1 Department of Electronics and
Informatics, Glucose effectiveness is an important
determinant of glucose tolerance that can be derived from minimal model
analysis of an intravenous glucose tolerance test (IVGTT). However,
recent evidence suggests that glucose effectiveness is overestimated by
minimal model analysis. Here we compare a new model-independent estimate of glucose effectiveness with the minimal model estimate by
reanalyzing published data in which insulin-dependent diabetic subjects
were each given IVGTTs under two conditions (Quon, M. J., C. Cochran,
S. I. Taylor, and R. C. Eastman.
Diabetes 43: 890-896, 1994). In
one case, a basal insulin level was maintained (BI-IVGTT). In the
second case, a dynamic insulin response was recreated (DI-IVGTT). Our
results show that minimal model glucose effectiveness is very similar
to the model-independent measurement during a BI-IVGTT but is three
times higher during a DI-IVGTT. To investigate the causes of minimal
model overestimation in the presence of a dynamic insulin response,
Monte Carlo simulation studies on a two-compartment model of glucose
kinetics with various insulin response patterns were performed. Results
suggest that minimal model overestimation is due to single-compartment
representation of glucose kinetics that results in a critical
oversimplification in the presence of increasingly dynamic insulin
secretion patterns.
intravenous glucose tolerance test; glucose kinetics
GLUCOSE EFFECTIVENESS, an important component of
glucose tolerance (1, 3), is defined as the ability of glucose to
promote its own disposal and inhibit its own production in the absence of an incremental insulin effect (i.e., when insulin is at basal levels) (2, 3).
An estimate of glucose effectiveness
(SG) can be obtained from the
minimal model analysis of an intravenous glucose tolerance test
(IVGTT). However, both indirect (6-8) and direct (10, 12)
experimental evidence indicates that
SG is an overestimate of the true
glucose effectiveness and that SG
estimation is influenced by the insulin profile during the IVGTT.
The overestimation of SG has been
suggested (7) to be due, in large part, to the single-compartment
approximation of glucose kinetics used by the minimal model.
Specifically, SG would incorporate a component reflecting the exchange kinetics between the accessible and
the inaccessible pool of the glucose system. The relationship between
SG and a reference index of
glucose effectiveness has recently been investigated with model
simulation studies (9, 11, 13), but conflicting results have been
produced. Whereas in Refs. 9 and 13 no correlation with the reference
index was found, in Ref. 11 an excellent concordance was obtained. Evidence has been provided (13), however, that the simulation conducted
in Ref. 11 poorly reflects real life (e.g., only one parameter at a
time is allowed to vary in the Monte Carlo runs).
The sensitivity of SG to the
insulin profile during the IVGTT has not been well characterized, and
many issues remain open to question (5). Does overestimation of
SG occur only during an IVGTT when
a dynamic insulin response is elicited but not during an optimal
protocol (i.e., during an IVGTT when insulin is clamped at its basal
value)? If so, what is the cause? Another issue that needs to be
addressed is the following: if SG
depends on insulin dynamics during the IVGTT, what is the role played
by the insulin response during the first 20 min of the test? This issue
has practical relevance because the insulin profile in the first 20 min
of the IVGTT is due exclusively to endogenous insulin secretion and may differ considerably among groups, whereas from 20 min on, the insulin
profile is made much more homogenous by exogenous insulin administration.
In the present study we sought to address these questions. To do so we
relied on both experimental and computer simulation studies. The data
base consists of a set of published data (12) in which subjects with
insulin-dependent diabetes mellitus (IDDM) were given an IVGTT on one
occasion with only basal insulin provided (BI-IVGTT) and on a second
occasion in the presence of a dynamic insulin response (DI-IVGTT), in
which a normal insulin response was recreated through a
computer-controlled infusion of insulin. In our reanalysis of this
data, we first calculated from BI-IVGTT data the model-independent
reference measure of glucose effectiveness, GE, recently proposed in
Ref. 1. We next used the minimal model to obtain estimates of
SG from both the BI-IVGTT and
DI-IVGTT. Our results show that SG
derived from the BI-IVGTT is similar and very well correlated to GE. In
contrast, SG derived from the DI-IVGTT is overestimated and correlates weakly with GE. Finally, to
investigate whether overestimation of
SG in the presence of an
incremental insulin response might be due to single-compartment undermodeling, we used a two-compartment model of glucose kinetics with
various patterns of dynamic insulin responses to generate computer
simulation results. We show that the effect of insulin dynamics on
SG is most likely a consequence of
undermodeling and that SG
decreases as insulin availability in the first 20 min of the test decreases.
