Oral Glucose Tolerance Test Minimal Model Indexes of ß-Cell Function and Insulin Sensitivity
Elena Breda,
Melissa K. Cavaghan,
Gianna Toffolo,
Kenneth S. Polonsky, and
Claudio Cobelli
From the Department of Electronics and Informatics (E.B., G.T., C.C.),
University of Padova, Padova, Italy; the Department of Medicine (M.K.C.), The
University of Chicago, Chicago, Illinois; and the Department of Medicine
(K.S.P.), Washington University School of Medicine, St. Louis, Missouri.
Address correspondence and reprint requests to Claudio Cobelli, Dipartimento
di Elettronica e Informatica, Via Gradenigo 6a, 35131 Padova, Italy. E-mail:
cobelli{at}dei.unipd.it
.
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ABSTRACT
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The simultaneous assessment of quantitative indexes of insulin secretion
and action in a single individual is important when quantifying their relative
role in the evolution of glucose tolerance in different physiopathological
states. Available methods quantify these indexes in relatively
nonphysiological conditions, e.g., during glucose clamps or intravenous
glucose tolerance tests. Here, we present a method based on a physiological
test applicable to large-scale genetic and epidemiologic studiesthe
oral glucose tolerance test (OGTT). Plasma C-peptide, insulin, and glucose
data from a frequently sampled OGTT with 22 samples throughout 300 min
(FSOGTT300-22) were analyzed in 11 subjects with various degrees of
glucose tolerance. In each individual, two indexes of pancreatic sensitivity
to glucose (
s [109 min-1] and
d [109]) and the insulin sensitivity index
(SI) (105 dl/kg per min per pmol/l) were
estimated by using the minimal model of C-peptide secretion and kinetics
originally proposed for intravenous graded glucose infusion and the minimal
model approach recently proposed for meal/OGTTs. The indexes obtained from
FSOGTT300-22 were used as a reference for internal validation of
OGTT protocols with reduced sampling schedules. Our results show that 11
samples in a 300-min period (OGTT300-11) is the test of choice
because the indexes it provides (
s = 36 ± 3 [means
± SE];
d = 710 ± 111; SI =
10.2 ± 2.4) show excellent correlation and are not statistically
different from those of FSOGTT300-22 (
s = 33
± 3;
d = 715 ± 120; SI =
10.1 ± 2.3). In conclusion, OGTT300-11, interpreted with
C-peptide and glucose minimal models, provides a quantitative description of
ß-cell function and insulin sensitivity in a single individual while
preserving the important clinical classification of glucose tolerance provided
by the standard 120-min OGTT.
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INTRODUCTION
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The simultaneous assessment of ß-cell function and insulin sensitivity
in a single individual is of primary importance when quantifying their
relative role in the evolution of glucose tolerance in different
physiopathological states. Two methods are available for this purpose: the
clamp technique, which uses a euglycemic-hyperinsulinemic and a
hyperglycemic-euinsulinemic clamp in the same individual
(1), and the intravenous
glucose tolerance test (IVGTT) interpreted by the minimal models of glucose
disposal (2) and C-peptide
kinetics and secretion (3).
Both these approaches give accurate and precise estimates of insulin
sensitivity and ß-cell function in a single individual, but plasma
glucose, C-peptide, and insulin concentrations achieved during these studies
are relatively nonphysiological. Recently, the need to quantify ß-cell
function and insulin sensitivity under more normal life conditions has
encouraged many investigators to use more physiological protocols, including
meal-like studies (4), graded
up and down glucose infusions
(5), meals
(6,7),
and oral glucose tolerance tests (OGTTs)
(8,9).
However, an approach to measure indexes of ß-cell function and insulin
action in a single individual based on a physiological test such as the OGTT
is not available. The ability to derive in a single individual important
information such as the clinical classification of oral glucose tolerance
while simultaneously quantifying his or her ß-cell function and insulin
sensitivity could provide a unique tool potentially applicable to large-scale
genetic and epidemiologic studies.
