Insulin Release in Impaired Glucose Tolerance: Oral Minimal Model Predicts Normal Sensitivity to Glucose but Defective Response Times

Studies were performed in nine healthy subjects with NGT (seven women and two men; age, 33 ± 3 years [mean ± SE]; BMI, 26 ± 2 kg/m²) and four subjects with IGT (two women and two men; age, 37 ± 3 years; BMI, 38 ± 3 kg/m²). This database includes the database used in Toffolo et al. (12).

Glucose tolerance was determined by using the American Diabetes Association Expert Committee criteria (14). All subjects had a normal screening blood count and chemistries and none were taking medications known to affect insulin secretion or action. All protocols were approved by the Institutional Review Board of the University of Chicago. Written informed consent was obtained from each subject.

All studies were performed in the Clinical Research Center at the University of Chicago, starting at 8:00 a.m., after an overnight fast. Intravenous cannulas were placed in a forearm vein for blood withdrawal, and the forearm was warmed to arterialize the venous sample. A second catheter was placed in the contralateral forearm for administration of glucose.

Subjects received graded glucose infusions at progressively increasing (step-up) and then decreasing (step-down) rates (4, 8, 16, 8, 4, and 0 mg · kg⁻¹ · min⁻¹). Each glucose infusion rate was administered for a total of 40 min. Glucose, C-peptide, and insulin levels were measured at 10-min intervals during a 40-min baseline period before the glucose infusion and throughout the 240-min glucose infusion.

Plasma glucose was measured immediately using a glucose analyzer (YSI Model 2300 STAT; Yellow Springs Instruments, Yellow Springs, OH). The coefficient of variation (CV) of this method is <2%. Serum insulin was assayed by a double-antibody technique (15) with a lower limit of sensitivity of 20 pmol/l and an average intra-assay CV of 6%. The cross-reactivity of proinsulin in the radioimmunoassay for insulin is ∼40%. Plasma C-peptide was measured as previously described (16). The lower limit of sensitivity of the assay is 0.02 nmol/l, and the intra-assay CV averaged 6%.

The insulin secretion rate (ISR) was derived by stochastic deconvolution of C-peptide levels (17). Mean ISR for each glucose infusion rate was then plotted against the corresponding mean glucose concentration to define the dose-response relationship (6–8). Indexes of β-cell function were derived by calculating the areas under the ISR curve (ISR AUC) over the entire protocol as well as during the step-up and the step-down parts only, and by normalizing them to the glucose AUC calculated for the same intervals. These three indexes of β-cell responsivity to glucose (Φ_DEC, Φ_{DEC UP}, and Φ_{DEC DOWN}) are thus independent of the glucose levels achieved.

Insulin secretion was also quantified from C-peptide and glucose levels by using the OMM (12,13). C-peptide kinetics are described by the well-known two-compartment model originally proposed by Eaton et al. (18):

where the overdot indicates time derivative; CP₁ and CP₂ (nmol/l) are C-peptide concentrations above basal in the accessible and peripheral compartments, respectively; k_ij (min⁻¹) are C-peptide kinetic parameters, and secretion rate (SR) (pmol · l⁻¹ · min⁻¹) is pancreatic secretion rate above basal, entering the accessible compartment, and normalized by the volume of distribution of compartment 1.

Pancreatic SR has been described as the sum of two components controlled by glucose concentration (static glucose control, SR_s) and its rate of increase (dynamic glucose control, SR_d):

SR_s is assumed to be equal to the provision of new insulin to the β-cells, Y (pmol · l⁻¹ · min⁻¹):

which is controlled by glucose according to the following equation:

Thus, SR_s is not linearly related to glucose concentration, but tends with a time constant 1/α (min) toward a steady-state value linearly related through parameter β to glucose concentration G above a threshold level h (mmol/l).

SR_d represents the secretion of insulin stored in the β-cells in a promptly releasable form (labile insulin), and is proportional to the rate of increase of glucose:

where:

According to Eqs. 6 and 7, the dynamic control is maximum when glucose increases just above its basal value G_b, then it decreases linearly with glucose concentration and vanishes when glucose concentration exceeds the threshold level G_t able to promote the secretion of all stored insulin. If G_t assumes an elevated value, k(G) approximates the constant k_d.

The profile of ISR (pmol/min) can be calculated as

where SR_b (pmol · min⁻¹ · l⁻¹) is insulin secretion in the basal state, CP_1b (nmol/l) is basal C-peptide concentration, and V₁ (l) is the distribution volume of the accessible compartment.

