Abstract
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 modelindependent estimate of glucose effectiveness with the minimal model estimate by reanalyzing published data in which insulindependent 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 (BIIVGTT). In the second case, a dynamic insulin response was recreated (DIIVGTT). Our results show that minimal model glucose effectiveness is very similar to the modelindependent measurement during a BIIVGTT but is three times higher during a DIIVGTT. To investigate the causes of minimal model overestimation in the presence of a dynamic insulin response, Monte Carlo simulation studies on a twocompartment model of glucose kinetics with various insulin response patterns were performed. Results suggest that minimal model overestimation is due to singlecompartment 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 (S_{G}) can be obtained from the minimal model analysis of an intravenous glucose tolerance test (IVGTT). However, both indirect (68) and direct (10, 12) experimental evidence indicates that S_{G} is an overestimate of the true glucose effectiveness and that S_{G}estimation is influenced by the insulin profile during the IVGTT.
The overestimation of S_{G} has been suggested (7) to be due, in large part, to the singlecompartment approximation of glucose kinetics used by the minimal model. Specifically, S_{G} would incorporate a component reflecting the exchange kinetics between the accessible and the inaccessible pool of the glucose system. The relationship between S_{G} 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 S_{G} to the insulin profile during the IVGTT has not been well characterized, and many issues remain open to question (5). Does overestimation of S_{G} 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 S_{G}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 insulindependent diabetes mellitus (IDDM) were given an IVGTT on one occasion with only basal insulin provided (BIIVGTT) and on a second occasion in the presence of a dynamic insulin response (DIIVGTT), in which a normal insulin response was recreated through a computercontrolled infusion of insulin. In our reanalysis of this data, we first calculated from BIIVGTT data the modelindependent reference measure of glucose effectiveness, GE, recently proposed in Ref. 1. We next used the minimal model to obtain estimates of S_{G} from both the BIIVGTT and DIIVGTT. Our results show that S_{G}derived from the BIIVGTT is similar and very well correlated to GE. In contrast, S_{G} derived from the DIIVGTT is overestimated and correlates weakly with GE. Finally, to investigate whether overestimation of S_{G} in the presence of an incremental insulin response might be due to singlecompartment undermodeling, we used a twocompartment 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 S_{G} is most likely a consequence of undermodeling and that S_{G}decreases as insulin availability in the first 20 min of the test decreases.
METHODS
The Data
The data we reanalyzed in this study were previously published, and we refer to the original publication for all details concerning subjects, experimental protocol, and assays (12). Briefly, seven subjects with IDDM, who had no detectable endogenous insulin secretion, each underwent a BIIVGTT and a DIIVGTT (0.3 g/kg). During the BIIVGTT, insulin was infused at a basal rate identical to that required to keep the patient euglycemic during an overnight fast. During the DIIVGTT, a computercontrolled insulin infusion was given in response to the intravenous glucose load to mimic a normal insulin secretory response.
Glucose Effectiveness: ModelIndependent Estimation
It has recently been shown (1) that glucose effectiveness can be calculated under very general assumptions, i.e., virtually in a modelindependent 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., BIIVGTT). 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 AUCbased index of glucose effectiveness is also equivalent, as shown in Ref. 1, to the analogous clampbased index. During a BIIVGTT, glucose effectiveness (GE) is
AUC[ΔG] 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 BIIVGTT glucose data ΔG are glucose decay data after a bolus injection, the natural candidate to describe them is a sum of decaying exponentials. We found that a twoexponential model was necessary and sufficient, according to Akaike’s criterion (4), to describe ΔG data
Glucose Effectiveness: Minimal Model Estimation
Glucose effectiveness was estimated with the minimal model of glucose disappearance (2) from both DI and BIIVGTT data. As usually done in minimal model identification, the first 10min glucose samples were ignored to favor the singlecompartment approximation of glucose kinetics.
DIIVGTT.
During the DIIVGTT, glucose disappearance was described by the classical minimal model equations (2)
Parameters S_{G}, V,p _{2}, andp _{3} were estimated by weighted nonlinear least squares (6) by assuming a 2% glucose measurement error with weights chosen optimally. Precision of parameter estimates was obtained from the inverse of the Fisher information matrix (4).
BIIVGTT.