Therefore, the aim of the present study was to investigate whether indexes
of ß-cell function and insulin sensitivity could be simultaneously
assessed in a single individual from OGTT data by extending to the OGTT the up
and down C-peptide minimal model
(5) and the insulin sensitivity
formula recently derived for a meal
(7). The database consisted of
a frequently sampled 300-min OGTT performed on 11 subjects with various
degrees of glucose tolerance. Indexes of insulin sensitivity and ß-cell
function based on OGTTs with reduced number of samples were also calculated
and compared with those derived from the frequently sampled OGTT to arrive at
a robust clinical protocol.
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RESEARCH DESIGN AND METHODS
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Subjects. Studies were performed on 11 nondiabetic subjects (4 men
and 7 women); 7 had normal glucose tolerance (NGT), and 4 had impaired glucose
tolerance (IGT). Mean age was 37 ± 3 years (means ± SE) (range
20-50), and BMI was 30.5 ± 2.1 kg/m2 (range 21.3-46.2)
(Table 1). Glucose tolerance was
determined by using the American Diabetes Association Expert Committee
criteria (10). All subjects
had a normal screening blood count and chemistries and took no medications
known to affect insulin secretion or action. All protocols were approved by
the Institutional Review Board of The University of Chicago. Written informed
consent was obtained from each subject.
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 catheters were placed into antecubital veins. The
blood sampling arm was heated to obtain arterialized venous samples. At time
0, subjects ingested a 75-g glucose load. Blood samples were collected at -15,
0, 10, 20, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225,
240, 255, 270, 285, and 300 min for measurement of glucose, insulin, and
C-peptide concentrations. This complete protocol, consisting of 22 blood
samples taken in a 300-min period after glucose ingestion, will be referred to
as frequently sampled OGTT (FSOGTT300-22).
OGTT protocols with reduced data sets were also considered
(Fig. 1): 1) sampling
schedule similar to the one commonly adopted for a standard OGTT, with samples
at 0, 30, 60, 90, 120, 150, 180, 240, and 300 min (9 samples throughout 300
min) referred to as OGTT300-9; 2) same as 1) with
two additional samples at 10 and 20 min (OGTT300-11); 3)
same as 2) but without the 300-min sample, thus shortening the
duration of the test from 5 to 4 h (OGTT240-10).
Assay. Plasma glucose was measured immediately using a glucose
analyzer (YSI Model 2300 STAT; Yellow Springs Instruments, Yellow Springs,
OH). The coefficient of variation (CV) of this method was <2%. Serum
insulin was assayed by a double-antibody technique
(11) with a lower limit of
sensitivity of 20 pmol/l and an average intra-assay CV of 6%. The
cross-reactivity of proinsulin in the radioimmunoassay for insulin was
40%. Plasma C-peptide was measured as previously described
(12). The lower limit of
sensitivity of the assay was 0.02 nmol/l and the intra-assay CV averaged
6%.
ß-Cell function. The minimal model of C-peptide secretion and
kinetics originally applied to intravenous glucose graded infusion data
(5) has been applied to assess
ß-cell secretion during an oral glucose perturbation.
C-peptide kinetics are described by using the well-known two-compartment
model originally proposed by Eaton et al.
(13):
where the overdot indicates time derivative; CP1 and CP2
(nmol/l) are C-peptide concentrations above basal in the accessible and
peripheral compartments, respectively; kij
(min-1) are C-peptide kinetic parameters; and SR (pmol ·
l-1 · min-1) is the pancreatic secretion above
basal, entering the accessible compartment, normalized by the volume of
distribution of compartment 1.
Pancreatic secretion SR has been described as the sum of two components
controlled respectively by glucose concentration (static glucose control
[SRs]) and by its rate of increase (dynamic glucose
control [SRd]):
SRs is assumed to be equal to the provision of new
insulin to the ß-cells (Y)
(pmol·l-1·min-1):
which is controlled by glucose according to the following:
It is worth noting that SRs is not linearly related to
glucose concentration but tends, with a time constant 1/
(min), toward
a steady-state value linearly related through the parameter ß to glucose
concentration G above a threshold level h (mmol/l).
