A discrete Single Delay Model for the Intra-Venous Glucose Tolerance Test

A glucose-insulin-glucagon coupled model of the isoglycemic intravenous glucose infusion experiment

Type 2 diabetes (T2D) is a pathophysiology that is characterized by insulin resistance, beta- and alpha-cell dysfunction. Mathematical models of various glucose challenge experiments have been developed to quantify the contribution of insulin and beta-cell dysfunction to the pathophysiology of T2D. There is a need for effective extended models that also capture the impact of alpha-cell dysregulation on T2D. In this paper a delay differential equation-based model is developed to describe the coupled glucose-insulin-glucagon dynamics in the isoglycemic intravenous glucose infusion (IIGI) experiment. As the glucose profile in IIGI is tailored to match that of a corresponding oral glucose tolerance test (OGTT), it provides a perfect method for studying hormone responses that are in the normal physiological domain and without the confounding effect of incretins and other gut mediated factors. The model was fit to IIGI data from individuals with and without T2D. Parameters related to glucagon action, suppression, and secretion as well as measures of insulin sensitivity, and glucose stimulated response were determined simultaneously. Significant impairment in glucose dependent glucagon suppression was observed in patients with T2D (duration of T2D: 8 (6-36) months) relative to weight matched control subjects (CS) without diabetes (k1 (mM)-1: 0.16 ± 0.015 (T2D, n = 7); 0.26 ± 0.047 (CS, n = 7)). Insulin action was significantly lower in patients with T2D (a1 (10 pM min)-1: 0.000084 ± 0.0000075 (T2D); 0.00052 ± 0.00015 (CS)) and the Hill coefficient in the equation for glucose dependent insulin response was found to be significantly different in T2D patients relative to CS (h: 1.4 ± 0.15; 1.9 ± 0.14). Trends in parameters with respect to fasting plasma glucose, HbA1c and 2-h glucose values are also presented. Significantly, a negative linear relationship is observed between the glucagon suppression parameter, k1, and the three markers for diabetes and is thus indicative of the role of glucagon in exacerbating the pathophysiology of diabetes (Spearman Rank Correlation: (n = 12; (-0.79, 0.002), (-0.73,.007), (-0.86,.0003)) respectively).

Examining Type 1 Diabetes Mathematical Models Using Experimental Data

Type 1 diabetes requires treatment with insulin injections and monitoring glucose levels in affected individuals. We explored the utility of two mathematical models in predicting glucose concentration levels in type 1 diabetic mice and determined disease pathways. We adapted two mathematical models, one with β-cells and the other with no β-cell component to determine their capability in predicting glucose concentration and determine type 1 diabetes pathways using published glucose concentration data for four groups of experimental mice. The groups of mice were numbered Mice Group 1–4, depending on the diabetes severity of each group, with severity increasing from group 1–4. A Markov Chain Monte Carlo method based on a Bayesian framework was used to fit the model to determine the best model structure. Akaike information criteria (AIC) and Bayesian information criteria (BIC) approaches were used to assess the best model structure for type 1 diabetes. In fitting the model with no β-cells to glucose level data, we varied insulin absorption rate and insulin clearance rate. However, the model with β-cells required more parameters to match the data and we fitted the β-cell glucose tolerance factor, whole body insulin clearance rate, glucose production rate, and glucose clearance rate. Fitting the models to the blood glucose concentration level gave the least difference in AIC of 1.2, and a difference in BIC of 0.12 for Mice Group 4. The estimated AIC and BIC values were highest for Mice Group 1 than all other mice groups. The models gave substantial differences in AIC and BIC values for Mice Groups 1–3 ranging from 2.10 to 4.05. Our results suggest that the model without β-cells provides a more suitable structure for modelling type 1 diabetes and predicting blood glucose concentration for hypoglycaemic episodes.

