Quantitative indexes of
-cell function during
graded up&down glucose infusion from C-peptide minimal
models
Gianna
Toffolo1,
Elena
Breda1,
Melissa K.
Cavaghan2,
David A.
Ehrmann2,
Kenneth S.
Polonsky3, and
Claudio
Cobelli1
1 Department of Electronics and Informatics, University of
Padova, 35131 Padova, Italy; 2 Department of Medicine, The
University of Chicago, Chicago, Illinois 60637; and
3 Department of Medicine, Washington University School
of Medicine, St Louis, Missouri 63110
 |
ABSTRACT |
Availability of
quantitative indexes of insulin secretion is important for definition
of the alterations in
-cell responsivity to glucose associated with
different physiopathological states. This is presently possible by
using the intravenous glucose tolerance test (IVGTT) in conjunction
with the C-peptide minimal model. However, the secretory response to a
more physiological slowly increasing/decreasing glucose stimulus may
uncover novel features of
-cell function. Therefore, plasma
C-peptide and glucose data from a graded glucose infusion protocol
(seven 40-min periods of 0, 4, 8, 16, 8, 4, and 0 mg · kg
1 · min
1) in eight
normal subjects were analyzed by use of a new model of insulin
secretion and kinetics. The model assumes a two-compartment description
of C-peptide kinetics and describes the stimulatory effect on insulin
secretion of both glucose concentration and the rate at which glucose
increases. It provides in each individual the insulin secretion profile
and three indexes of pancreatic sensitivity to glucose:
s,
d, and
b, related,
respectively, to the control of insulin secretion by the glucose level
(static control), the rate at which glucose increases (dynamic
control), and basal glucose. Indexes (means ± SE) were
s = 18.8 ± 1.8 (109
min
1),
d = 222 ± 30 (109), and
b = 5.2 ± 0.4 (109 min
1). The model also allows one to
quantify the
-cell times of response to increasing and decreasing
glucose stimulus, equal to 5.7 ± 2.2 (min) and 17.8 ± 2.0 (min), respectively. In conclusion, the graded glucose infusion
protocol, interpreted with a minimal model of C-peptide secretion and
kinetics, provides a quantitative assessment of pancreatic function in
an individual. Its application to various physiopathological states
should provide novel insights into the role of insulin secretion in the
development of glucose intolerance.
insulin secretion;
-cell sensitivity; mathematical model; kinetics
 |
INTRODUCTION |
SEVERAL
PROTOCOLS are currently in use to define the alterations in
-cell responsivity to glucose associated with different physiopathological states, including the intravenous glucose tolerance test (IVGTT), the hyperglycemic clamp, the graded glucose infusion, and
the oscillatory glucose infusion. In view of the importance of
-cell
dysfunction in the physiopathology of type 2 diabetes, these tests play
an important role in our understanding of this condition. All these
tests are based on the assumption that the major defects in
-cell
function result in reduced or absent secretory response to glucose. On
the other hand, the inability to sense a fall in glucose and to
suppress insulin secretion appropriately should also be considered as a
possible defect in
-cell dysfunction.
An advantage of the graded glucose infusion protocol is its
ability to characterize the dose-response relationship between glucose
and secretion rate during a physiological perturbation, first by
reconstructing the insulin secretion rate (ISR) by deconvolution, and
then by plotting the average ISR against the corresponding average
glucose level during each glucose infusion period (4, 5,
7). The value of the graded glucose infusion as a measure of
-cell function could be greatly enhanced if it were possible to
obtain, in addition to ISR, quantitative indexes describing
-cell
sensitivity to glucose, similar to what is available for the IVGTT,
interpreted with a C-peptide minimal model (14, 15).
The aim of the present study was to investigate whether a detailed
characterization of
-cell function can also be obtained from a more
physiological slowly increasing/decreasing glucose infusion protocol
(up&down graded infusion) by using a model to interpret glucose and
C-peptide data.
 |
MATERIALS AND METHODS |
Selection and Definition of Study Subjects
Studies were performed in eight healthy nondiabetic subjects (7 females and 1 male). Mean age was 34 ± 3 (SE) yr, and body mass
index was 26.1 ± 1.7 kg/m2 . Glucose tolerance was
determined by World Health Organization criteria during an oral glucose
tolerance test (17). All subjects had a normal screening
blood count and chemistries and took no medications known to affect
glucose metabolism. All fasting plasma glucose levels were <98 mg/dl
(5.4 mM), and glycosylated hemoglobin values were normal. The study
protocol was approved by the Institutional Review Board at the
University of Chicago, and all subjects gave written informed consent.