SRd, on the other hand, represents the secretion of insulin
stored in the ß-cells in a promptly releasable form (labile insulin) and
is proportional to the rate of increase of glucose:
where:
According to Eqs. 6 and 7, the dynamic control is at its maximum when
glucose increases just above its basal value Gb, then it
decreases linearly with glucose concentration and vanishes when glucose
concentration exceeds the threshold level Gt, which is
able to promote the secretion of all stored insulin. If parameter
Gt assumes an elevated value, k(G)
approximates the constant Kd.
Indexes. The model allows the estimation of two indexes of
ß-cell function: the static and the dynamic sensitivity to glucose. In
addition, a single global index of ß-cell sensitivity to glucose, which
suitably combines both the static and the dynamic control indexes, can be
calculated.
Static. The static sensitivity (
s
[109 min-1]) is a measure of the effect of glucose on
ß-cell secretion and is the ratio between SR and glucose concentrations
(above the threshold level h) at steady state:
Dynamic. The dynamic sensitivity (
d
[109]) is a measure of the stimulatory effect of the rate at which
glucose increases upon the secretion of stored insulin. It is defined 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, normalized by the glucose increase
Gmax Gb:
If parameter Gt assumes an elevated value,
d approximates the constant
Kd·Global. In addition to
s and
d, which give a detailed
portrait of ß-cell function, it is also useful to define and derive a
single global index of ß-cell sensitivity to glucose (
[109 min-1]), which suitably combines both the static
(
s) and the dynamic (
d) control
indexes. This is particularly advantageous in calculating the so-called
disposition index, i.e., ß-cell function x insulin sensitivity (see
below).
The global index of ß-cell sensitivity to glucose is defined as the
average increase above basal of pancreatic secretion (Eq. 3) over the average
glucose stimulus above the threshold level h:
(see APPENDIX) can be calculated from model parameter h,
model indexes
s and
d, and the area
under the curve of G above the threshold level h:
has been derived from the model, but it can also be calculated by a
virtually model-independent formula (see APPENDIX):
Insulin secretion rate. The model also provides the profile
of the insulin secretion rate (ISR [pmol/min]) during the OGTT:
where SRb (pmol·min-1·l-1) is
insulin secretion in the basal state, and V1 (liters) is
the distribution volume of the accessible compartment.
Insulin sensitivity. To calculate insulin sensitivity, we have
applied to the OGTT the formula for a meal recently proposed by Caumo et al.
(7). As detailed in this study,
insulin sensitivity index (SI [105 dl/kg per
min per pmol/l]) can be calculated with an area under the curve formula:
where G is plasma glucose concentration;
G and
I are glucose and insulin concentrations above basal,
respectively; AUC denotes the area under the curve calculated from time 0 to
t
; GE is glucose effectiveness (dl ·
kg-1 · min-1); DOGTT is the
dose of ingested glucose per unit of body weight (mg/kg); and f is
the fraction of ingested glucose that actually appears in the systemic
circulation. When glucose falls below basal, a slightly different formula
needs to be used (we refer to Eq. 7 in Caumo et al.
(7) for details). Calculation
of SI requires insertion of values for GE and f.
Here we used the values proposed by Caumo et al.: GE = 0.024 dl ·
kg-1 · min-1 and f = 0.8.
Disposition index. The so-called disposition index (DI), i.e.,
ß-cell function x insulin sensitivity
(14,15),
is a parsimonious and effective way to express ß-cell function in
relation to the degree of insulin resistance. To this end, it is convenient to
use the single global ß-cell function index
. The DI is thus defined
as follows:
Numerical identification. All C-peptide model parameters are a
priori uniquely identifiable. However, numerical identification of the model
requires the knowledge of C-peptide kinetics. Standard kinetic parameters were
calculated by using the method proposed by Van Cauter et al.
(16).
