## Abstract

We investigated β-cell function and its relationship to insulin sensitivity in 17 normal volunteers. For insulin secretion (derived by C-peptide deconvolution), a mathematical model was applied to 24-h triple-meal tests (MT) as well as oral glucose tolerance tests (OGTT); insulin sensitivity was assessed by the euglycemic insulin clamp technique. The β-cell model featured a glucose concentration-insulin secretion dose response (characterized by secretion at 5 mM glucose and slope), a secretion component proportional to the glucose concentration derivative, and a time-dependent potentiation factor (modulating the dose response and accounting for effects of sustained hyperglycemia and incretins). The β-cell dose-response functions estimated from the whole 24-h MT, the first 2 h of the MT, and the OGTT differed systematically, because a different potentiation factor was involved. In fact, potentiation was higher than average during meals (1.6 ± 0.1-fold during the first meal) and had a different time course in the MT and OGTT. However, if potentiation was accounted for, the 24- and 2-h MT and the OGTT yielded similar dose responses, and most β-cell function parameters were intercorrelated (*r* = 0.50–0.86, *P* ≤ 0.05). The potentiation factor was found to be related to plasma glucose-dependent insulin-releasing polypeptide concentrations (*r* = 0.49, *P*< 0.0001). Among β-cell function parameters, only insulin secretion at 5 mM glucose from MT correlated inversely with insulin sensitivity (24-h MT: *r* = −0.74, *P* < 0.001; 2-h MT: *r* = −0.52, *P* < 0.05), whereas the dose-response slope and the OGTT parameters did not. In nine other subjects, reproducibility of model parameters was evaluated from repeated MTs. Coefficients of variation were generally ∼20%, but the derivative component was less reproducible. We conclude that our model for the multiple MT yields useful information on β-cell function, particularly with regard to the role of potentiation. With cautious interpretation, a 2-h MT or a standard OGTT can be used as surrogates of 24-h tests in assessing spontaneous β-cell function.

- insulin secretion
- glucose-induced insulin release
- potentiation of glucose-induced insulin release
- insulin sensitivity

the assessment of insulin secretionwith experiments mimicking physiological conditions is most often based on clinical tests of relatively short duration, such as the 2- to 4-h oral glucose tolerance test (OGTT) or a single-meal test. From these data, various empirical parameters of β-cell function are often calculated, but indexes obtained by modeling are also available (1, 6). Short tests are obviously useful in clinical investigation, but they may be insufficient to reveal aspects of β-cell function emerging from longer observation periods. For example, in recent studies using modeling (9, 10), we have shown that meal-related potentiation of insulin secretion plays an important role in determining the daily profile of insulin release.

The extent to which indexes of β-cell function derived from short and long tests agree with each other is not known, nor is it clear which characteristics of β-cell function a short test may miss. Furthermore, although compensation between insulin secretion and insulin sensitivity is a well established phenomenon, little is known about which of the parameters of β-cell function derived from oral tests best reflects this compensatory mechanism.

This study aimed at elucidating these aspects of β-cell function. Using the modeling analysis of 24-h multiple-meal experiments as a reference for insulin secretion, we examined the extent to which the key parameters of β-cell function can be retrieved from a 2-h meal test (the first 2 h of the 24-h studies) or a 2-h OGTT. Furthermore, we examined which β-cell function parameters are related to insulin sensitivity, as determined by the euglycemic insulin clamp, and what impact the test format has on this relationship. Finally, we evaluated the reproducibility of the model parameters from repeated-meal tests.

## MATERIALS AND METHODS

### Insulin Secretion Protocol

Data for the study of insulin secretion and insulin sensitivity were taken from a previously reported study (12). Seventeen healthy subjects (9 males, 8 females; age 33 ± 2 yr; body mass index 25 ± 1 kg/m^{2}) underwent an OGTT, a 24-h triple-meal test, and a hyperinsulinemic euglycemic glucose clamp on three separate days, as previously described in detail (12).

