Nonlinear mixed effects to improve glucose minimal model parameter estimation: a simulation study in intensive and sparse sampling

Quantifying the effect of nutritional interventions on metabolic resilience using personalized computational models

The manifestation of metabolic deteriorations that accompany overweight and obesity can differ greatly between individuals, giving rise to a highly heterogeneous population. This inter-individual variation can impede both the provision and assessment of nutritional interventions as multiple aspects of metabolic health should be considered at once. Here, we apply the Mixed Meal Model, a physiology-based computational model, to characterize an individual's metabolic health in silico. A population of 342 personalized models were generated using data for individuals with overweight and obesity from three independent intervention studies, demonstrating a strong relationship between the model-derived metric of insulin resistance (ρ = 0.67, p < 0.05) and the gold-standard hyperinsulinemic-euglycemic clamp. The model is also shown to quantify liver fat accumulation and β-cell functionality. Moreover, we show that personalized Mixed Meal Models can be used to evaluate the impact of a dietary intervention on multiple aspects of metabolic health at the individual level.

Nonlinear Mixed Effects Modeling of Glucagon Kinetics in Healthy Subjects

OBJECTIVE

To date, the lack of a model of glucagon kinetics precluded the possibility of estimating and studying glucagon secretion in vivo, e.g., using deconvolution, as done for other hormones like insulin and C-peptide. Here, we used a nonlinear mixed effects technique to develop a robust population model of glucagon kinetics, able to describe both the typical population kinetics (TPK) and the between-subject variability (BSV), and relate this last to easily measurable subject characteristics.

METHODS

Thirty-four models of increasing complexity (variably including covariates and correlations among random effects) were identified on glucagon profiles obtained from 53 healthy subjects, who received a constant infusion of somatostatin to suppress endogenous glucagon production, followed by a continuous infusion of glucagon (65 ng/kg/min). Model selection was performed based on its ability to fit the data, provide precise parameter estimates, and parsimony criteria.

RESULTS

A two-compartment model was the most parsimonious. The model was able to accurately describe both the TPK and the BSV of model parameters as function of body mass and body surface area. Parameters were precisely estimated, with central volume of distribution V1 = 5.46 L and peripheral volume of distribution V2 = 5.51 L. The introduction of covariates resulted in a significant shrinkage of the unexplained BSV and considerably improved the model fit.

CONCLUSION

We developed a robust population model of glucagon kinetics.

SIGNIFICANCE

This model provides a deeper understanding of glucagon kinetics and is usable to estimate glucagon secretion in vivo by deconvolution of plasma glucagon concentration data.

Personalized computational model quantifies heterogeneity in postprandial responses to oral glucose challenge

Plasma glucose and insulin responses following an oral glucose challenge are representative of glucose tolerance and insulin resistance, key indicators of type 2 diabetes mellitus pathophysiology. A large heterogeneity in individuals' challenge test responses has been shown to underlie the effectiveness of lifestyle intervention. Currently, this heterogeneity is overlooked due to a lack of methods to quantify the interconnected dynamics in the glucose and insulin time-courses. Here, a physiology-based mathematical model of the human glucose-insulin system is personalized to elucidate the heterogeneity in individuals' responses using a large population of overweight/obese individuals (n = 738) from the DIOGenes study. The personalized models are derived from population level models through a systematic parameter selection pipeline that may be generalized to other biological systems. The resulting personalized models showed a 4-5 fold decrease in discrepancy between measurements and model simulation compared to population level. The estimated model parameters capture relevant features of individuals' metabolic health such as gastric emptying, endogenous insulin secretion and insulin dependent glucose disposal into tissues, with the latter also showing a significant association with the Insulinogenic index and the Matsuda insulin sensitivity index, respectively.

Population insulin sensitivity from sparsely sampled oral glucose tolerance tests

OBJECTIVE

This work aimed to estimate population-level insulin sensitivity (S_I) from 2-hour oral glucose tolerance tests (OGTT) with less than 7 samples.

RESEARCH DESIGN AND METHODS

The current methodology combines the OGTT mathematical model developed by Dalla Man et al., with nonlinear multilevel (NLML) statistical model to estimate population-level insulin sensitivity (S_I) from sparsely sampled datasets (3 or 4 samples per subject obtained in 120 min). To validate our novel methodology of population S_I estimation, we simulated 50 virtual subjects. We simulated 10 observations per subject over 240 minutes. After estimating their S_I using the OGTT model, the virtual subjects were split into two groups, subjects with S_I above the average and ones with below average. Subsequently, the simulated data were analyzed using statistical software and employing a t-test. The mean estimates of population S_I for the two groups of virtual subjects and their respective 95% CI were compared to the estimates obtained with our novel NLML group S_I estimates obtained using the 3 and 4 time points per subject. To further validate the performance of the novel NLML model, a set of 34 prediabetic and 30 diabetic subjects with T2D was used. As outlined above for the in-silico subjects, differences between the prediabetic and T2D subjects in regard to S_I was assessed using the classical two-stage approach (individual S_I estimation followed by statistical comparison of the two groups). The average estimates obtained with the classical two-stage approach were compared to the group estimated obtained with the NLML approach using 3 (0, 60, and 120 minutes) points per subject obtained in 120 minutes.

