Identification of an integrated mathematical model of standard oral glucose tolerance test for characterization of insulin potentiation in health

Two new formulations, respectively denominated INT_M1 and INT_M2, of an integrated mathematical model to describe the glycemic and insulinemic responses to a 75 g oral glucose tolerance test (OGTT) are proposed and compared. The INT_M1 assumes a single compartment for the intestine and the derivative of a power exponential function for the gastric emptying rate, while, in the INT_M2, a nonlinear three-compartment system model is adopted to produce a more realistic, multiphase gastric emptying rate. Both models were implemented in a Matlab-based, two-step procedure for estimation of seven adjustable coefficients characterizing the gastric emptying rate and the incretin, insulin and glucose kinetics. Model behaviour was tested vs. mean plasma glucagon-like peptide 1 (GLP-1), glucose-dependent insulinotropic polypeptide (GIP), glucose and insulin measurements from two different laboratories, where glycemic profiles observed during a 75 g OGTT were matched in healthy subjects (HC1- and HC2-group, respectively) by means of an isoglycemic intravenous glucose (I-IVG) infusion. Under the hypothesis of an additive effect of GLP-1 and GIP on insulin potentiation, our results demonstrated a substantial equivalence of the two models in matching the data. Model parameter estimates showed to be suitable markers of differences observed in the OGTT and matched I-IVG responses from the HC1-group compared to the HC2-group. Model implementation in our two-step parameter estimation procedure enhances the possibility of a prospective application for individualization of the incretin effect in a single subject, when his/her data are plugged in.     

Estimating postprandial glucose fluxes using hierarchical Bayes modelling

A new stochastic computational method was developed to estimate the endogenous glucose production, the meal-related glucose appearance rate (R(a meal)), and the glucose disposal (R(d)) during the meal tolerance test. A prior probability distribution was adopted which assumes smooth glucose fluxes with individualized smoothness level within the context of a Bayes hierarchical model. The new method was contrasted with the maximum likelihood method using data collected in 18 subjects with type 2 diabetes who ingested a mixed meal containing [U-(13)C]glucose. Primed [6,6-(2)H(2)]glucose was infused in a manner that mimicked the expected endogenous glucose production. The mean endogenous glucose production, R(a meal), and R(d) calculated by the new method and maximum likelihood method were nearly identical. However, the maximum likelihood gave constant, nonphysiological postprandial endogenous glucose production in two subjects whilst the new method gave plausible estimates of endogenous glucose production in all subjects. Additionally, the two methods were compared using a simulated triple-tracer experiment in 12 virtual subjects. The accuracy of the estimates of the endogenous glucose production and R(a meal) profiles was similar [root mean square error (RMSE) 1.0±0.3 vs. 1.4±0.7μmol/kg/min for EGP and 2.6±1.0 vs. 2.9±0.9μmol/kg/min for R(a meal); new method vs. maximum likelihood method; P=NS, paired t-test]. The accuracy of R(d) estimates was significantly increased by the new method (RMSE 5.3±1.9 vs. 4.2±1.3; new method vs. ML method; P<0.01, paired t-test). We conclude that the new method increases plausibility of the endogenous glucose production and improves accuracy of glucose disposal compared to the maximum likelihood method.     

A method to measure the degree of control per se in the oral glucose tolerance test

Using the glucose and insulin values from a 5-hr oral glucose tolerance test, nine quantitative measures have been developed to separate normal, "flat-curve," and non-insulin-dependent diabetic (NIDD) patients. The purpose of these measures is to quantify per se the degree of control operating in glucose homeostasis. A control index which is based upon Swan's minimizing principle and uses only the glucose values was successful in assessing the degree of control operating in glucose homeostasis.     

