1Department of Information Engineering, University of Padova, I-35131 Padova; 2San Raffaele Scientific Institute, 20131 Milano, Italy; and 3Division of Endocrinology, Diabetes, Metabolism, and Nutrition, Department of Internal Medicine, Mayo Clinic and Foundation, Rochester, Minnesota 55905
Submitted 10 July 2003 ; accepted in final form 30 April 2004
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ABSTRACT |
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glucose kinetics; rate of appearance of glucose; tracer; meal; insulin resistance
Unfortunately, the estimation of insulin sensitivity after an oral glucose perturbation is not an easy task, the major obstacle being that the rate of appearance of glucose absorption is unknown. The most reliable, probably the reference method for estimating insulin sensitivity during a meal, is a two-step procedure. First, the rate of appearance of absorbed glucose is reconstructed as accurately as possible with a tracer method. A reliable method uses a tracer-to-tracee clamp technique in conjunction with a model of glucose kinetics in nonsteady state (14). Two tracers are necessary, one given intravenously, mimicking the rate of appearance of glucose absorption, and one together with the meal. Once the rate of appearance of absorbed glucose is known, to estimate insulin sensitivity one has to explain the meal plasma glucose measurements with a model describing insulin action on glucose production and disposal. This model already exists and is the classic single-compartment minimal model. In fact, when glucose and insulin change smoothly as during an oral test, the single-compartment description of glucose kinetics has proven to be sufficiently accurate (2).
In summary, the state-of-the-art method for the estimation of insulin sensitivity during an oral test entails the use of a two-tracer protocol in conjunction with a model of non-steady-state glucose kinetics to obtain the rate of appearance of glucose absorption (Ra mealref), and then the use of the minimal model to extract insulin sensitivity (SIref) from plasma glucose and insulin concentrations. Since this is an expensive and labor-intensive procedure, an alternative approach capable of reducing the cost and the effort is required to favor a widespread application of the oral test for the estimation of insulin sensitivity. Recently, a nontracer method based on a new model, the oral minimal model (OMM), has been proposed to estimate insulin sensitivity (SI) together with the glucose rate of appearance (Ra meal) from plasma glucose and insulin concentrations measured after a meal or an oral glucose tolerance test (OGTT; see Ref. 9). The method has obvious advantages over the reference tracer method discussed above, but its performance in recovering both insulin sensitivity and rate of appearance of glucose absorption has never been compared with other methods, albeit some reassuring indirect validation results are available: in two studies (3, 9) insulin sensitivity was compared with that obtained from IVGTT performed in the same subjects, and correlation was satisfactory. It is thus important to establish whether or not the "nontracer" Ra meal and SI provided by OMM compare well with the "tracer" Ra mealref and SIref provided by the reference method.
This is precisely the aim of the present contribution. We do this comparison on a database consisting of 88 subjects who underwent the triple-tracer meal protocol described (4).
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DATABASE AND PROTOCOL |
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Plasma samples were collected at times (t) 120, 30, 20, 10, 0, 5, 10, 15, 20, 30, 40, 50, 60, 75, 90, 120, 150, 180, 210, 240, 260, 280, 300, 360, and 420 min. Figure 1 shows the mean glucose and insulin plasma concentration curves.
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METHODS |
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Reconstruction of Ra mealref.
An accurate estimation of Ra mealref was obtained by applying the single-compartment model of Steele et al. (14) to the clamped tracer-to-tracee ratio (TTR) = tracer III/Gex:
![]() | (1) |
Estimation of SIref .
By knowing Ra mealref, it was possible to identify the classic minimal model in each subject by using Ra mealref as the known exogenous input. The model is described by
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Nontracer Method
A new model has been recently developed to estimate insulin sensitivity together with the glucose rate of appearance during an oral test (9). It couples the single-compartment minimal model with a parametric description of the glucose rate of appearance:
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The parametric description of Ra meal proposed (9) is a piecewise-linear function with known break point ti and unknown amplitude i:
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Parameter Estimation
Tracer method.