#### OGTT.

A 75-g OGTT was performed in all subjects at 8:30 AM, with blood samples drawn at 0, 30, 60, 90, and 120 min following glucose ingestion for the measurement of glucose, insulin, and C-peptide concentrations. The OGTT was performed at a distance of 2–4 wk from the 24-h test and the clamp.

#### Twenty-four-hour meal tests.

The subjects arrived at the clinical research unit at 7:30 AM after an overnight fast. An intravenous cannula was placed in an antecubital vein for blood sampling. Three meals were served: breakfast at 8:00 AM, lunch at 12:00 noon, and dinner at 6:00 PM. Total energy intake was ∼10 MJ for men and ∼8 MJ for women (30% breakfast, 35% lunch and 35% dinner). Distribution of energy intake (carbohydrates-fat-protein) was 50–37–13% for breakfast, 38–49–13% for lunch, and 50–33–17% for dinner. Blood samples for the measurement of glucose, insulin, C-peptide, and glucose-dependent insulin-releasing polypeptide (GIP) were drawn every 30 min corresponding to the meals and every hour for the remaining periods for a total of 24 h.

#### Glucose clamp.

Euglycemic insulin clamps were performed in all subjects in the morning on the day after the 24-h test. The insulin infusion rate was 1 mU · min^{−1} · kg^{−1}, and plasma glucose was clamped at ∼5 mM. Insulin-stimulated glucose uptake was calculated as the average glucose infusion rate between 120 and 150 min and was expressed in micromoles per minute per kilogram of lean body mass (μmol · min^{−1} · kg_{LBM}
^{−1}), as measured by bioelectrical impedance.

In all studies, plasma glucose, insulin, C-peptide, and GIP concentrations were measured as previously reported (12).

### Reproducibility Protocol

Reproducibility of the indexes of insulin secretion was assessed in a different group of nine healthy male subjects (age 24 ± 1 yr; body mass index 22 ± 1 kg/m^{2}), who underwent two four-meal tests on two consecutive days. The subjects were admitted to the hospital 1 or 2 days before the study and consumed their last meal at 7:00 PM on the day before the study. On the study day, four meals were served: breakfast at 8:00 AM, lunch at 12:00 AM, afternoon snack at 4:00 PM, and dinner at 8:00 PM. Total energy intake was ∼9 MJ (30% breakfast, 31% lunch, 9% snack, and 30% dinner). Meal composition was 50% carbohydrate, 20% protein, and 30% fat for the three main meals. Blood sampling started before breakfast (*time 0*) and continued for 15 h for the measurement of plasma glucose, insulin, and C-peptide concentrations. Samples were drawn at 15- to 60-min intervals corresponding to the meals (no blood sampling was performed between 4:00 and 8:00 PM). Written informed consent was obtained from all subjects, and the protocol was approved by the local Ethics Committee.

Reproducibility of the model indexes was expressed as an average coefficient of variation, calculated as the ratio between the index standard deviation and the mean index value in the group. The index standard deviation was calculated as the sample standard deviation of the difference between the indexes obtained from the two tests divided by two.

### Modeling Analysis

#### Model of β-cell function.

Parameters of β-cell function were determined by modeling, as previously described (10). In brief, the model consists of three units: *1*) a model for fitting the glucose data, the purpose of which is to smooth and interpolate plasma glucose concentrations; *2*) a β-cell model describing the dependence of insulin (or C-peptide) secretion on glucose concentration; and *3*) a model of C-peptide kinetics, i.e., the two-exponential model proposed by Van Cauter et al. (14), in which the model parameters are individually adjusted to the subject's anthropometric data.