Experimental Protocol
All studies were performed in the Clinical Research Center at
the University of Chicago, starting at 0800 in the morning after an
overnight fast. Intravenous cannulas were placed in a forearm vein for
blood withdrawal, and the forearm was warmed to arterialize the venous
sample. A second catheter was placed in the contralateral forearm for
administration of glucose.
Subjects received graded glucose infusions at progressively increasing
and then decreasing rates (0, 4, 8, 16, 8, 4, 0 mg · kg
1 · min
1). Each
glucose infusion rate was administered for a total of 40 min. Glucose
and C-peptide levels were measured at 10-min intervals during a 40-min
baseline period before the glucose infusion and throughout the 240-min
glucose infusion.
Assay
Plasma glucose was measured immediately by the glucose oxidase
technique (Yellow Springs Instrument analyzer, Yellow Springs, OH). The
coefficient of variation of this method is <2%. Plasma C-peptide was
measured as previously described (10). The lower limit of
sensitivity of the assay is 0.02 pmol/ml, and the average intra- and
interassay coefficients of variation are 6 and 8%, respectively.
Glycosylated hemoglobin was measured by boronate affinity
chromatography, with an intra-assay coefficient of variation of 4%
(Bio-Rad Laboratories, Hercules, CA).
Models of C-peptide Secretion and Kinetics
Because the secretion model is assessed from C-peptide
measurements taken in plasma, it must be integrated into a model of whole body C-peptide kinetics. The well validated model, originally proposed in Ref. 9, has been assumed (Fig.
1): compartment 1, accessible
to measurement, represents plasma and rapidly equilibrating tissues;
compartment 2 represents tissues in slow exchange with plasma. Model equations are
|
(1)
|
where the overdot indicates time derivative;
CP1 (pmol/l) is C-peptide concentration (above basal) in
compartment 1; CP2 (pmol/l) is the equivalent
concentration in compartment 2 (above basal), equal to the
C-peptide mass in compartment 2 divided by the volume of the
accessible compartment; k12 and
k21 (min
1) are transfer rate
parameters between compartments; k01
(min
1) is the irreversible loss; and SR
(pmol · l
1 · min
1) is the
pancreatic secretion (above basal) entering the accessible compartment,
normalized to the volume of distribution of compartment 1.
As for the IVGTT model (14), the functional relationship between insulin secretion and plasma glucose concentration is derived
from a previously proposed model (11, 12) based on the
packet storage hypothesis of insulin secretion. SR is described as the
sum of two components controlled, respectively, by glucose concentration (static glucose control) and by the rate of change of
glucose concentration (dynamic glucose control)
|
(2)
|
SRs is assumed to be equal to Y
(pmol · l
1 · min
1), the
provision of new insulin to the
-cells
|
(3)
|
which is controlled by glucose according to the following
equation
|
(4)
|
i.e., in response to an elevated glucose level, Y and thus
SRs tend with a time constant 1/
(min) toward a
steady-state value linearly related via parameter
(min
1) to glucose concentration G (mmol/l) above its
basal level Gb (static glucose control). Parameter
describes the static control of glucose on
-cells.

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Fig. 1.
Model of C-peptide kinetics. CP1 and
CP2 (pmol/l) are C-peptide concentrations in the accessible
and peripheral compartments, respectively; kij
(min 1) are kinetic parameters; SR
(pmol · l 1 · min 1) is the
pancreatic secretion normalized to the volume of distribution of
compartment 1, and y is the C-peptide concentration
measurement.
|
|
SRd is assumed to represent the secretion of insulin stored
in the
-cells in a promptly releasable form (labile insulin). Labile
insulin is not homogeneous with respect to the glucose stimulus: for a
given glucose step, only a fraction of labile insulin is mobilized, so
that more insulin can be rapidly released in response to a subsequent
more elevated glucose step. It is first assumed that the amount of
released insulin (dQ) in response to a glucose increase from G to G+dG
is proportional to the glucose increase dG
|
(5)
|
The flux of insulin secretion, SRd, is then
proportional to the derivative of glucose
|
(6)
|
Parameter kd describes the dynamic
control of glucose on insulin secretion, i.e., the effect of the rate
of change of glucose on insulin secretion when glucose concentration is
increasing (dG/dt positive).