C-peptide model secretory parameters were estimated, together with a
measure of their precision, by nonlinear least squares
(17,18)
using SAAM II software (19).
When
was elevated and estimated with poor precision, the Bayesian
approach implemented in SAAM II was used. Measurement error of C-peptide
concentration has been assumed to be independent and gaussian, with zero mean
with a constant but unknown variance. Glucose concentration, linearly
interpolated between data, and its time derivative have been assumed as
error-free model inputs. Area under the curve in the global index and insulin
sensitivity formulas was calculated using the trapezoidal rule.
To evaluate the precision of SI, Monte Carlo methods
(20) have been applied to Eq.
14 (or, when appropriate, to Eq. 7 from Caumo et al.
[7]), taking into account both
measurement errors of glucose and insulin concentrations (assumed to be
independent and gaussian, with zero mean and a constant CV of 2 and 6%,
respectively) and population variability of f and glucose
effectiveness (assumed to be gaussian with a CV of 10 and 25%, respectively
[7]).
Statistical
analyses. Results are given as means ± SE. The statistical
significance of differences between the same parameters calculated from
different sampling schedules has been calculated using the Wilcoxon's
signed-rank test. Linear regression and Spearman rank correlation analyses
were used to examine the relationship between parameters. Significance was
declared at P < 0.05.
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RESULTS
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FSOGTT300-22. Mean plasma glucose, C-peptide, and insulin
concentrations during the FSOGTT300-22 are shown in
Fig. 2. The C-peptide minimal
model well describes experimental data, as shown by the weighted residual plot
(Fig. 3). Average ß-cell
sensitivity indexes were as follows:
d = 715 ± 120 and
s = 33 ± 3 (means ± SE). They were estimated
with good precision for all subjects with an average CV of 25 ± 4 and 9
± 1%, respectively. The ISR profile was also reconstructed and is shown
in Fig. 4. Average
SI was 10.1 ± 2.3, and its precision averaged 12
± 1%.

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FIG. 2. Average (mean ± SE, n = 11) concentration of plasma glucose,
C-peptide, and insulin obtained during the 75-g
FSOGTT300-22.
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OGTT300-9. The C-peptide minimal model is obviously able
to fit the reduced data set (Fig.
3), but
d values (1,114 ± 224, CV 44
± 9%) were statistically different from those estimated by using the
FSOGTT300-22 (Fig.
5) and were not correlated with them
(Fig. 6). The early portion of
the ISR profiles (not shown) was not superimposable to that calculated using
the FSOGTT300-22. These results indicate that OGTT300-9
does not accurately describe the early portion of the data, where the dynamic
control of glucose is active. The values of
s (35 ± 4,
CV 15 ± 4%) and SI (10.1 ± 2.4, CV 13
± 1%), on the other hand, did not significantly change (Figs.
5 and
6).
OGTT300-11. All the indexes, including
d,
were not different from and were well correlated with those obtained from the
FSOGTT300-22 (
d = 710 ± 111, CV 39 ±
5%;
s = 36 ± 3, CV 13 ± 3%;
SI = 10.2 ± 2.4, CV 13 ± 1%) (Figs.
5 and
6). Individual values of
d,
s, and SI are
summarized in Table 2, together
with their precision. The average ISR profiles estimated using the two
protocols were also very similar (Fig.
4), thus indicating that the more accurate description of the
early portion of the experiment by OGTT300-11 with respect to
OGTT300-9 is essential to obtain results similar to those of the
FSOGTT300-22. The weighted residual plot of the C-peptide minimal
model is shown in Fig. 3.
OGTT240-10. ß-Cell sensitivity indexes
d (710 ± 126, CV 40 ± 7%),
s (35
± 3, CV 12 ± 1%), and SI (8.9 ± 2.1,
CV 15 ± 1%) were well correlated with those obtained during the
FSOGTT300-22 (Fig.
