IVGTT glucose minimal model covariate selection by nonlinear mixed-effects approach

Measuring and estimating insulin resistance in clinical and research settings

The article discusses how to measure insulin resistance in muscle, liver, and adipose tissue in human participants. The most frequently used methodologies to evaluate insulin resistance are described in detail starting from the gold standard, that is, the euglycemic hyperinsulinemic clamp, to the intravenous glucose tolerance test, surrogate indices based on fasting measurements, or dynamic tests (such as oral glucose or mixed meal tolerance tests). The accuracy, precision, and reproducibility of the tests as well as cutoff values are reported.

Modeling venous plasma samples in [18F] FDG PET studies: a nonlinear mixed-effects approach

The gold-standard approach to quantifying dynamic PET images relies on using invasive measures of the arterial plasma tracer concentration. An attractive alternative is to employ an image-derived input function (IDIF), corrected for spillover effects and rescaled with venous plasma samples. However, venous samples are not always available for every participant. In this work, we used the nonlinear mixed-effects modeling approach to develop a model which infers venous tracer kinetics by using venous samples obtained from a population of healthy individuals and integrating subject-specific covariates. Population parameters (fixed effects), their between-subject variability (random effects), and the effects of covariates were estimated. The selected model will allow to reliably infer venous tracer kinetics in subjects with missing measurements. Clinical relevance - The derived model will be relevant for fully noninvasive dynamic FDG PET quantification using image-derived input functions in both healthy and patient populations when hemodynamics is not impaired.

Variability in Estimated Modelled Insulin Secretion

BACKGROUND

The ability to measure insulin secretion from pancreatic beta cells and monitor glucose-insulin physiology is vital to current health needs. C-peptide has been used successfully as a surrogate for plasma insulin concentration. Quantifying the expected variability of modelled insulin secretion will improve confidence in model estimates.

METHODS

Forty-three healthy adult males of Māori or Pacific peoples ancestry living in New Zealand participated in an frequently sampled, intravenous glucose tolerance test (FS-IVGTT) with an average age of 29 years and a BMI of 33 kg/m². A 2-compartment model framework and standardized kinetic parameters were used to estimate endogenous pancreatic insulin secretion from plasma C-peptide measurements. Monte Carlo analysis (N = 10 000) was then used to independently vary parameters within ±2 standard deviations of the mean of each variable and the 5th and 95th percentiles determined the bounds of the expected range of insulin secretion. Cumulative distribution functions (CDFs) were calculated for each subject for area under the curve (AUC) total, AUC Phase 1, and AUC Phase 2. Normalizing each AUC by the participant's median value over all N = 10 000 iterations quantifies the expected model-based variability in AUC.

RESULTS

Larger variation is found in subjects with a BMI > 30 kg/m², where the interquartile range is 34.3% compared to subjects with a BMI ≤ 30 kg/m² where the interquartile range is 24.7%.

CONCLUSIONS

Use of C-peptide measurements using a 2-compartment model and standardized kinetic parameters, one can expect ~±15% variation in modelled insulin secretion estimates. The variation should be considered when applying this insulin secretion estimation method to clinical diagnostic thresholds and interpretation of model-based analyses such as insulin sensitivity.

Modeling Between-Subject Variability in Subcutaneous Absorption of a Long-Acting Insulin Glargine 100 U/mL by a Nonlinear Mixed Effects Approach

Subcutaneous insulin absorption is well-known to vary significantly both between and within subjects (BSV and WSV, respectively). This variability considerably obstacles the establishing of a reproducible and effective insulin therapy. Some models exist to describe the subcutaneous kinetics of both fast and long-acting insulin analogues; however, none of them account for the BSV. The aim of this study is to develop a nonlinear mixed effects model able to describe the BSV observed in the subcutaneous absorption of a long-acting insulin glargine 100 U/mL. Four stochastic models of the BSV were added to a previously validated model of subcutaneous absorption of insulin glargine 100 U/mL. These were assessed on a database of 47 subjects with type 1 diabetes. The best model was selected based on residual analysis, precision of the estimates and parsimony criteria. The selected model provided good fit of individual data, precise population parameter estimates and allowed quantifying the BSV of the insulin glargine 100 U/mL pharmacokinetics. Future model development will include the description of the WSV of long- acting insulin absorption.

