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Chapter 29 Whole-body and cellular models of glucose-stimulated insulin secretion Gianna Toffolo, Morten Gram Pedersen, Claudio Cobelli Abstract Models of glucose stimulated insulin secretion are commonly used to measure beta-cell function and to gain insight into the biological mechanisms of insulin release. Depending on the scope, the complexity of the model must be chosen appropriately. We present two models of minimal complexity, able to assess beta-cell function in an individual during intravenous and oral glucose perturbations and a comprehensive model of insulin secretion, describing intracellular events. We show how comparison of cellular and minimal models provides insight into the mechanisms underlying the different aspects of the minimal models, and yields biological meaning to their indices. Keywords beta-cell, insulin secretion, glucose control, mathematical models 29.1 Introduction Dipartimento di Ingegneria dell’Informazione, Università di Padova, via Gradenigo 6A, 35131 Padova (Italy) E-mail: toffolo@dei.unipd.it, pedersen@dei.unipd.it, cobelli@dei.unipd.it. 2 Mechanistic, physiologically based models have been widely used to describe the control exerted by glucose on insulin secretion. They range from minimal (coarse) models which describe the key components of the system at whole-body level, aiming to measure beta-cell function in an individual (Toffolo et al., 1995, Hovorka et al., 1998, Breda et al., 2001, Cretti et al., 2001, Mari et al., 2002) to maximal (fine-grain) models (Grodsky 1972, Bertuzzi et al., 2007, Chen et al., 2008, Pedersen et al., 2008, Pedersen and Sherman, 2009) which include a comprehensive description of the process at cellular level, mainly for simulation purposes. In general terms, model complexity depends on the question being asked: minimal models are intended to quantify processes which are not directly measurable. The rationale is to link the accessible variables, usually plasma concentrations, to the nonaccessible fluxes/parameters of interest, identified on dynamic data measured during to be a perturbation. The system is described at whole-body level, but the model is not a large scale one: not every known substrate/hormone needs to be included because the macro-level response of the system would be relatively insensitive to many micro-level relationships. In addition, because it is not possible to estimate the values of many system parameters from a limited number of in vivo dynamic data, many of the unit processes must be lumped together. Therefore, desirable features of this class of models include: i) physiologically based; ii) parameters that can be estimated with reasonable precision from a single dynamic response of the system; iii) parameters that vary within physiologically plausible ranges; and iv) ability to describe the dynamics of the system with the smallest number of identifiable parameters. In contrast to minimal, maximal (fine-grain) models are comprehensive descriptions attempting to implement the body of knowledge about a process at cellular or even subcellular level. This class of models is not intended to be identified, since without massive experimental investigation on a single individual it is not possible to relate with confidence alterations in the dynamics of bloodborne substances to specific changes in parameters of a comprehensive model. This means that these models are not generally useful for the quantification of specific metabolic relationships – their utility is in their ability to formalize the 2 3 available knowledge, introduce some new hypothesis, investigate the role of individual components via simulation studies. To illustrate how these two different modeling approaches provide different insight regarding glucose-stimulated insulin secretion, we present two models of minimal complexity, able to assess beta-cell function in an individual during i.v. and oral glucose perturbations and a comprehensive model of insulin secretion, describing intracellular events. 29.2 Modeling issues in assessing beta-cell function Assessment of beta-cell function in humans under physiologic conditions has been a challenge due to the feedback nature of the glucose-insulin system, so that plasma insulin and glucose data reflect not only insulin secretion but also insulin action on hepatic glucose production and glucose utilization by peripheral tissues. Other key insulin processes are also involved, such as hepatic insulin extraction and whole-body insulin kinetics, and all these processes should be assessed under physiologic conditions, using a single, simple and physiologic test. We will briefly discuss these issues, with reference to the minimal modeling strategy. Glucose-insulin feedback loop While several techniques have been propose to “open” the feedback loop experimentally, such as the glucose clamp technique, the model-based solution is to maintain the glucose-insulin feedback mechanisms active but open the loop mentally, by partitioning the whole system into two subsystems (Figure 29.1) linked by the measurable variables, insulin and glucose concentration. The two subsystems are then modeled separately: for the insulin secretion model, glucose is the (known) input and insulin the output, while for the model of insulin action on glucose production and utilization, insulin is the (known) input and glucose the output. beta-cell function is estimated from the insulin secretion model, but then interpreted relative to the prevailing level of insulin action, as discussed in Section 29.4. 