6). However,
d and
s did not
significantly change, whereas SI was statistically
different from values calculated during the FSOGTT300-22
(Fig. 5), thus indicating the
importance of the 300-min sample for an accurate estimation of insulin
sensitivity. The weighted residual plot of the C-peptide minimal model is
shown in Fig. 3.
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DISCUSSION
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In this study, minimal model indexes of ß-cell function and insulin
sensitivity have been proposed and successfully assessed during an 11-sample
300-min OGTT in individuals with various degrees of glucose tolerance. While
maintaining the possibility of estimating indexes in the single individual,
the OGTT300-11 offers a number of additional advantages with
respect to available approaches based on the clamp or IVGTT techniques: the
oral perturbation reproduces a physiological condition, and the test is easy
to perform, with potential application to large scale genetic and
epidemiologic studies. These features make it an appealing tool because it may
exhibit a larger statistical power than population-oriented indexes such as
those proposed by Matthews et al.
(21) and Matsuda and DeFronzo
(22).
Minimal models, through a parsimonious description of glucose/C-peptide and
glucose/insulin relationships, provide reliable indexes of ß-cell
function and insulin sensitivity. More precisely, ß-cell sensitivity
indexes have been calculated by extending to the OGTT data the C-peptide
minimal model recently developed for intravenous glucose graded up and down
infusion (5). The model
assumptions are particularly favorable to an OGTT protocol, i.e., a situation
where C-peptide and glucose concentrations show slow dynamics compared with
the IVGTT, where excursions are rapid and large. The model has already been
successfully used to assess ß-cell function during physiological tests in
normal subjects (5) and in
individuals with various degrees of glucose tolerance
(23). It incorporates the
assumption that glucose stimulates pancreatic insulin secretion by exerting
both a static (dependent on glucose concentration) and a dynamic (dependent on
glucose rate of change) control. The model provides the insulin secretion
profile and indexes of ß-cell function: the static
s, the
dynamic
d, and the global
sensitivity to glucose.
SI has been calculated by using a formula recently
proposed for a meal glucose tolerance test
(7). The approach associates a
parsimonious parametric representation of splanchnic glucose absorption with
the minimal model description of glucose disposal. This method has been
validated by comparing its SI estimates to those provided
by an insulin-modified IVGTT performed on the same group of 10 normal
subjects. The significant correlation between the two SI
indexes (
s = 0.89, P < 0.01) indicates that a
reliable measure of insulin sensitivity can be derived from either an oral
test or an IVGTT.
To derive reliable estimates of ß-cell function and insulin
sensitivity during an OGTT, a rich database was initially used, with 22
samples taken in a 300-min OGTT (FSOGTT300-22). Then, because our
objective was to propose a protocol sufficiently simple but still robust, we
examined reduced sampling schedule OGTTs by using the indexes obtained from
the FSOGTT300-22 as a reference for internal validation. The
results indicate that 11 samples in a 300-min OGTT (OGTT300-11) are
sufficient to obtain results similar to those from the
FSOGTT300-22, since the two sets of indexes show high correlation
(Fig. 6) and are not
statistically different (Fig.
5). Also, the ISR profiles are virtually superimposable
(Fig. 4). A 300-min experiment
is necessary to obtain such results because the values estimated for
SI during a shorter experiment (OGTT240-10) are
slightly lower than those estimated during a 300-min experiment
(FSOGTT300-22, OGTT300-9, and OGTT300-11).
The accurate description of the early portion of the C-peptide and glucose
curves, provided by the 10, 20, 30 min samples after the oral glucose
ingestion, is also necessary to obtain reliable estimates of the index
d.