In particular, with regard to the β-cell model, the relationship between insulin release and plasma glucose concentrations is represented as the sum of two components. The first component [S_{g}(*t*)] expresses a static relationship between insulin secretion and glucose concentration and embodies a β-cell dose-response function. This dose-response function is, however, modulated by a time-varying factor, expressing a potentiation effect upon insulin secretion
Equation 1where Q(*t*) is a function of time such that e^{Q(t)} is 1 on average during the experiment and*f*(G) is the function of glucose concentration representing the β-cell dose response. Q(*t*) is represented in discrete form as a piecewise linear function over 10-min intervals.

The dose response has the following properties (10): *1*) *f*(G) is positive for G > 0; *2*) *f*(G) is quasi-linear for G below or above a given glucose threshold; and *3*) the transition between the two quasi-linear portions can be smooth or sharp. *f*(G) is determined by four parameters: the initial and final slopes, the threshold glucose level at which the change in slope occurs, and a parameter determining the smoothness of the change.

The factor P(*t*) =*e*
^{Q(t)} is termed the potentiation factor and is greater than 1 if Q(*t*) > 0, and less than 1 if Q(*t*) < 0. P(*t*) is constrained to be 1 on average to obtain a unique P(*t*) and*f*(G) [otherwise *k*P(*t*) and*f*(G)/*k*, for any *k*, would yield the same S_{g}(*t*)]. The potentiation factor includes true potentiation, i.e., a daytime potentiation of insulin secretion due to exposure to meal-related hyperglycemia and the secretory influence of gut incretin hormones and other secretory phenomena such as circadian rhythms and pulsatility of insulin secretion.

The dose-response function *f*(G) is modified during the experiment by the potentiation factor P(*t*). As P(*t*) averages 1, *f*(G) represents the average static relationship between glucose concentration and insulin secretion.

The second insulin secretion component [S_{d}(*t*)] represents a dynamic dependence of insulin secretion on the rate of change of glucose concentration. S_{d}(*t*) is proportional to the derivative of glucose concentration when the derivative is positive and is zero otherwise
Equation 2Total insulin secretion, S(*t*), is the sum of the two components described above
Equation 3S(*t*) is calculated every 10 min for the whole 24-h period. Insulin secretion and all related parameters are expressed in picomoles per minute per square meter of body surface area. Estimation of the model parameters in each subject [the 4 parameters of*f*(G), Q(*t*), and*k*
_{d}] was performed by fitting the model to the glucose and C-peptide data by use of regularized least-squares methods, as described previously (10).

In our approach, insulin secretion S(*t*) is the input to the standard C-peptide model (14). Because P(*t*) ensures that S(*t*) has the necessary flexibility to fit C-peptide accurately (10), the whole approach can be regarded as a complex way to perform deconvolution [S(*t*) is the input function subject to the usual smoothness constraint]. Thus total insulin secretion is virtually independent of the assumptions of *Eqs. 1-3
*, and S(*t*) is equivalent to that obtained by classical C-peptide deconvolution.

#### Model parameters of β-cell function.

From the estimated model parameters, other indexes describing β-cell function were calculated. Twenty-four-hour insulin output (ISR_{24h}, nmol/m^{2}) was calculated as the integral of total insulin secretion. Mean nighttime insulin output (ISR_{n}, pmol · min^{−1} · m^{−2}) was computed as the mean insulin secretion during the 8 h preceding breakfast. For all tests, fasting pretest insulin secretion (ISR_{0}, pmol · min^{−1} · m^{−2}) was calculated as insulin secretion at *time 0*. The insulin secretion value corresponding to a glucose concentration of 5 mM (ISR_{5}, pmol · min^{−1} · m^{−2}) was calculated from the dose-response function; this parameter quantifies insulin secretion at, or around, normal basal plasma glucose values. The slope of the dose-response function at 5 mM glucose concentration (Sl_{5}, pmol · min^{−1} · m^{−2} · mM^{−1}) or the average slope for the glucose range 5–7 mM (Sl_{5–7}, pmol · min^{−1} · m^{−2} · mM^{−1}) was also obtained; these parameters quantify the sensitivity of β-cells to glucose concentration changes in the physiological range. The excursions of the potentiation factor were quantified using ratios between mean values at different time intervals [e.g., P(50–70 min)/P(0–20 min)].