As will be detailed in RESULTS, the model described so far,
hereafter indicated as model M1, is able to describe the
C-peptide data of most, but not all subjects. We therefore tested a
second model, called model M2, which differs from
M1 in that it incorporates a more flexible description of
the dynamic control (Fig. 2):
SRd is still proportional to the derivative of glucose, but
the proportionality factor is allowed to vary with glucose
concentration
|
(7)
|
According to Eq. 7, the dynamic control is maximum
when glucose increases just above its basal value; then 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, i.e., an additional
increase of glucose above Gt has no effect on insulin
secretion. M2 is a generalization of M1: in fact,
for elevated Gt, the term
approximates
1, and M2 reduces to M1.

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Fig. 2.
Parameter k(G) of the glucose dynamic control,
equal to the ratio between secretion rate of stored insulin and the
rate of change of glucose, for model M1 (A) and
model M2 (B).
|
|
Model Assessment of Insulin Secretion
Insulin secretion profile.
Models M1 and M2 allow one to reconstruct the
profile of insulin secretion ISR (pmol/min) during the up&down graded
infusion as
|
(8)
|
|
(9)
|
where SRb is insulin secretion in the basal state,
and V1 (in liters) is the C-peptide volume of distribution
in the accessible compartment.
Sensitivity indexes.
Three sensitivity indexes can be defined.
STATIC.
The static sensitivity to glucose
s (min
1)
measures the stimulatory effect of a glucose stimulus on
-cell
secretion at steady state. For both models
|
(10)
|
DYNAMIC.
The dynamic sensitivity to glucose measures the stimulatory effect of
the rate of change of glucose on secretion of stored insulin. To
calculate this index, it is useful to define first the parameter
X0 (pmol/l) as the amount of insulin (per unit of C-peptide
distribution volume) released in response to the maximum glucose
concentration Gmax achieved during the experiment
|
(11)
|
For model M1, X0 is simply
|
(12)
|
For model 2, two situations must be considered. If
Gt > Gmax, i.e., the dynamic control of
glucose on insulin secretion is active in the entire rising portion of
the curve, then
|
(13)
|
If Gt < Gmax, then the dynamic
glucose control is active as long as G < Gt, and
X0 becomes
|
(14)
|
By normalizing X0 to the glucose increase, the
dynamic sensitivity to glucose
d (dimensionless) can be
derived
|
(15)
|
BASAL.
The basal sensitivity index
b (min
1)
measures basal insulin secretion rate over basal glucose concentration
|
(16)
|
Response times.
The models also allow one to quantify the
-cell response times (min)
to a glucose stimulus. For both models, the
-cell response time to a
decreasing glucose stimulus (Tdown) is simply
|
(17)
|
because in this case, secretion equals provision Y, which is
described by Eq. 4, with 1/
as time constant. When
glucose increases, the additional amount X0 of insulin
secreted due to the dynamic control of glucose accelerates the
-cell
response. As detailed in the APPENDIX, this is equivalent
to reduction in the
-cell response time now indicated as
Tup
|
(18)
|
Model Identification
For both models M1 and M2, all parameters
are a priori uniquely identifiable (6, 8), i.e., kinetic
parameters k01, k21, k12 , and secretory parameters
,
,
kd for M1 or
,
,
kd, Gt for M2.