A pharmacodynamic approach to optimizing insulin therapy

We have developed a program for simulation and optimization of insulin therapy in patients with insulin-dependent diabetes. The program, denoted GLUCOJECT, is based on a physiologic model of minimal complexity, which describes the pharmacokinetics of absorption and clearance of subcutaneous insulin and the dynamics of glucose utilization as dependent on both prevailing glucose and insulin levels. With self-monitored glucose values and insulin doses collected with one of several commercially available memory meters, GLUCOJECT reconstructs an average or 'typical' daily plasma glucose and insulin profile and displays them in a graph. The program then calculates the expected rate of glucose utilization, which permits calculation of the rate of glucose entry into plasma from both endogenous (hepatic) and exogenous (dietary) sources. In turn, this allows one to calculate an 'ideal' plasma insulin profile required to maintain a relatively constant 'ideal' plasma glucose level. GLUCOJECT can evaluate several different insulin regimens involving various combinations of short-, intermediate- and long-acting insulins, and select the one(s) most closely approximating the ideal or optimal insulin profile, using a least-squares criterion. For any optimized insulin regimen, GLUCOJECT calculates and displays the predicted time course of plasma glucose. These features make the program attractive as an educational tool for both patients and health care professionals and could potentially assist in the management of patients with insulin-dependent diabetes.     

Estimation of insulin sensitivity in diabetic Göttingen Minipigs

In patients with type 1 diabetes mellitus, insulin sensitivity is a parameter which strongly affects insulin therapy. Due to its time-dependent variation, this parameter can improve the strategy for automatic closed-loop blood glucose control. The aim of this work is to estimate the insulin sensitivity of patients with type 1 diabetes mellitus based on measured blood glucose concentrations. For this, an Extended Kalman Filter is used, based on a simplified version of the well-known Sorensen model. The compartment model of Sorensen was adapted to the glucose metabolic behaviour in diabetic Göttingen Minipigs by means of experimental data and reduced by neglecting unobservable state variables. Here, the Extended Kalman Filter is designed for simultaneous state and parameter estimation of insulin sensitivity using the insulin infusion rate and the meal size as input signals, and measurements of blood glucose concentration as output signal. The performance of the Extended Kalman Filter was tested in in silico studies using the minipig model, and is analysed by comparing the output signal of the filter with measurement data from the animal trials.     

MINMOD: a computer program to calculate insulin sensitivity and pancreatic responsivity from the frequently sampled intravenous glucose tolerance test

Insulin sensitivity and pancreatic responsivity are the two main factors controlling glucose tolerance. We have proposed a method for measuring these two factors, using computer analysis of a frequently-sampled intravenous glucose tolerance test (FSIGT). This 'minimal modelling approach' fits two mathematical models with FSIGT glucose and insulin data: one of glucose disappearance and one of insulin kinetics. MINMOD is the computer program which identifies the model parameters for each individual. A nonlinear least squares estimation technique is used, employing a gradient-type of estimation algorithm, and the first derivatives (not known analytically) are computed according to the 'sensitivity approach'. The program yields the parameter estimates and the precision of their estimation. From the model parameters, it is possible to extract four indices: SG, the ability of glucose per se to enhance its own disappearance at basal insulin, SI, the tissue insulin sensitivity index, phi 1, first phase pancreatic responsivity, and phi 2, second phase pancreatic responsivity. These four characteristic parameters have been shown to represent an integrated metabolic portrait of a single individual.     

Experimental validation of a glucose-insulin control model to stimulate patterns in glucose turnover

To verify a structured model of the glucose-insulin system, metabolic measurements were compared with model-based simulations in insulin-dependent diabetic dogs which had been previously identified in terms of model parameters. Glycaemia, glucosuria, plasma insulin, and the rates of appearance Ra and disappearance Rd of glucose (kinetics of double-labelled glucose, evaluated according to Steele's equation in its non-steady-state version) were observed under the following conditions, starting from normoglycaemia during glucose-controlled insulin infusion (GCII): (I) insulin withdrawal, (II) insulin withdrawal and glucose infusion, (III) constant i.v. infusion of glucose and insulin, (IV) glucose infusion during GCII. After fitting the patterns of glycaemia, simulations of the other state variables were accomplished, employing the individual model parameters, the preset experimental inputs, and the GCII control constants (test IV only). Under nearly all conditions, correspondence was excellent between measured and simulated data. There were, however, the following exceptions: incomplete representation by the model of kinetics in glucose utilisation after interruption of insulin supply, overestimation of glucosuria by the model in the presence of insulin. It is concluded that the model provides a reasonable representation of metabolic processes which are of importance in the treatment of insulin-dependent diabetes mellitus and that it might thus appropriately simulate the outcome of metabolic regimens.     

