1Department of Information Engineering, University of Padova, Padua; 2San Raffaele Scientific Institute, Milan, Italy; and 3Division of Endocrinology, Diabetes, Metabolism, and Nutrition Department of Internal Medicine, Mayo Clinic and Foundation, Rochester, Minnesota
Submitted 18 July 2004 ; accepted in final form 14 June 2005
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ABSTRACT |
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insulin action; glucose utilization; tracer-to-tracee activity clamp; intravenous glucose tolerance test; oral glucose tolerance test
The aim of this study is to develop and validate an oral labeled minimal model (OMM*). We show that OMM* provides reliable estimates of disposal and Ra meal. To validate OMM*, we took advantage of a unique data set (11) containing 88 individuals who underwent a triple-tracer labeled meal, as well as a labeled IVGTT (IVGTT*). The triple-tracer labeled meal, thanks to the use of the tracer-to-tracee clamp technique, provided a model-independent reference for the appearance rate of ingested glucose (
) (4).
was then used as a known input of a model of labeled glucose kinetics. This model, denoted as reference tracer model (RM*), was identified from labeled meal data and yielded a reference measure of disposal insulin sensitivity, S
. OMM* is validated by comparing OMM*-based estimates of
and Ra meal to
andR
, respectively. Validation is further strengthened by comparing OMM*
with the IVGTT*
measured in the same subjects with the traditional labeled minimal model.
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METHODS |
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The database consisted of 88 normal subjects [46 males and 42 females: age = 58 ± 2 yr (range 1987); body wt = 77 ± 2 kg (range 53129); BMI = 26.71 ± 0.1 kg/m2 (range 2035); fasting glucose = 92.07 ± 0.7 mg/dl (range 77.29105.12)] who received both a triple-tracer mixed meal and an IVGTT*.
Labeled mixed meal.
The triple-tracer mixed meal (10 kcal/kg: 45% carbohydrate, 15% protein, 40% fat) contained 1 ± 0.02 g/kg glucose. The meal was labeled with [1-13C]glucose (G*), thus allowing us to derive the exogenous, i.e, coming from the meal, glucose (Gmeal) as
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Figure 1, AC, left, shows mean glucose, exogenous glucose, and insulin plasma concentration curves. Beginning at time 0, [6-3H]glucose was infused intravenously at a variable rate, mimicking Ra meal. [6,6-2H2]glucose was also infused as part of a separate protocol. Because the ratio in plasma between [6-3H]glucose and [1-13C]glucose was maintained almost constant (Fig. 2A), Steele's model provided an essentially model-independent estimate of the Ra of ingested tracer (R) (4); total R
was then calculated from R
(Fig. 2B) as
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Models
Oral minimal models.
For the sake of clarity, a brief description of the "cold" (i.e., unlabeled) oral minimal models (11, 12) precedes the presentation of the new "hot" (i.e., labeled) OMM*. This is done not only because the two oral minimal models hinge on the same monocompartmental structure of glucose kinetics, but also because they share the exogenous glucose input, i.e, Ra meal. Denoting by G the total plasma glucose concentration, the rate of glucose disappearance (Rd) and the net hepatic glucose balance (NHGB) (Fig. 3, left) model equation following Ref. 5 is
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| (5) |
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Labeled oral minimal model.
OMM is unable to distinguish the individual contribution of glucose production and disposal. To overcome this limitation, a glucose tracer is administered orally and unlabeled glucose and plasma concentrations of the glucose tracer are measured. OMM* relies on Gmeal (eq. 1), and model equation is
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| (8) |
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Because OMM* and OMM share Ra meal, the two models were identified simultaneously (see Identification).
Reference-labeled model.
The validation of OMM* estimates of Ra meal and was accomplished by using the following rationale. As mentioned in Data, during the labeled meal, two additional tracers were infused intravenously. In particular, [6-3H]glucose was infused to clamp the ratio between concentrations in plasma of [6-3H]glucose and ingested glucose tracer. This allowed us to derive reliable and virtually model-independent estimates of Ra meal (4). This estimate, denoted R
, was used not only as a reference with which OMM* estimate of Ra meal was compared but also as a known input of a model with the same structure of OMM* (Fig. 3, right):
![]() | (10) |
![]() | (11) |
By identifying this model, denoted as reference-labeled model (RM*), from Gmeal and insulin data we were able to obtain reference values for OMM* parameters (indicated by ref), in particular for (see Identification). Comparison between
, estimated with RM*, and
, provided by OMM*, allowed OMM* validation.
IVGTT* minimal models.
