating diabetes, more investigations of glucose effectiveness in other disease conditions can be conducted using the hierarchical modeling framework reported herein.

The brain is responsible for maintaining whole-body energy homeostasis by changing energy input and availability. The hypothalamus and dorsal vagal complex (DVC) are the primary sites of metabolic control, able to sense both hormones and nutrients and adapt metabolism accordingly. The mitochondria respond to the level of nutrient availability by fusion or fission to maintain energy homeostasis; however, these processes can be disrupted by metabolic diseases including obesity and type II diabetes (T2D). Mitochondrial dynamics are crucial in the development and maintenance of obesity and T2D, playing a role in the control of glucose homeostasis and whole-body metabolism across neurons and glia in the hypothalamus and DVC.

Glucose effectiveness, defined as the ability of glucose itself to increase glucose utilization and inhibit hepatic glucose production, is an important mechanism maintaining normoglycemia. We conducted a minimal modeling analysis of glucose effectiveness at zero insulin (GEZI) using intravenous glucose tolerance test data from subjects with type 2 diabetes (T2D, n=154) and non-diabetic (ND) subjects (n=343). A hierarchical statistical analysis was performed, which provided a formal mechanism for pooling the data from all study subjects, to yield a single composite population model that quantifies the role of subject specific characteristics such as weight, height, age, sex, and glucose tolerance. Based on the resulting composite population model, GEZI was reduced from 0.021 min–1 (standard error – 0.00078 min–1) in the ND population to 0.011 min–1 (standard error – 0.00045 min–1) in T2D. The resulting model was also employed to calculate the proportion of the non–insulin-dependent net glucose uptake in each subject receiving an intravenous glucose load. Based on individual parameter estimates, the fraction of total glucose disposal independent of insulin was 72.8% ± 12.0% in the 238 ND subjects over the course of the experiment, indicating the major contribution to the whole-body glucose clearance under non-diabetic conditions. This fraction was significantly reduced to 48.8% ± 16.9% in the 30 T2D subjects, although still accounting for approximately half of the total in the T2D population based on our modeling analysis. Given the potential application of glucose effectiveness as a predictor of glucose intolerance and as a potential therapeutic target for treating diabetes, more investigations of glucose effectiveness in other disease conditions can be conducted using the hierarchical modeling framework reported herein. [ABSTRACT FROM AUTHOR]

The brain is responsible for maintaining whole-body energy homeostasis by changing energy input and availability. The hypothalamus and dorsal vagal complex (DVC) are the primary sites of metabolic control, able to sense both hormones and nutrients and adapt metabolism accordingly. The mitochondria respond to the level of nutrient availability by fusion or fission to maintain energy homeostasis; however, these processes can be disrupted by metabolic diseases including obesity and type II diabetes (T2D). Mitochondrial dynamics are crucial in the development and maintenance of obesity and T2D, playing a role in the control of glucose homeostasis and whole-body metabolism across neurons and glia in the hypothalamus and DVC. [ABSTRACT FROM AUTHOR]

In this research, we present a mathematical model of diabetes mellitus showing how glucose metabolism in the body is related to pancreatic insulin. We looked at the breakdown of glucose due to direct injection of insulin into the vein, and the breakdown of glucose due to the uptake of tissues such as the brain and neurons, and also considered the increased glucose level coming from consuming food and eating glucose directly. From this behavior, we can summarize it as a mathematical model of the system of glucose. Then, we used linear and nonlinear mathematical theorem analysis by Lyapunov's method with conditional and local stability and global stability cases respectively. Finally, a numerical approach along with a graph presentation was used to confirm the conclusive analysis of the role that insulin works in the process of regulating the blood sugar level in the human body. [ABSTRACT FROM AUTHOR]

A mathematical model for glucose-insulin regulatory system is presented. The glucose-insulin regulatory system has been introduced with a new variable which is ingested glucose. The ingested glucose is the external source of glucose that is coming from source of food and assumed to follow the logistic growth model. In addition, we analyze glucose constant value consumed from medicine. With the introduction of ingested glucose a three variable model is established. The stability of the model is analyzed by construction of Lyapunov function and conditions for stability have been derived. Numerical simulations are used to validate and describe the stability of the proposed model. [ABSTRACT FROM AUTHOR]

This paper presents an identifier based intelligent adaptive fuzzy control scheme with regulating blood glucose concentration in normoglycemic level of 70 mg/dl for type 1 diabetic patients. The identifier is built with fuzzy neural network (FNN) to predict the blood glucose concentration of the diabetic patient. The fuzzy based controller with generic operating regimes which cluster all the adaptive control rules is designed to robustly reject the multiple meal disturbances resulting from food intake and deal with the parametric uncertainties in model and measurement noise. All the parameters of the FNN and of the fuzzy logic system are tuned by backpropagation (BP), to achieve the control objectives. The numerical simulations are performed to show that the set point tracking, meal disturbances and measurement noise rejection can be realized within this method. [ABSTRACT FROM AUTHOR]

The insulin injection rate in type-I diabetic patients is a complex control problem. The mathematical dynamics for the insulin/glucose metabolism can be different for various patients who undertake different activities, have different lifestyles, and have other illnesses. In this study, a robust regulation system on the basis of generalized type-2 (GT2) fuzzy-logic systems (FLSs) is designed for the regulation of the blood glucose level. Unlike previous studies, the dynamics of glucose–insulin are unknown under high levels of uncertainty. The insulin-glucose metabolism has been identified online by GT2-FLSs, considering the stability criteria. The learning scheme was designed based on the Lyapunov approach. In other words, the GT2-FLSs are learned using adaptation rules that are concluded from the stability theorem. The effect of the dynamic estimation error and other perturbations, such as patient activeness, were eliminated through the designed adaptive fuzzy compensator. The adaptation laws for control parameters, GT2-FLS rule parameters, and the designed compensator were obtained by using the Lyapunov stability theorem. The feasibility and accuracy of the designed control scheme was examined on a modified Bergman model of some patients under different conditions. The simulation results confirm that the suggested controller has excellent performance under various conditions.

