Objetivo Específico Nro 04

The homeostasis between glucose, insulin, and -cell mass plays an important role in the study of diabetes, which is characterized by hyperglycaemia, insulin resistance, and the persistence of -cell dysfunction. Several mathematical models have been formulated to study glucose-insulin dynamics, in order to contribute to the understanding of the aetiology of diabetes. The main idea of formulating mathematical models is to preserve biologically relevance by referring to experimental data and subsequently testing the biological hypothesis by subjecting the predictive results from mathematical simulations to further experimentation. A mathematical model can also help the design of experiments by defining specific parameters which were ill-defined numerically by previous experiments. A simple mathematical model of the insulin-glucose feedback loop was proposed by Bolie (1961), describing the relationship between glucose and insulin by means of ordinary differential equations (ODEs). Based on this glucose-insulin fundamental feedback loop, the model can be extended. A three-ODE model of glucose-insulin dynamics, which is also known as the standard minimal model, was built by Bergman et al. (1981). It further extended the two-compartment glucose-insulin model into a three-compartment model, i.e. glucose, insulin, and remote insulin. Other physiological parameters relating to glycaemic homeostasis, such as glucagon, NEFAs (non-esterified fatty acids), TAGs (triacylglycerides), and leptin, can be also be incorporated into more extensive mathematical models (Pattaranit and van den Berg, 2008).

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or T2DM. More details of -cell mass regulation have been described in Section 1.1.5. Mathematical models of -cell turnover have been introduced to simulate the phenomenon as well. Finegood et al. (1995) formulated a mathematical model to stimulate -cell mass of adult rats between the ages of one and three month using the relationship between -cell number, size, proliferation, neogenesis, and apoptosis. Using the human IAPP (islet amyloid polypeptide) transgenic (HIP) rat, Manesso et al. (2009) presented the dynamics of -cell turnover, which was detected by immunostaining of Ki67, which is a marker for proliferation, and the terminal deoxynucleotidyl (TdT)-mediated dUTP nick-end labeling (TUNEL), which is a marker for apoptosis, and suggested that -cell regeneration is contributed by neogenesis other than -cell replication.

Apart from those physiological parameters, other perturbations of computational modelling can be implemented to suit the actual physiological situation, for example, introducing a time delay to describe insulin secretion in response to glucose stimulation or hepatic glycogenesis in response to insulin secretion (Bennett and Gourley, 2004; Giang et al., 2008; Li et al., 2006; Tolic et al., 2000).

As more experimental data were obtained, more sophisticated and robust mathematical models could be constructed to address biological questions. Several models were formulated to understand the relationship between glycaemic control and insulinemia within a limited time duration of the object‘s lifespan, i.e. the model is only representing a short-term simulation of the experimental period (Bergman et al., 1979; Bolie, 1961; Li et al., 2006; Panunzi et al., 2007). Thus, mathematical models combing long-term effect, such as -cell regulation, with glycaemic controls, i.e. glucose level and insulin level, have been published as well to study the

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aetiology of diabetes (Chen et al., 2010; de Winter et al., 2006; Hamren et al., 2008; Ribbing et al., 2010; Topp et al., 2000).

Neuroendocrine control systems, such as glucose homeostasis, respond rapidly to physiological challenges while concurrently undergoing adaptation on much slower time scales, such as -cells adaption or decompensation during the progress of diabetes, e.g. Schmidt and Thews (1989) and Frayn (1996). Moreover, an interplay prevails between processes at disparate scales: the slow adaptation is dependent on the fast events. Such an interplay is a pervasive characteristic of many biological systems.

As is well known, the dynamics of such systems can be analysed by considering separate and distinct dynamical systems that correspond to the biological system as it operates on two or more ―time scales‖. In such procedures, the approximation usually is exact if the slower component is ―infinitely slower‖ than the faster component, see Keener and Sneyd (1998) for examples and applications. On a given time scale, the slower variables are ―frozen‖ and appear as constant parameters, whereas the faster ones can often be treated using a quasi-steady state approximation (up to boundary or transition layers) in which the fast variable is essentially replaced by a function which relates its (fast-time system) equilibrium value to the prevailing values of the slower variables. Similarly, in those cases where the fast-time system does not equilibrate but tends to a periodic solution (e.g., a limit cycle or the stationary response to periodic forcing), intuition would suggest that the fast variables might be replaced by a long-term average. However, this intuition need not be correct. This thesis outlines a general method to tackle such problems on two time scales. The type of system describes the dynamics of an organism's physiology (or

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the relevant part thereof) together with the dynamics of the (neuro)endocrine system that regulates this physiology. The method is applied to a well-established model of the regulation of glucose concentration in the blood plasma, which was extended with a slow component, viz. the dynamics of the mass of endocrine cells.