A quasi-equilibrium reduced model of pancreatic insulin secretion

Much attention has been devoted in the last few decades to mathematical models of insulin secretion, in order to better understand the regulation of glycemia and its derangements. The glucose-insulin homeostatic mechanism is so complex and gives rise to such diverse behavior following perturbations that different models had been published, which reproduced the results of single experiments. More recently, a unifying model of pancreatic insulin secretion was proposed, which is able to account, with a single value of the (meta)parameters, for the wide array of clinically observed behavior. This model explicitly represented the pulsatile nature of the many pancreatic hormone-secreting firing units: the price to pay for its flexibility and performance is the very high dimensionality (hundreds of thousand equations) of the corresponding dynamical system. Clearly, it would be desirable to reduce this model to a much simpler form while retaining its power to reproduce heterogeneous phenomena. The present work reviews the qualitative behavior of this pancreas pulsatile model and offers some insight into its reduction in equilibrium and quasi-equilibrium conditions, also considering single-shot (non-repeated) glucose jumps from an approximately resting condition (such as would occur in standard Intra-Venous bolus dosing of glucose during diabetes diagnostic maneuvers). The resulting quasi-steady-state model can be further endowed with additional lower-order dynamics to also approximate transient behavior. Although a more accurate reduction of the original pulsatile model is left to further investigation, numerical results confirm the biomedical applicability of the formulation already obtained.

A mathematical model of food intake

The metabolic, hormonal and psychological determinants of the feeding behavior in humans are numerous and complex. A plausible model of the initiation, continuation and cessation of meals taking into account the most relevant such determinants would be very useful in simulating food intake over hours to days, thus providing input into existing models of nutrient absorption and metabolism. In the present work, a meal model is proposed, incorporating stomach distension, glycemic variations, ghrelin dynamics, cultural habits and influences on the initiation and continuation of meals, reflecting a combination of hedonic and appetite components. Given a set of parameter values (portraying a single subject), the timing and size of meals are stochastic. The model parameters are calibrated so as to reflect established medical knowledge on data of food intake from the National Health and Nutrition Examination Survey (NHANES) database during years 2015 and 2016.

A revised Sorensen model: Simulating glycemic and insulinemic response to oral and intra-venous glucose load

In 1978, Thomas J. Sorensen defended a thesis in chemical engineering at the University of California, Berkeley, where he proposed an extensive model of glucose-insulin control, model which was thereafter widely employed for virtual patient simulation. The original model, and even more so its subsequent implementations by other Authors, presented however a few imprecisions in reporting the correct model equations and parameter values. The goal of the present work is to revise the original Sorensen's model, to clearly summarize its defining equations, to supplement it with a missing gastrio-intestinal glucose absorption and to make an implementation of the revised model available on-line to the scientific community.

A novel fast-slow model of diabetes progression: Insights into mechanisms of response to the interventions in the Diabetes Prevention Program

Several models for the long-term development of T2DM already exist, focusing on the dynamics of the interaction between glycemia, insulinemia and β-cell mass. Current models consider representative (fasting or daily average) glycemia and insulinemia as characterizing the compensation state of the subject at some instant in slow time. This implies that only these representative levels can be followed through time and that the role of fast glycemic oscillations is neglected. An improved model (DPM15) for the long-term progression of T2DM is proposed, introducing separate peripheral and hepatic (liver and kidney) insulin actions. The DPM15 model no longer uses near-equilibrium approximation to separate fast and slow time scales, but rather describes, at each step in slow time, a complete day in the life of the virtual subject in fast time. The model can thus represent both fasting and postprandial glycemic levels and describe the effect of interventions acting on insulin-enhanced tissue glucose disposal or on insulin-inhibited hepatic glucose output, as well as on insulin secretion and β-cell replicating ability. The model can simulate long-term variations of commonly used clinical indices (HOMA-B, HOMA-IR, insulinogenic index) as well as of Oral Glucose Tolerance or Euglycemic Hyperinsulinemic Clamp test results. The model has been calibrated against observational data from the Diabetes Prevention Program study: it shows good adaptation to observations as a function of very plausible values of the parameters describing the effect of such interventions as Placebo, Intensive LifeStyle and Metformin administration.

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.

Modeling the dynamics of glucose, insulin, and free fatty acids with time delay: The impact of bariatric surgery on type 2 diabetes mellitus