#### Empirical parameters of β-cell function.

For all tests, the following empirical parameters of β-cell function were also calculated: *1*) the ratio of fasting insulin secretion to fasting glucose level (ISR_{0}/G_{0});*2*) the ratio of the integral of insulin secretion to the integral of glucose concentration over the first 2 h (ISR_{2h}/G_{2h}); *3*) the ratio of insulin concentration to glucose concentration at 30 min (I_{30}/G_{30}); and *4*) the ratio of the insulin concentration increment to the glucose concentration increment (above basal) at 30 min postglucose load (∂I_{30}/∂G_{30}). These parameters empirically attempt to normalize insulin secretion to glucose levels.

### Statistical Analysis

All data are presented as means ± SE. Randomness of the model residuals was tested using the runs test (5). The Wilcoxon signed rank test was used to compare various indexes in the same group. Linear regression analysis was carried out using standard techniques.

## RESULTS

Figure 1 shows the mean 24-h profiles of glucose, insulin, and C-peptide concentrations in 17 healthy subjects. As can be appreciated from Fig. 1, the model fit (solid line) was good. The model residuals in each subject did not show systematic deviations, as assessed by the runs test. On average, the model residuals were not different from zero (by one-sample*t*-test) at most time points, and at all time points the mean residual error did not exceed 1% for glucose and 3.7% for C-peptide. The 2-h test and the OGTT gave similar results.

Figure 2 shows the average insulin secretion profile with its components as resolved by the model. From these data, total 24-h insulin secretion was calculated to be 156 ± 41 nmol/m^{2}, whereas the mean nocturnal secretion rate was 57 ± 16 pmol · min^{−1} · m^{−2}.

Figure 3
*A* shows that the dose response obtained from the 24-h test is shifted downward of the dose response obtained from the first 2 h of the same test. This difference is largely due to potentiation. In fact, when the original 24-h dose-response function is scaled by the average potentiation factor of the first 2 h (which was 1.6 ± 0.08), the difference is greatly reduced, if not abolished. The model indexes characterizing the dose response of the different tests are given in Table 1, and their intercorrelations are reported in Table 2.

Figure 4 compares the potentiation profile obtained from the 24-h test with that obtained from the first 2 h of the same test. Because both are constrained to average 1, but on different time intervals (24 vs. 2 h), the original 24-h potentiation factor is higher than the 2-h potentiation factor during the first 2 h of the meal test. By scaling the 24-h potentiation factor by its average value during the first 2 h, the two profiles are brought into full coincidence.

Figure 5 shows the dose response obtained from the 2-h meal test compared with that obtained from the OGTT. Table1 reports the model indexes for the OGTT. The two dose responses are similar although not identical. Most of the parameters obtained from the OGTT are significantly correlated with the 2-h meal test parameters (Table 2).

Figure 6 shows the potentiation factor obtained from the 2-h meal test vs. that obtained from the OGTT. Although by definition the average value of both variables is 1, the two time courses are significantly different. In fact, the ratio of the potentiation factor between 50 and 70 min (peak value for the meal) to that between 0 and 20 min (initial value) was almost twofold higher with the meal than with the OGTT (3.0 ± 0.4 vs. 1.7 ± 0.2,*P* < 0.005). This indicates that a mixed meal induced a stronger potentiation of insulin secretion than oral glucose within an equal time frame.

On the euglycemic clamp, insulin-mediated glucose disposal averaged 62 ± 3 μmol · min^{−1} · kg_{LBM}
^{−1}. The correlation coefficients between this measure of insulin sensitivity and the model indexes of β-cell function are shown in Table 3. The strongest correlation was with insulin secretion at 5 mM glucose, in a reciprocal fashion as expected. The correlation was weaker for the 2-h meal test and did not reach statistical significance for the OGTT. In contrast, correlations with the insulinogenic indexes I_{30}/G_{30} and ∂I_{30}/∂G_{30} were positive and fully statistically significant only for the OGTT.