To gain insight into complex biological endocrine glucose-insulin regulatory system where the interactions of components of the metabolic system and time-delay inherent in the biological system give rise to complex dynamics. The modeling has increased interest and importance in physiological research and enhanced the medical treatment protocols. This brief contains a new model using time delay differential equations, which give an accurate result by utilizing two explicit time delays. The bifurcation analysis has been conducted to find the main system parameters bifurcation values and corresponding system behaviors. The results found consistent with the biological experiments results. [ABSTRACT FROM AUTHOR]

Prediction of physiological mechanics are important in medical practice because interventions are guided by predicted impacts of interventions. But prediction is difficult in medicine because medicine is complex and difficult to understand from data alone, and the data are sparse relative to the complexity of the generating processes. Computational methods can increase prediction accuracy, but prediction with clinical data is difficult because the data are sparse, noisy and nonstationary. This paper focuses on predicting physiological processes given sparse, non-stationary, electronic health record data in the intensive care unit using data assimilation (DA), a broad collection of methods that pair mechanistic models with inference methods. A methodological pipeline embedding a glucose–insulin model into a new DA framework, the constrained ensemble Kalman filter (CEnKF) to forecast blood glucose was developed. The data include tube-fed patients whose nutrition, blood glucose, administered insulins and medications were extracted by hand due to their complexity and to ensure accuracy. The model was estimated using an individual's data as if they arrived in real-time, and the estimated model was run forward producing a forecast. Both constrained and unconstrained ensemble Kalman filters were estimated to compare the impact of constraints. Constraint boundaries, model parameter sets estimated, and data used to estimate the models were varied to investigate their influence on forecasting accuracy. Forecasting accuracy was evaluated according to mean squared error between the model-forecasted glucose and the measurements and by comparing distributions of measured glucose and forecast ensemble means. The novel CEnKF produced substantial gains in robustness and accuracy while minimizing the data requirements compared to the unconstrained ensemble Kalman filters. Administered insulin and tube-nutrition were important for accurate forecasting, but including glucose in IV medication delivery did not increase forecast accuracy. Model flexibility, controlled by constraint boundaries and estimated parameters, did influence forecasting accuracy. Accurate and robust physiological forecasting with sparse clinical data is possible with DA. Introducing constrained inference, particularly on unmeasured states and parameters, reduced forecast error and data requirements. The results are not particularly sensitive to model flexibility such as constraint boundaries, but over or under constraining increased forecasting errors. [Display omitted] • Data assimilation (DA) is applied to forecast the glucose-insulin system. • A new constrained DA method is introduced. • The new DA method reduces error and requires less data. • The results generalize any physiological system estimated with sparse clinical data. [ABSTRACT FROM AUTHOR]

The article discusses mathematical models which may be considered in the study of diabetes. Topics explored include the use of ordinary differential equations (ODE) in the evaluation of diagnostic tests for diabetes such as oral glucose tolerance test and meal glucose tolerance test, published studies which focused on the ODE model to understand concentration and regulation of both insulin and glucose, and the availability of various computer software for the numerical simulation of data.

Abstract: An overview of some of the mathematical models appearing in the literature for use in the glucose-insulin regulatory system in relation to diabetes is given, enhanced with a survey on available software. The models are in the form of ordinary differential, partial differential, delay differential and integro-differential equations. Some computational results are also presented. [Copyright &y& Elsevier]

The insulin injection rate in type-I diabetic patients is a complex control problem. The mathematical dynamics for the insulin/glucose metabolism can be different for various patients who undertake different activities, have different lifestyles, and have other illnesses. In this study, a robust regulation system on the basis of generalized type-2 (GT2) fuzzy-logic systems (FLSs) is designed for the regulation of the blood glucose level. Unlike previous studies, the dynamics of glucose–insulin are unknown under high levels of uncertainty. The insulin-glucose metabolism has been identified online by GT2-FLSs, considering the stability criteria. The learning scheme was designed based on the Lyapunov approach. In other words, the GT2-FLSs are learned using adaptation rules that are concluded from the stability theorem. The effect of the dynamic estimation error and other perturbations, such as patient activeness, were eliminated through the designed adaptive fuzzy compensator. The adaptation laws for control parameters, GT2-FLS rule parameters, and the designed compensator were obtained by using the Lyapunov stability theorem. The feasibility and accuracy of the designed control scheme was examined on a modified Bergman model of some patients under different conditions. The simulation results confirm that the suggested controller has excellent performance under various conditions. [ABSTRACT FROM AUTHOR]

This paper proposes a delayed fractional-order model of glucose–insulin interaction in the sense of the Caputo fractional derivative with incommensurate orders. This fractional-order model is developed from the first-order model of glucose–insulin interaction. Firstly, we investigate the non-negativity and the boundedness of solutions of the fractional-order model. Secondly, the stability and the bifurcation of the model are studied by separating the associated characteristic equation of the model into its real and imaginary parts and taking a time delay as the bifurcation parameter. The asymptotic stability and the Hopf bifurcation are discussed via the condition of creation of the bifurcation. Furthermore, it is shown that the onset of the bifurcation is related to the fractional orders of the model. Finally, some numerical simulations of the model using the Adam–Bashforth–Moulton predictor corrector scheme are demonstrated to support our obtained theoretical results. [ABSTRACT FROM AUTHOR]