Glucose production during an IVGTT by deconvolution:
validation with the tracer-to-tracee clamp technique
Paolo
Vicini1,
Jeffrey J.
Zachwieja2,3,
Kevin E.
Yarasheski2,
Dennis M.
Bier4,
Andrea
Caumo5, and
Claudio
Cobelli1
1 Department of Electronics and
Informatics, University of Padova, 35131 Padova;
5 Scientific Institute San
Raffaele, 20132 Milano, Italy;
2 Metabolism Division, Washington
University School of Medicine, Saint Louis, Missouri 63110;
3 Pennington Biomedical Research
Center, Baton Rouge, Louisiana 70808; and
4 Children's Nutrition Research
Center, Baylor College of Medicine, Houston, Texas 77030-2600
 |
ABSTRACT |
Recently, a new
method, based on a two-compartment minimal model and deconvolution
[A. Caumo and C. Cobelli. Am. J. Physiol 264 (Endocrinol.
Metab.. 37): E829-E841, 1993; P. Vicini, G. Sparacino, A. Caumo, and C. Cobelli. Comput. Meth.
Prog. Biomed. 52: 147-156, 1997], has been
proposed to estimate endogenous glucose production (EGP) from labeled
intravenous glucose tolerance test (IVGTT) data. Our aim here is to
compare this EGP profile with that independently obtained with the
reference method, based on the tracer-to-tracee ratio (TTR) clamp. An
insulin-modified (0.03 U/kg body wt infused over 5 min)
[6,6-2H2]glucose-labeled
IVGTT (0.33 g/kg of glucose) was performed in 10 normal subjects. A
second tracer
([U-13C]glucose) was
also infused during the test in a variable fashion to clamp endogenous
glucose TTR. The TTR clamp was quite successful. As a result, the EGP
profile, reconstructed from
[U-13C]glucose data
with the models of Steele and Radziuk, were almost superimposable. The
deconvolution-obtained EGP profile, calculated from
[6,6-2H2]glucose
data, showed remarkable agreement with that obtained from the TTR
clamp. Some differences between the two profiles were noted in the
estimated basal EGP and in the initial modalities of EGP inhibition. A
high interindividual variability was also observed with both methods in
the resumption of EGP to baseline; variability was high in both the
timing and the extent of resumption. In conclusion, the use of the
two-compartment minimal model of the IVGTT and deconvolution allows the
estimation of a profile of EGP that is in very good agreement with that
independently obtained with a TTR clamp.
nonsteady state; mathematical model; tracer-to-tracee ratio; specific activity clamp
 |
INTRODUCTION |
THE LABELED INTRAVENOUS GLUCOSE TOLERANCE TEST (IVGTT)
is a powerful tool to estimate parameters describing glucose metabolism in vivo, in normal and disease states. In particular, it has recently been shown that, if a two-compartment minimal model (2CMM) of labeled
glucose kinetics is used to describe the impulse response of the
glucose system, it is possible to obtain, via deconvolution, a
physiologically plausible profile of endogenous glucose production (EGP) in normal humans during the IVGTT nonsteady state (4, 24), thus
eliminating the anomalies previously encountered when a
single-compartment minimal model was used (5). This approach was also
recently used with success in normal subjects by Overkamp et al. (19).
The ability of reliably measuring EGP in nonsteady state is crucial to
investigate the modalities of glucose metabolism in normal and disease
states, and in particular to elucidate to what extent abnormal
regulation of EGP contributes to the fasting hyperglycemia and
carbohydrate intolerance associated with non-insulin-dependent diabetes
mellitus, obesity, and advancing age.
Validation of this deconvoluted profile of EGP by means of direct
measurement, i.e., the arteriovenous (a-v) balance technique, is
extremely difficult because the a-v technique is easily applicable only
in steady state (26), where the Fick principle (13) is known to hold.
In all generality, assessment of EGP in nonsteady state from tracer
data must rely on a model of glucose kinetics. However, modeling
glucose kinetics in nonsteady state is difficult. General-purpose
simplistic models are in use (20, 22); however, as shown by
non-steady-state theory (6, 17), they introduce errors. However, theory
(6, 17, 18) also shows that, if the glucose specific activity [or
the tracer-to-tracee ratio (TTR) for a stable isotope tracer (9)]
is kept constant (or, more realistically, its variations are
minimized), one can obtain a reliable estimate of EGP regardless of the
model used to describe non-steady-state glucose kinetics (6, 18).
Clamping glucose specific activity or TTR during an IVGTT is very
difficult, because the time course of EGP, besides being unknown, is
likely to present noticeable and at least partly unpredictable
variations. A pioneering attempt along this line was proposed in the
dog by Cowan and Hetenyi (11), a study in which the glucose system was
prelabeled with a primed, continuous infusion of tracer that was
initiated 100 min before the labeled IVGTT and maintained thereafter;
in this way, specific activity reached a constant value before the
labeled IVGTT, and its changes during the test were thus reduced.