However, numerical identification of the models requires knowledge of
C-peptide kinetics. Kinetic parameters were fixed to standard values by
following the method proposed in Ref. 16. Their average
values (means ± SE) were k01 = 0.0600 ± 0.0006 min
1;
k21 = 0.0559 ± 0.0017 min
1; k12 = 0.0492 ± 0.0002 min
1; and V1 = 4.06 ± 0.06 liters. The secretory parameters of both models were then estimated for
each subject, together with a measure of their precision, by applying
weighted nonlinear least square methods (6, 8) to
C-peptide data by using the SAAMII software (3). Weights
were chosen optimally, i.e., equal to the inverse of the variance of
the measurement errors, which were assumed to be independent, gaussian,
and zero mean with a constant standard deviation, which has been
estimated a posteriori. Glucose concentration, linearly interpolated
between data, and its time derivative, calculated by means of a spline
function interpolation of glucose data, have been assumed as error-free
model inputs. The comparison between models was made on the basis of
criteria such as independence of residuals, precision of the estimates,
and the principle of parsimony as implemented by the Akaike Information
Criterion (AIC) (6, 8).
Statistical Analysis
Values are reported as means ± SE. The statistical
significance of differences has been calculated by the two-tailed
Student's t-test. The independence of residuals has been
assessed by use of the runs test (2). P < 0.05 was considered statistically significant.
 |
RESULTS |
Mean plasma glucose and C-peptide concentration values during the
up&down graded glucose infusion protocol are shown in Fig. 3.

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Fig. 3.
Mean plasma glucose and C-peptide concentration during
the up&down graded glucose infusion (n = 8).
|
|
Individual secretion parameters of models M1 and
M2 are summarized in Table 1,
together with their precision. The ability of model M1 to
fit the individual data is shown in Fig.
4. From Table 1, precise estimates are
obtained with M1 in all of the eight subjects. With
M2, precise estimates of all parameters are obtained only in
subjects 5, 7, and 8. In these subjects,
model M2 performs better than M1, as indicated by
a lower AIC value (Table 2). In
particular, it performs notably better than M1 in
subjects 5 and 8, for whom M1 produces
a systematic underestimation of the initial portion of the data (Fig.
4). In these subjects, residuals are independent with M2 but
not with M1 (Fig. 5). In subject 7, M2 performance slightly improves,
because residuals are independent for both models, but AIC is lower
with M2. However, M2 cannot be resolved in
subjects 1, 2, 3, 4, and 6, because
Gt estimates are very high and affected by poor precision
(Table 1) with no improvement in model fit, i.e., M2 tends
to reduce to M1. Therefore, insulin secretion has been
assessed by using M1 for subjects 1, 2, 3, 4, and
6 and M2 for subjects 5, 7, and 8; the mean profile of
-cell secretion (Eqs. 8 and 9) is shown in Fig. 6;
sensitivity indexes and response times are reported in Table
3.

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Fig. 5.
Weighted residuals of model M2 (continuous
line) against those of model M1 (dashed line) in
subjects 5 and 8.
|
|

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Fig. 6.
Mean -cell secretion during the up&down graded glucose
infusion predicted by use of models M1 (for subjects
1, 2, 3, 4, and 6) and M2 (subjects 5, 7, and 8). ISR, insulin secretion rate.
|
|
 |
DISCUSSION |
The C-peptide minimal modeling approach, which has been
successfully applied to IVGTT data (14, 15), has been used
here to assess
-cell secretion during a more physiological glucose perturbation, in which a rising followed by a falling glucose concentration is produced by an exogenous intravenous glucose infusion.
A novel version of the model is proposed, which incorporates the
assumption that glucose stimulates pancreatic insulin secretion by
exerting both a static control, i.e., proportional to its
concentration, and a dynamic control, i.e., proportional to its rate of
change. Similar assumptions are not new in modeling hormone secretory processes. In the present study, they have been used to interpret the
data mechanistically, because they have been derived by building on
specific assumptions about the physiology of insulin secretion, first
formulated in the classical packet storage insulin secretion model
(11, 12) and then incorporated in the minimal model of
insulin secretion and kinetics during IVGTT (14, 15). More specifically, the model assumes the presence in the
-cells of a pool
of promptly releasable insulin, which can be rapidly secreted when
glucose increases above its basal value, and an insulin provision process, which accounts for a slower component of secretion by allowing
the formation of new insulin from insulin precursors and/or conversion
of insulin from a storage to a labile form.
The Static Control of Glucose on Insulin Secretion
It is assumed that insulin provision under steady-state conditions
is proportional, through parameter
, to the glucose stimulus, with a
delay with respect to the glucose profile represented by 1/
.