To estimate Ra mealref, we used the single-compartment model of Steele et al. (14). Because the tracer-to-tracee ratio was well clamped, a two-compartment model gave virtually the same results (4). The tracer-to-tracee ratio profile was smoothed using WINSTODEC (13), which also provides the first derivative of the smoothed signal. Finally, the minimal model of glucose kinetics was numerically identified by nonlinear least squares (7, 8), as implemented in SAAM II (Simulation Analysis and Modeling software; see Ref. 1). Measurement error on glucose data was assumed to be independent, Gaussian, with zero mean and known SD [coefficient of variation (CV) = 2%]. Insulin concentration is the model-forcing function and was assumed to be known without error. The model provided estimates of SGref, Vref, p2ref, and SIref . Knowledge of Ra mealref allowed us to calculate fref [the fraction of the ingested dose (D) that is actually absorbed] as
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Nontracer method.
OMM was numerically identified with the same strategy described above. However, OMM identification requires a number of assumptions that were discussed in detail (9). Briefly, to ensure its a priori identifiability, one has to assume values for V and SG. Here, we fixed them to the median obtained with the tracer method, i.e., V = Vm-ref, SG = SGm-ref (median was preferred to mean values since parameters are not normally distributed, see RESULTS). To improve numerical identifiability of the remaining parameters, p2, p3, and i (i = 1... 8), a Gaussian bayesian prior was considered on the square root of p2ref (SQRp2ref), which is normally distributed (see RESULTS): SQRp2 = mean SQRp2ref; SD = 10%. Finally, a constraint was imposed to guarantee that the area under the estimated Ra meal equals the total amount of ingested glucose, D, multiplied by the fraction of the ingested dose that is actually absorbed, f (fixed to the median of reference values f = fm-ref): this constraint provides an additional relationship among the unknown parameters
i, thus reducing the number of unknowns by one. In summary, OMM provides estimates of SI, p2, and
i (i.e., the Ra meal profile).
Statistical Analysis
Data are presented as means ± SE. Two-sample comparisons were done by Wilcoxon signed-rank test, and Shapiro-Wilk test was used to verify if parameters are normally distributed (significance level set to 5%). Pearson's correlation was used to evaluate univariate correlation. To investigate how sensitive the OMM estimate of SI was to the assumptions on V, SG, f, and p2, we investigated the relationship between the percent deviation of SI, SI%, and the percent deviation of the various parameters that were fixed (namely
V%,
SG%,
f%, and
p2%) using regression analysis. The percent deviation of each parameter was calculated as the difference between the reference and OMM parameter value divided by the reference. We used single-regression analysis to determine the association of
SI% with
V%,
SG%,
f%, and
p2% separately. Next, we used forward and backward stepwise multiple regression (using F ratio-to-remove = 4 and F ratio-to-enter = 3.996) to assess the relevance of the fixed parameters as predictors of the error affecting SI.
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RESULTS |
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Ra mealref obtained using the tracer-to-tracee ratio clamp technique is shown in Fig. 2A.
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The mean profile of Ra meal provided by the model is shown in Fig. 2A. The fit of OMM was satisfactory (Fig. 3B). Insulin sensitivity was estimated with good precision: SI = 11.68 ± 0.73 x 104 dl·kg1·min1·µU1·ml, CV = 7 ± 0.3%; parameters i were also estimated with good precision, the less precise parameter being
8, which is expected to be very close to zero:
1 = 5.36 ± 0.33 mg·kg1·min1 (CV = 12 ± 2%),
2 = 7.78 ± 0.24 mg·kg1·min1 (CV = 10 ± 0.4%),
3 = 6.00 ± 0.28 mg·kg1·min1 (CV = 27 ± 9%),
4 = 5.05 ± 0.22 mg·kg1·min1 (CV = 11 ± 0.5%),
5 = 4.77 ± 0.28 mg·kg1·min1 (CV = 30 ± 15%),
6 = 3.52 ± 0.19 mg·kg1·min1 (CV = 12 ± 2%),
7 = 2.09 ± 0.09 mg·kg1·min1 (CV = 6 ± 0.5%), and
8 = 0.34 ± 0.05 mg·kg1·min1 (CV = 42 ± 4%). The bayesian estimate of p2 gives p2 = 0.011 ± 0.0005 min1 (CV = 15 ± 0.7%).