The temporal pattern of GIP concentrations resembled that of the potentiation factor (Fig. 7). In the whole group, the potentiation factor was positively correlated with GIP levels on a minute-by-minute basis (i.e.*,* potentiation factor at the time points when GIP was measured against GIP level in a multiple regression model with separate coefficients for each subject,*r* = 0.49, *P* < 0.0001). This correlation was also significant in ∼50% of the individual subjects (9 of 17, *r* = 0.44–0.78, *P* < 0.05). Furthermore, in the whole group, the ratio of the integral of the potentiation factor after the third meal (6 PM to 1 AM) to the integral of the potentiation factor after the second meal (12 noon to 6 PM) was positively correlated to the corresponding ratio of the integral GIP concentrations (*r* = 0.51,*P* < 0.05).

Table 4 shows the results of the reproducibility test carried out in nine other healthy volunteers. None of the parameters differed between the two tests to a statistically significant extent.

## DISCUSSION

This study shows that potentiation is a key element in the analysis of β-cell function on several accounts. The inclusion of the potentiation factor in the model is essential to explain the changes in the relationship between glucose concentration and insulin secretion in the 24-h experiments, as pointed out previously (9, 10). Without potentiation, some of the features of this relationship, apparent in Figs. 1 and 2, could not be explained. In fact, Fig. 1shows that, during the first meal, the insulin peak is higher than during the last meal despite slightly lower glucose concentrations, whereas Fig. 2 shows that, if a constant dose response is assumed (broken line of Fig. 2), neither the amplitude nor the shape of the secretory peaks can be satisfactorily described.

This study extends previous findings obtained with a 15-h protocol (10) by showing that the potentiation factor is plausibly related to plasma GIP concentrations on a minute-by-minute basis (Fig.7). This suggests cause-effect mechanisms, which are known to exist (4) but have never been evaluated before on a modeling basis. Furthermore, we find that a mixed meal produces a stronger stimulation of potentiation than a pure glucose load (Fig. 6) despite higher glucose concentrations during the OGTT. This result indicates that a different potentiation stimulus is at play with a mixed meal vs. a glucose load. This difference can be regarded as a consequence of a different enteroinsular effect (4) or the influence of secretagogues other than glucose. Because we did not measure plasma GIP levels during the OGTT, we could not discriminate between these two mechanisms nor can we single out the role of glucose-dependent potentiation per se (11). Nevertheless, our approach offers a tool to quantify potentiation under different conditions and to explore physiological correlates of this key feature of β-cell response.

The need to account for changes in the dose-response *f*(G) by means of the time-dependent potentiation factor P(*t*) introduces a problem of reference point for P(*t*), as P(*t*) and *f*(G) can be rescaled to any size without affecting the static secretion component (see *Modeling Analysis*, *Eq. 1
*). We have chosen to constrain P(*t*) to be 1 on average during the experimental period. In this way, the corresponding dose response represents the average dose response for the experiment and P(*t*) a relative potentiation factor: P(*t*) < 1 indicates potentiation below average (not real inhibition or depotentiation of insulin secretion) and P(*t*) > 1 potentiation above average.

Alternatively, P(*t*) could be set to 1 in the nocturnal or pretest period and the dose response scaled accordingly: P(*t*) would thus be >1 during the meals and represent actual potentiation above the pretest period. Ideally, the resulting*f*(G) would not be affected by the specific time course of the potentiation factor during the test, as is the case when P(*t*) averages 1 and would be test independent. This choice might appear physiologically more appropriate, but it has some limitations. First, the resulting *f*(G) does not represent the average dose response; i.e., *f*(G) underestimates the insulin secretion rates corresponding to the glucose levels observed during the test. Second, in the 2-h tests, only a single concentration value is available for the determination of the pretest potentiation factor, which is therefore estimated with limited precision. Last, at the present stage of model development, this approach does not result in a fully test-independent*f*(G). In fact, if the dose responses of Fig. 3 are normalized to the potentiation factor at *time 0* instead of the average in the first 2 h, the two dose responses are still in good agreement but not fully coincident (results not shown).