IVGTT* data were interpreted with the classic single-compartment IVMM (5) and with the labeled two-compartment minimal models (IVMM*) (14), thus obtaining an estimate of SI and in the same subjects. This, in addition to providing a one-to-one comparison of IVGTT*
vs. OMM*
, allows us to examine the relationship between SI and
in IVGTT and meal.
Identification
Identifiability.
Because OMM* (like OMM) is a priori nonidentifiable, the a priori knowledge necessary for its identification was obtained from RM*. OMM* was thus identified by fixing V* and to the mean values obtained with RM*, i.e., V* = V* ref,
=
. Mean values can safely be used because they are normally distributed (see RESULTS). At variance with OMM, where a Bayesian prior on p2 was needed to improve numerical identifiability, OMM* takes advantage of the fact that it shares Ra meal with OMM. The simultaneous identification of OMM* and OMM from two measurements (Gmeal and G) relaxes the necessity of using Bayesian priors for p2 and
. A constraint (11, 12) was imposed to guarantee that the area under the estimated Ra meal equals the total amount of ingested glucose, D, multiplied by the fraction that is actually absorbed, f. Because f-values estimated with RM* were not normally distributed (RESULTS), f was fixed to the median of RM*: f = fm-ref. Finally, oral tracer measurements provided information as to when Ra meal began to rise in each subject. If tracer concentration is zero up to time ti and is different from zero at time
, then one can safely assume that Ra meal is zero up to ti.
Parameter estimation. All models were numerically identified by nonlinear least squares (7, 9), as implemented in SAAM II [Simulation Analysis and Modeling software (2)]. Measurement error was assumed to be independent, gaussian, with zero mean and known constant standard deviation. Insulin concentration is the model-forcing function and is assumed to be known without error.
Statistical Analysis.
Data are presented as means ± SE. Two-sample comparisons were done by Wilcoxon signed rank test, and a Shapiro-Wilk test was used to verify whether parameters were normally distributed (significance level set to 5%). Pearson's correlation was used to evaluate univariate correlation.
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RESULTS |
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RM* parameters were estimated with good precision, and their mean values are S= 0.0118 min1, V*ref = 1.60 dl/kg, p
= 0.039 min1, and S
= 9.24 104 dl·kg1·min1 per µU/ml. The fraction of ingested glucose that reaches plasma, calculated as the ratio between area under R
and ingested dose, was fref = 0.89. Parameters S
, V*ref were normally distributed (P values not significant by Shapiro-Wilk test), although fref was not (median value fm-ref = 0.90).
OMM* and OMM.
The mean profile of Ra meal, reconstructed by simultaneously identifying OMM* and OMM, is shown in Fig. 4. Of note, Ra meal was virtually superimposable (data not shown) to that reconstructed by identifying OMM alone (11).
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IVMM* and IVMM.
and SI were 11.59 and 6.91 104 dl·kg1·min1 per µU/ml, respectively.
OMM* vs. RM*.
A good agreement was found between OMM* Ra meal and (Fig. 4).
provided by OMM* and
were well correlated (r = 0.80, P < 0.0001; Fig. 5A), and their mean values were not significantly different (9.64 vs. 9.24 x 104 dl·kg1·min1 per µU/ml). To quantify how sensitive the OMM/OMM* estimate of SI and
to the assumptions made on V, V*, SG,
, and f were, we used multiple regression analysis between the percentage deviation of SI (
) and the percentage deviation of V, SG, and f (V*,
, f). We found that the percentage deviations of f and SG explain the deviation in SI estimate (0.685, P < 0.0001), although the deviation of V doesn't contribute significantly to the regression. Conversely, the percentage deviations of f,
, and V* explain the deviation in
estimate (0.747, P < 0.0001).
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Correlation between IVMM* and OMM* was significant (r = 0.67, P < 0.0001; Fig. 5B) but their values were significantly different (9.64 vs. 11.59 104 dl·kg1·min1 per µU/ml).
SI vs .
The relationship between SI and is different during meal and IVGTT. With the oral minimal models, one had 12.24 vs. 9.64 x 104 dl·kg1·min1 per µU/ml, withSI >
in 81% of the subjects, although for the IVGTT SI was lower than
on average (SI = 6.91 vs.
= 11.59 x 104 dl·kg1·min1 per µU/ml) and was so in
90% of the subjects.
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DISCUSSION |
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The Ra meal profile obtained with OMM* was very similar to (Fig. 4). This is an important result, because due to the experimental conditions (i.e., a tracer-to-tracee clamp), R
provided an essentially model-independent measure of the glucose Ra.
obtained by OMM* was also very similar to reference values:
= 9.64 ± 0.80 x 104 vs.
= 9.24 ± 0.63 x 104 dl·kg1·min1 per µU/ml, with a satisfactory correlation between the two (r = 0.80, P < 0.0001; Fig. 5A). However, from Fig. 5, differences exist at the individual level, thus indicating that OMM* is more robust in population rather than individual studies.