Comparison
A good agreement was found between the OMM Ra meal and Ra mealref profiles (mean curves are shown in Fig. 2A). We calculated the relative deviation of OMM Ra meal estimates from the reference Ra mealref in each breakpoint of the piecewise-linear function as:
![]() | (8) |
The temporal profile of the mean Ra meal is shown in Fig. 2B and does not present systematic deviation from zero. Moreover, SI provided by OMM and SIref were not significantly different (SI = 11.68 ± 0.73 104 dl·kg1·min1·µU1·ml, SIref = 11.55 ± 0.68 x 104 dl·kg1·min1·µU1·ml) and well correlated (r = 0.86, P < 0.0001; Fig. 5). To quantify the agreement between OMM and reference SI in each individual, we calculated the absolute percentage deviation between the two measurements as:
![]() | (9) |
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DISCUSSION |
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Recently, a new OMM has been proposed that is able to estimate insulin sensitivity SI in a given individual from plasma glucose and insulin concentration measured after an oral glucose perturbation, by simultaneously reconstructing also the rate of appearance of absorbed glucose (Ra meal). In this study, we have validated the OMM against what can be considered today the state-of-art reference method for estimating insulin sensitivity during a meal or an OGTT. The reference method relies on tracers and comprises the following two steps: first, a two-tracer protocol is used in conjunction with a model of non-steady-state glucose kinetics to obtain the rate of glucose absorption; second, the calculated rate of glucose absorption is used in conjunction with the classic minimal model to measure insulin sensitivity from the measured plasma glucose and insulin concentrations.
To test the nontracer OMM against the tracer method, we analyzed with both methods 88 subjects who underwent the triple-tracer meal protocol described (4). The comparison between the Ra meal profile obtained with the OMM and Ra mealref indicates a good agreement between the two (Fig. 2). The temporal plot of the deviations of OMM Ra meal estimates from the reference in each breakpoint of the piecewise-linear function does not present systematic deviation from zero. However, some differences between the reference profile and OMM prediction can be seen at the individual level. A likely explanation for these discrepancies is that OMM requires us to fix some parameters. In particular, the use of a population value for the fraction of absorbed dose (f) forces the area under the curve of Ra meal to take on a value that is correct on average but differs slightly at the individual level. The comparison between insulin sensitivity indexes obtained by OMM and the reference model indicates an excellent agreement in average between the estimates of insulin sensitivity (SIref = 11.55 ± 1 x 104 dl·kg1·min1·µU1·ml, SI = 11.68 ± 2 x 104 dl·kg1· min1·µU1·ml); correlation between the two indexes is satisfactory (r = 0.86, P < 0.0001; Fig. 5).
The ability of the OMM to provide results that are in good agreement with those provided by the more complex and expensive tracer method hinge on a simple but effective parametric modeling of the rate of glucose absorption and on the assumption that some model parameters such as V, SG, and f equal their population values. In addition, the model uses a bayesian prior on p2 to improve the numerical identification properties of the model. The novelty of the present study is that the reference model provides values of kinetic parameters during a meal. The large number of subjects studied in the present experiments permitted the distributions to be determined for all parameters. As is evident in Fig. 4, kinetic parameters were not normally distributed (as confirmed by the Shapiro-Wilk normality test, P < 0.01). Therefore, the median rather than mean values (Vm-ref = 1.45 dl/kg, SGm-ref = 0.025 min1, fm-ref = 0.90) were used as reference for OMM to better describe nonnormality of the distributions. SAAM II software can only provide Gaussian priors for the Bayesian estimation. Because SQRp2, but not p2, is normally distributed (P = 0.35 and P < 0.0001, respectively, by Shapiro-Wilk test), the model was parameterized to make SQRp2 explicit and use the Gaussian prior on it (Fig. 4). With the use of these reference values, OMM provided an accurate estimate of insulin sensitivity not significantly different from SIref (SI = 11.68 ± 0.73 vs. SIref = 11.55 ± 0.68 x 104 dl·kg1·min1·µU1·ml) also when insulin sensitivity is expressed per unit of distribution volume (SI = 8.05 ± 0.50 vs. SIref = 8.56 ± 0.60 x 104 min·µU1·ml). In previous studies (3, 9), mean population values derived from IVGTT and clamp studies were used [V = 1.7 dl/kg; SG = 0.014 min1, p2 = 0.03 min1, f = 0.86 (4)]. As seen above, mean values of parameters during a meal are different from those of the IVGTT; in particular, volume of distribution and p2 are lower (Vref = 1.50 dl/kg, p2ref = 0.013 min1), whereas glucose effectiveness is higher (SGref = 0.031 min1). However, the OMM method is quite robust on average: if intravenous values of kinetic parameters are used for OMM identification, insulin sensitivity (SIiv = 11.82 ± 0.66 x 104 dl·kg1·min1· µU1·ml; see Ref. 3) is not statistically different from SIref (insulin sensitivity per unit of distribution volume is different, SIiv = 6.95 ± 0.39 x 104 min·µU1·ml vs. SIref = 8.56 ± 0.60 x 104 min·µU1·ml, P = 0.0005 because of differences in the distribution volume) and correlation between OMM and the reference model remains very good (r = 0.83, P < 0.001).