During the BIIVGTT, the above basal insulin action X is identically null, and the model reduces to
Monte Carlo Simulation
To test the hypothesis that singlecompartment undermodeling is responsible for the discrepancies between estimates of glucose effectiveness obtained by modelindependent 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 twocompartment model of glucose kinetics with endogenous glucose production described by the same glucoseinsulin 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 twocompartment model and, for each insulin profile, 200 noisy IVGTT glucose data sets were generated. In addition to the standard (see Fig. 3 A) and the basal insulin IVGTT (see Fig. 3 F), profiles from an insulinmodified IVGTT showing a progressively decreasing firstphase insulin response (from normal in Fig.3 C to no response in Fig.3 E) 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 10min glucose samples).
Statistical Analysis
Data in the text and Figs. 13 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 ttest was used to compare different measures of glucose effectiveness made in the same subject.P values <0.05 were considered statistically significant.
RESULTS
Experimental Data
The time courses of plasma glucose (top) and insulin (bottom) concentrations during the BIIVGTT and DIIVGTT are shown in Fig. 1(left andright, respectively). During the BIIVGTT, the insulin level was relatively constant, and the glucose profile approached the baseline ∼6 h after the glucose injection. During the DIIVGTT, the glucose profile was similar to the one that is commonly observed in subjects with normal glucose tolerance.
ModelIndependent GE
The twoexponential model fit was very good and is shown in Fig.2. The modelindependent GE was calculated from precisely estimated parameters. For example, α = 0.205 ± 0.022 min^{−1}, with a mean precision of 17% (range 11–30%), and β = 0.0040 ± 0.0009 min^{−1}, with a mean precision of 21% (range 3–24%). GE was 0.0099 ± 0.0017 dl ⋅ min^{−1} ⋅ kg^{−1}, with a mean precision of 3% (Table 1).
Minimal Model Analysis of BIIVGTT and DIIVGTT Data
The minimal model fit of BIIVGTT data was as good as that provided by the twoexponential model from 10 min on (Fig. 2). Parameters S_{G} and V were estimated with good precision. The mean values of S_{G}and V were, respectively, 0.0044 ± 0.0007 (min^{−1}) and 2.76 ± 0.25 (dl/kg) with a mean precision of 5% (range 2–12%) and 2% (range 1–4%). Of note is that S_{G} (0.0044 ± 0.0007 min^{−1}) virtually coincided with the slowest component β (0.0044 ± 0.0009 min^{−1}) of the twoexponential model and that there was an excellent correlation between them (r = 0.99,P < 0.001).
In the case of DIIVGTT, the minimal model fit assessed in terms of residuals was good, and all the parameters S_{G}, V,p _{2}, andp _{3} were estimated with satisfactory precision. In particular, the mean value of S_{G} was 0.0152 ± 0.0029 min^{−1}, 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 modelindependent and minimal model analysis of this data are shown in Table 1. Because S_{G} measures fractional glucose effectiveness (i.e., per unit of glucose distribution volume), it was multiplied by V to obtain a minimalmodel measure of glucose effectiveness, S_{G}V, comparable with GE (1, 5). Precision of S_{G}V was obtained by error propagation. S_{G}V estimated from BIIVGTT data (0.0112 ± 0.0012 dl ⋅ min^{−1} ⋅ kg^{−1}, mean precision 5%) was virtually identical to GE. In addition, S_{G}V was highly correlated with GE (r = 0.97,P < 0.001), with the regression line not statistically different from the identity line. In contrast, S_{G}V from DIIVGTT data (0.0294 ± 0.0050 dl ⋅ min^{−1} ⋅ kg^{−1}) was three times higher than GE, and the two measures were poorly correlated (r = −0.4).
The agreement between S_{G}V derived from the BIIVGTT and GE was remarkable in each individual except forsubject no. 3, where S_{G}V 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 20min glucose samples. S_{G}V in subject no. 3 decreased from 0.0069 to 0.0033 min^{−1}, 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 S_{G}V 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 S_{G} estimation are shown in Fig.3. The results are reported in Table2. They show that, when the minimal model is used to interpret glucose data generated by a more complex twocompartment model, S_{G}estimation is markedly influenced by insulin dynamics. Glucose effectiveness was virtually the same with the standard and the insulinmodified IVGTT, i.e., the DIIVGTT (profiles A and B) but markedly decreased with the early insulin response (profiles C, D, andE). When insulin remained at the basal level throughout the test (profile F), as during the BIIVGTT, the lowest value of glucose effectiveness was obtained.