Adaptive control strategy for regulation of blood glucose levels in patients with type 1 diabetes

Current insulin therapy for patients with type 1 diabetes often results in high variability in blood glucose concentrations and may cause hyperglycemic/hypoglycemic episodes. Closing the glucose control loop with a fully automated electro-mechanical pancreas will improve the quality of life for insulin-dependent patients. An adaptive control algorithm is proposed to keep glucose concentrations within normoglycemic range and dynamically respond to glycemic challenges. A model-based control strategy is used to calculate the required insulin infusion rate, while the model parameters are recursively tuned. The algorithm handles delays associated with insulin absorption, time-lag between subcutaneous and blood glucose concentrations, and variations in inter/intra-subject glucose–insulin dynamics. Simulation results for simultaneous meal and physiological disturbances are demonstrated for subcutaneous insulin infusion.     

Nonlinear glucose–insulin control considering delays—Part II: Control algorithm

The closed loop control of blood glucose levels in a nonlinear glucose–insulin regulatory system is considered in this paper. Based on the subcutaneous glucose sensor readings, a control algorithm is designed and implemented. A mathematical model characterizing the ultradian oscillatory nature of the glucose–insulin regulatory system of diabetic patients is considered and an estimation based model predictive control scheme with physiological and actuator constraints is implemented. An in silico preclinical testing is done to corroborate the control algorithm using the UVa/Padova virtual patient software.     

Continuous-time interval model identification of blood glucose dynamics for type 1 diabetes

While good physiological models of the glucose metabolism in type 1 diabetic patients are well known, their parameterisation is difficult. The high intra-patient variability observed is a further major obstacle. This holds for data-based models too, so that no good patient-specific models are available. Against this background, this paper proposes the use of interval models to cover the different metabolic conditions. The control-oriented models contain a carbohydrate and insulin sensitivity factor to be used for insulin bolus calculators directly. Available clinical measurements were sampled on an irregular schedule which prompts the use of continuous-time identification, also for the direct estimation of the clinically interpretable factors mentioned above. An identification method is derived and applied to real data from 28 diabetic patients. Model estimation was done on a clinical data-set, whereas validation results shown were done on an out-of-clinic, everyday life data-set. The results show that the interval model approach allows a much more regular estimation of the parameters and avoids physiologically incompatible parameter estimates.     

Application of the SAAM modeling program to minimal model analysis of intravenous glucose tolerance test data

The minimal model approach to analysis of intravenous glucose tolerance tests (IVGTT) yields estimates of parameters representing insulin sensitivity, glucose-mediated glucose disposal and pancreatic responsiveness. The precision of these estimates can deteriorate if the glucose and insulin data lack well-defined structure or freedom from data noise (random error). The precision of parameter estimates can be enhanced if data sets from two or more IVGTTs, obtained under different experimental conditions in the same subject, are analysed together in one data file. Following initial fitting using CONSAM, the conversational version of the modeling program SAAM, those parameters whose estimates remain at the same value under the different experimental conditions are constrained. This effectively reduces the number of adjustable parameters, and their estimates can then be fine-tuned with enhanced precision using the batch version of SAAM.     