The Data
ABSTRACT
Top
Abstract
Introduction
Methods
Results
Discussion
References
INTRODUCTION
Top
Abstract
Introduction
Methods
Results
Discussion
References
METHODS
Top
Abstract
Introduction
Methods
Results
Discussion
References
Glucose Effectiveness: Model-Independent Estimation
It has recently been shown (1) that glucose effectiveness can be calculated under very general assumptions, i.e., virtually in a model-independent way, from an experiment in which exogenous glucose is administered to produce a transient excursion of glucose above basal levels while insulin remains at basal levels (e.g., BI-IVGTT). Under these conditions, the assessment of glucose effectiveness does not require any structural modeling of the glucose system and is simply given by the ratio between the amount of exogenous glucose administered and the area under the curve (AUC) of the glycemic excursion above basal levels. It is worth remarking that this AUC-based index of glucose effectiveness is also equivalent, as shown in Ref. 1, to the analogous clamp-based index. During a BI-IVGTT, glucose effectiveness (GE) is
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(1) |
AUC[G] can be evaluated with the trapezoidal rule or by
fitting a parametric function to the glucose data, e.g., a sum of polynomials or exponentials, and deriving the AUC from the estimated parameter values. The latter approach is statistically more robust, because it allows one to assess not only the value of GE but also its
precision. Because BI-IVGTT glucose data
G are glucose decay data
after a bolus injection, the natural candidate to describe them is a
sum of decaying exponentials. We found that a two-exponential model was
necessary and sufficient, according to Akaike's criterion (4), to
describe
G data
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(2) |
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(3) |
Glucose Effectiveness: Minimal Model Estimation
Glucose effectiveness was estimated with the minimal model of glucose disappearance (2) from both DI- and BI-IVGTT data. As usually done in minimal model identification, the first 10-min glucose samples were ignored to favor the single-compartment approximation of glucose kinetics.DI-IVGTT. During the DI-IVGTT, glucose disappearance was described by the classical minimal model equations (2)
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(4) |
BI-IVGTT. During the BI-IVGTT, the above basal insulin action X is identically null, and the model reduces to
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Monte Carlo Simulation
To test the hypothesis that single-compartment undermodeling is responsible for the discrepancies between estimates of glucose effectiveness obtained by model-independent and minimal model analysis, we resorted to Monte Carlo simulation such as in Refs. 9 and 13. The details of the Monte Carlo simulation ingredients, e.g., model structure, parameter values, and noise level, are fully described in Ref. 13. Briefly, a two-compartment model of glucose kinetics with endogenous glucose production described by the same glucose-insulin relationship embodied in the minimal model (2) was used as a reference. Normal parameter values were chosen. Six different insulin profiles (see Fig. 3 in RESULTS) were used as input to the two-compartment model and, for each insulin profile, 200 noisy IVGTT glucose data sets were generated. In addition to the standard (see Fig. 3A) and the basal insulin IVGTT (see Fig. 3F), profiles from an insulin-modified IVGTT showing a progressively decreasing first-phase insulin response (from normal in Fig. 3C to no response in Fig. 3E) were also used. This allowed us to evaluate the sensitivity of the minimal model glucose effectiveness to insulin dynamics in the initial 20 min of the test. The generated plasma glucose and insulin time courses were then used to estimate glucose effectiveness with the minimal model (assuming known the basal glucose and insulin concentrations and ignoring, as usual, the first 10-min glucose samples).Statistical Analysis
Data in the text and Figs. 1-3 are given as means ± SE. Linear regression analysis was used to evaluate the relationship between GE and the minimal model assessment of glucose effectiveness. The paired Student's t-test was used to compare different measures of glucose effectiveness made in the same subject. P values <0.05 were considered statistically significant. ![]() |
RESULTS |
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Experimental Data
The time courses of plasma glucose (top) and insulin (bottom) concentrations during the BI-IVGTT and DI-IVGTT are shown in Fig. 1 (left and right, respectively). During the BI-IVGTT, the insulin level was relatively constant, and the glucose profile approached the baseline ~6 h after the glucose injection. During the DI-IVGTT, the glucose profile was similar to the one that is commonly observed in subjects with normal glucose tolerance.
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Model-Independent GE
The two-exponential model fit was very good and is shown in Fig. 2. The model-independent GE was calculated from precisely estimated parameters. For example,
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Minimal Model Analysis of BI-IVGTT and DI-IVGTT Data
The minimal model fit of BI-IVGTT data was as good as that provided by the two-exponential model from 10 min on (Fig. 2). Parameters SG and V were estimated with good precision. The mean values of SG and V were, respectively, 0.0044 ± 0.0007 (minIn the case of DI-IVGTT, the minimal model fit assessed in terms of
residuals was good, and all the parameters
SG, V,
p2, and
p3 were estimated
with satisfactory precision. In particular, the mean value of
SG was 0.0152 ± 0.0029 min1, with a precision of
34% (range 9-74%), and that of V was 2.01 ± 0.17 dl/kg, with
a precision of 3% (range 2-5%).