Our choice to refer the dose response to a potentiation factor that is set to average 1 over the entire observation period is natural and physiologically appropriate in a 24-h test: if the relationship between glucose concentration and insulin secretion is not constant in time, it is logical to use the 24-h average. However, if a shorter test is used, the observed relationship between glucose concentration and insulin secretion is necessarily the one prevailing during the test, as its evolution over a longer time period is not seen. Because the potentiation factor changes with time and is higher than average during the first meal, the 2-h test yields a dose response that is shifted upward of the corresponding function derived from the 24-h test (Fig.3
*A*). When the latter is rescaled by the average value of the potentiation factor during the first 2 h, the difference is greatly reduced (Fig. 3
*B*). This is not a model artifact: the average insulin secretion for a 2-h test is higher than during a whole day when related to the corresponding glucose levels. Quantitatively, the ratio of mean insulin secretion to mean glucose concentration in the 2-h test is twice that in the 24-h test.

These effects of potentiation have the important consequence that the dose-response function estimated with the present model (and, probably, with others) is critically dependent on the duration and format of the experiment. Thus dose-response comparisons are legitimate only if they are obtained from experiments of similar kind and duration. Furthermore, the observed differences in dose responses obtained from short tests may in reality be due to differences in the level of potentiation operative during the given experimental period rather than to real dose-response effects (as they would be estimated from a 24-h protocol). Stated otherwise, the inherent drawback of short tests is that they cannot precisely discriminate between dose response and potentiation.

In any case, accounting for the role of potentiation makes the analysis of a 2-h test in substantial agreement with that of the 24-h protocol. In particular, the time course of potentiation is essentially superimposable between the two tests (Fig. 4
*B*), the corresponding dose-response curves are similar (Fig. 3
*B*), the two sets of model indexes of β-cell function are generally intercorrelated (Table 2), and the correlation between insulin sensitivity and β-cell function is preserved (Table 3). Although the concordance is not complete (Fig. 3
*B*; Table 1), the results are remarkably consistent, given that the 2-h test features a single glucose peak and only five blood samples and that a single meal cannot unfold all the complexity of the relationship between glucose concentration and insulin secretion observed in a 24-h study. Therefore, albeit with the limitations discussed above, the analysis of a 2-h meal test with the present model yields physiologically meaningful parameters of β-cell function.

The inclusion of potentiation makes our model not directly comparable with models based on different principles and short tests (1, 6,13). Although the need to account for potentiation has been recognized (see discussion in Ref. 10), a comparison of the performance of different models with the same data set is outside the scope of this work.

The inverse relationship between insulin secretion and insulin sensitivity (Table 3) is an expected result (e.g., Ref.7). However, the present work is, to our knowledge, the first to examine this relationship by considering multiple parameters of β-cell function derived from a mathematical model. The use of the model indexes has the advantage that insulin secretion is not viewed in absolute terms but in relation to glucose. Thus, although absolute secretion may differ as a consequence of differences in glucose levels, changes in model indexes reflect the real adaptation of β-cell glucose sensitivity to the prevailing degree of insulin resistance.