Pharmacometrics Approaches and its Applications in Diabetes: An Overview

Type 2 diabetes mellitus is the most prevalent and progressive in nature. As the time progress, the multifaceted complications and comorbidities associated to diabetes worsen in the form of macrovascular or microvascular or both. Pharmacometrics modeling is a step forward in minimizing the risk or at least understanding the factors associated to its progression with the passage of time. These models investigate diabetes treatments effects and the progression factors with different viewpoints incorporating insulin-glucose dynamics, dose-response and pharmacokinetics, and pharmacodynamics relationships. Pharmacometrics modeling is an innovative approach in a sense that it is taking us away from the conventional analysis by providing all the opportunities in improving the decision-making in health sector. It has been suggested that we can achieve greater statistical power for determining drug effects through model-based evaluation than through traditional evaluations. The main aim of this review was to evaluate pharmacometrics approaches used in modeling diabetes progression through time and also the integrated models describing glucose-insulin dynamics.

Modeling Between-Subject Variability in Subcutaneous Absorption of a Fast-Acting Insulin Analogue by a Nonlinear Mixed Effects Approach

Despite the great progress made in insulin preparation and titration, many patients with diabetes are still experiencing dangerous fluctuations in their blood glucose levels. This is mainly due to the large between- and within-subject variability, which considerably hampers insulin therapy, leading to defective dosing and timing of the administration process. In this work, we present a nonlinear mixed effects model describing the between-subject variability observed in the subcutaneous absorption of fast-acting insulin. A set of 14 different models was identified on a large and frequently-sampled database of lispro pharmacokinetic data, collected from 116 subjects with type 1 diabetes. The tested models were compared, and the best one was selected on the basis of the ability to fit the data, the precision of the estimated parameters, and parsimony criteria. The selected model was able to accurately describe the typical trend of plasma insulin kinetics, as well as the between-subject variability present in the absorption process, which was found to be related to the subject’s body mass index. The model provided a deeper understanding of the insulin absorption process and can be incorporated into simulation platforms to test and develop new open- and closed-loop treatment strategies, allowing a step forward toward personalized insulin therapy.

An Analysis of Glucose Effectiveness in Subjects With or Without Type 2 Diabetes via Hierarchical Modeling

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.

Modelling the effects of glucagon during glucose tolerance testing

Background

Glucose tolerance testing is a tool used to estimate glucose effectiveness and insulin sensitivity in diabetic patients. The importance of such tests has prompted the development and utilisation of mathematical models that describe glucose kinetics as a function of insulin activity. The hormone glucagon, also plays a fundamental role in systemic plasma glucose regulation and is secreted reciprocally to insulin, stimulating catabolic glucose utilisation. However, regulation of glucagon secretion by α-cells is impaired in type-1 and type-2 diabetes through pancreatic islet dysfunction. Despite this, inclusion of glucagon activity when modelling the glucose kinetics during glucose tolerance testing is often overlooked. This study presents two mathematical models of a glucose tolerance test that incorporate glucose-insulin-glucagon dynamics. The first model describes a non-linear relationship between glucagon and glucose, whereas the second model assumes a linear relationship.

Results

Both models are validated against insulin-modified and glucose infusion intravenous glucose tolerance test (IVGTT) data, as well as insulin infusion data, and are capable of estimating patient glucose effectiveness (s_G) and insulin sensitivity (s_I). Inclusion of glucagon dynamics proves to provide a more detailed representation of the metabolic portrait, enabling estimation of two new diagnostic parameters: glucagon effectiveness (s_E) and glucagon sensitivity (δ).

Conclusions

The models are used to investigate how different degrees of pax‘tient glucagon sensitivity and effectiveness affect the concentration of blood glucose and plasma glucagon during IVGTT and insulin infusion tests, providing a platform from which the role of glucagon dynamics during a glucose tolerance test may be investigated and predicted.

Translation Between Two Models; Application with Integrated Glucose Homeostasis Models

PURPOSE

For some biological systems, there exist several models with somewhat different features and perspectives. We propose an evaluation method for NLME models by analyzing real and simulated data from the model of main interest using a structurally different, but similar, NLME model. We showcase this method using the Integrated Glucose Insulin (IGI) model and the Integrated Minimal Model (IMM). Additionally, we try to map parameters carrying similar information between the two models.