Comparison Among Different Estimates of Glucose Effectiveness
Estimates of glucose effectiveness derived from both model-independent and minimal model analysis of this data are shown in Table 1. Because SG measures fractional glucose effectiveness (i.e., per unit of glucose distribution volume), it was multiplied by V to obtain a minimal-model measure of glucose effectiveness, SGV, comparable with GE (1, 5). Precision of SGV was obtained by error propagation. SGV estimated from BI-IVGTT data (0.0112 ± 0.0012 dl · minThe agreement between SGV derived
from the BI-IVGTT and GE was remarkable in each individual except for
subject no. 3, where SGV was three times higher than
GE. To ascertain if this discrepancy could be due to the presence in
this subject of a fast component still playing an important role after
10 min, we repeated the identification in all subjects by ignoring the
first 20-min glucose samples. SGV
in subject no. 3 decreased from 0.0069 to 0.0033 min1, thus
approaching GE = 0.0021 min
1, whereas no
appreciable modifications were noted in the other six subjects. As a
result, the mean value of SGV
became 0.0105 ± 0.0016 min
1, and the correlation
with GE improved (r = 0.999, P < 0.000001).
Monte Carlo Simulation
The six insulin profiles used for the Monte Carlo simulations to assess the effect of insulin dynamics on SG estimation are shown in Fig. 3. The results are reported in Table 2. They show that, when the minimal model is used to interpret glucose data generated by a more complex two-compartment model, SG estimation is markedly influenced by insulin dynamics. Glucose effectiveness was virtually the same with the standard and the insulin-modified IVGTT, i.e., the DI-IVGTT (profiles A and B) but markedly decreased with the early insulin response (profiles C, D, and E). When insulin remained at the basal level throughout the test (profile F), as during the BI-IVGTT, the lowest value of glucose effectiveness was obtained.
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DISCUSSION |
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The IVGTT minimal model method provides, in addition to an index of insulin sensitivity, an index of glucose effectiveness that measures the ability of glucose to favor its own disappearance from plasma by promoting its own utilization and inhibiting its own endogenous production when insulin is at basal levels. This index has been shown to characterize several pathophysiological states as well as to have predictive power (see recent review in Ref. 3). However, recent experimental evidence (10, 12) has shown that glucose effectiveness estimated from a standard or an insulin-modified IVGTT (when a dynamic incremental insulin response is present) is overestimated when compared with values derived from an IVGTT in which the potentially confounding effect of hyperinsulinemia is eliminated by maintaining insulin at its basal level throughout the test. However, until recently it was not possible to easily investigate the mechanism underlying this discrepancy, because a minimal model-independent measure of glucose effectiveness was needed to relate minimal model estimates to a reference measure.
In this study we have used the recently proposed model-independent measure of glucose effectiveness (1) to assess the domain of validity of the minimal model measurement. Our results indicate that when the minimal model index of glucose effectiveness (expressed as the product SGV) is estimated from an IVGTT in which insulin is kept constant at the basal level (BI-IVGTT), its results are virtually identical to the glucose effectiveness index GE obtained in a model-independent way from AUC calculations. In contrast, when insulin changes dynamically (DI-IVGTT), the minimal model overestimates the model-independent measure of glucose effectiveness by a factor of three.
The excellent concordance between the minimal model estimate of glucose effectiveness obtained from the BI-IVGTT and GE is in contrast with the findings of Finegood and Tzur (10), who reported no correlation between the SG derived from the BI-IVGTT and SG(clamp), i.e., the clamp-based index of glucose effectiveness. A likely explanation of this discrepancy is related to the fact that glucose effectiveness spans a relatively narrow range in many different metabolic states. This makes the correlation analysis between different estimates of glucose effectiveness extremely sensitive to measurement errors and day-to-day variability. Note that in this study SGV and GE have been calculated from the same BI-IVGTT data, whereas in Ref. 10 the SG and SG(clamp) were estimated with different experimental approaches on different days.