The role of free fatty acids (FFA) on Type 2 diabetes mellitus (T2DM) progression has been studied extensively with prior studies suggesting that individuals with shared familial genetic predisposition to metabolic-related diseases may be vulnerable to dysfunctional plasma FFA regulation. A harmful cycle arises when FFA are not properly regulated by insulin contributing to the development of insulin resistance, a key indicator for T2DM, since prolonged insulin resistance may lead to hyperglycemia. We introduce a hypothesis-driven dynamical model and use it to evaluate the role of FFA on insulin resistance progression that is mathematically constructed within the context of individuals that have genetic predisposition to dysfunctional plasma FFA. The dynamics of the nonlinear interactions that involve glucose, insulin, and FFA are modeled by incorporating a fixed-time delay with the corresponding delay-differential equations being studied numerically. The results of computational studies, that is, extensive simulations, are compared to the known minimal ordinary differential equations model. Parameter estimation and model validation are carried out using clinical data of patients who underwent bariatric surgery. These estimates provide a quantitative measure that is used to evaluate the regulation of lipolysis by insulin action measured by insulin sensitivity, within a metabolically heterogeneous population (non-diabetic to diabetic). Results show that key metabolic factors improve after surgery, such as the effect of insulin inhibition of FFA on insulin and glucose regulation, results that do match prior clinical studies. These findings indicate that the reduction in weight or body mass due to surgery improve insulin action for the regulation of glucose, FFA, and insulin levels. This reinforces what we know, namely, that insulin action is essential for regulating FFA and glucose levels and is a robust effect that can be observed not only in the long-term, but also in the short-term; thereby preventing the manifestation of T2DM.

Consistency of compact and extended models of glucose-insulin homeostasis: The role of variable pancreatic reserve

Published compact and extended models of the glucose-insulin physiologic control system are compared, in order to understand why a specific functional form of the compact model proved to be necessary for a satisfactory representation of acute perturbation experiments such as the Intra Venous Glucose Tolerance Test (IVGTT). A spectrum of IVGTT’s of virtual subjects ranging from normal to IFG to IGT to frank T2DM were simulated using an extended model incorporating the population-of-controllers paradigm originally hypothesized by Grodsky, and proven to be able to capture a wide array of experimental results from heterogeneous perturbation procedures. The simulated IVGTT’s were then fitted with the Single-Delay Model (SDM), a compact model with only six free parameters, previously shown to be very effective in delivering precise estimates of insulin sensitivity and secretion during an IVGTT. Comparison of the generating, extended-model parameter values with the obtained compact model estimates shows that the functional form of the nonlinear insulin-secretion term, empirically found to be necessary for the compact model to satisfactorily fit clinical observations, captures the pancreatic reserve level of the simulated virtual patients. This result supports the validity of the compact model as a meaningful analysis tool for the clinical assessment of insulin sensitivity.

Oscillatory dynamics of an intravenous glucose tolerance test model with delay interval

Type 2 diabetes mellitus (T2DM) has become prevalent pandemic disease in view of the modern life style. Both diabetic population and health expenses grow rapidly according to American Diabetes Association. Detecting the potential onset of T2DM is an essential focal point in the research of diabetes mellitus. The intravenous glucose tolerance test (IVGTT) is an effective protocol to determine the insulin sensitivity, glucose effectiveness, and pancreatic β-cell functionality, through the analysis and parameter estimation of a proper differential equation model. Delay differential equations have been used to study the complex physiological phenomena including the glucose and insulin regulations. In this paper, we propose a novel approach to model the time delay in IVGTT modeling. This novel approach uses two parameters to simulate not only both discrete time delay and distributed time delay in the past interval, but also the time delay distributed in a past sub-interval. Normally, larger time delay, either a discrete or a distributed delay, will destabilize the system. However, we find that time delay over a sub-interval might not. We present analytically some basic model properties, which are desirable biologically and mathematically. We show that this relatively simple model provides good fit to fluctuating patient data sets and reveals some intriguing dynamics. Moreover, our numerical simulation results indicate that our model may remove the defect in well known Minimal Model, which often overestimates the glucose effectiveness index.

Simulation of Trauma Incidents : Modelling the Evolution of Patients and Resources

Mathematical modeling and simulation with medical applications has gained much interest in the last few years, mainly due to the widespread availability of low-cost technology and computational power. This paper presents an integrated platform for the in-silico simulation of trauma incidents, based on a suite of interacting mathematical models. The models cover the generation of a scenario for an incident, a model of physiological evolution of the affected individuals, including the possible effect of the treatment, and a model of evolution in time of the required medical resources. The problem of optimal resource allocation is also investigated. Model parameters have been identified according to the expertise of medical doctors and by reviewing some related literature. The models have been implemented and exposed as web services, while some software clients have been built for the purpose of testing. Due to its extendability, our integrated platform highlights the potential of model-based simulation in different health-related fields, such as emergency medicine and personal health systems. Modifications of the models are already being used in the context of two funded projects, aiming at evaluating the response of health systems to major incidents with and without model-based decision support.