METHODS

A bootstrap of real data and simulated datasets from both the IMM and IGI models were analyzed with the two models. Important parameters of the IMM were mapped to IGI parameters using a large IMM simulated dataset analyzed under the IGI model.

RESULTS

Comparison of the parameters estimated from real data and data simulated with the IMM and analyzed with the IGI model demonstrated differences between real and IMM-simulated data. Comparison of the parameters estimated from real data and data simulated with the IGI model and analyzed with the IMM also demonstrated differences but to a lower extent. The strongest parameter correlations were found for: insulin-dependent glucose clearance (IGI) ~ insulin sensitivity (IMM); insulin-independent glucose clearance (IGI) ~ glucose effectiveness (IMM); and insulin effect parameter (IGI) ~ insulin action (IMM).

CONCLUSIONS

We demonstrated a new approach to investigate models' ability to simulate real-life-like data, and the information captured in each model in comparison to real data, and the IMM clinically used parameters were successfully mapped to their corresponding IGI parameters.

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

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

Optimum designs for non-linear mixed effects models in the presence of covariates

In this article, we present a new method for optimizing designs of experiments for non-linear mixed effects models, where a categorical factor with covariate information is a design variable combined with another design factor. The work is motivated by the need to efficiently design preclinical experiments in enzyme kinetics for a set of Human Liver Microsomes. However, the results are general and can be applied to other experimental situations where the variation in the response due to a categorical factor can be partially accounted for by a covariate. The covariate included in the model explains some systematic variability in a random model parameter. This approach allows better understanding of the population variation as well as estimation of the model parameters with higher precision.

Modeling changes in glucose and glycerol rates of appearance when true basal rates of appearance cannot be readily determined

Advancing diabetes care requires accurate physiological assessments. Hyperinsulinemic clamps with stable isotope tracers can simultaneously measure insulin's ability to suppress lipolysis and hepatic glucose release. Traditionally, these methods require an assessment of basal glucose and glycerol rate of appearance (Ra). Basal Ra is challenging to measure in insulin-dependent diabetes, where exogenous insulin required to maintain normoglycemia can raise peripheral insulin concentrations sufficiently to suppress basal Ra. Thus we identified two alternative statistical approaches to describe changes in glucose and glycerol Ra that are less reliant on basal assessments. Sixteen youths (4 type 1 diabetic, 4 type 2 diabetic, 4 lean controls, and 4 obese nondiabetic) underwent a four-phase ("basal" and 10, 16, and 80 mU·m(2)·min(-1)) hyperinsulinemic euglycemic clamp with glucose and glycerol tracers. Glucose and glycerol Ra were calculated per phase. A statistical method, the standard two-stage (STS) algorithm, was applied to the individual log insulin vs. Ra curves to calculate a single predicted Ra value. A population-based mixed-effects model (MEM) compared the group average Ra with log insulin curves and described individual deviations from group means and was used to calculate individual predicted Ra. Both models were applied to the participant data, and predicted Ras at the mean insulin concentration per phase (10 for glycerol, 16 for glucose) were calculated, with good agreement between observed and predicted values. In our data set, the MEM was better able to detect group differences. Both STS and MEM can model lipolysis and endogenous glucose release in insulin-dependent states when basal Ra cannot be accurately measured.

Glucose Homeostatic Law: Insulin Clearance Predicts the Progression of Glucose Intolerance in Humans

Homeostatic control of blood glucose is regulated by a complex feedback loop between glucose and insulin, of which failure leads to diabetes mellitus. However, physiological and pathological nature of the feedback loop is not fully understood. We made a mathematical model of the feedback loop between glucose and insulin using time course of blood glucose and insulin during consecutive hyperglycemic and hyperinsulinemic-euglycemic clamps in 113 subjects with variety of glucose tolerance including normal glucose tolerance (NGT), impaired glucose tolerance (IGT) and type 2 diabetes mellitus (T2DM). We analyzed the correlation of the parameters in the model with the progression of glucose intolerance and the conserved relationship between parameters. The model parameters of insulin sensitivity and insulin secretion significantly declined from NGT to IGT, and from IGT to T2DM, respectively, consistent with previous clinical observations. Importantly, insulin clearance, an insulin degradation rate, significantly declined from NGT, IGT to T2DM along the progression of glucose intolerance in the mathematical model. Insulin clearance was positively correlated with a product of insulin sensitivity and secretion assessed by the clamp analysis or determined with the mathematical model. Insulin clearance was correlated negatively with postprandial glucose at 2h after oral glucose tolerance test. We also inferred a square-law between the rate constant of insulin clearance and a product of rate constants of insulin sensitivity and secretion in the model, which is also conserved among NGT, IGT and T2DM subjects. Insulin clearance shows a conserved relationship with the capacity of glucose disposal among the NGT, IGT and T2DM subjects. The decrease of insulin clearance predicts the progression of glucose intolerance.