In the present work, we use the TTR clamp technique to validate the
deconvolution method for reconstructing EGP during an IVGTT. To do so,
two tracers,
[6,6-2H2]glucose
and [U-13C]glucose,
were simultaneously injected in the same subject and used to provide,
respectively, the deconvolution and the TTR clamp EGP profile.
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EXPERIMENTAL DESIGN AND METHODS |
Subjects and Protocol
Subjects.
Five young men and five young women [age: 27 ± (SE) 7 yr, body
weight: 62 ± 3 kg] were studied after an overnight fast. All were healthy, nonmedicated, nondiabetic, in the normal range for height
and weight, and free of cardiovascular disease. Before participation,
the subjects received instructions from a research dietitian on how to
follow a weight-maintaining diet that provided
250 g of carbohydrate
on each of the 3 days leading up to an experiment. During the day
before an experiment, no exercise or organized sport was allowed. All
subjects received a detailed description (both verbal and written) of
the experimental protocol, they were informed of potential risks, and
written informed consent was obtained. This study was approved by the
Washington University School of Medicine Institutional Review Board.
Protocol
On the morning of an experiment the subjects were weighed and, starting
at 0800, a Teflon catheter was inserted into a forearm vein of both
arms, one for blood sampling and the second for infusion of glucose and
glucose isotopes. Patency of the catheter used for blood sampling was
maintained with a slow drip of sterile saline. The study lasted a total
of 6 h for each subject (Fig. 1). After a
baseline blood sample was drawn, the subjects received a 2-h
(0-120 min) primed (7 mg/kg), constant intravenous infusion (70 µg · kg
· min
1)
of [U-13C]glucose to
assess basal EGP. The pump used was a Harvard variable-rate microprocessor-controlled motor-driven syringe infusion pump (Pump 22, South Natick, MA). We calibrated by pumping water at a set rate for a
measured time period and measured the weight and volume of the water
pumped. This agreed with the rate set on the pump. Also, during the
tracer infusion experiment, we marked the syringe volume and clock time
each time we changed the infusion rate. We could then calculate the
volume delivered over a measured period of time (rate) during the
actual experiment.

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Fig. 1.
Experimental protocol. Each subject received simultaneously a variable
[U-13C]glucose
infusion (A) and a
[6,6-2H2]glucose-labeled
intravenous glucose tolerance test (IVGTT,
B).
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The second part of the experiment consisted of a 4-h (120-360 min)
IVGTT (330 mg/kg body wt of exogenous glucose were administered as a
bolus at 120 min) labeled with the stable isotope
[6,6-2H2]glucose
(the concentration of pure
[6,6-2H2]glucose
in the bolus was 10% of the unlabeled, natural glucose content) and
modified with insulin infusion. In humans, the plasma glucose and
insulin patterns after glucose injection are quite similar. Therefore,
it is difficult to differentiate between the effects of glucose and
insulin during an IVGTT, and this ultimately results in increased
variability in the minimal model parameter estimates of insulin
sensitivity and glucose effectiveness (25). Thus, we modified the
stable labeled IVGTT protocol to include a short-term insulin infusion,
so that the temporal patterns of glucose and insulin became less
similar. To do this, 0.03 U/kg of regular human insulin (Novolin, Novo
Nordisk, Princeton, NJ) was infused for 5 min starting at
minute 20 of the IVGTT. During the
IVGTT, [U-13C]glucose
infusion continued, but the infusion rate was this time adjusted in a
stepwise fashion to mimic the predicted pattern of EGP, thus keeping
the TTR of
[U-13C]glucose (and
thus of endogenous glucose) constant. A few pilot experiments were
allowed to adjust the stepwise infusion (initially based on the results
of Ref. 4), and the following schedule was used (in percentages of
basal rate): basal period, 100% of basal infusion rate; 0-2 min,
100%; 2-4 min, 80%; 4-6 min, 60%; 6-8 min, 40%;
8-10 min, 20%; 10-15 min, 10%; 15-30 min, 5%;
30-40 min, 20%; 40-50 min, 30%; 50-55 min, 40%;
55-60 min, 60%; 60-70 min, 80%; 70-80 min, 100%;
80-100 min, 110%; 100-120 min, 100%; 120-160 min,
90%; 160-240 min, 80%. In the event that a blood sample and
change in infusion rate occurred at the same time point, blood sampling
preceded the change in infusion rate.
Blood samples were obtained at 0, 2, 3, 5, 8, 10, 15, 20, 30, 45, 60,
75, 90, 100, 110, and 120 min during the first part of the experiment
and at 122, 123, 124, 125, 126, 128, 130, 132, 134, 136, 139, 142, 144, 146, 148, 150, 155, 160, 165, 170, 175, 180, 190, 200, 220, 240, 260, 280, 300, 330, and 360 min during the IVGTT. Total glucose, plasma
insulin, and the complete mass spectra for
[U-13C]glucose and
[6,6-2H2]glucose
were measured from each sample.
Methods
Measurements.