Parameter
thus represents the sensitivity
s (static
sensitivity index) of
-cells to the glucose stimulus, because it
measures the relation between secretion rate (above basal) at steady
state and the glucose stimulus (above basal). Its value, 18.8 ± 1.8, can be compared with the sensitivity in the basal state,
b = 5.2 ± 0.4, because they are both
steady-state secretory indexes. Our results (
s
significantly higher than
b) indicate that a separate
assessment of
-cell function in the basal state and during a glucose
stimulus is important, because
-cells are more sensitive to a
suprabasal glucose stimulus than to the basal glucose level.
The Dynamic Control of Glucose on Insulin Secretion
The assumption of a static glucose control is not sufficient to
provide a reliable description of the C-peptide data when the glucose
infusion rate is first increased and then decreased; the model fit
obtained by coupling the model of C-peptide kinetics (Eq. 1)
with a secretion rate coming from provision only, i.e., SR(t) = SRs (Eqs. 3 and 4) produces a systematic underestimation, especially in the
rising portion of C-peptide data, as shown in Fig.
7. These findings suggest the existence
of an additional secretion term that is active when glucose increases
and represents the counterpart of the IVGTT first-phase secretion
observed immediately after the glucose bolus injection. However, the
increase in glucose concentrations from basal to maximum levels during
the up&down graded infusion protocol (120 min) is much slower than
during the IVGTT (2-3 min). The description adopted for the
up&down graded infusion was therefore different from that used for the
IVGTT, albeit based on similar assumptions, namely the packet storage hypothesis of insulin secretion (11, 12). According to
this hypothesis, a bulk of insulin is stored in the
-cells in a
promptly releasable form and is secreted, when glucose exceeds its
basal level, with a nonhomogeneous response: for a given increase in glucose concentration, only a portion of labile insulin is secreted, so
that subsequent more elevated glucose concentration steps are able to
stimulate the secretion of additional insulin. By assuming that the
amount of insulin secreted in a given period of time depends on the
glucose increase in that period, one finds that insulin secretion is
controlled by the glucose rate of change through a proportionality
constant k(G), which in principle depends upon G. Two
different descriptions have been tested for k(G), thus
leading to two different versions of the minimal model of C-peptide
secretion during the up&down glucose infusion, denoted as models
M1 and M2, respectively. In the former, it has been assumed simply that k(G) is constant, k(G) = kd, i.e., it does not depend on G. This means
that an increase
G in glucose concentration, from G1 to
G2 = G1+
G, promotes the secretion of an
amount of insulin proportional to
G but independent of the glucose
levels G1 and G2. Parameter
kd represents the sensitivity
d
(dynamic sensitivity index) of
-cells to the glucose rate of change.
The product of kd and the total increase in
glucose concentration in the rising portion of the data measures the
total amount X0 (pmol/l) of insulin stored in the
-cells
before the experiment and thus released during the experiment.

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Fig. 7.
Inadequacy of a model that simply assumes a static
glucose control on insulin secretion, shown as its mean fit against
mean C-peptide concentration.
|
|
Model M1 was able to accurately describe the C-peptide data
of all except two subjects, where it produced a systematic
underestimation of the initial portion of the data. A preliminary
analysis of data obtained from the up&down graded glucose infusion
protocol in physiopathological states, i.e., severe obesity and
impaired glucose tolerance (unpublished observations), confirmed the
inadequacy of M1 to reproduce C-peptide data of a portion of
subjects and suggested the use of a more flexible description of
k(G). Therefore, model 2 was introduced, with
k(G) linearly dependent on G, i.e., an increase
G in
glucose concentration promotes the secretion of an amount of insulin
dependent not only on
G but also on the glucose levels
G1 and G2. Model M2 assumes that the
sensitivity of the dynamic glucose control is maximal when G varies
(increases) around basal, and then decreases with higher G so as to
vanish at the threshold glucose level Gt able to promote
the secretion of the totality of stored insulin. k(G) is
then described by two parameters, the maximal sensitivity at basal
glucose, kd, and the threshold glucose
concentration Gt. M2 is a
generalization of M1, because M2 reduces to
M1 when the threshold value Gt becomes very
large. This is confirmed by our results: M2 significantly improves upon M1 in those subjects for whom M1
was not adequate and reduces to M1 in the other subjects
(Fig. 2). As with M1, the
-cell dynamic sensitivity index
d and the total amount X0 of stored insulin
can be measured from M2 parameters.