RESULTS

Unique and identifiable individual estimates of S_I were obtained for all virtual subjects. In comparison to the subjects with above average S_I (n=25), the subjects with simulated below average S_I (n=25) exhibited significantly lower insulin sensitivity (P<0.001). Our novel NLML population model confirmed these findings (4-point OGTT: P<0.001; 3-point OGTT: P<0.001). In a similar fashion to the one outlined for the virtual subjects, the median insulin sensitivities estimated with the classical two-stage approach were different between the prediabetic (n=34) and T2D subjects (n=32, P=0.004). Using 3 points per subject, our novel NLML model confirmed these findings (P<0.001).

CONCLUSIONS

The population estimates of S_I from OGTT data is an effective tool to assess population insulin sensitivity and assess differences that may not be possible when calculating individual S_I or when less than 7 samples are available.

Kinetic Studies to Elucidate Impaired Metabolism of Triglyceride-rich Lipoproteins in Humans

To develop novel strategies for prevention and treatment of dyslipidemia, it is essential to understand the pathophysiology of dyslipoproteinemia in humans. Lipoprotein metabolism is a complex system in which abnormal concentrations of various lipoprotein particles can result from alterations in their rates of production, conversion, and/or catabolism. Traditional methods that measure plasma lipoprotein concentrations only provide static estimates of lipoprotein metabolism and hence limited mechanistic information. By contrast, the use of tracers labeled with stable isotopes and mathematical modeling, provides us with a powerful tool for probing lipid and lipoprotein kinetics in vivo and furthering our understanding of the pathogenesis of dyslipoproteinemia.

Improved Estimation of Human Lipoprotein Kinetics with Mixed Effects Models

Context

Mathematical models may help the analysis of biological systems by providing estimates of otherwise un-measurable quantities such as concentrations and fluxes. The variability in such systems makes it difficult to translate individual characteristics to group behavior. Mixed effects models offer a tool to simultaneously assess individual and population behavior from experimental data. Lipoproteins and plasma lipids are key mediators for cardiovascular disease in metabolic disorders such as diabetes mellitus type 2. By the use of mathematical models and tracer experiments fluxes and production rates of lipoproteins may be estimated.

Results

We developed a mixed effects model to study lipoprotein kinetics in a data set of 15 healthy individuals and 15 patients with type 2 diabetes. We compare the traditional and the mixed effects approach in terms of group estimates at various sample and data set sizes.

Conclusion

We conclude that the mixed effects approach provided better estimates using the full data set as well as with both sparse and truncated data sets. Sample size estimates showed that to compare lipoprotein secretion the mixed effects approach needed almost half the sample size as the traditional method.

A non-linear mixed effect modelling approach for metabolite correction of the arterial input function in PET studies

Quantitative PET studies with arterial blood sampling usually require the correction of the measured total plasma activity for the presence of metabolites. In particular, if labelled metabolites are found in the plasma in significant amounts their presence has to be accounted for, because it is the concentration of the parent tracer which is required for data quantification. This is achieved by fitting a Parent Plasma fraction (PPf) model to discrete metabolite measurements. The commonly used method is based on an individual approach, i.e. for each subject the PPf model parameters are estimated from its own metabolite samples, which are, in general, sparse and noisy. This fact can compromise the quality of the reconstructed arterial input functions, and, consequently, affect the quantification of tissue kinetic parameters. In this study, we proposed a Non-Linear Mixed Effect Modelling (NLMEM) approach to describe metabolite kinetics. Since NLMEM has been developed to provide robust parameter estimates in the case of sparse and/or noisy data, it has the potential to be a reliable method for plasma metabolite correction. Three different PET datasets were considered: [11C]-(+)-PHNO (54 scans), [11C]-PIB (22 scans) and [11C]-DASB (30 scans). For each tracer both simulated and measured data were considered and NLMEM performance was compared with that provided by individual analysis. Results showed that NLMEM provided improved estimates of the plasma parent input function over the individual approach when the metabolite data were sparse or contained outliers.

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.

A graphical method for practical and informative identifiability analyses of physiological models: a case study of insulin kinetics and sensitivity

BACKGROUND

Derivative based a-priori structural identifiability analyses of mathematical models can offer valuable insight into the identifiability of model parameters. However, these analyses are only capable of a binary confirmation of the mathematical distinction of parameters and a positive outcome can begin to lose relevance when measurement error is introduced. This article presents an integral based method that allows the observation of the identifiability of models with two-parameters in the presence of assay error.

METHODS

The method measures the distinction of the integral formulations of the parameter coefficients at the proposed sampling times. It can thus predict the susceptibility of the parameters to the effects of measurement error. The method is tested in-silico with Monte Carlo analyses of a number of insulin sensitivity test applications.

RESULTS

The method successfully captured the analogous nature of identifiability observed in Monte Carlo analyses of a number of cases including protocol alterations, parameter changes and differences in participant behaviour. However, due to the numerical nature of the analyses, prediction was not perfect in all cases.

