Prediction of postprandial blood glucose under uncertainty and intra-patient variability in type 1 diabetes: A comparative study of three interval models

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Abstract

The behavior of three insulin action and glucose kinetics models was assessed for an insulin therapy regime in the presence of patient variability. For this purpose, postprandial glucose in patients with type 1 diabetes was predicted by considering intra- and inter-patient variability using modal interval analysis. Equations to achieve optimal prediction are presented for models 1, 2 and 3, which are of increasing complexity. The model parameters were adjusted to reflect the “same” patient in the presence of variability. The glucose response envelope for model 1, the simplest insulin–glucose model assessed, included the responses of the other two models when a good fit of the model parameters was achieved. Thus, under variability, simple glucose–insulin models may be sufficient to describe patient dynamics in most situations.

Introduction

Type 1 diabetes mellitus (T1DM) is an autoimmune disease characterized by elevated plasma glucose levels corresponding to acute or chronic hyperglycemia. This is caused by destruction of insulin-producing β-cells in the islets of Langerhans in the pancreas by cells of the immune system. Untreated hyperglycemia can lead to serious complications that include cardiovascular diseases, kidney failure, blindness and stroke. Hypoglycemia (low blood glucose) can cause confusion, clumsiness, or fainting if untreated. Severe hypoglycemia can lead to seizures, coma, and even death.

Since the Diabetes Control and Complications Trial [1], euglycemia has been recommended as the control objective for T1DM patients unless a contraindication exists. However, there is still no universal, efficient and safe system able to normalize glucose levels in patients. The intensive insulin therapy required to achieve glucose control based on injection of basal and bolus insulin to emulate its physiological secretion can increase the risk of severe hypoglycemia with all its consequences.

Prediction of glucose excursions is key in decision-aid systems for insulin therapy optimization in T1DM [2], [3] and glucose control strategies [4]. However, there is large intra- and inter-individual variability in patients behavior. Moreover, food intake (amount of carbohydrates) is another important source of uncertainty since accurate estimates are difficult for a mixed meal. Thus, the development of prediction tools able to consider different sources of uncertainty (input, parameters, initial state) is necessary.

Mathematical models of glucose regulation have been studied over the past 50 years. Makroglou et al. [5] presented an overview of the glucose–insulin regulatory models available in the literature. More recently, the engineering effort in modeling of the insulin–glucose system through the last 50 years was presented by Cobelli et al. [6], starting with the minimal model and following with subsequent models. The most well-known model is the so-called minimal model, which contains a minimal number of parameters [7] and is widely used in physiological research work to estimate glucose effectiveness and insulin sensitivity in the intravenous glucose tolerance test.

Different simulation models of glucose regulation for T1DM patients have been reviewed [8] but comparative studies are not available, in contrast to the situation for Intensive Care Unit glucose models [9]. Here we focus on models for postprandial glucose prediction.

There is no consensus on the degree of complexity a glucoregulatory model should use to describe physiological phenomena considering the large intra-patient variability observed that can jeopardize any attempt to obtain accurate model predictions after a few hours. For this reason, here we compare three postprandial insulin action and glucose kinetics models in the presence of patient variability (insulin sensitivity) and uncertainty in food intake estimation. The models used are those proposed by Bergman et al. [7], Hovorka et al. [10] and Dalla Man et al. [11], which are of increasing complexity. These models are combined with a shared insulin pharmacokinetic and glucose intestinal absorption submodels. The influence of the complexity of insulin action and glucose kinetics submodels on postprandial glucose excursions under uncertainty is then analyzed. The kinetic parameters for each model are adjusted based on data for ten adults from the educational version of the University of Virginia (UVa) simulator for comparison.

The variability and uncertainty is represented here by interval values. Simulation of a model involving interval values produces an envelope that represents the evolution of each state variable over time. A way to compute this envelope is using interval analysis. One of the main problems in interval computations is the existence of multiple instances of the same variable in the expression to be evaluated, leading to an overestimation of the result since each instance is considered independently. To address this problem in the present study, modal interval analysis (MIA) [12] was applied, which permits to avoid, under some conditions, the problem of overestimation. MIA provides a strong theoretical background for dealing with problems involving uncertainty and logical quantifiers. Using MIA in the prediction of plasma glucose considering uncertainty in food intake and patient parameters such as insulin sensitivities, upper and lower bounds that define an envelope for all possible glucose excursions suffered by the patient are predicted. The problem of tackling uncertainty in the prediction of postprandial blood glucose was analyzed by Calm et al. in [13] comparing the Monte Carlo simulation (MCS) and MIA approaches. They concluded that in contrast to MIA, MCS cannot guarantee that the actual response for a given model is within the bunch of possible postprandial responses. In addition, a much less computational time is required for interval simulation using MIA than for MCS.

Section snippets

Interval models of physiological subsystems of glucose regulation

Prediction of postprandial glycemia involves modeling of subcutaneous insulin absorption, carbohydrate digestion and absorption, insulin pharmacokinetics and pharmacodynamics (PK/PD), and glucose metabolism. Here, a model of intestinal glucose absorption, a subcutaneous insulin PK model and three different models for insulin action and glucose kinetics were studied considering different sources of uncertainty. The aim was to analyze the influence of the complexity of this latter submodel when

Parameter adjustment

The main objective of this study was to compare the plasma glucose dynamics of three models for insulin therapy for a specific patient. This requires adjustment of the parameters of each model considering intra-patient variability to reflect the “same” patient.

The glucose kinetic parameters of models 1 and 2 were adjusted for a suite of virtual patients from the UVa simulator using model 3 [11]. The simulator provides a set of virtual subjects based on real individual data, a simulated sensor

Results

Glucose concentrations for ten adults from the UVa simulator were simulated in a open loop over a period of 1000 min. For each patient, the following typical scenario was assumed: food intake, 80 g; bolus insulin, 5 IU; and an insulin rate to achieve basal glucose of 100 mg/dL. The glucose kinetic parameters of models 1 and 2 were adjusted according to the methodology presented in the previous section. Fig. 3 shows a representative fit of the models for a set of glucose data for adult patient 3

Conclusion

In this work, three insulin action and glucose kinetics models were studied using MIA to consider different sources of uncertainty. Equations to achieve optimal prediction of postprandial glucose in patients with T1DM for each model were presented.

The behavior of models proposed by Bergman et al., Hovorka et al. and Dalla Man et al. for a given insulin therapy was analyzed in detail. The plasma glucose dynamics were compared by adjusting the model parameters considering intra-patient

Conflict of interest

None declared.

Acknowledgements

This work was partially supported by the Spanish Ministry of Science and Innovation through Grant DPI-2010-20764-C02, and by the Autonomous Government of Catalonia through Grant SGR 523.

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