Visual Predictive Check in Models with Time-Varying Input Function

The nonlinear mixed effects models are commonly used modeling techniques in the pharmaceutical research as they enable the characterization of the individual profiles together with the population to which the individuals belong. To ensure a correct use of them is fundamental to provide powerful diagnostic tools that are able to evaluate the predictive performance of the models. The visual predictive check (VPC) is a commonly used tool that helps the user to check by visual inspection if the model is able to reproduce the variability and the main trend of the observed data. However, the simulation from the model is not always trivial, for example, when using models with time-varying input function (IF). In this class of models, there is a potential mismatch between each set of simulated parameters and the associated individual IF which can cause an incorrect profile simulation. We introduce a refinement of the VPC by taking in consideration a correlation term (the Mahalanobis or normalized Euclidean distance) that helps the association of the correct IF with the individual set of simulated parameters. We investigate and compare its performance with the standard VPC in models of the glucose and insulin system applied on real and simulated data and in a simulated pharmacokinetic/pharmacodynamic (PK/PD) example. The newly proposed VPC performance appears to be better with respect to the standard VPC especially for the models with big variability in the IF where the probability of simulating incorrect profiles is higher.

Advancing our understanding of the glucose system via modeling: a perspective

The glucose story begins with Claude Bernard's discovery of glycogen and milieu interieur, continued with Banting's and Best's discovery of insulin and with Rudolf Schoenheimer's paradigm of dynamic body constituents. Tracers and compartmental models allowed moving to the first quantitative pictures of the system and stimulated important developments in terms of modeling methodology. Three classes of multiscale models, models to measure, models to simulate, and models to control the glucose system, are reviewed in their historical development with an eye to the future.

Nutrient infusion bypassing duodenum-jejunum improves insulin sensitivity in glucose-tolerant and diabetic obese subjects

The mechanisms of type 2 diabetes remission after bariatric surgery is still not fully elucidated. In the present study, we tried to simulate the Roux-en-Y gastric bypass with a canonical or longer biliary limb by infusing a liquid formula diet into different intestinal sections. Nutrients (Nutrison Energy) were infused into mid- or proximal jejunum and duodenum during three successive days in 10 diabetic and 10 normal glucose-tolerant subjects. Plasma glucose, insulin, C-peptide, glucagon, incretins, and nonesterified fatty acids (NEFA) were measured before and up to 360 min following. Glucose rate of appearance (Ra) and insulin sensitivity (SI), secretion rate (ISR), and clearance were assessed by mathematical models. SI increased when nutrients were delivered in mid-jejunum vs. duodenum (SI × 10⁴ min⁻¹·pM⁻¹: 1.11 ± 0.44 vs. 0.62 ± 0.22, P < 0.015, in controls and 0.79 ± 0.34 vs. 0.40 ± 0.20, P < 0.05, in diabetic subjects), whereas glucose Ra was not affected. In controls, Sensitivity of NEFA production was doubled in mid-jejunum vs. duodenum (2.80 ± 1.36 vs. 1.13 ± 0.78 × 10⁶, P < 0.005) and insulin clearance increased in mid-jejunum vs. duodenum (2.05 ± 1.05 vs. 1.09 ± 0.38 l/min, P < 0.03). Bypass of duodenum and proximal jejunum by nutrients enhances insulin sensitivity, inhibits lipolysis, and increases insulin clearance. These results may further our knowledge of the effects of bariatric surgery on both insulin resistance and diabetes.