OMM also provides indexes of β-cell function. The static sensitivity index Φ_s (10⁹ min⁻¹) equals parameter β and measures the effect of glucose on β-cell secretion at steady state:

The dynamic sensitivity index Φ_d (10⁹) is a measure of the stimulatory effect of the rate at which glucose increases upon the secretion of stored insulin. It is defined as the amount of insulin (per unit of C-peptide distribution volume) released in response to the maximum glucose concentration (G_max) achieved during the experiment, normalized by the glucose increase G_max - G_b:

If G_t assumes an elevated value, Φ_d approximates the constant k_d.

Finally, the basal sensitivity index Φ_b (10⁹ min⁻¹) measures basal SR over basal glucose concentration:

The OMM also allows one to quantify the β-cell response times (min) to both a decreasing (T_down) and an increasing (T_up) glucose stimulus. When glucose decreases,

since in this case secretion equals provision (described by Eq. 5) with 1/α as time constant. When glucose increases, the additional amount X₀ of insulin secreted due to the dynamic control of glucose accelerates the β-cell response. As detailed in Toffolo et al. (12), this is equivalent to a reduction in the β-cell response time:

From the model parameters, a global index of β-cell sensitivity to glucose, Φ (10⁹ min⁻¹), similar to Φ_DEC, can also be measured:

where T (min) is the time at which the system returns to steady-state conditions after the perturbation. Equation 14 is slightly different from the global index already defined in Breda et al. (13), because it considers total pancreatic secretion, like Φ_DEC, and not secretion above basal.

The glucose minimal model (19) was applied to glucose and insulin concentrations, and an estimate of insulin sensitivity (S_I [10⁵ min⁻¹ per pmol/l]) was derived for each subject.

Deconvolution was performed by using the program WINSTODEC (STOchastic DEConvolution) (20). OMM and glucose minimal model parameters were estimated, together with a measure of their precision, by nonlinear least squares (21,22) using SAAM II software (23). Population parameters for C-peptide kinetics (24) were used in both DEC and OMM approaches. Measurement errors have been assumed to be independent and Gaussian, with zero mean. Errors in C-peptide measurements were assumed with a constant but unknown variance; errors in glucose measurement with a CV of 2%. AUCs were calculated by using the trapezoidal rule.

The significance of differences was determined by either the Mann-Whitney U test or the Wilcoxon signed-rank test, as appropriate. For all analyses, a two-tailed P value of < 0.05 was considered to indicate statistical significance. All results are expressed as mean ± SE. Statistical analysis was performed using StatView 5.0 (SAS Institute, Cary, NC).

Mean plasma glucose, C-peptide, and insulin concentrations achieved by NGT and IGT subjects during the graded up-and-down glucose infusion are shown in Fig. 1. IGT subjects achieved significantly higher glucose, C-peptide, and insulin levels than subjects with NGT. However, while glucose levels returned to basal before the end of the experiment in both NGT and IGT, C-peptide and insulin levels remained very high in IGT.

Mean ISR profiles obtained by deconvolution for both NGT and IGT subjects are shown in Fig. 2 and the dose-response relationship between glucose and ISR throughout the infusion protocol is shown in Fig. 3. This relationship shows a hysteresis during decreasing glucose steps, i.e., β-cell response to glucose appears to be higher than during increasing glucose steps, and this is more pronounced in IGT subjects.

Both ISR AUC (10⁻³ pmol) and glucose AUC (10⁻³ mmol · l⁻¹ · min) were significantly higher in IGT compared with NGT (ISR AUC: 212 ± 34 vs. 114 ± 11, P = 0.013; glucose AUC: 3.3 ± 0.2 vs. 2.2 ± 0.1, P = 0.0055). As a result, Φ_DEC (10⁹ l · min⁻¹) was not different between IGT and NGT subjects (NGT, 52 ± 5; IGT, 64 ± 10) (Fig. 4). Φ_{DEC UP} (10⁹ l · min⁻¹) was not different between the groups either (NGT, 51 ± 5; IGT, 48 ± 9), whereas Φ_{DEC DOWN} (10⁹ l · min⁻¹) was significantly higher in IGT than in NGT (NGT, 55 ± 5; IGT, 75 ± 10). These results suggest that β-cell response to glucose in IGT is similar to that in NGT during increasing steps but is significantly higher during decreasing steps. Alternatively, when comparing the step-up versus the step-down phase, Φ_{DEC DOWN} was significantly higher than Φ_{DEC UP} in IGT but not in NGT subjects (P < 0.05), suggesting that β-cell response to glucose in IGT is significantly higher during decreasing than during increasing glucose steps, consistent with the results observed from the dose-response curve.