OGTT minimal model indexes: NGT versus IGT. The standard 120-min
OGTT, with blood samples usually drawn at 0, 30, 60, 90, and 120 min, provides
the important clinical classification of glucose tolerance
(10). The
OGTT300-11 we propose here includes the standard samples, thus
still allowing one to perform the standard classification of glucose tolerance
but also enabling one to estimate indexes of ß-cell function and insulin
sensitivity, which help to better characterize glucose tolerance in a single
individual. From our data, when OGTT300-11 indexes in NGT subjects
(n = 7) were compared with those obtained in IGT subjects (n
= 4), no difference was found in the pancreatic sensitivity indexes (NGT:
d = 728 ± 124,
s = 37 ± 4,
= 48 ± 6; IGT:
d = 678 ± 241,
s
= 34 ± 6,
= 42 ± 9), but a statistically significant
difference was found in both the SI (NGT 14.0 ±
2.7, IGT 3.4 ± 1.0) and the DI (NGT 678 ± 178, IGT 160 ±
80).
These results confirm that subjects with IGT are characterized by an
inadequate insulin secretory response for the degree of insulin resistance
(24) or, in other words, that
IGT is characterized by a relative, rather than absolute, insulin
deficiency.
OGTT versus intravenous glucose infusions. Pancreatic indexes
s and
d estimated from OGTT300-11
were also compared with the same indexes estimated during either IVGTT or
graded up and down glucose infusions. The IVGTT counterparts of
d and
s are respectively the first- and
second-phase sensitivity indexes
1 and
2. The
values obtained during an insulin-modified IVGTT (300 mg/kg) in normal
subjects (n = 15) were
2 = 10.9 ± 1.4 and
1 = 191 ± 29
(25); ß-cell indexes
obtained during graded up and down glucose infusions (0, 4, 8, 16, 8, 4, and 0
mg·kg-1·min-1) in nondiabetic subjects
(n = 8) were
s = 18.8 ± 1.8 and
d = 222 ± 30
(5). Both
s and
d are threefold higher during OGTT than during intravenous
tests. This can probably be ascribed to the presence of the well-known
insulin-stimulating gastrointestinal hormones known as the incretin effect,
which are secreted in response to oral but not intravenous glucose
administration (26). It will
be of interest in future studies to compare ß-cell indexes and
SI obtained from OGTTs with values obtained in the same
subjects during intravenous glucose infusionspossibly during a graded
up and down glucose infusion, which better simulates an OGTT. Preliminary
results are available on only six of the eleven subjects studied here (four
with NGT and two with IGT) who underwent both an OGTT and a graded up and down
glucose infusion. The results confirmed our expectations. Indexes of
ß-cell function (
d and
s) were markedly
higher during the oral perturbation (OGTT:
d = 677 ±
154,
s = 39 ± 5; up and down:
d = 105
± 43,
s = 17 ± 3, P < 0.05). The
OGTT SI was higher than the graded up and down
SI (OGTT SI = 13.1 ± 3.6; up
and down SI = 8.8 ± 2.3, P < 0.05) and
showed a high correlation with it (
s = 0.83)a trend
already observed during meal tolerance tests
(7).
Importance of the dynamic control of glucose on insulin secretion.
Indexes of ß-cell function have been recently proposed in the literature
based on modeling analyses of glucose and C-peptide data during a meal
(6) and a 120-min OGTT
(9). Both models assume a
control of glucose, but not of its rate of change, on insulin secretion. This
is a gross simplification because the importance of dynamic control of glucose
on insulin secretion (active when glucose concentration increases) has been
shown both in previous studies where graded up and down glucose infusion data
were analyzed (5) and in the
present study. If a model similar to the OGTT minimal model but not accounting
for the dynamic glucose control is used to analyze the data, systematic
deviations occur in the early portion of the OGTT
(Fig. 7).

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FIG. 7. Inadequacy of a C-peptide model that simply assumes a static glucose
control on insulin secretion. Mean weighted residuals show a systematic
deviation in the first 60 min.
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The approach proposed by Cretti et al.