A Unifying Organ Model of Pancreatic Insulin Secretion

The secretion of insulin by the pancreas has been the object of much attention over the past several decades. Insulin is known to be secreted by pancreatic β-cells in response to hyperglycemia: its blood concentrations however exhibit both high-frequency (period approx. 10 minutes) and low-frequency oscillations (period approx. 1.5 hours). Furthermore, characteristic insulin secretory response to challenge maneuvers have been described, such as frequency entrainment upon sinusoidal glycemic stimulation; substantial insulin peaks following minimal glucose administration; progressively strengthened insulin secretion response after repeated administration of the same amount of glucose; insulin and glucose characteristic curves after Intra-Venous administration of glucose boli in healthy and pre-diabetic subjects as well as in Type 2 Diabetes Mellitus. Previous modeling of β-cell physiology has been mainly directed to the intracellular chain of events giving rise to single-cell or cell-cluster hormone release oscillations, but the large size, long period and complex morphology of the diverse responses to whole-body glucose stimuli has not yet been coherently explained. Starting with the seminal work of Grodsky it was hypothesized that the population of pancreatic β-cells, possibly functionally aggregated in islets of Langerhans, could be viewed as a set of independent, similar, but not identical controllers (firing units) with distributed functional parameters. The present work shows how a single model based on a population of independent islet controllers can reproduce very closely a diverse array of actually observed experimental results, with the same set of working parameters. The model's success in reproducing a diverse array of experiments implies that, in order to understand the macroscopic behaviour of the endocrine pancreas in regulating glycemia, there is no need to hypothesize intrapancreatic pacemakers, influences between different islets of Langerhans, glycolitic-induced oscillations or β-cell sensitivity to the rate of change of glycemia.

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.

Mathematical modeling of renal tubular glucose absorption after glucose load

A partial differential Progressive Tubular Reabsorption (PTR) model, describing renal tubular glucose reabsorption and urinary glucose excretion following a glucose load perturbation, is proposed and fitted to experimental data from five subjects. For each subject the Glomerular Filtration Rate was estimated and both blood and urine glucose were sampled following an Intra-Venous glucose bolus. The PTR model was compared with a model representing the conventional Renal Threshold Hypothesis (RTH). A delay bladder compartment was introduced in both formulations. For the RTH model, the average threshold for glycosuria varied between 9.90 ± 4.50 mmol/L and 10.63 ± 3.64 mmol/L (mean ± Standard Deviation) under different hypotheses; the corresponding average maximal transport rates varied between 0.48 ± 0.45 mmol/min (86.29 ± 81.22 mg/min) and 0.50 ± 0.42 mmol/min (90.62 ± 76.15 mg/min). For the PTR Model, the average maximal transports rates varied between 0.61 ± 0.52 mmol/min (109.57 ± 93.77 mg/min) and 0.83 ± 0.95 mmol/min (150.13 ± 171.85 mg/min). The time spent by glucose inside the tubules before entering the bladder compartment varied between 1.66 ± 0.73 min and 2.45 ± 1.01 min. The PTR model proved much better than RTH at fitting observations, by correctly reproducing the delay of variations of glycosuria with respect to the driving glycemia, and by predicting non-zero urinary glucose elimination at low glycemias. This model is useful when studying both transients and steady-state glucose elimination as well as in assessing drug-related changes in renal glucose excretion.

Routine OGTT: a robust model including incretin effect for precise identification of insulin sensitivity and secretion in a single individual