The addition of a tracer to the meal, besides allowing segregation of exogenous component, Gmeal, of plasma glucose concentration G, and thus estimation of insulin sensitivity on glucose disposal, also has beneficial effects on the numerical identifiability of OMM* (and OMM). In fact, OMM*, which is based on Gmeal data, shares Ra meal with OMM, which is based on G data. Thus, for its identification, the unlabeled plasma glucose concentration data G can also be exploited by simultaneously identifying OMM. By doing so, the number of available data doubles, although the number of parameters increases by only two (i.e., from 10 parameters with OMM* alone to 12 with both OMM* and OMM). The improvement in numerical identifiability allowed estimation of and p2 in each individual without having to resort to Bayesian priors as in Refs. 11 and 12. However, parameters V, V*, SG,
, and f still need to be fixed to population values derived from the reference model. To quantify how sensitive the OMM/OMM* estimate of SI and
to the assumptions made on these parameters were, we investigated, by multiple regression analysis, the relationship between their percentage deviation and that of SI and
. We found that the percentage deviations of f and SG explain the deviation in SI estimate (0.685, P < 0.0001), whereas the deviation of V doesn't contribute significantly to the regression. The percentage deviations of f,
, and V* from the fixed values explain the deviation in
estimate (0.747, P < 0.0001).
It is of interest to compare with OMM* and IVGTT*. The two estimates showed a significant correlation, r = 0.67, P < 0.0001 (Fig. 5B), although the latter measure was significantly higher: 11.59 vs. 9.64 x 104 dl·kg1·min1 per µU/ml.
The relationship between the cold and hot estimates of SI in IVGTT vs. meal is worth commenting on. The IVGTT results observed in the present 88 subjects are consistent with those previously reported in smaller-size studies (8, 10, 14). SI was lower than in
90% of the subjects: SI = 6.91 x 104 vs.
= 11.59 x 104 dl·kg1·min1 per µU/ml. This relationship is clearly unphysiological, because SI measures the overall effect of insulin on both glucose disposal and production, whereas
measures only the effect of insulin on disposal; by definition, SI should be equal to or greater than
. Possible reasons for this unexpected pattern have been discussed in Ref. 8. The relationship between SI and
improves dramatically with the OMM, SI = 12.24 x 104 vs.
= 9.64 x 104 dl·kg1·min1 per µU/ml, with SI >
in 81% of the subjects. However, SI was still less than
in 19% of the subjects. It is likely that the improved performance of the oral minimal models stems from the fact that the minimal model assumptions are more tenable during the gentle meal than during the massive IVGTT perturbation, particularly those concerning the description of how NHGB is controlled by glucose and insulin, which are embodied in both cold models. However, the finding that
was greater than SI in 19% of the subjects also calls for a revision of this functional description for a meal perturbation.
A comment on the relationship between p2 and in IVGTT and meal, as well as on the difference between intravenous and oral values, is also in order. During both IVGTT and meal, p2 is approximately one-third of
(Table 1). This means that insulin action on the liver has a slower dynamic than insulin action on glucose utilization, and, in all likelihood, the underlying assumption of the classic minimal model, that insulin action on glucose production has the same dynamics of insulin action on glucose utilization, is probably not entirely correct. Moreover, the difference found in p2 and
values during IVGTT and meal can be explained by considering the differences between the two tests. During IVGTT, glucose and insulin explore a wider range of values than during the meal (G: 90300 vs. 90170 mg/dl; I: 4120 vs. 460 µU/ml), thus possibly uncovering some parameter nonlinearities (1, 13).
In conclusion, OMM* provides a means of assessing bothRa meal and after the ingestion of a carbohydrate-containing meal. Because OMM is simultaneously identified, this approach also permits assessment of the overall effect of insulin on glucose production and disposal (SI). Furthermore, when the labeled and unlabeled oral minimal models are combined with the oral C-peptide minimal model (6, 15), insulin secretion and
-cell function indexes can also be measured at the same time. However, although in the present study good performance of the method was observed in a wide range of glucose tolerance (SI: 1.52 ÷ 30.40 x 104 dl·kg1·min1 per µU/ml), further studies in diabetic individuals with abnormalities in insulin secretion and action of various degrees of severity are needed to better define the domain of validity of the model. Finally, because insulin action appears to be dependent on the pattern of insulin (1, 13), future studies will be required to determine whether the ability of insulin to stimulate glucose uptake and suppress glucose production in the presence of the continuously changing insulin concentrations observed after a meal is the same as that observed in the presence of different insulin profiles (e.g., during a hyperinsulinemic clamp).
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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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