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-cell function during
graded up&down glucose infusion from C-peptide minimal
models
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
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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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 · min1) 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
min1), d = 222 ± 30 (109), and b = 5.2 ± 0.4 (109 min1). 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
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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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.
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MATERIALS AND METHODS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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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 · kg1 · min1). 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
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(1) |
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1) are transfer rate
parameters between compartments; k01
(min1) is the irreversible loss; and SR
(pmol · l1 · min1) 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)
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(2) |
1 · min1), the
provision of new insulin to the -cells
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(3) |
![<A><AC>Y</AC><AC>˙</AC></A>(<IT>t</IT>)<IT>=</IT>−<IT>&agr;</IT>{Y(<IT>t</IT>)<IT>−&bgr;</IT>[G(<IT>t</IT>)<IT>−</IT>G<SUB>b</SUB>]} Y(<IT>0</IT>)<IT>=0</IT>](/content/vol280/issue1/fulltext/E2/img006.gif)
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(4) |
(min) toward a
steady-state value linearly related via parameter (min1) 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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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
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(5) |
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(6) |
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
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(7) |
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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
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(8) |
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(9) |
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Sensitivity indexes. Three sensitivity indexes can be defined.
STATIC. The static sensitivity to glucoses (min1)
measures the stimulatory effect of a glucose stimulus on -cell
secretion at steady state. For both models
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(10) |
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(11) |
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(12) |
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(13) |
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(14) |
d (dimensionless) can be
derived
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(15) |
b (min1)
measures basal insulin secretion rate over basal glucose concentration
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(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
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(17) |
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
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(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 min1;
k21 = 0.0559 ± 0.0017 min1; k12 = 0.0492 ± 0.0002 min1; 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 |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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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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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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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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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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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 · kg1 · min1)
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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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
![ISR<SUB>ss</SUB><IT>=</IT>{SR<SUB>b</SUB><IT>+&bgr;</IT>[G<SUB>ss</SUB><IT>−</IT>G<SUB>b</SUB>]}V<SUB><IT>1</IT></SUB>](/content/vol280/issue1/fulltext/E2/img026.gif)
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(19) |
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 indexess 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.
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APPENDIX |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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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) |
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) |
-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
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(A3) |
-cell response time
from 1/ to 1/ X0/Ymax.
In conclusion, the -cell response time Tdown during a
decreasing glucose stimulus is simply
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(A5) |
-cell response time Tup becomes
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(A6) |
(Gmax Gb). When this approximation
is used for Ymax, and Eq. 15 is used for
X0, Eq. A6 becomes
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(A7) |
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ACKNOWLEDGEMENTS |
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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.
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FOOTNOTES |
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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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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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