A program to estimate insulin sensitivity and pancreatic responsivity from an IVGTT using the minimal modeling technique

A user-friendly program coded in PASCAL for the IBM PC has been developed to determine the etiology of impaired glucose tolerance using an intravenous glucose tolerance test (IVGTT). It makes use of the "minimal modeling technique," a method that has been shown to be adequate for the quantitative determination of insulin sensitivity and insulin resistance. Two models are used, the minimal model of glucose disappearance and the minimal model of insulin kinetics. The first model is described by two nonlinear ordinary differential equations (ODEs) which are solved numerically, and which yield the insulin sensitivity index SI. The second model is described by an ODE for which an explicit solution was obtained, and which yields the pancreatic responsivity parameters phi 1 and phi 2. The product SI.phi 2 can be used to segregate subjects into "good" and "low" tolerance types. The program provides best-fit plots along with numerical values of the parameters and their uncertainties, and requires little intervention from the user. The fact that it requires a noninvasive IVGTT as input and that it has been written for the ubiquitous IBM PC are added advantages.     

Proteomic analysis of human proximal tubular cells exposed to high glucose concentrations

Hyperglycemia is a major key factor in the pathogenesis of microvascular complications of diabetes, including diabetic nephropathy (DN). Most studies to date have focused on the glomerular abnormalities found in DN. However, nephromegaly in the early stages of diabetes and the correlation of tubulointerstitial pathology rather than glomerular pathology with declining renal function in DN suggests the involvement of the tubulointerstitium. The etiology of the tubulointerstitial pathology in DN, however, is not fully understood. In this study, to understand the DN pathways, we constructed an initial 2-DE reference map for primitively cultured human proximal tubule (HK-2) cell in the presence of 5?mM and 25?mM glucose, which correspond to blood glucose concentrations during the normal and hyperglycemia conditions, respectively. Differentially expressed HK-2 cell cellular proteins at the high glucose concentration were identified via ESI-Q-TOF MS/MS and confirmed by Western blotting; enolase 1 (up-regulated) and lactate dehydrogenase (down-regulated). The regulation of these proteins will help in understanding DN mechanism through the glycolysis metabolic pathways in high glucose stimulated HK-2 cells.     

A model of the endogenous glucose balance incorporating the characteristics of glucose transporters

This paper describes the development and preliminary test of a model of the endogenous glucose balance that incorporates the characteristics of the glucose transporters GLUT1, GLUT3 and GLUT4. In the modeling process the model is parameterized with nine parameters that are subsequently estimated from data in the literature on the hepatic- and endogenous- balances at various combinations of blood glucose and insulin levels. The ability of the resulting endogenous balance to fit blood glucose measured from patients was tested on 20 patients. The fit obtained with this model compared favorably with the fit obtained with the endogenous balance currently incorporated in the DIAS system.     

Pharmacokinetics of insulin lispro in type 2 diabetes during closed-loop insulin delivery

Insulin pharmacokinetics is not well understood during continuous subcutaneous insulin infusion in type 2 diabetes (T2D). We analyzed data collected in 11 subjects with T2D [6 male, 9 white European and two of Indian ethnicity; age 59.7(12.1) years, BMI 30.1(3.9) kg/m², fasting C-peptide 1002.2(365.8) pmol/l, fasting plasma glucose 9.6(2.2) mmol/l, diabetes duration 8.0(6.2) years and HbA1c 8.3(0.8)%; mean(SD)] who underwent a 24-h study investigating closed-loop insulin delivery at the Wellcome Trust Clinical Research Facility, Cambridge, UK. Subcutaneous delivery of insulin lispro was modulated every 15 min according to a model predictive control algorithm. Two complementary insulin assays facilitated discrimination between exogenous (lispro) and endogenous plasma insulin concentrations measured every 15–60 min. Lispro pharmacokinetics was represented by a linear two-compartment model whilst parameters were estimated using a Bayesian approach applying a closed-form model solution. The time-to-peak of lispro absorption (t_max) was 109.6 (75.5–120.5) min [median (interquartile range)] and the metabolic clearance rate (MCR_I) 1.26 (0.87–1.56) × 10⁻² l/kg/min. MCR_I was negatively correlated with fasting C-peptide (r_s = −0.84; P = .001) and with fasting plasma insulin concentration (r_s = −0.79; P = .004). In conclusion, compartmental modelling adequately represents lispro kinetics during continuous subcutaneous insulin infusion in T2D. Fasting plasma C-peptide or fasting insulin may be predictive of lispro metabolic clearance rate in T2D but further investigations are warranted.     