Minimal Model Indexes vs. Quasi-Steady-State Analysis
In the literature, the low-dose (glucose doses = 2, 3, 4, 6, and 8 mg · kg
1 · min
1)
graded glucose infusion experiments were used to explore the relationship between glucose stimulus and insulin secretion response in
various physiopathological states (4, 5, 7). In those studies, the pancreatic secretion profile (ISR) was reconstructed by
deconvolution from plasma C-peptide data by assuming the
two-compartment model of C-peptide kinetics (Fig. 1), with parameters
either derived (4) from a bolus intravenous C-peptide
injection performed in the same subjects or fixed (5, 7)
to standard values that follow the method proposed in Ref.
16. During each glucose infusion period, average ISR was
calculated and plotted against the corresponding average glucose level
to describe the dose-response relation between the two variables. These
studies demonstrated a linear relationship across glucose
concentrations spanning the glucose physiological range, i.e., up to
10-12 mmol/l in normal subjects and 18-20 mmol/l in
non-insulin-dependent diabetes mellitus patients. This is confirmed by
our data, because the relationship between average ISR derived by
deconvolution and the corresponding average glucose concentration (Fig.
8) is approximately linear during
increasing glucose steps. During decreasing glucose steps, the
relationship shows an hysteresis, i.e., ISR appears to be higher than
with increasing glucose steps. However, it is worth noting that the use
of a quasi-steady-state method of data analysis to interpret a
non-steady-state situation, like the one between plasma glucose and
C-peptide concentration during the graded glucose infusion, is not
entirely accurate, and particularly so with the protocol adopted in
this study, because average glucose concentration and average ISR
calculated during each step underestimate the steady-state values
during the increasing steps and overestimate them during the decreasing
steps.

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Fig. 8.
Relationship between average deconvolution-derived
insulin secretion rate (ISR) and average glucose concentration during
the up&down graded infusion experiment. The model-predicted
relationship is shown by the dashed line.
|
|
The minimal model approach overcomes these problems because model
equations describe the non-steady-state relationships between glucose
concentration and ISR during the graded infusion protocol. The model
can also be used as a simulation tool to predict the steady-state
relationship between glucose concentration and ISR, as if an ideal
up&down graded infusion experiment were performed in which each glucose
infusion step lasts until glucose and then ISR reach their steady-state
levels. By denoting steady state with the subscript ss, the
model-derived relationship, also shown in Fig. 8, is
|
(19)
|
From Eq. 19 it is evident that the minimal model
assumes a linear steady-state relationship between glucose stimulus and
ISR but provides reliable estimates of its parameters from
non-steady-state data, such as those measured during an up&down graded
glucose infusion experiment: index
s =
, when
multiplied by V1, is the slope of this relation, and
(SRb
Gb)V1 is the intercept.
The minimal model also allows one to estimate the
-cell response
times Tdown and Tup during a decreasing and an
increasing glucose step. The former coincides with the time constant of
insulin provision, whereas the second is an equivalent parameter that also takes into account the ability of the dynamic glucose control to
accelerate the rate with which
-cells respond to an increasing glucose stimulus. In normal subjects, the
-cell response time Tup during an increasing glucose step is 5.7 ± 2.2 (min), lower than the
-cell response time during a decreasing
glucose step, Tdown = 17.8 ± 2.0 (min), because
of the dynamic control of glucose on the secretion of stored insulin.
Up&Down Graded Infusion vs. IVGTT
Pancreatic indexes
s and
d estimated
with the up&down graded glucose infusion (Table 3) can be compared with
their IVGTT counterparts, the second-phase sensitivity
2
and the first-phase sensitivity
1, obtained in normal
subjects:
2 = 11.3 ± 1.1, 10.5 ± 0.6, 10.9 ± 1.4 from, respectively, standard IVGTT at 500 mg/kg dose
(14), standard IVGTT at 300 mg/kg dose (1, 13, 18), and insulin-modified IVGTT at 300 mg/kg dose
(15);
1 = 92 ± 15, 156 ± 18, 191 ± 29 in the same three groups. Both
s and
d are significantly higher than the IVGTT indexes
2 and
1. However, both the profile and
the range of glucose, and thus of C-peptide concentrations, are
markedly different and higher on average in the up&down graded infusion
experiment compared with IVGTT, thus indicating an effect of the
glucose perturbation pattern and/or glucose range on static and dynamic
glucose control. In particular, these results suggest that
-cells
are more sensitive to a slow glucose increase, as observed during the
graded glucose infusion protocol, than to the brisk rise in glucose
concentration observed after an IVGTT.
