Insights to the minimal model of insulin secretion through a mean-field beta cell model

doi:10.1016/j.jtbi.2005.04.023

Journal of Theoretical Biology

Volume 237, Issue 4, 21 December 2005, Pages 382-389

https://doi.org/10.1016/j.jtbi.2005.04.023 Get rights and content

Abstract

The present work introduces an extension of the original minimal model of second phase insulin secretion during the intravenous glucose tolerance test (IVGTT), which can provide both physiological and mathematical insights to the minimal model. The extension is named the mean-field beta cell model since it returns the average response of a large number of nonlinear secretory entities. Several secretion models have been proposed for the IVGTT, and we shall identify two fundamentally different theoretical features of these models. Both features can play a central role during the IVGTT, including the one presented in the mean-field beta cell model.

Introduction

The intravenous glucose tolerance test (IVGTT) is widely used in order to estimate parameters that constitute the so-called metabolic portrait of the test subject. The insulin sensitivity and glucose effectiveness provide the glucose kinetics part of the metabolic portrait, while the first- and second-phase insulin secretion indices are measures of pancreatic $β$ -cell function. The parameters are embedded in two different minimal models, one describing the glucose kinetics (Bergman et al., 1979) while the other describes the insulin secretion (Toffolo et al., 1980). The minimal models are extensively used and implemented in published computer programs (Pacini and Bergman, 1986) and (Vega-Catalan, 1990). Here we shall concentrate upon the minimal model of insulin secretion (hereafter named MM), as applied e.g. in Pacini and Cobelli (1990), Marchesini et al. (1990), and Piccardo et al. (1994).

MM simply states that during an IVGTT, the second phase insulin secretion into plasma is proportional to the time since the glucose bolus was administrated and to the glucose concentration above some threshold value. MM is based on data, and does not provide any physiological arguments for why the secretion rate rises linearly in time, which subsequently might cast doubt on whether the physiology is satisfactory described by the model. Furthermore, the explicit time dependence causes mathematical problems, specifically when the two minimal models are unified (Gaetano and Arino, 2000).

Based on data analysis, other statistical models have been suggested to replace the minimal model of insulin secretion (Toffolo et al., 1995), (called M1 and M2 in the present paper, just as in the original paper). These models can be argued from theoretically more comprehensive models (Licko and Silvers, 1975), which make them physiologically and mathematically more appealing. However, the physiologic assumptions and the theoretical feature behind these models are not the only reasonable ones, which is subject for further elaboration in the following.

The primary goal of the present paper is to introduce a theoretical and mathematical extension of the original MM that clarifies the physiologic background and describes the theoretical feature that has made MM a success. The added structure introduced to clarify the physiology solves the unboundedness of the secretion, which was the more crucial of the mathematical problems.

The theoretical idea behind the suggested model is to describe the collective secretory response of all the beta cells as a single object, a mean-field beta cell. The dynamics of the mean-field beta cell is different from the individual entities it is composed of, but we can give a physiological understanding of the mean-field model based on the behavior described by an appropriate model for the individual entities.

The secondary purpose of the present text is to compare M1 to the mean-field beta cell model in a theoretical context motivated by the distributed threshold model (Grodsky, 1972), which enables us to identify the two separate theoretical model features of the original MM and M1. Finally the models are compared in a short data analysis, to demonstrate that both theoretical features can be seen in the IVGTT experiment. This data analysis deals exclusively with the theoretical features, that could be identified only for a small number of subjects, whereas other complications of a full scale data analysis were completely ignored. The present model is thus not proposed as a competitive model for the IVGTT, but as a tool to illustrate the theoretical features that may be necessary to consider for new insulin secretion models. Especially when more mechanistic models are pursued for a coherent description of different challenges of the beta cell.

All models discussed in the present text are summarized in the appendix, for easy reference.

Section snippets

Insulin synthesis and release

Insulin secretion in response to an abrupt increase in blood glucose concentration can to a large extent be described by two phases, a rapid first phase followed by a slowly rising second phase. These phases are related to the pleiotropic effects that glucose induce on the beta cell, ranging over regulation of insulin biosynthesis, movement of insulin within the beta cell, and insulin release.

The duration of the IVGTT is only a few hours, which is a brief period when dealing with insulin

The mean-field beta cell model

The first mean-field beta cell model (MFM1 in the following) is a description of the above-basal second phase secretion during the IVGTT. The first phase secretion is not modelled, and could be included either by using a dirac delta function, or simply by letting the plasma insulin concentration start at the maximal insulin concentration $I_{\max}$ .

We shall quantify the physiological argument that the second phase secretion must be a combination of the provision of new insulin and the heterogeneity

Motivating models for the IVGTT from the distributed threshold hypothesis

The distributed threshold hypothesis (Grodsky, 1972) has played a central role in the development of a suitable model to estimate the secretion indices during an IVGTT. The original experiments on the perfused rat pancreas used to quantify the distributed threshold model was in fact used as motivation for MM (Toffolo et al., 1980). More directly the distributed threshold model has motivated another model for the IVGTT (Licko and Silvers, 1975), which in a slightly altered form has provided an

Analysis

It is important to note that a regular performance comparison of the presented models is not provided and believed to be out of the scope of the present paper. Data has been included merely to illustrate the theoretical point that the inhomogeneity of the individual secretory entities may be an important interpretation for the plasma insulin profiles for some individuals following the IVGTT. As a tool to demonstrate this interpretation, we have provided the least-squares fit of the relevant

Conclusion

We have presented a physiological interpretable mean-field beta cell model for insulin secretion during the IVGTT, and showed that this model is very close to the original MM. In fact, it was shown that the original MM reappears in some limit of the mean-field model and that we may be close to this limit for some subjects. The model was extended with an extra compartment of readily releasable insulin, which enabled us to understand and describe the first phase secretion.

Theoretical insights of

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Insights to the minimal model of insulin secretion through a mean-field beta cell model

Abstract

Introduction

Section snippets

Insulin synthesis and release

The mean-field beta cell model

Motivating models for the IVGTT from the distributed threshold hypothesis

Analysis

Conclusion

Anal. Biochem.

FEBS Lett.

Math. Biosci.

Gastroenterology

J. Biol. Chem.

Comput. Methods Programs Biomed.

Comput. Methods Programs Biomed.

Metabolism

Comput. Biomed. Res.

Quantitative estimation of insulin sensitivity

Am. J. Physiol.