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. Such an
understanding of the underlying mechanisms can help interpreting differences in
14
15
beta-cell sensitivity indices between different populations or patient groups, or
give insight in the physiologically most important steps regulated by for example
GLP-1 (Dalla Man et al., revised).
15
16
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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).
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