Conversely, the
-cell response time to a decreasing glucose
stimulus, estimated from the up&down graded glucose infusion, varies in
a range (11-28 min) similar to the one observed with the standard IVGTT.
In conclusion, the dynamic insulin secretory responses to increasing
and decreasing glucose concentrations can be modeled using
modifications of the minimal model approach. The new models allow the
characterization of both basal and dynamic insulin secretory responses
as well as parameters of
-cell sensitivity. The application of this
model to various physiopathological states associated with alterations
in insulin secretion and/or action should provide novel insights into
the role of these processes in the development of glucose intolerance.
 |
APPENDIX |
The purpose here is to define the
-cell response time by
considering both secretion components: secretion from provision, controlled by glucose (static control), and secretion of stored insulin, controlled by the glucose rate of change (dynamic control).
For insulin provision Y (Eq. 4), the
-cell response time
is simply 1/
, which represents the time at which Y approximates its
steady-state level [Yss =
(Gmax
Gb)] by 1/e = 63%, in response to a glucose step
increase from basal (G = Gb) to an elevated level
(G = Gmax). Under these experimental conditions, the
-cell response time causes a reduction in the amount of secreted
insulin, which can be evaluated by integrating Eq. 4 from
time 0 to a time t1, at which Y well
approximates its steady-state level
|
(A1)
|
In Eq. A1,
Ysst1 represents the amount of
insulin that would be secreted (above basal) in the
0-t1 interval if the response were immediate,
and Yss/
is the reduction of this amount due to the
-cell response time.
A relation similar to Eq. A1 also holds for the up&down
protocol, where glucose and Y increase from basal
[G(0) = Gb, Y(0) = 0]
to elevated levels [G(t1) = Gmax, Y(t1) = Ymax] with time-varying patterns, because by integrating
Eq. 4 one has
|
(A2)
|
where the first term of the right hand side still represents the
amount of insulin that would be secreted (above basal) in the
0-t1 interval from provision Y if the response
were immediate. As before, the
-cell response time 1/
determines
a reduction in the total amount of secreted insulin that is
proportional to this time and to the maximum value of provision Y.
The dynamic control of insulin secretion by glucose causes the
additional secretion of an amount X0 of stored insulin.
Therefore, the total amount of secreted insulin is
|
(A3)
|
By comparing Eq. A3 with Eq. A2, the
additional insulin secreted due to the dynamic control of glucose
causes a reduction in the delay between the glucose stimulus and the
insulin response equivalent to a reduction of
-cell response time
from 1/
to 1/
X0/Ymax.
In conclusion, the
-cell response time Tdown during a
decreasing glucose stimulus is simply
|
(A5)
|
because only the static control is active. During an increasing
glucose stimulus, when both the static and the dynamic controls are
active, the
-cell response time Tup becomes
|
(A6)
|
Tup can be expressed as a function of sensitivity
indexes if the system approximates steady-state conditions at
time t1, so that
Y(t1) = Ymax
(Gmax
Gb). When this approximation
is used for Ymax, and Eq. 15 is used for
X0, Eq. A6 becomes
|
(A7)
|
With our data, the use of Eq. A7 instead of
A6 results in a modest overestimation of Tup,
<10% as an average.
 |
ACKNOWLEDGEMENTS |
This work was partially supported by National Institute of Diabetes
and Digestive and Kidney Diseases Grants DK-31842, DK-20595, and
DK-02742, and by the Blum Kovler Foundation.
 |
FOOTNOTES |
Address for reprint requests and other correspondence: C. Cobelli, Dipartimento di Elettronica e Informatica, Via Gradenigo 6a,
35131 Padova, Italy (E-mail: cobelli{at}dei.unipd.it).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 24 February 2000; accepted in final form 24 August 2000.
 |
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