Blood samples for glucose and insulin were collected in heparinized
tubes. All samples were immediately placed on ice, and after
centrifugation, plasma was stored at
80°C for insulin and glucose tracer analysis. Plasma glucose concentration was determined at
bedside by the glucose oxidase method using a Beckman Glucose Analyzer
(Beckman Instruments, Fullerton, CA), and plasma insulin was determined
by radioimmunoassay (16). A portion of each plasma sample was
deproteinized in cold acetone. After centrifugation, the supernatant
was removed and evaporated, and the pentaacetate derivative of glucose
was formed by the addition of 100 µl of acetic anhydride-pyridine
(1:1). Glucose was separated by gas chromatography at 180°C on a
3% OV 101 column, and
2H and
13C isotopic abundance was
measured by positive chemical ionization mass spectrometry by use of
selected ion monitoring of peak intensities at mass-to-charge ratios
331, 332, 333, 334, 335, 336, and 337. The
[U-13C]glucose tracer
(99 atom%) was manufactured by Isotec (Miamisburg, OH), and the
[6,6-2H2]glucose
tracer (99 atom%) was manufactured by MassTrace (Woburn, MA). The gas
chromatography-mass spectrometer (GC-MS) characteristics and
manufacturer were Finnigan 3300 GC-quadrupole-MS (Sunnyvale, NY).
The TTR of
[6,6-2H2]glucose
[ZI(t)]
and [U-13C]glucose
[ZII(t)]
at each sampling time t were
calculated from the peak ratio data by using an extension of the case
of multiple labels of the formulas in Refs. 8-10. From the TTRs,
ZI(t)
and
ZII(t)
and total glucose G(t), one can
derive the concentrations of exogenous glucose injected with the
[6,6-2H2]glucose-enriched
bolus in the sample
|
(1)
|
and
of exogenous glucose injected with the
[U-13C]glucose-enriched
bolus
|
(2)
|
In
the following explanation, we will refer to
[6,6-2H2]glucose
and [U-13C]glucose
concentrations, meaning by this the "total exogenous concentration" injected with these tracers, in keeping with an established formalism (1, 9, 10, 23, 24).
Endogenous glucose, i.e., the portion of plasma glucose concentration
attributable to EGP, was calculated from
ZI(t),
ZII(t), and total glucose concentration G(t)
(7, 9)
|
(3)
|
The
measurement error associated with all these variables was directly
derived via error propagation by use of an extension of the formulas
described in Refs. 1 and 10.
EGP estimation from the TTR clamp.
From non-steady-state theory (6, 17, 18) we can see that, if the TTR in
the accessible compartment is kept constant, ZII(t) = ZII, all peripheral compartments
that exchange material with the accessible compartment will have the
same TTR as the accessible compartment and EGP can be calculated in a
model-independent way as follows
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(4)
|
EGP(t) is thus equal to the
known time-varying tracer infusion rate
(t),
apart from the scale factor given by the constant TTR. It is, however,
quite difficult, in practice, to maintain the TTR perfectly constant
throughout the experiment, and
EGP(t) estimation still requires a
model. Nevertheless, by keeping the variations of
ZII(t)
small, one greatly enhances the accuracy of the model reconstruction of
EGP and makes it much less dependent on the validity of the model used.
Because in our experiment the TTR clamp was quite good (in Fig.
2 it can be seen how
[U-13C]glucose
concentration profile in plasma closely followed that of endogenous
glucose), we were confident that the non-steady-state error was
minimized and that determination of EGP was almost model independent.
As a matter of fact, the estimates of EGP provided by models of Steele
(22) and Radziuk et al. (20), i.e., the two models that are widely used
to perform non-steady-state analysis, were almost identical (as we will
show later on). We chose the EGP profile calculated with Radziuk's
two-compartment model because it is more accurate than Steele's model
(6, 20). Specifically, we adopted the version of Radziuk's model with
only one time-varying irreversible loss from the accessible compartment
(20), because it allows us to express the unknown EGP with a
simple formula (see APPENDIX A for
details)
|
(5)
|
where
V1 is the volume of the accessible
pool (dl/kg), and
q2 denotes the
endogenous glucose mass in the second compartment (mg/kg). Still,
although all of the elements in Eq. 5
can be directly evaluated from data, which makes it particularly
appealing, experimental noise can jeopardize some calculations,
especially the first derivative of
ZII(t),
which is evaluated at the midpoint of each sampling integral via finite
differences. For this reason, both endogenous glucose and TTR data were
smoothed by using a three-samples moving average. In this fashion, we
set forth to calculate for each subject a time course of EGP that is
likely to be reliable and can then be used as a reference for
validation purposes. Note that
EGP(t) is also evaluated at the
midpoint of each sampling interval.

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Fig. 2.
Mean tracer-to-tracee ratio (TTR) clamp plasma data.
A: endogenous glucose concentration;
B:
[U-13C]glucose
concentration; C: endogenous glucose
TTR (ZII).
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EGP estimation by deconvolution.