The concordance between SGV
obtained from the BI-IVGTT and GE indicates that the single-compartment
minimal model is adequate to measure glucose effectiveness when insulin
remains basal during the IVGTT. This occurs despite the fact that the
minimal model SGV hinges on a
single-pool description of the glucose system and is calculated by
ignoring the first 10-min glucose samples, whereas GE is based on much
broader assumptions about the glucose system and is calculated by
relying on the whole glucose data set from 0 to 180 min. The reason why
this happens is that, during a BI-IVGTT, the fast component of glucose
disappearance, which is not accounted for by the minimal model, quickly
fades away (within the initial 20 min of the test). From that time on,
glucose decay is well described by the slow component only. This can
easily be seen by referring to the parameters of the two-exponential function used to calculate AUC[G], as in
Eq. 3. Because
A/
<< B/
(see
RESULTS), for the calculation of
AUC[
G], and thus of GE, the (slowest) exponential
function is enough, or, in other terms, the contribution to
AUC[
G] of the fast exponential is negligible. Now, if
one sees the minimal model in its exponential version
(Eq. 6), it is clear why the
model-independent GE and the minimal model
SGV give the same results. Note
that this reasoning would have been much less transparent if the
trapezoidal method had been used to calculate AUC[
G],
and thus GE. One additional observation is that the equivalence between
SGV and GE, which holds for a
BI-IVGTT, cannot be taken for granted in other experimental conditions,
even when insulin is maintained at the basal level. In fact, there may
be cases in which, due to a format of glucose administration with
relatively rapid and frequent changes, the behavior of the
glucose system cannot be well approximated by a single-compartment model.
Our result, that the minimal model glucose effectiveness obtained from
a DI-IVGTT is three times higher than that estimated from a BI-IVGTT,
is clearly a symptom of model error. The sensitivity of
SG to the IVGTT insulin profile
has also recently been observed in dog studies by Finegood and Tzur
(10), but no mechanistic explanation of why this happens was offered in
that study. In the present study, we resorted to Monte Carlo simulation
to clarify whether single-compartment undermodeling can explain
SG sensitivity to insulin
dynamics. A physiologically based two-compartment model was used to
simulate IVGTT glucose data in the presence of different insulin
profiles. We reasoned that, if undermodeling plays a role in making
SG sensitive to insulin dynamics
during the IVGTT, the minimal model
SG estimated from simulated
BI-IVGTT and DI-IVGTT data should exhibit the same trend observed with
real data. As a matter of fact, similarly to Finegood and Tzur (10) and
to the experimental results obtained in the present study,
SG estimated from a simulated
DI-IVGTT (Fig. 3, profiles A and
B) was higher than that estimated
from a simulated BI-IVGTT (profile
F). Moreover, SG
progressively decreased with the early insulin response (Fig. 3,
profiles C, D, and
E), thus corroborating the
suggestion of Finegood and Tzur that caution must be exercised in the
interpretation of differences in the estimates of
SG between subject groups with significant differences in -cell function. Of note is that the value
of SGV estimated from simulated
BI-IVGTT (profile F) was close to
the glucose effectiveness of the reference two-compartment model
(0.0023 vs. 0.0021 dl · min
1 · kg
1).
This result obtained from simulated data confirms that the minimal
model yields an accurate estimate of glucose effectiveness only
when insulin remains at the basal level during the IVGTT.
All in all, the experimental results of this study, those of Finegood and Tzur (10), and our Monte Carlo simulations suggest that the single-pool description is reasonably adequate when the glucose system is studied at basal insulin but becomes critical when insulin is elevated in the initial portion of the IVGTT. A possible explanation is related to the fact that, at variance with the BI-IVGTT when glucose decay is dictated by glucose effectiveness only, during a DI-IVGTT the minimal model has to distinguish between the individual contributions of glucose and insulin action to glucose disappearance. During a DI-IVGTT, SG is mainly estimated in the initial portion of the test, when glucose concentration is high and insulin action, albeit increasing, is still low. As a result, SG assumes a value that reflects both the fast and slow components of glucose disappearance per se. The value taken on by SG progressively decreases with the early insulin response, because the portion of the IVGTT crucial for its estimation (i.e., when glucose is high and insulin action is low) becomes wider and wider. As a consequence, SG reflects a combination of the two components in which the role played by the fast component becomes less and less important. In particular, during a BI-IVGTT, SG gets close to the slow component because SG is estimated from the entire 180-min glucose data set.
In conclusion, the results of the present study show that the minimal model estimate of glucose effectiveness is very similar to a model-independent measurement when insulin is kept at basal level, but not when it exhibits the dynamic pattern traditionally observed during an IVGTT. Monte Carlo simulation results suggest that single-compartment undermodeling can explain SG sensitivity to insulin dynamics and that the early insulin response during the IVGTT markedly influences the value assumed by SG.
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ACKNOWLEDGEMENTS |
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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." It was presented in abstract form at the 57th Meeting of the American Diabetes Association, Boston, MA, June 1997.
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FOOTNOTES |
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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. §1734 solely to indicate this fact.
Address for reprint requests: C. Cobelli, Dept. of Electronics and Informatics, Via Gradenigo 6/A, Univ. of Padova, 35131 Padova, Italy.
Received 7 May 1998; accepted in final form 17 August 1998.
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