The necessity of using some population a priori knowledge for OMM numerical identification can affect SI estimates in the single individual: the absolute percentage deviation between reference and OMM measurements is | SI | = 29 ± 3% on average (range 0.42160%). The discrepancy between the estimates of insulin sensitivity obtained with the reference model and with the OMM arises, at least in part, from the assumptions that parameters such as V, SG, f, and p2 take on the same value in all subjetcs (p2 is constrained to a bayesian prior). To investigate how sensitive the OMM estimate of SI was to the assumptions made on V, SG, f, and p2, we investigated the relationship existing between the percent deviation of SI,
SI%, and the percent deviation on the various parameters that were fixed (namely
V%,
SG%,
f%, and
p2%) using multiple-regression analysis. We found that the percentage deviation of f, SG, and p2 from the fixed values explains the deviation in SI estimate (0.752, P < 0.0001), whereas the deviation of V does not give a significant contribution to the regression. We conclude that the OMM is quite robust and well applicable in population studies, although it is less accurate in estimating the individual values of SI.
Until now, it has been possible to compare the SI provided by OMM with the analogous index obtained from the IVGTT (3, 9). Although SI values estimated during IVGTT were 40% lower than OMM ones, they correlated well with each other (r = 0.62, P < 0.001). Whereas these results were reassuring, one source of concern was that the experimental conditions during the IVGTT were markedly different from those during the oral test both in terms of the route of glucose administration and in the pattern of change of plasma glucose and insulin concentrations. Also, in the previous study (3, 9), the IVGTT studies and the oral tests were not performed on the same days, adding a potential source of variability to the comparison. Thus the present validation of the OMM estimates of Ra meal and SI against an independent "oral reference technique" adds credibility to the OMM and provides the necessary prerequisite for a more general validation of this novel method against the hyperinsulinemic-euglycemic clamp: this important issue may be addressed in a separate study, but the available preliminary results (6) are reassuring.
Finally, it is also of interest to compare OMM and reference SI with other indexes, such as the Homeostasis Model Assessment (HOMA) (15), the Quantitative Insulin Sensitivity Check Index (QUICKI) (11), and the insulin sensitivity index (ISI) of Matsuda-De Fronzo (12), frequently used to measure insulin sensitivity during an OGTT or meal. Correlation between SIOMM and ISI index is r = 0.63, whereas correlation with HOMA or QUICKI is much lower (r = 0.43 and 0.46, respectively). This was expected because HOMA and QUICKI indexes are derived from glucose and insulin concentration in the basal state, whereas OMM and ISI indexes are both estimated using the 420-min data. Similar results were found if SIref was compared with the three indexes (HOMA: r = 0.43; QUICKI: r = 0.44; ISI: r = 0.59).
In conclusion, we have validated the nontracer OMM method by comparing its Ra meal and SI measurements with those obtained by the tracer reference method. Our results indicate that the OMM provides an accurate estimation of both the Ra meal profile and insulin sensitivity. Thus the OMM candidates as a simple, cost-effective, and reliable tool to measure both the rate of glucose absorption and insulin sensitivity from oral glucose tests without employing tracers.
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GRANTS |
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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. Section 1734 solely to indicate this fact.
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REFERENCES |
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