The disposition index: from individual to population approach

To correctly evaluate the glucose control system, it is crucial to account for both insulin sensitivity and secretion. The disposition index (DI) is the most widely accepted method to do so. The original paradigm (hyperbolic law) consists of the multiplicative product of indices related to insulin sensitivity and secretion, but more recently, an alternative formula has been proposed with the exponent α (power function law). Traditionally, curve-fitting approaches have been used to evaluate the DI in a population: the algorithmic implementations often introduce some critical issues, such as the assumption that one of the two indices is error free or the effects of the log transformation on the measurement errors. In this work, we review the commonly used approaches and show that they provide biased estimates. Then we propose a novel nonlinear total least square (NLTLS) approach, which does not need to use the approximations built in the previously proposed alternatives, and show its superiority. All of the traditional fit procedures, including NLTLS, account only for uncertainty affecting insulin sensitivity and secretion indices when they are estimated from noisy data. Thus, they fail when part of the observed variability is due to inherent differences in DI values between individuals. To handle this inevitable source of variability, we propose a nonlinear mixed-effects approach that describes the DI using population hyperparameters such as the population typical values and covariance matrix. On simulated data, this novel technique is much more reliable than the curve-fitting approaches, and it proves robust even when no or small population variability is present in the DI values. Applying this new approach to the analysis of real IVGTT data suggests a value of α significantly smaller than 1, supporting the importance of testing the power function law as an alternative to the simpler hyperbolic law.

Identification of the glucose minimal model by stochastic nonlinear-mixed effects methods

The nonlinear mixed effects models (NLMEM) are widespread modeling techniques in PKPD analysis and epidemiological studies because they can produce a description of not only the individual but also of the population features. Moreover, they are able to deal with individual data sparseness by borrowing the lack of information from the entire population. In this way, the NLMEM do not fail where instead other techniques, such as the traditional individual weighted least squares (WLS), sometimes do. The NLME approach relies on the maximization of a likelihood function that due to model parametric nonlinearity not always has an explicit solution. Various techniques have been proposed to solve this problem including the first order (FO) and the first order conditional (FOCE) estimation methods that approximate the likelihood function through a linearization; the expectation maximization algorithm (EM) that maximize the exact likelihood; the Bayesian estimation method where a third stage of variability, the distribution of the population parameters, is taken into account [1]. Recently, new estimation methods that rely on the EM algorithm have been implemented in the last release of the population software NONMEM [2]. These methods are: the iterative two stage (ITS), Monte Carlo importance sampling EM (IMP), Monte Carlo importance sampling EM assisted by Mode a Posteriori estimation (IMPMAP) and the Stochastic Approximation EM (SAEM). Moreover, another new method is available, the Markov Chain Monte Carlo Bayesian Analysis (BAYES), next to the more known FO and FOCE. With this article we want to complete the Denti et al [3] simulation study by evaluating the newest population methods applied on the IVGTT glucose minimal model.

Intestinal transit of a glucose bolus and incretin kinetics: a mathematical model with application to the oral glucose tolerance test

The rate of appearance (R(a)) of exogenous glucose in plasma after glucose ingestion is presently measured by tracer techniques that cannot be used in standard clinical testing such as the oral glucose tolerance test (OGTT). We propose a mathematical model that represents in a simple way the gastric emptying, the transport of glucose along the intestinal tract, and its absorption from gut lumen into portal blood. The model gives the R(a) time course in terms of parameters with a physiological counterpart and provides an expression for the release of incretin hormones as related to glucose transit into gut lumen. Glucose absorption was represented by assuming two components related to a proximal and a distal transporter. Model performance was evaluated by numerical simulations. The model was then validated by fitting OGTT glucose and GLP-1 data in healthy controls and type 2 diabetic patients, and useful information was obtained for the rate of gastric emptying, the rate of glucose absorption, the R(a) profile, the insulin sensitivity, and the glucose effectiveness. Model-derived estimates of insulin sensitivity were well correlated (r = 0.929 in controls and 0.886 in diabetic patients) to data obtained from the euglycemic hyperinsulinemic clamp. Although the proposed OGTT analysis requires the measurement of an additional hormone concentration (GLP-1), it appears to be a reasonable choice since it avoids complex and expensive techniques, such as isotopes for glucose R(a) measurement and direct assessment of gastric emptying and intestinal transit, and gives additional correlated information, thus largely compensating for the extra expense.