In order to provide a method for precise identification of insulin sensitivity from clinical Oral Glucose Tolerance Test (OGTT) observations, a relatively simple mathematical model (Simple Interdependent glucose/insulin MOdel SIMO) for the OGTT, which coherently incorporates commonly accepted physiological assumptions (incretin effect and saturating glucose-driven insulin secretion) has been developed. OGTT data from 78 patients in five different glucose tolerance groups were analyzed: normal glucose tolerance (NGT), impaired glucose tolerance (IGT), impaired fasting glucose (IFG), IFG+IGT, and Type 2 Diabetes Mellitus (T2DM). A comparison with the 2011 Salinari (COntinuos GI tract MOdel, COMO) and the 2002 Dalla Man (Dalla Man MOdel, DMMO) models was made with particular attention to insulin sensitivity indices IS_COMO, IS_DMMO and k_xgi (the insulin sensitivity index for SIMO). ANOVA on k_xgi values across groups resulted significant overall (P<0.001), and post-hoc comparisons highlighted the presence of three different groups: NGT (8.62×10⁻⁵±9.36×10⁻⁵ min⁻¹pM⁻¹), IFG (5.30×10⁻⁵±5.18×10⁻⁵) and combined IGT, IFG+IGT and T2DM (2.09×10⁻⁵±1.95×10⁻⁵, 2.38×10⁻⁵±2.28×10⁻⁵ and 2.38×10⁻⁵±2.09×10⁻⁵ respectively). No significance was obtained when comparing IS_COMO or IS_DMMO across groups. Moreover, k_xgi presented the lowest sample average coefficient of variation over the five groups (25.43%), with average CVs for IS_COMO and IS_DMMO of 70.32% and 57.75% respectively; k_xgi also presented the strongest correlations with all considered empirical measures of insulin sensitivity. While COMO and DMMO appear over-parameterized for fitting single-subject clinical OGTT data, SIMO provides a robust, precise, physiologically plausible estimate of insulin sensitivity, with which habitual empirical insulin sensitivity indices correlate well. The k_xgi index, reflecting insulin secretion dependency on glycemia, also significantly differentiates clinically diverse subject groups. The SIMO model may therefore be of value for the quantification of glucose homeostasis from clinical OGTT data.

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.

Evaluation of a novel continuous glucose monitoring-based method for mealtime insulin dosing--the iBolus--in subjects with type 1 diabetes using continuous subcutaneous insulin infusion therapy: a randomized controlled trial

OBJECTIVE

Prandial insulin dosing is an empirical practice associated frequently with poor reproducibility in postprandial glucose response. Based on continuous glucose monitoring (CGM), a method for prandial insulin administration (iBolus) is presented and evaluated for people with type 1 diabetes using CSII therapy.

SUBJECTS AND METHODS

An individual patient's model for a 5-h postprandial period was obtained from 6-day ambulatory CGM and used for iBolus calculation in 12 patients with type 1 diabetes. In a double-blind, crossover study each patient underwent four meal tests with 40 g or 100 g of carbohydrates (CHOs), both on two occasions. For each meal, the iBolus or the traditional bolus (tBolus) was given before mealtime (t(0)) in a randomized order. We measured the postprandial glycemic response as the area under the curve of plasma glucose (AUC-PG(0-5h)) and variability as the individual coefficient of variation (CV) of AUC-PG(0-5h). The contribution of the insulin-to-CHO ratio, CHO, plasma glucose at t(0) (PG(t0)), and insulin dose to AUC-PG(0-5h) and its CV was also investigated.

RESULTS

AUC-PG(0-5h) was similar with either bolus for 40-g (iBolus vs. tBolus, 585.5±127.5 vs. 689.2±180.7 mg/dL·h) or 100-g (752.1±237.7 vs. 760.0±263.2 mg/dL·h) CHO meals. A multiple regression analysis revealed a significant model only for the tBolus, with PG(t0) being the best predictor of AUC-PG(0-5h) explaining approximately 50% of the glycemic response. Observed variability was greater with the iBolus (CV, 16.7±15.3% vs. 10.1±12.5%) but independent of the factors studied.

CONCLUSIONS

A CGM-based algorithm for calculation of prandial insulin is feasible, although it does not reduce unpredictability of individual glycemic responses. Causes of variability need to be identified and analyzed for further optimization of postprandial glycemic control.

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.

Advantages of the single delay model for the assessment of insulin sensitivity from the intravenous glucose tolerance test

Background

The Minimal Model, (MM), used to assess insulin sensitivity (IS) from Intra-Venous Glucose-Tolerance Test (IVGTT) data, suffers from frequent lack of identifiability (parameter estimates with Coefficients of Variation (CV) less than 52%). The recently proposed Single Delay Model (SDM) is evaluated as a practical alternative.

Methods

The SDM was applied to 74 IVGTTs from lean (19), overweight (22), obese (22) and morbidly obese (11) subjects. Estimates from the SDM (K_xgI) were compared with the corresponding MM (S_I), 1/HOMA-IR index and Euglycemic-Hyperinsulinemic Clamp (M-EHC over 7 subjects) estimates.

Results

K_xgI was identifiable in 73 out of 74 subjects (CV = 69% in the 74^th subject) and ranged from 1.25 × 10^-5 to 4.36 × 10^-4min^-1pM^-1; S_I CV was >52% in 36 subjects (up to 2.36 × 10⁹%) and presented 18 extreme values (≤ 1.5 × 10^-12 or ≥ 3.99).