3 4 (FIGURE 29.1) Hepatic extraction When inferred from plasma insulin concentrations, insulin secretion cannot be isolated from hepatic insulin extraction since plasma data reflect the fraction of pancreatic secretion which appears in plasma, denoted as posthepatic insulin secretion and approximately equal to 50% of pancreatic secretion. This problem can be bypassed if C-peptide concentration is measured during the perturbation and used to estimate insulin secretion, since C-peptide is secreted equimolarly with insulin (Zawalich and Zawalich, 2002) but it is extracted by the liver to a negligible extent (Polonsky et Rubenstein, 1984). Plasma C-peptide concentration thus reflects C-peptide plasma rate of appearance which, apart from the rapid liver dynamics, is a good measure of C-peptide pancreatic secretion which in turn coincides with insulin pancreatic secretion. Whole-body kinetics To be identified on plasma C-peptide measurements, the secretion model must be integrated into a model of whole-body C-peptide kinetics. The widely used model (Eaton et al., 1980) is shown in Figure 29.2: compartment 1, accessible to measure, represents plasma and rapidly equilibrating tissues, compartment 2 represents tissues in slow exchange with plasma. Model equations are conveniently expressed in terms of C-peptide concentration above basal in the two compartments, denoted as CP1 and CP2 (pmol/l): d CP1(t)/dt = - [k01+k21] CP1(t) + k12 CP2(t) + ISR(t) , CP1(0)=0 , (1) d CP2(t)/dt = k21 CP1(t) - k12 CP2(t), CP2(0)=0 , (2) where k01, k21, k12 (min-1) are transfer rate parameters and ISR (pmol l-1 min-1) is secretion above basal, normalized by the volume of distribution of compartment 1, to be described according to the models presented in the following section. 4 5 Parameters of C-peptide kinetics are usually determined, without loss of accuracy, with the population approach (Van Cauter et al., 1992) (FIGURE 29.2) Intravenous and oral glucose tests Either intravenous glucose tolerance test, IVGTT, or ingestion of glucose, e.g., an oral glucose tolerance test, OGTT, or a mixed meal are used to perturb the system. The oral perturbations are no doubt more physiological than the intravenous ones with the incretin effect in operation and with the meal being superior to OGTT due to presence of nutrients, i.e., proteins and fat. Glucose and insulin profiles during the various tests are markedly different: with IVGTT, glucose increases rapidly (within 2-5 minutes) to the maximum level and then declines to basal, thus rendering evident the biphasic nature of insulin secretion while with oral tests glucose increases in the first 60-90 minutes and then decreases, with a smoother profile, and the two phases are not clearly separable. Therefore, different models of insulin secretion were developed for the two types of perturbation, which include the same basic ingredients but adapt them to address the different aspects of secretion mechanisms assessed during the two experimental conditions. 29.3 Minimal models of insulin secretion Intravenous glucose tolerance test The IVGTT model (Toffolo et al., 1995), shown in Fig. 29.3, assumes that insulin secretion ISR, appearing as input in the C-peptide kinetic model eq. 1-2, is proportional to the amount of insulin in the secretory granules, X (pmol l-1), which results from the balance between ISR and provision/docking of new insulin secretory granules, Y (pmol min-1) ISR(t) = m X(t) , dX(t)/dt = - m X(t) + Y(t) , (3) X(0) = X0 . 5 (4) 6 Due to the rapid turnover of compartment X (1/m~2 min), initial condition X0 is responsible of 1st phase secretion likely representing exocytosis of previously primed insulin secretory granules (commonly called readily releasable). This rapid component of insulin secretion is due to a derivative glucose control, since it is elicited by the rate of increase of glucose from basal up to the maximum occurring after the glucose bolus. The slower 2nd phase derives from Y, that occurs in response to a given (i.e. proportional to) glucose concentration, according to the following equation dY(t)/dt = - Y(t) - [G(t) -h] , Y(0) = 0, (5) i.e. in response to an elevated glucose level, Y and thus ISR, tends with a delay (1/~10 min in normal subjects) towards a steady state value linearly related via parameter (min-1) to glucose concentration G (mmol/l) above a threshold value h. 1/presumably represents the time required for new “readily releasable” granules to dock, be primed then exocytosed. The meaning of model parameters X0, m, and are readily envisioned by making reference to a thought experiment of an above-basal step increase of glucose: the rise of glucose induces a first phase secretion, modeled as a monoexponential decay with time constant 1/m from a maximum value X0 . The total amount of insulin secreted during this phase, obtained by integrating first phase secretion over time, coincides with X0. This is followed by a second phase secretion due to provision, which tends with time constant 1/ towards a steady state which is linearly related to the glucose step size through parameter . Given the above mechanistic interpretation of model parameters, it is immediate to define 1st phase responsivity to glucose, 1(dimensionless), as X0 normalized to the maximum glucose increment G (mmol/l) and 2nd phase responsivity to glucose, 2(min-1), as parameter 1=X0/G (6) 2= (7) 6 7 To complete the picture, a basal responsivity index, b (min-1) can be defined as secretion normalized to glucose concentration b=ISRb/Gb = k01C1b/Gb (8) where Gb is the end-test glucose concentration. (Figure 29.3) Oral glucose tests All of the previous model ingredients employed in the IVGTT model were necessary to describe the data, with the exception of the fast turning over insulin releasable pool which is not evident under these conditions (Breda et al., 2001, Breda et al., 2002). However, at variance with IVGTT where the first phase contributes only during the first 2-5 minutes, in the oral tests glucose concentration gradually increases during the first 60-90 minutes, thus requiring a secretion component proportional to the rate of glucose increase. The oral model, shown in Figure 29.3, thus features a dynamic component, ISRd, that senses the rate of increase of glucose concentration, and a static component, ISRs, that represents the release of insulin that, after a delay, occurred in proportion to prevailing glucose concentration: ISR(t)=ISRd(t)+ISRs(t) (9) ISRd represents the secretion of insulin stored in the beta-cells in a promptly releasable form and is proportional to the rate of increase of glucose: SR d t = { k G ⋅ Ġ t 0 Ġ t 0 Ġ t ≤ 0 where: 7 } (10) 8 { k d⋅ 1− k G = G t − Gb Gt − Gb G b ≤ G t <G t otherwise 0 } (11) According to eq. 11, the dynamic control is maximum when glucose increases just above its basal value, it decreases linearly with glucose concentration and vanishes when glucose concentration exceeds the threshold level Gt able to promote the secretion of all stored insulin. For elevated Gt, k(G) approximates the constant kd. ISRs is assumed to be equal to the provision of new insulin to the -cells, Y (pmol L-1 min-1): ISRs(t)=Y(t) (12) which is controlled by glucose according to the same equation as for the IVGTT model (eq.5). The dynamic responsivity d (109), which is the counterpart of IVGTT first phase responsivity, is equal to the total amount of insulin released in response to the glucose rate of increase normalized to the maximal increase Gmax-Gb: G max ∫ Φ d= k G dG Gb G max − G b { k d [1− = G max − G b 2⋅ G t − G b k d Gt − G b 2⋅ G max − G b ] if if G t G max G t ≥ G max } (13) For elevated Gt, d kd. The static responsivity s (109 min-1), which is the counterpart of IVGTT second phase responsivity, still equals parameter : s = (14) 8 9 In addition to oral tests, the model described by eq.9-12 is also able to describe insulin secretion during i.v. tests characterized by smooth glucose profiles such as up&down glucose infusions (Toffolo et al., 2001) and hyperglycemic clamp (Steil et al., 2004). 9 10 1.5 Minimal models of insulin action and hepatic insulin extraction Due to feedback nature of the glucose-insulin system, beta-cell function is not sufficient to evaluate the efficiency of glucose homeostasis in a given individual. To this purpose, beta-cell function needs to be interpreted in light of the prevailing insulin action, measured by insulin sensitivity, as formulated by the disposition index paradigm (Bergman et al., 1981), which assumes that glucose tolerance of an individual is related to the product of beta-cell function and insulin sensitivity. Thanks to its intuitive and reasonable grounds, this measure of beta-cell functionality, which was first introduced for IVGTT, has become the method of choice also with other tests, like clamp and OGTT, as reviewed in (Cobelli et al., 2007) where some recent developments, related to formulation and practical implementation of the disposition index are also discussed. Since the effect of insulin on peripheral tissues is determined not only by the biologic effect of insulin but also by the amount of insulin to which the tissue is exposed, hepatic insulin extraction provide the third dimension to the metabolic status of an individual. Not only beta-cell function but also