CONCLUSIONS

Thus although the current method has valuable and significant capabilities in terms of study or test protocol design, additional developments would further strengthen the predictive capability of the method. Finally, the method captures the experimental reality that sampling error and timing can negate assumed parameter identifiability and that identifiability is a continuous rather than discrete phenomenon.

A three-compartment model of the C-peptide-insulin dynamic during the DIST test

Dynamic insulin sensitivity (SI) tests often utilise model-based parameter estimation. This research analyses the impact of expanding the typically used two-compartment model of insulin and C-peptide kinetics to incorporate a hepatic third compartment. The proposed model requires only four C-peptide assays to simulate endogenous insulin production (uen), greatly reducing the cost and clinical burden. Sixteen subjects participated in 46 dynamic insulin sensitivity tests (DIST). Population kinetic parameters are identified for the new compartment. Results are assessed by model error versus measured data and repeatability of the identified SI. The median C-peptide error was 0% (IQR: -7.3, 6.7)%. Median insulin error was 7% (IQR: -28.7, 6.3)%. Strong correlation (r=0.92) existed between the SI values of the new model and those from the original two-compartment model. Repeatability in SI was similar between models (new model inter/intra-dose variability 3.6/12.3% original model -8.5/11.3%). When frequent C-peptide samples may be available, the added hepatic compartment does not offer significant diagnostic, repeatability improvement over the two-compartment model. However, a novel and successful three-compartment modelling strategy was developed which provided accurate estimation of endogenous insulin production and the subsequent SI identification from sparse C-peptide data.

Measurement of human surfactant protein-B turnover in vivo from tracheal aspirates using targeted proteomics

We describe a method to measure protein synthesis and catabolism in humans without prior purification and use the method to measure the turnover of surfactant protein-B (SP-B). SP-B, a lung-specific, hydrophobic protein essential for fetal-neonatal respiratory transition, is present in only picomolar quantities in tracheal aspirate samples and difficult to isolate for dynamic turnover studies using traditional in vivo tracer techniques. Using infusion of [5,5,5-(2)H(3)] leucine and a targeted proteomics method, we measured both the quantity and kinetics of SP-B tryptic peptides in tracheal aspirate samples of symptomatic newborn infants. The fractional synthetic rate (FSR) of SP-B measured using the most abundant proteolytic fragment, a 10 amino acid peptide from the carboxy-terminus of proSP-B (SPTGEWLPR), from the circulating leucine pool was 0.035 +/- 0.005 h(-1), and the fractional catabolic rate was 0.044 +/- 0.003 h(-1). This technique permits high-throughput and sensitive measurement of turnover of low abundance proteins with minimal sample preparation.

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

Population approaches, traditionally employed in pharmacokinetic-pharmacodynamic studies, have shown value also in the context of glucose-insulin metabolism models by providing more accurate individual parameters estimates and a compelling statistical framework for the analysis of between-subject variability (BSV). In this work, the advantages of population techniques are further explored by proposing integration of covariates in the intravenous glucose tolerance test (IVGTT) glucose minimal model analysis. A previously published dataset of 204 healthy subjects, who underwent insulin-modified IVGTTs, was analyzed in NONMEM, and relevant demographic information about each subject was employed to explain part of the BSV observed in parameter values. Demographic data included height, weight, sex, and age, but also basal glycemia and insulinemia, and information about amount and distribution of body fat. On the basis of nonlinear mixed-effects modeling, age, visceral abdominal fat, and basal insulinemia were significant predictors for SI (insulin sensitivity), whereas only age and basal insulinemia were significant for P2 (insulin action). The volume of distribution correlated with sex, age, percentage of total body fat, and basal glycemia, whereas no significant covariate was detected to explain variability in SG (glucose effectiveness). The introduction of covariates resulted in a significant shrinking of the unexplained BSV, especially for SI and P2 and considerably improved the model fit. These results offer a starting point for speculation about the physiological meaning of the relationships detected and pave the way for the design of less invasive and less expensive protocols for epidemiological studies of glucose-insulin metabolism.

Diabetes: Models, Signals, and Control

The control of diabetes is an interdisciplinary endeavor, which includes a significant biomedical engineering component, with traditions of success beginning in the early 1960s. It began with modeling of the insulin-glucose system, and progressed to large-scale in silico experiments, and automated closed-loop control (artificial pancreas). Here, we follow these engineering efforts through the last, almost 50 years. We begin with the now classic minimal modeling approach and discuss a number of subsequent models, which have recently resulted in the first in silico simulation model accepted as substitute to animal trials in the quest for optimal diabetes control. We then review metabolic monitoring, with a particular emphasis on the new continuous glucose sensors, on the analyses of their time-series signals, and on the opportunities that they present for automation of diabetes control. Finally, we review control strategies that have been successfully employed in vivo or in silico, presenting a promise for the development of a future artificial pancreas and, in particular, discuss a modular architecture for building closed-loop control systems, including insulin delivery and patient safety supervision layers. We conclude with a brief discussion of the unique interactions between human physiology, behavioral events, engineering modeling and control relevant to diabetes.