Our analysis shows that the strongest correlation is the one between insulin sensitivity and insulin secretion at a given, near-basal glucose concentration (ISR_{5}). Although the lack of correlation with the other parameters (Sl_{5}, Sl_{5–7}, *k*
_{d}) may be due, in part, to their lower reproducibility (particularly for*k*
_{d}), our results do suggest that physiological β-cell adaptation to insulin resistance occurs principally by modulation of the basal secretory tone. This work also shows that the analysis of β-cell function requires appropriate indexes and that modeling can be helpful in this respect. For instance, if the traditional insulinogenic indexes (I_{30}/G_{30} and ∂I_{30}/∂G_{30}) are used, the correlation between insulin sensitivity and β-cell function is contrary to expectation (Table 3). Furthermore, using the empirical indexes of β-cell function that do show the expected correlation with insulin sensitivity, such as ISR_{2h}/G_{2h}, one cannot establish which function of the β-cell adapts to insulin resistance.

The analysis of the OGTT reveals some similarities with, but also some differences from, the 2-h meal test. The dose responses obtained from these tests are similar (Fig. 5), and the model indexes of β-cell function are correlated (Table 2). However, as discussed above, the potentiation factor differs (Fig. 6); i.e., a stronger potentiation is observed with a mixed meal than after a glucose load, notwithstanding the higher glucose concentrations attained during the OGTT. Another difference with the OGTT is that all correlations between insulin sensitivity and indexes of β-cell function are lost (Table 3). This finding, which is in disagreement with other studies, may be due to several reasons. First, because the OGTT and the clamp were performed some weeks apart, changes in insulin sensitivity may have occurred, thereby loosening the relationship between insulin secretion and sensitivity. Second, it is possible that, in the OGTT, the precision of the estimated β-cell function parameters is limited due to the small number of samples. Third, the number of subjects is limited, as is also the span in their body mass index and insulin sensitivity. Last, because insulin sensitivity is compared with β-cell glucose sensitivity and not with absolute insulin secretion, it is possible that the OGTT is a less effective test compared with the 2-h meal test; with the latter, in fact, the correlation is preserved not only with the model-based indexes but also with an empirical index (Table 3).

The reproducibility studies proved that, in a 15-h multiple-meal test, which is an intermediate condition between the 2-h and the 24-h test, the model-based indexes have an acceptable reproducibility (Table 4). The coefficients of variation of the β-cell function parameters (∼20%) are larger than the coefficients of variation of the β-cell function indexes derived from more controlled intravenous tests (2) but in the same range as those of other tests of insulin secretion, such as the arginine test (8). The parameter of the derivative control takes exception, being definitely less reproducible. The purpose of the derivative component of insulin secretion (or rate sensitivity) is to describe the anticipation of insulin secretion observed at the onset of hyperglycemia. Although this component is essential to describe the early phase of insulin secretion during a meal or an OGTT [not only in this work but also in other studies using our model or other models (1, 9, 10, 13)], it cannot be precisely quantified. Indeed, whether anticipation of insulin release during a meal or an OGTT is related to the so-called first-phase insulin secretion is still unclear, and the cellular mechanisms underlying this process are poorly understood (3,7). Thus a physiologically based mathematical description for this phenomenon is not available. Nevertheless, with the present model it has been possible to detect a significant impairment in the parameter of the derivative control in diabetic subjects (10), as well as an increase after treatment with nateglinide, an insulin secretagogue (Ferrannini and Mari, unpublished observations). This suggests that this parameter has some physiological significance despite its limitations.

In conclusion, we have characterized β-cell function by modeling 24-h insulin secretion during normal living conditions, and we have comparatively evaluated the performance of a 2-h meal test and the OGTT. We have found that the model indexes of β-cell function derived from a 2-h meal test (and, to a lesser extent, from an OGTT) give a good approximation of those obtained from 24-h experiments but that some potentially important information on potentiation of insulin secretion is diluted out.

## Acknowledgments

We are indebted to the Institut de Recherches Internationales Servier (Courbevoie, France) for providing a research grant in partial support of this work.

## Footnotes

Address for reprint requests and other correspondence: A. Mari, LADSEB-CNR, corso Stati Uniti 4, 35127 Padua, Italy (E-mail:mari{at}ladseb.pd.cnr.it).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

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