insulin sensitivity and hepatic insulin extraction can be measured at whole-body level, during an IVGTT or oral tests, by employing glucose (Bergman et al., 1981, Dalla Man et al., 2002) and insulin (Toffolo et al., 2006, Campioni et al., 2009) minimal models. Minimal models of insulin secretion, action and hepatic extraction have been used in a number of pathophysiological studies, including the effect of age and gender on glucose metabolism (Basu et al., 2006); the effect of anti-aging drugs (Nair et al., 2006); the influence of ethnicity (Petersen et al., 2006); insulin sensitivity and beta-cell function in nondiabetic (Sunehag et al., 2009) and obese (Cali et al., 2009) adolescents; the pathogenesis of prediabetes (Bock et al., 2006 and 2007) and type 2 diabetes (Dalla Man et al., 2008, Basu et al., 2009). 1.6 Cellular model of insulin secretion In relation to the clinical interest described in the previous sections, and considering that the physiological task of the beta-cell is to secrete insulin, it 10 11 might be surprising how little work there has been done on modeling insulin secretion, compared to the focus on other aspects of beta-cell physiology such as bursting electrical activity and oscillatory calcium levels and insulin secretion, where mathematical modeling has contributed significantly to the understanding of the generation of these rhythmic patterns (for reviews, see (Bertram et al., 2007; Pedersen, 2009)). However, already in the 1970's, Grodsky (1972), Cerasi et al. (1974), and others, did model the pancreatic insulin response to various kinds of glucose stimuli. Due to the limited knowledge of the beta-cell biology at that time, these models were phenomenological. Only recently has our knowledge of the control of the movement and fusion of insulin granules increased to a level where we have started to formulate mechanistically based models. Grodsky (1972) proposed that insulin was located in "packets", plausibly the insulin containing granules, but also possibly entire beta-cells. Some of the insulin was stored in a reserve pool, while other insulin packets were located in a labile pool, ready for release in response to glucose. The labile pool is responsible for the first phase of insulin secretion (Grodsky, 1972), while the reserve pool is responsible for a creating a sustained second phase. This basic distinction has been at least partly confirmed when the packets are identified with granules (Daniel et al., 1999; Olofsson et al., 2002). Grodsky (1972) moreover assumed that the labile pool is heterogeneous in the sense that the packets in the pool have different thresholds with respect to glucose beyond which they release their content. This assumption was necessary for explaining the so-called staircase experiment, where the glucose concentration was stepped up, each step giving rise to a peak of insulin. There has been no support of granules having different thresholds (Nesher and Cerasi, 2002), but already Grodsky (1972) mentioned that cells apparently have different thresholds based on electrophysiological measurements. Later, Jonkers and Henquin (2001) showed that the number of active cells is a sigmoidal function of the glucose concentration, as assumed by Grodsky (1972) for the threshold distribution. Recently, we have showed how to unify the threshold distribution for cells with the pool description for granules (Pedersen et al., 2008), thus providing an 11 12 updated version of Grodsky's model, which takes into account more of the recent knowledge of beta-cell biology. An overview of the model is given in Fig. 4. (Figure 29.4) It includes mobilization of secretory granules from a reserve pool to the cell periphery, where they attach to the plasma membrane (docking). The granules can mature further (priming), thus entering the 'readily releasable pool' (RRP). Calcium influx triggers membrane fusion, and subsequent insulin release. We included the possibility of so-called kiss-and-run exocytosis, where the fusion pore reseals before the granule cargo is released. The glucose-dependent increase in the number of cells showing a calcium signal (Jonkers and Henquin, 2001) was included by distinguishing between readily releasable granules in silent and active cells. Therefore, the RRP is heterogeneous in the sense that only granules residing in cells with a threshold for calcium activity below the ambient glucose concentration are allowed to fuse. Our model thus provides a biologically founded explanation for the heterogeneity assumed by Grodsky (1972), and is able to simulate the characteristic biphasic insulin secretion pattern in response to a step in glucose stimulation, as well as the secretory profile of the staircase stimulation protocol. The equations of the model are as follows: dM(t)/dt = [ M(G) - M(t) ]/, (15) dD(t)/dt = M(G,t) – r D(t) – p+ D(t) + p- ∫0 h(g,t)dg , (16) where M is the mobilization flux, D is the insulin in the docked pool, and r is the rate of reinternalization. The RRP is modelled by a time-varying density function h(g,t) indicating the amount of insulin in the RRP in beta-cells with a threshold between g and g+dg, described by the equation dh(g,t)/dt= p+ D(t) (g) – p-h(g,t) – f+ h(g,t) (G-g). 