We refer to APPENDIX A for details on
the 2CMM and the deconvolution approach to estimate EGP. Briefly, the
relation between EGP (input) and endogenous glucose concentration
Ge (output) during an IVGTT can be
described by the integral equation
|
(6)
|
where
Gb is baseline glucose
concentration, and h(t,
) is the
time-varying kernel of the glucose system described by the 2CMM
identified from
[6,6-2H2]glucose
data. Details of the procedure used for estimation of EGP are reported
in Refs. 12 and 24. We wish to point out that Eq. 6 is a Fredholm integral equation of the first kind, but we use here the term "deconvolution" [rigorously
applicable only when the kernel
h(t,
) is time invariant] both
for the sake of simplicity and in keeping with the existing literature
(4, 24). EGP is estimated by the following formula
|
(7)
|
where
H is the system matrix,
F is an appropriately chosen
regularization matrix (chosen according to the desired degree of
smoothness of the EGP estimate; see Refs. 12, 21, and 24 for details),
Ge is the deviation of
endogenous glucose from baseline, B is
the (diagonal) matrix of measurement errors associated with endogenous
glucose samples, EGPb is basal EGP, and
is a
regularization parameter.
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RESULTS AND DISCUSSION |
Basal pretest plasma glucose was 88 ± 6 (SE) mg/dl, and basal pretest
plasma insulin concentration was 4 ± 1 µU/ml. Mean plasma [U-13C]glucose
concentration (CII) and
endogenous glucose TTR (ZII) are
shown in Fig. 2, together with endogenous glucose concentration (Ge). Mean plasma glucose (G),
insulin (I), and
[6,6-2H2]glucose
concentrations (CI) are shown in
Fig. 3.
TTR Clamp
Model parameters (V1,
k21,
k12,
k01) were
estimated for each subject (Table 1) by
weighted nonlinear least squares from the data of the primed, constant
infusion of
[U-13C]glucose during
the baseline period of 0-120 min. Weights were chosen optimally,
i.e., equal to the inverse of the measurement error variance (3). The
measurement error associated with the tracer measurements was assumed
to be independent, white, Gaussian, with mean zero and a variance
generated by error propagation from the peak ratio measurement error
variance (10) and ranged between 6 and 10% (higher for lower
concentration values). Peak ratio standard deviations were determined
via replicate measurements. Equation 5
was then applied to the data, and the average profile of EGP is shown
in Fig. 4.
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Table 1.
Parameter values of model of Radziuk et al. (20) identified from data
of primed, constant infusion of [U-13C]glucose during
baseline period
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Fig. 4.
TTR clamp estimates of endogenous glucose production (EGP) obtained
with 2-compartment model of Radziuk et al. (20) ( ) and with
single-compartment model of Steele et al. (22) ( ).
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A rapid and almost complete suppression of EGP was seen only 10-15
min after the glucose challenge and between 20 and 40 min EGP increased
rapidly. In this period (120-160 min of the experimental protocol), the TTR clamp was quite good, and we believe that this constitutes evidence that our estimate of EGP is substantially correct,
because it means that the shape of EGP during that time closely agreed
with that of the variable
[U-13C]glucose
infusion. Clamping TTR between 200 and 360 min was, however, much more
difficult, because the time of resumption of EGP varied among the
subjects. Moreover, when both tracer and endogenous glucose
concentrations are very low, such as during maximal inhibition, the
time course of the TTR is very sensitive to changes in both tracer and
glucose concentration. Therefore, if a small increase in endogenous
glucose concentration is not accompanied by a concomitant increase in
the tracer concentration (as dictated off-line by the exogenous
infusion), this would induce a noticeable decrease in the TTR. Now, the
problem is to determine in some way to what extent these problems
related to the TTR clamp introduced an error in our estimate of EGP
between 200 and 360 min. As a possible approach to at least partially
assess the magnitude of this error, we compared EGP calculated with the
model of Steele (22) (setting the volume of the glucose compartment
equal to 1.3 dl/kg body wt and individualizing the basal value of the
irreversible, time-varying loss from this volume and plasma clearance
rate) with that of the model of Radziuk et al. (20). In fact, because theory shows (6) that the error affecting EGP estimation depends on
both the model used for non-steady-state glucose kinetics and the rate
of change of the TTR, the difference between the models of Radziuk et
al. and Steele (i.e., between a good and a poor model of
non-steady-state glucose kinetics) provides some insight on the
influence of the TTR changes on the estimate of EGP. The time courses
of EGP thus derived were almost superimposable (Fig. 4); therefore,
despite an imperfect clamp of specific activity, it is possible to
conclude that the EGP estimate calculated between 200 and 360 min is reliable.
The sudden up-and-down performance of EGP observed with both models in
the first 10 min of the IVGTT is unlikely to be representative of a
true physiological occurrence. Rather, it is probably a symptom of the
marked ill-conditioning affecting EGP estimation in that portion of the
IVGTT. We speculate that, even in the presence of a good TTR clamp and
data smoothing, when sampling is very frequent and the tracer infusion
rate is frequently changed, as in the initial part of the IVGTT, even
very small errors in the evaluation of the TTR derivative are
uncontrollably amplified and determine spurious oscillations in the
deconvolved profile.