A circulatory model for calculating non-steady-state glucose fluxes. Validation and comparison with compartmental models

This study presents a circulatory model of glucose kinetics for application to non-steady-state conditions, examines its ability to predict glucose appearance rates from a simulated oral glucose load, and compares its performance with compartmental models. A glucose tracer bolus was injected intravenously in rats to determine parameters of the circulatory and two-compartment models. A simulated oral glucose tolerance test was performed in another group of rats by infusing intravenously labeled glucose at variable rates. A primed continuous intravenous infusion of a second tracer was given to determine glucose clearance. The circulatory model gave the best estimate of glucose appearance, closely followed by the two-compartment model and a modified Steele one-compartment model with a larger total glucose volume. The standard one-compartment model provided the worst estimate. The average relative errors on the rate of glucose appearance were: circulatory, 10%; two-compartment, 13%; modified one-compartment, 11%; standard one-compartment, 16%. Recovery of the infused glucose dose was 93+/-2, 94+/-2, 92+/-2 and 85+/-2%, respectively. These results show that the circulatory model is an appropriate model for assessing glucose turnover during an oral glucose load.     

Modified minimal model using a single-step fitting process for the intravenous glucose tolerance test in Type 2 diabetes and healthy humans

The classical minimal model with single compartment was modified by the assumption that the insulin decay rate is not always a first-order process, and a mathematical function for describing the insulin infusion rate is introduced. The modified model was used to study four sets of published data including healthy humans and Type 2 diabetes with different types of insulin infusion rates. The single-step fitting process took the glucose-insulin system as a dynamic integrated physiological system and generated the real optimized model parameters from the experimental data using the modified model. It also avoided the errors from the interpolation or extrapolation for taking measured insulin points as inputs, which were mostly employed in publications when using the single or multi-compartments minimal model. The averaged R(2) value between measured and calculated plasma concentrations for these four cases is 0.977, which indicates excellent agreement.     

Positive sliding mode control for blood glucose regulation

Biological systems involving positive variables as concentrations are some examples of so-called positive systems. This is the case of the glycemia–insulinemia system considered in this paper. To cope with these physical constraints, it is shown that a positive sliding mode control (SMC) can be designed for glycemia regulation. The largest positive invariant set (PIS) is obtained for the insulinemia subsystem in open and closed loop. The existence of a positive SMC for glycemia regulation is shown here for the first time. Necessary conditions to design the sliding surface and the discontinuity gain are derived to guarantee a positive SMC for the insulin dynamics. SMC is designed to be positive everywhere in the largest closed-loop PIS of plasma insulin system. Two-stage SMC is employed; the last stage SMC2 block uses the glycemia error to design the desired insulin trajectory. Then the plasma insulin state is forced to track the reference via SMC1. The resulting desired insulin trajectory is the required virtual control input of the glycemia system to eliminate blood glucose (BG) error. The positive control is tested in silico on type-1 diabetic patients model derived from real-life clinical data.     

A circulatory model for the estimation of insulin sensitivity

Insulin sensitivity is commonly assessed by analysing intravenous glucose tests with a simple model, called the minimal model. However, the physiological meaning of the insulin sensitivity indices estimated using this model is not transparent. To overcome this problem, a circulatory model was developed to analyse intravenous glucose tests in which a tracer was injected together with glucose. Physiological parameters, such as glucose volume, clearance and production, were estimated. Insulin sensitivity was defined in terms of glucose clearance. The parameter values estimated in 5 normal subjects were in good agreement with the values reported in previous independent studies.