12 (17) 13 Here (G-g) is the Heaviside unit step function, which is 1 for G>g and zero otherwise, indicating that fusion only occurs when the threshold is reached. The priming flux p+ D distributes among cells according to the fraction of cells with threshold g described by the time-constant function (g). The secretion rate is expressed as SR(t) = m F(t) + SRb, (18) where SRb is basal secretion, m is the rate constant of release and F is the size of the fused pool, which follows dF/dt = - (m+k)F + f +H(G, t), (19) where f+ is the rate constant of fusion, k is the kiss-and-run rate, and H(G,t)=∫0Gh(g, t) dg represents the amount of insulin in the RRP in cells with a threshold below G. For further details of the model, we refer to the original article (Pedersen et al., 2008). 1.7 Cellular modeling: insight into minimal models Modeling of intracellular events helps in understanding the role of different mechanisms of insulin secretion, both on cellular and whole-body levels. We have recently shown that the secretion rate SR (Eq. (18)) of the cellular, mechanistic model (Pedersen et al., 2008) contains implicitly the three main ingredients of the OGTT minimal models: (i) a static term, which includes (ii) a delay τ due to the delayed refilling of the docked pool D, and (iii) a dynamic term proportional to dG/dt (Pedersen et al., to appear). The latter derivative control is due to the cellular activation thresholds (Jonkers and Henquin, 2001). The comparison of cellular models to minimal models provides insight into the mechanisms underlying the different aspects of the minimal models, and in a sense justifies them by giving a mechanistic underpinning. Model comparison 13 14 also provides a link between the secretion indices of the minimal model to cell biological events, thus yielding biological meaning to the indices. Other recent models go into greater details of the regulation and properties of different pools of granules in single cells (Bertuzzi et al., 2007; Chen et al., 2008, Pedersen and Sherman, 2009). Such details allowed connecting recent imaging experiments (Ohara-Imaizumi et al., 2007) with granule properties, such as a highly calcium-sensitive pool (Wan et al., 2004; Yang and Gillis, 2004), and the investigation of the so-called amplifying pathway of glucose stimulated insulin secretion (Henquin, 2000). We note that although these models describe the dynamics and control of the secretory granules in great details they are unable to reproduce the crucial staircase experiment, because they do not have any threshold distribution in the sense of Grodsky (1972), and in contrast to our recent model (Pedersen et al., 2008). 1.8 Conclusions Models of minimal complexity provide simultaneous assessment of beta-cell function, hepatic insulin extraction and insulin sensitivity under physiologic conditions using a simple protocol. Minimal model complexity is an essential property, since reliable estimates of model parameters need to be derived from a limited number of data collected from an individual. The amount of information they provide is appealing, since it conveys novel insights regarding the regulation of fasting and postprandial glucose metabolism in diabetic and non-diabetic humans. However, in addition to simplicity of the minimal models, it is also desirable that they are physiologic, i.e., that they are linked to the underlying biology of the insulin secreting beta-cells. We have recently (Pedersen et al., to appear) presented a way to make such a connection using a recent model (Pedersen et al., 2008) describing intracellular mechanisms. This analysis showed how the three main components of oral minimal secretion models, derivative control, proportional control and delay, are related to subcellular events, thus providing mechanistic underpinning of the assumptions of the minimal models. 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Effects of glucose, exogenous insulin, and carbachol on C-peptide and insulin secretion from isolated perifused rat islets. J Biol Chem 277: 2623326237. 19 20 FIGURE LEGENDS Figure 1. beta-cell function is causally related to insulin sensitivity since the glucose system is negatively feedback regulated. Figure 2. The two compartment model of C-peptide kinetics Figure 3. The C-peptide minimal models which allow to estimate beta-cell responsivity from an IVGTT (left) and an OGTT & meal (right). Figure 4. Schematic representation of the mechanistic model (Pedersen et al., 2008). The RRP has been divided into readily releasable granules located in silent cells with no calcium influx, exocytosis and release (open circles) and readily releasable granules located in triggered cells (filled circles). 20