Deconvolution
The uniquely identifiable parameters of the 2CMM were estimated from
[6,6-2H2]glucose
data by weighted nonlinear least squares as described in Ref. 24. The
model was identified in all subjects (Table 2). The measurement error coefficient of
variation (CV) of
[6,6-2H2]glucose
concentration data ranged from 2.3 to 12.6% on average, with lower
precision associated with lower concentration values. The mean EGP time
course estimated by deconvolution is shown in Fig.
5 together with that obtained with the TTR
clamp (mean ± SE). We can see that, on average, there is a very good
accordance between the two estimates. EGP is almost completely
inhibited already at 140 min, and early in the test it clearly shows a
bimodal pattern, probably caused by the insulin injection between 140 and 145 min [a plot of the average insulin action
x(t) estimated by the model is shown
in Fig. 6]. The exogenous
insulin administration results then in a new suppression of EGP, even
if EGP is already almost completely inhibited by the glucose bolus
alone. Resumption to baseline is accompanied by a consistent overshoot
of EGP, well above basal, which is probably driven by the
counterregulatory hormones. A comparison of the timing of EGP dynamics
evaluated with the two approaches and of the extent of EGP overshoot is given in Table 3. It should be noted that
resumption to baseline is rather variable (both in timing and extent)
among individuals, so that endogenous glucose TTR could not always be
kept precisely constant at the time of resumption (see also Fig. 3). An
experimental design adjustable from subject to subject would in
principle have been valuable, but mass spectrometry measurements are
not feasible on-line.
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Table 2.
Two-compartment minimal model parameters identified from IVGTT data of
[6,6-2H2]glucose data with assumption of
insulin and glucose plasma concentrations as known inputs
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Fig. 5.
EGP estimated by deconvolution (thick line) and by TTR clamp and
2-compartment model of Radziuk et al. (20) ( ). Standard errors are
shown by thin lines and error bars.
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Fig. 6.
Mean insulin action
(min 1) in remote insulin
compartment estimated by 2-compartment minimal model. Error bars are
standard deviations.
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It is of interest to note that the 2CMM accurately assesses the
dynamics of EGP during the IVGTT but provides an estimate of basal EGP
that is lower than that independently calculated from
[U-13C]glucose data in
the steady state before the IVGTT
(EGPb = 2.47 ± 0.15 vs. 1.89 ± 0.19 mg · kg
1 · min
1,
P < 0.05 from paired, two-tailed
t-test). Because basal EGP is the
product of basal glucose clearance times basal glucose concentration,
the underestimation of EGP might reflect an underestimation of basal
glucose clearance by the 2CMM analysis. Given that the estimation of
basal glucose clearance by the 2CMM hinges on the final part of the
tracer disappearance curve (when insulin action is almost negligible
and glucose is close to its basal level), a preliminary question must
be addressed before concluding that the 2CMM provides a biased estimate
of glucose clearance: is it true that glucose clearance in the final
part of the test is similar to that in the pretest? An answer to this
question comes from the analysis of
[U-13C]glucose data:
the mean value of glucose clearance that is assessed from the model of
Radziuk et al. (20) in the final part of the IVGTT [calculated as
the product
k01(t) · V1]
is not different from that measured with the same tracer before the
IVGTT (2.90 ± 0.56 vs. 2.47 ± 0.46, not significant). This result
allows us to conclude that for some reason the 2CMM identified from
[6,6-2H2]glucose
data underestimates the glucose clearance achieved at the end of the
IVGTT. The next step is to localize the source of error: is there a
problem in the model or in the
[6,6-2H2]glucose
data? We have come to the conclusion that the problem is in the
[6,6-2H2]glucose
concentration data. This was proven by taking the last five to six data
points of the
[6,6-2H2]glucose
disappearance curve and fitting these data with a single exponential.
We reasoned that, because insulin action is low in that part of the
test and glucose is close to its basal level, fitting a single
exponential through the latter part of the test is a feasible method
to estimate the slowest eigenvalue
(
2) of glucose kinetics, the
one that is mainly responsible for the clearance. We then compared this
data-driven assessment of
2 to
the slow eigenvalue of the 2CMM. We reasoned that if the 2CMM
underestimates glucose clearance because of some model error, the
model-based
2 should be lower
than the data-driven
2. We
found that the two values of
2 were virtually identical:
2 from the single-exponential fitting was 7.9 ± 1.6 × 10
2/min and from the 2CMM
was 7.2 ± 0.8 × 10
2/min,
r2 = 0.78. This
result indicates both that the 2CMM provides an estimate of glucose
clearance that simply reflects the final IVGTT [6,6-2H2]glucose
data and that
[6,6-2H2]glucose
data provide an estimate of glucose clearance lower than that provided
by [U-13C]glucose data.
A hypothesis that could account for this observation is that there is a
spurious fraction of
[6,6-2H2]glucose
that is recirculating back from the liver, possibly due to the large
resumption of EGP in the second part of the IVGTT. Glucose-carbon
recycling is not a viable option, because similar values of plasma
clearance rate (PCR) and EGP were estimated with protocols
in absence of carbon tracers.