DISCUSSION
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 insulinmodified 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 modelindependent measure of glucose effectiveness was needed to relate minimal model estimates to a reference measure.
In this study we have used the recently proposed modelindependent 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 S_{G}V) is estimated from an IVGTT in which insulin is kept constant at the basal level (BIIVGTT), its results are virtually identical to the glucose effectiveness index GE obtained in a modelindependent way from AUC calculations. In contrast, when insulin changes dynamically (DIIVGTT), the minimal model overestimates the modelindependent measure of glucose effectiveness by a factor of three.
The excellent concordance between the minimal model estimate of glucose effectiveness obtained from the BIIVGTT and GE is in contrast with the findings of Finegood and Tzur (10), who reported no correlation between the S_{G} derived from the BIIVGTT and S_{G(clamp)}, i.e., the clampbased 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 daytoday variability. Note that in this study S_{G}V and GE have been calculated from the same BIIVGTT data, whereas in Ref. 10 the S_{G} and S_{G(clamp)} were estimated with different experimental approaches on different days.
The concordance between S_{G}V obtained from the BIIVGTT and GE indicates that the singlecompartment minimal model is adequate to measure glucose effectiveness when insulin remains basal during the IVGTT. This occurs despite the fact that the minimal model S_{G}V hinges on a singlepool description of the glucose system and is calculated by ignoring the first 10min 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 BIIVGTT, 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 twoexponential function used to calculate AUC[ΔG], as inEq. 3 . BecauseA/α <<B/β (seeresults), for the calculation of AUC[ΔG], and thus of GE, the (slowest) exponential function is enough, or, in other terms, the contribution to AUC[ΔG] of the fast exponential is negligible. Now, if one sees the minimal model in its exponential version (Eq. 6), it is clear why the modelindependent GE and the minimal model S_{G}V give the same results. Note that this reasoning would have been much less transparent if the trapezoidal method had been used to calculate AUC[ΔG], and thus GE. One additional observation is that the equivalence between S_{G}V and GE, which holds for a BIIVGTT, 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 singlecompartment model.
Our result, that the minimal model glucose effectiveness obtained from a DIIVGTT is three times higher than that estimated from a BIIVGTT, is clearly a symptom of model error. The sensitivity of S_{G} 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 singlecompartment undermodeling can explain S_{G} sensitivity to insulin dynamics. A physiologically based twocompartment 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 S_{G} sensitive to insulin dynamics during the IVGTT, the minimal model S_{G} estimated from simulated BIIVGTT and DIIVGTT 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, S_{G} estimated from a simulated DIIVGTT (Fig. 3, profiles A andB) was higher than that estimated from a simulated BIIVGTT (profile F). Moreover, S_{G}progressively decreased with the early insulin response (Fig. 3,profiles C, D, andE), thus corroborating the suggestion of Finegood and Tzur that caution must be exercised in the interpretation of differences in the estimates of S_{G} between subject groups with significant differences in βcell function. Of note is that the value of S_{G}V estimated from simulated BIIVGTT (profile F) was close to the glucose effectiveness of the reference twocompartment 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 singlepool 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 BIIVGTT when glucose decay is dictated by glucose effectiveness only, during a DIIVGTT the minimal model has to distinguish between the individual contributions of glucose and insulin action to glucose disappearance. During a DIIVGTT, S_{G} 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, S_{G} assumes a value that reflects both the fast and slow components of glucose disappearance per se. The value taken on by S_{G} 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, S_{G} 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 BIIVGTT, S_{G} gets close to the slow component because S_{G} is estimated from the entire 180min 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 modelindependent 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 singlecompartment undermodeling can explain S_{G} sensitivity to insulin dynamics and that the early insulin response during the IVGTT markedly influences the value assumed by S_{G}.
Acknowledgments
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.
Footnotes

Address for reprint requests: C. Cobelli, Dept. of Electronics and Informatics, Via Gradenigo 6/A, Univ. of Padova, 35131 Padova, Italy.

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 Copyright © 1998 the American Physiological Society