The OMM well describes experimental data in both NGT and IGT subjects (Fig. 5). OMM predictions of ISR are shown in Fig. 2 and are not different from ISR obtained by deconvolution.

Index of basal β-cell sensitivity to glucose was higher in IGT compared with NGT (Φ_b [10⁹ min⁻¹] NGT 5.0 ± 0.4; IGT 8.5 ± 1.8), but the difference just failed to reach statistical significance (P = 0.08). Indexes of static, dynamic, and global β-cell sensitivity to glucose did not differ between groups (Φ_s [10⁹ min⁻¹] NGT 18.4 ± 1.7; IGT 17.8 ± 1.6; Φ_d [10⁹] NGT 279 ± 64; IGT 250 ± 36; Φ [10⁹ min⁻¹] NGT 13 ± 1; IGT 16 ± 1) (Fig. 4). However, the ability of the β-cell to respond to either rising or falling glucose levels was substantially disordered in subjects with IGT, as reflected in the response time parameters T_up (min) and T_down (min), which were significantly higher in IGT (37.8 ± 15.0 and 51.7 ± 15.4, respectively) compared with NGT (1.3 ± 4.8 and 17.4 ± 1.8, respectively).

The index of insulin sensitivity S_I (10⁵ min⁻¹ per pmol/l) derived by minimal model analysis was significantly higher in NGT (4.9 ± 0.8) than in IGT (1.0 ± 0.4) (Fig. 4).

In this study, we have used both deconvolution and the OMM approaches to study β-cell response to a physiological glucose stimulus in IGT. The difference between these two approaches is not in the reconstruction of the ISR profile, but in the indexes of β-cell function. In particular, OMM analysis allows the estimation of β-cell response times T_up and T_down, which are significantly higher in IGT than in NGT.

Deconvolution analysis of C-peptide levels revealed that subjects with IGT have characteristic alterations in insulin secretion that distinguish them from subjects with NGT. The dose-response relationship (Fig. 3) shows a hysteresis during decreasing glucose steps, which is much more pronounced in subjects with IGT, suggesting that β-cell sensitivity to glucose in IGT is higher during a decreasing than during an increasing glucose stimulus. Consistently, Φ_{DEC DOWN} was 58% higher than Φ_{DEC UP} in IGT (P < 0.05), but only 11% higher in NGT subjects (NS). Alternatively, when comparing NGT versus IGT, Φ_DEC and Φ_{DEC UP} were not different between groups, while Φ_{DEC DOWN} was significantly higher in IGT.

However, the use of a quasi-steady-state method to interpret a non-steady-state situation, like the one between plasma glucose and C-peptide concentration during the graded glucose infusion, is not entirely accurate and particularly so with the protocol adopted in this study. In fact, ISR AUC reflects not only pancreatic sensitivity to glucose, but also the β-cell response time to a glucose stimulus; it thus underestimates the steady-state value during the increasing steps and overestimates it during the decreasing steps.

Various indexes of β-cell function have been recently proposed in the literature. They are based on modeling analyses of glucose and C-peptide data during protocols characterized by slowly increasing and then decreasing glucose and C-peptide concentrations, such as a meal (9), a 120-min OGTT (10), free living conditions (11), graded up-and-down glucose infusion protocols (12), and a 300-min OGTT (13). In particular, the model proposed in Hovorka et al. (9) simply assumes a linear immediate control of glucose on insulin secretion, while the model proposed in Cretti et al. (10) assumes a control of glucose on insulin secretion similar to the static glucose control of OMM, and thus characterized by a delay between glucose stimulus and β-cell response. The model proposed in Mari et al. (11) assumes a (nonlinear) control of glucose and a control of glucose rate of change on insulin secretion, but no delay between glucose stimulus and β-cell response. A third secretion term is also incorporated into the model, which is not constrained to a specific functional form but is allowed to take on arbitrary smooth zero-mean time course. This term is introduced to obtain a good fit to the data, and thus a good ISR profile, but makes it difficult to assess the assumptions of the model structure against the data. Finally, the model proposed in Toffolo et al. (12) and Breda et al. (13), here called OMM, assumes controls of both glucose and glucose rate of change on insulin secretion, as well as a delay between glucose stimulus and β-cell response.

Given this controversy in the assumptions built into the available models, a purpose of this study was to test whether the OMM assumptions on the mechanisms of insulin secretion (i.e., controls of glucose and of glucose rate of change as well as delay between glucose stimulus and β-cell response) are really necessary to adequately describe C-peptide data in NGT and IGT subjects during graded up-and-down glucose infusions.