To gain further understanding of the 2CMM validity, we attempted to
validate its description of the glucose system by the following
procedure. We applied the known
[U-13C]glucose
infusion as a known input to the 2CMM identified from [6,6-2H2]glucose
data, and then we compared the model output with the measured
[U-13C]glucose
concentration data. We reasoned that, if the model identified from the
[6,6-2H2]glucose
dynamics was able to account for the observed
[U-13C]glucose data,
this would support its validity. It can be seen from the results
reported in Fig. 7 that the match between
data and model is quite good except for the final part of the IVGTT, where the model prediction tends to overestimate the observed data.
This results in additional evidence that the 2CMM underestimates glucose clearance in the final part of the IVGTT.

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Fig. 7.
Results of applying the known
[U-13C]glucose
infusion as a known input to the 2-compartment minimal model (2CMM)
identified from
[6,6-2H2]glucose
data. Model prediction ( ) is shown together with experimental
[U-13C]glucose
concentration data ( with SD error bars).
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Our results also provide some new insight into the control of glucose
and insulin on EGP suppression. In the first 40 min after the start of
the IVGTT, EGP shows a bimodal pattern, that is, it is almost totally
inhibited already 15-20 min after the glucose challenge, then
slightly but quickly resumes, and is again inhibited at ~30-35
min (see Table 3). This second nadir is in all likelihood due to the
exogenous insulin infusion. It is interesting to note, however, that
EGP is already almost completely suppressed before the insulin
injection; thus, it is likely that the insulin injection pushes further
a process that is already fully inhibited. There is also a very high
interindividual variability in both the timing and the entity of EGP
resumption to baseline, as demonstrated by Table 3 and the error bars
in Fig. 5, and this is confirmed by both methods.
Finally, the time course of EGP allows us to qualitatively answer the
question of whether EGP inhibition is triggered directly by plasma
glucose concentration. Apart from the values at 122 and 123 min, which
apparently show a fast EGP inhibition and a quick return to basal but
are in all likelihood spurious (see earlier discussion), EGP is
suppressed slowly with respect to the time course of plasma glucose
concentration, and this rules out the possibility that EGP inhibition
is directly proportional to plasma glucose. It therefore seems more
likely that some delayed glucose signal is responsible for EGP inhibition.
Conclusions
We have validated the estimate of EGP during an IVGTT obtained
by deconvolution against a model-independent estimate obtained with the
TTR clamp. Both approaches demonstrated almost complete suppression of
EGP within 20 min and a high interindividual variability in the
resumption to baseline. The time course of EGP calculated with both
approaches compared very well. Thus the 2CMM, in addition to providing
metabolic indexes of glucose effectiveness and insulin sensitivity,
also recovers, when used in conjunction with deconvolution, EGP time
course during the IVGTT; it therefore is a potentially very useful tool
to assess glucose metabolism in vivo.
 |
APPENDIX A. EGP Estimate from TTR Clamp |
Equations of the model of Radziuk et al. (20) for endogenous glucose
(tracee) kinetics during the IVGTT are
|
(A1)
|
where q1 and
q2 denote
endogenous glucose masses (mg/kg) in the first and second compartments,
respectively, V1 is the volume of
the accessible pool (dl/kg), Gb is
basal (pretest) glucose concentration, and
k21
(min
1),
k12
(min
1), and
k01
(min
1) are rate
parameters describing glucose kinetics.
Equations for the tracer
[U-13C]glucose
(CII) kinetics are
|
(A2)
|
where the superscript * denotes tracer-related variables, and
CII(0) is different from zero,
because by experimental design [U-13C]glucose has
already reached a constant level in plasma before the initiation of the IVGTT.
A priori identifiable model parameters are
V1,
k21,
k12, and the
basal value of
k01(t),
k01. Glucose
plasma clearance rate PCR and basal EGP,
EGPb, can be directly calculated
from the model parameters as
|
(A3)
|
and
|
(A4)
|
respectively.
Let us calculate now EGP from the model of Radziuk et al. (20).
Rearranging the tracee equations, we can derive an expression for
EGP(t)
|
(A5)
|
and
from the definition of TTR
|
(A6)
|
Substituting
in Eq. A5 the expression for
k01 obtained from
Eqs. A1 and A2, we obtain
|
(A7)
|
from
which, rearranging, we obtain the final expression for EGP
|
(A8)
|
Now,
(t)
is the mass of tracer glucose in the accessible pool, and
q2(t)
is the mass of tracee glucose in the second, nonaccessible pool.
Whereas
q1(t)
(
) was calculated at
each sampling point by multiplying the endogenous (exogenous) glucose
concentration by V1, i.e., the
volume of the accessible pool,
q2(t)
(
) was calculated at
each sampling point by solving the first-order differential equation with constant parameters that describe the kinetics of the Radziuk model's second pool. This differential equation has
q1(t)
(
) as forcing input.