K_xgI correlated well with 1/HOMA-IR (r = 0.56, P < 0.001), whereas the correlations K_xgI-S_I and 1/HOMA-IR-S_I were high (r = 0.864 and 0.52 respectively) and significant (P < 0.001 in both cases) only in the non-extreme S_I sub-sample (56 subjects). Correlations K_xgI vs. M-EHC and S_I vs. M-EHC were positive (r = 0.92, P = 0.004 and r = 0.83, P = 0.02 respectively). K_xgI decreased for higher BMI's (P < 0.001), S_I significantly so only over the non-extreme-S_I sub-sample. The Acute Insulin Response Index was also computed and the expected inverse (hyperbolic) relationship with the K_xgI observed.

Conclusions

Precise estimation of insulin sensitivity over a wide range of BMI, stability of all other model parameters, closer adherence to accepted physiology make the SDM a useful alternative tool for the evaluation of insulin sensitivity from the IVGTT.

An integrated model for the glucose-insulin system

The integrated glucose-insulin model was originally developed on a variety of intravenous glucose provocation experiments in healthy volunteers and type 2 diabetic patients. The model, which simultaneously describes time-courses of glucose and insulin based on mechanism-based components for production, elimination and homeostatic feedback, has been further extended to oral glucose provocations, meal tests and insulin administration. The model has been used to describe experiments ranging from 4 to 24 hr. Applications of the integrated glucose-insulin model include the clinical assessment of the mechanism of action of anti-diabetic drugs and the magnitude of their effects. Finally, the model was used for optimizing the design of provocation experiments.

MATHEMATICAL MODELS OF SUBCUTANEOUS INJECTION OF INSULIN ANALOGUES: A MINI-REVIEW

In the last three decades, several models relevant to the subcutaneous injection of insulin analogues have appeared in the literature. Most of them model the absorption of insulin analogues in the injection depot and then compute the plasma insulin concentration. The most recent systemic models directly simulate the plasma insulin dynamics. These models have been and/or can be applied to the technology of the insulin pump or to the coming closed-loop systems, also known as the artificial pancreas. In this paper, we selectively review these models in detail and at point out that these models provide key building blocks for some important endeavors into physiological questions of insulin secretion and action. For example, it is not clear at this time whether or not picomolar doses of insulin are found near the islets and there is no experimental method to assess this in vivo. This is of interest because picomolar concentrations of insulin have been found to be effective at blocking beta-cell death and increasing beta-cell growth in recent cell culture experiments.

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

It is well known that Michaelis–Menten kinetics is suitable for the response function in chemical reaction, when the reaction rate does not increase indefinitely when an excess of resource is available. However, the existing models for insulin therapies assume that the response function of insulin clearance is proportional to the insulin concentration. In this paper, we propose a new model for insulin therapy for both type 1 and type 2 diabetes mellitus, in which the insulin degradation rate assumes Michaelis–Menten kinetics. Our analysis shows that it is possible to mimic pancreatic insulin secretion by exogenous insulin infusions, and our numerical simulations provide clinical strategies for insulin–administration practices.

Mathematical models of diabetes progression

Few attempts have been made to model mathematically the progression of type 2 diabetes. A realistic representation of the long-term physiological adaptation to developing insulin resistance is necessary for effectively designing clinical trials and evaluating diabetes prevention or disease modification therapies. Writing a good model for diabetes progression is difficult because the long time span of the disease makes experimental verification of modeling hypotheses extremely awkward. In this context, it is of primary importance that the assumptions underlying the model equations properly reflect established physiology and that the mathematical formulation of the model give rise only to physically plausible behavior of the solutions. In the present work, a model of the pancreatic islet compensation is formulated, its physiological assumptions are presented, some fundamental qualitative characteristics of its solutions are established, the numerical values assigned to its parameters are extensively discussed (also with reference to available cross-sectional epidemiologic data), and its performance over the span of a lifetime is simulated under various conditions, including worsening insulin resistance and primary replication defects. The differences with respect to two previously proposed models of diabetes progression are highlighted, and therefore, the model is proposed as a realistic, robust description of the evolution of the compensation of the glucose-insulin system in healthy and diabetic individuals. Model simulations can be run from the authors' web page.