(9) is simplistic but
particularly appealing because it only requires five samples throughout 120
min. To compare our approach with that proposed by Cretti et al., we applied
the latter to our data by adopting the sampling schedule proposed in
(9), i.e., 0, 30, 60, 90, and
120 min. Model fit and residual plots showed a systematic underestimation of
the data between 15 and 75 min, thus indicating that the model proposed by
Cretti et al. is too simplistic to describe OGTT data. The apparent glucose
threshold
(9) assumed
values (often
0) far from basal glucose, and ß-cell sensitivity
(9) was different when
compared with the minimal model counterpart (
s). This may
reflect the fact that
incorporates both the static and the dynamic
glucose controls, whereas
s describes the static glucose
control only but is more likely the consequence of numerical compensations due
to differences in the estimates of threshold
in the Cretti et al.
model (often
0) and h in our model (always
Gb). In conclusion, care must be exercised in
adopting simplistic but appealing methods because structural errors can lead
to compensations among parameters and consequently to inaccurate ß-cell
portraits. We think that the protocol and the methods proposed here, which
enable one to obtain a precise description of both ß-cell function and
insulin action in a single individual, are a good compromise between
model/protocol simplicity and accuracy.
In conclusion, by extending to the OGTT the recently developed C-peptide
minimal model during intravenous glucose graded up and down infusion
(5) and by applying to the OGTT
the minimal model formula recently proposed for a meal glucose tolerance test
(7), we have shown that it is
possible to simultaneously assess individual parameters of ß-cell
function and insulin sensitivity from an 11-sample 300-min OGTT. Of note is
that OGTT300-11 preserves the important clinical classification of
glucose tolerance provided by a standard 120-min OGTT. The detailed
description of ß-cell function and insulin action thus available in a
single individual, together with the ease of execution of the protocol, should
make this approach a powerful tool for measuring changes in insulin secretion
and action that would also be applicable to large-scale genetic and
epidemiologic studies. The present study is the first attempt to
simultaneously assess insulin sensitivity and ß-cell function in the
single individual during an OGTT and shows encouraging results in subjects
with various degrees of glucose tolerance. However, this study is definitely
unfinished and further work needs to be performed to define the domain of
validity of this approach throughout the whole range of glucose tolerance,
including patients with diabetes.
 |
APPENDIX
|
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The purpose of this section is to derive the global index of ß-cell
sensitivity to glucose,
.
Model dependent formula. Pancreatic secretion SR is the sum of a
static (SRs) and a dynamic (SRd) component (Eq. 3).
SRs is described in Eqs. 4 and 5. The integral from time 0 to
of SRs can thus be calculated by integrating Eq. 5:
Because Y(
) = Y(0), i.e., the system returns for
t
to the basal steady state, the expression for
and thus for
becomes the following:
SRd is described in Eqs. 6 and 7. The integral from time 0 to
of SRd can thus be calculated by integrating Eq. 6:
then
By using the definition of
d (Eq. 9), the integral from
time 0 to
of SRd can be expressed as follows:
It is thus possible to calculate the global index of ß-cell
sensitivity to glucose as follows:
Model independent formula. By integrating Eqs. 1 and 2 from 0 to
:
Because CP1(
) = CP1(0) and
CP2(
) = CP2(0), i.e., the system returns for
t
to the basal steady state, the following holds:
By substituting Eq. A10 in Eq. A9, Eq. A9 becomes:
Finally, by substituting Eq. A11 in Eq. 10,
is given by:
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ACKNOWLEDGMENTS
|
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These studies were supported in part by U.S. National Institutes of Health
Grants DK-02742, DK-31842, and DK-20595 and General Clinical Research Center
Grant M01 RR00055.
The authors are indebted to Jacqueline Imperial, RN, at the Clinical
Research Center at the University of Chicago for her expert care of the
subjects who participated in the study. They also wish to thank Dr. Andrea
Caumo for his constructive and helpful advice in the development of the
manuscript.
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FOOTNOTES
|
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CV, coefficient of variation; DI, disposition index;
FSOGTT300-22, frequently sampled OGTT with 22 samples throughout
300 min; IGT, impaired glucose tolerance; ISR, insulin secretion rate; IVGTT,
intravenous glucose tolerance test; NGT, normal glucose tolerance; OGTT, oral
glucose tolerance test; SI, insulin sensitivity index.
Received for publication June 12, 2000
and accepted in revised form September 8, 2000
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