The time course (continuous) of
q1(t)
(
) was calculated by
linear interpolation.
 |
APPENDIX B. EGP Estimate by Deconvolution |
The 2CMM is described by Refs. 5, 10, and 24
|
(B1)
|
where
and
denote
[6,6-2H2]glucose
masses in the first (accessible pool) and second (slowly-equilibrating) compartments, respectively (mg/kg),
x(t) is insulin action
(min
1),
I(t) and
Ib are plasma insulin and basal
(endtest) insulin, respectively (µU/ml),
Gb is basal (endtest) glucose
concentration (mg/dl),
q1(t)
is cold glucose mass in the accessible pool (mg/dl), CI(t)
is plasma
[6,6-2H2]glucose
concentration (mg/dl), D* is the
hot glucose dose (mg/kg), V1 is
the volume of the accessible pool (dl/kg), and
k21
(min
1),
k12
(min
1),
k02
(min
1),
p2
(min
1), and
sk
(µU · ml
1 · min
1)
are rate parameters describing glucose kinetics and insulin action.
Briefly, the model structure assumes that insulin-independent glucose
disposal takes place in the accessible pool and is the sum of two
components, one constant (due to, e.g., central nervous system and red
blood cells) and the other proportional to glucose mass. This brings us
to the rate constant describing the irreversible loss from the
accessible pool
|
(B2)
|
where
G(t) is the glucose concentration in
the accessible pool of volume V1,
Rd,0
(mg · kg
1 · min
1)
is the constant component of glucose disposal, and
kp
(min
1) accounts for the
proportional-to-mass term.
To arrive at a priori uniquely identifiable parameters, it is assumed
that 1) in the basal steady state,
insulin-independent glucose disposal (from compartment
1) is three times insulin-dependent glucose disposal
(from compartment 2), and
2)
Rd,0 is fixed to the
experimentally determined value (2) of 1 mg · kg
1 · min
1;
this brings us to the following relation among the model parameters
|
(B3)
|
The
uniquely identifiable parameters of the 2CMM are six:
V1,
k21,
k12,
k02,
p2, and
sk. From the
model parameters, it is possible to calculate basal EGP
|
(B4)
|
The
estimate of EGP during the IVGTT nonsteady state was obtained by
describing with the 2CMM the impulse response
h(t,
) of the glucose system, and by
applying a deconvolution algorithm to invert the integral relation of
Eq. 6. The deconvolution approach used
here is nonparametric, is described in Refs. 12 and 24, and was
previously used to reconstruct insulin secretion during an IVGTT (21).
We refer to these works for details. An advantage of the method
consists in the availability of a new statistically sound criterion,
based on maximum likelihood, for the choice of the regularization parameter.
 |
ACKNOWLEDGEMENTS |
The authors thank Dr. Gianna M. Toffolo, of the Department of
Electronics and Informatics of the University of Padova, for helpful
collaboration in the interpretation of the mass spectrometry measurements.
 |
FOOTNOTES |
This work was partially supported by a grant from the Italian Ministero
della Università e della Ricerca Scientifica e Tecnologica (MURST
40%) on "Biosistemi e Bioinformatica," by Grant 9300457PF40 from
the Italian National Research Council (CNR) on "Aging," and by
National Institutes of Health Grants RR-02176, RR-11095, RR-00954, RR-00036, and DK-49393.
Present address of P. Vicini: Department of Bioengineering, University
of Washington, Seattle, WA 98195.
Address for reprint requests: C. Cobelli, Dept. of Electronics and
Informatics, Univ. of Padova, Via Gradenigo 6/A, 35131 Padova, Italy.
Received 16 December 1997; accepted in final form 9 October 1998.
 |
REFERENCES |
1.
Avogaro, A.,
P. Vicini,
A. Valerio,
A. Caumo,
and
C. Cobelli.
The hot but not the cold minimal model allows precise assessment of insulin sensitivity in NIDDM subjects.
Am. J. Physiol.
270 (Endocrinol. Metab. 33):
E532-E540,
1996[Abstract/Free Full Text].
2.
Best, J. D.,
J. Taborsky, Jr.,
J. B. Halter,
and
D. Porte, Jr.
Glucose disposal is not proportional to plasma glucose level in man.
Diabetes
30:
847-850,
1981[Abstract].
3.
Carson, E. R.,
C. Cobelli,
and
L. Finkelstein.
The Mathematical Modeling of Metabolic and Endocrine Systems. New York: Wiley, 1983.
4.
Caumo, A.,
and
C. Cobelli.
Hepatic glucose production during the labeled IVGTT: estimation by deconvolution with a new minimal model.
Am. J. Physiol.
264 (Endocrinol. Metab. 27):
E829-E841,
1993[Abstract/Free Full Text].
5.
Caumo, A.,
A. Giacca,
M. Morgese,
G. Pozza,
P. Micossi,
and
C. Cobelli.
Minimal model of glucose disappearance: lessons from the labeled IVGTT.
Diab. Med.
8:
822-832,
1991[Medline].
6.
Cobelli, C.,
A. Mari,
and
E. Ferrannini.
Non-steady state: error analysis of Steele's model and developments for glucose kinetics.
Am. J. Physiol.
252 (Endocrinol. Metab. 15):
E679-E689,
1987[Abstract/Free Full Text].
7.
Cobelli, C.,
and
G. Toffolo.
Constant specific activity input allows reconstruction of endogenous glucose concentration in non-steady state.
Am. J. Physiol.
258 (Endocrinol. Metab. 21):
E1037-E1040,
1990[Abstract/Free Full Text].
8.
Cobelli, C.,
G. Toffolo,
D. M. Bier,
and
R. Nosadini.
Models to interpret kinetic data in stable isotope tracer studies.
Am. J. Physiol.
253 (Endocrinol. Metab. 16):
E551-E564,
1987[Abstract/Free Full Text].
9.
Cobelli, C.,
G. Toffolo,
and
D. M. Foster.
Tracer-to-tracee ratio for analysis of stable isotope tracer data: link with radioactive kinetic formalism.
Am. J. Physiol.
262 (Endocrinol. Metab. 25):
E968-E975,
1992[Abstract/Free Full Text].
10.
Cobelli, C.,
P. Vicini,
G. Toffolo,
and
A. Caumo.
The hot IVGTT minimal models: simultaneous assessment of disposal indices and endogenous glucose production.
In: The Minimal Model Method and Determinants of Glucose Tolerance, edited by J. Lovejoy,
and R. N. Bergman. Baton Rouge, LA: Louisiana State University Press, 1998, p. 202-239. (Pennington Nutr. Ser.)
11.
Cowan, J. R.,
and
G. Hetenyi, Jr.
Glucoregulatory responses in normal and diabetic dogs recorded by a new tracer method.
Metab. Clin. Exp.
20:
360-372,
1971.[Medline]
12.
De Nicolao, G.,
G. Sparacino,
and
C. Cobelli.
Nonparameteric input estimation in physiological systems: problems, methods, case studies.
Automatica
33:
851-870,
1997.
13.
Fick, A.
Über die Messung des Blutquantums in den Herzventrikeln.
Verhandl Phys. Med. Ges. Wurzburg
2:
XVI,
1870.
14.
Finegood, D. T.,
R. N. Bergman,
and
M. Vranic.
Estimation of endogenous glucose production during hyperinsulinemic-euglycemic glucose clamps: comparison of unlabeled and labeled exogenous glucose infusates.
Diabetes
36:
914-924,
1987[Abstract].
15.
Finegood, D. T.,
I. M. Hramiak,
and
J. Duprè.
A modified protocol for estimation of insulin sensitivity with the minimal model of glucose kinetics in patients with insulin dependent diabetes.
J. Clin. Endocrinol. Metab.
70:
1538-1548,
1990[Abstract].
16.
Hales, C.,
and
P. J. Randle.
Immunoassay of insulin with insulin antibody precipitate.
Biochem. J.
88:
137-146,
1963.
17.
Jacquez, J. A.
Theory of production rate calculations in steady and non-steady states and its application to glucose metabolism.
Am. J. Physiol.
262 (Endocrinol. Metab. 25):
E779-E790,
1992[Abstract/Free Full Text].
18.
Norwich, K. N.
Measuring rates of appearance in systems which are not in steady state.
Can. J. Physiol. Pharmacol.
51:
91-101,
1973[Medline].
19.
Overkamp, D.,
J. F. Gautier,
W. Renn,
A. Pickert,
A. J. Scheen,
R. M. Schmulling,
M. Eggstein,
,
and
P. J. Lefebvre.
Glucose turnover in humans in the basal state and after intravenous glucose: a comparison of two models.
Am. J. Physiol.
273 (Endocrinol. Metab. 36):
E284-E296,
1997[Abstract/Free Full Text].
20.
Radziuk, J.,
K. H. Norwich,
and
M. Vranic.
Experimental validation of measurements of glucose turnover in nonsteady state.
Am. J. Physiol.
234 (Endocrinol. Metab. Gastrointest. Physiol. 3):
E84-E93,
1978[Abstract/Free Full Text].
21.
Sparacino, G.,
and
C. Cobelli.
A stochastic deconvolution method to reconstruct insulin secretion rate after a glucose stimulus.
IEEE Trans. Biomed. Eng.
435:
512-529,
1996.
22.
Steele, R.
Influence of glucose loading and of injected insulin on hepatic glucose output.
Ann. NY Acad. Sci.
82:
420-430,
1959.
23.
Vicini, P.,
A. Caumo,
and
C. Cobelli.
The hot IVGTT two compartment minimal model: indexes of glucose effectiveness and insulin sensitivity.
Am. J. Physiol.
273 (Endocrinol. Metab. 36):
E1024-E1032,
1997[Medline].
24.
Vicini, P.,
G. Sparacino,
A. Caumo,
and
C. Cobelli.
Estimation of EGP after a glucose perturbation by nonparametric stochastic deconvolution.
Comput. Methods Prog. Biomed.
52:
147-156,
1997[Medline].
25.
Yang, Y. J.,
J. H. Youn,
and
R. N. Bergman.
Modified protocols improve insulin sensitivity estimation using the minimal model.
Am. J. Physiol.
253 (Endocrinol. Metab. 16):
E595-E602,
1987[Abstract/Free Full Text].
26.
Zierler, K. L.
Theory of the use of arteriovenous concentration differences for measuring metabolism in steady and nonsteady states.
J. Clin. Invest.
40:
2111-2125,
1961.
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