When using a model similar to the OMM, but not accounting for the delay between glucose-stimulus and β-cell response, we obtained model predictions shifted to the left with respect to real C-peptide data (Fig. 5) with residuals that were no longer independent, showing that such a delay is real and necessary to appropriately describe C-peptide data in subjects with NGT or IGT.

The importance of a control of both glucose and glucose rate of change on insulin secretion during physiological glucose perturbations has already been shown in previous studies (12,13) and is confirmed by the results of the present study. The model fit obtained by coupling the model of C-peptide kinetics with a secretion rate controlled only by glucose and not by glucose rate of change, i.e., SR(t) = SR_s, produces a systematic underestimation, especially in the rising portion of C-peptide data, as shown in Fig. 5.

OMM overcomes the problems associated with quasi-steady-state approaches, since model equations describe the non-steady-state relationships between glucose concentration and ISR. It assumes a linear steady-state relationship between glucose concentration and ISR and estimates the slope β of this relationship from non-steady-state data, such as those measured during a graded up-and-down glucose infusion experiment. The sensitivity β is thus the same during both the step-up and the step-down phases (Fig. 3). The difference between phases observed by the deconvolution approach is indeed a consequence of the β-cell response times T_down and T_up. The former coincides with the time constant of insulin provision, whereas the second is an equivalent parameter, which also takes into account the ability of the dynamic glucose control to accelerate the rate with which β-cells respond to an increasing glucose stimulus (12). IGT subjects have similar static and dynamic β-cell sensitivities to glucose compared with NGT, but they have statistically higher response times T_up and T_down. This means that the substantially higher insulin secreted in the down portion of the protocol by IGT does not reflect a higher β-cell sensitivity to glucose (as the hysteresis in the dose-response curve and the higher Φ_{DEC DOWN} could suggest), but simply longer response times than NGT. This quantitative difference in the response times is evident visually on inspection of the glucose and C-peptide curves during the protocol (Fig. 1).

When insulin sensitivity was calculated by the glucose minimal model, it was evident that IGT subjects, besides being characterized by a higher delay between glucose stimulus and β-cell response, have significantly lower insulin sensitivity indexes compared with NGT subjects. These results suggest that IGT subjects have a relative defect in the ability to secrete insulin to adequately compensate for their insulin resistance.

We also tested and used the OMM on 10 NGT and 6 IGT subjects during a frequently sampled 300-min OGTT (unpublished observations). Also in this situation, all OMM assumptions (static and dynamic control of glucose on insulin secretion, as well as a delay between glucose stimulus and β-cell response) were necessary to adequately describe C-peptide data. The basal index of β-cell function Φ_b (10⁹ min⁻¹) was higher (even if not significantly) in IGT than in NGT (8.4 ± 1.8 vs. 5.7 ± 0.5), whereas indexes of β-cell function during the glucose stimulus were not different between the two groups (e.g., Φ_d [10⁹] was 604 ± 97 in NGT and 515 ± 106 in IGT; Φ_s [10⁹ min⁻¹] was 31 ± 3 in NGT and 28 ± 4 in IGT). Response time parameters were higher (even if not significantly) in IGT (e.g., T_down [min] was 6.7 ± 2.1 in NGT and 16.4 ± 6.5 in IGT). Eight (two IGT and six NGT) of these 16 subjects also underwent a graded up-and-down glucose infusion protocol. Pancreatic indexes Φ_d (10⁹) and Φ_s (10⁹ min⁻¹) during the oral glucose perturbation were significantly higher than during the intravenous glucose infusion (OGTT: Φ_d = 587 ± 125, Φ_s = 33 ± 4; up-and-down: Φ_d = 114 ± 54, Φ_s = 17 ± 3). This can probably be ascribed to the presence of the insulin-stimulating gastrointestinal hormones, which are secreted in response to oral but not intravenous glucose administration (incretin effect) (25).

In conclusion, we have shown that the recently proposed OMM adequately describes C-peptide data during physiological glucose perturbations in subjects with NGT or IGT. It overcomes the problems associated with quasi-steady-state data analysis of non-steady-state situations and provides a quantitative assessment of pancreatic function in an individual. Its application has provided novel insights into the mechanisms of insulin secretion in IGT. Also, the simultaneous assessment of insulin sensitivity in a single individual should make the C-peptide and glucose minimal model approach a powerful tool to measure changes in insulin secretion and action under physiological conditions. Further work needs to be performed to better define the domain of validity of this approach throughout the whole range of glucose tolerance, including patients with overt diabetes.

The authors wish to thank Dr. Melissa K. Cavaghan (Indiana University School of Medicine) and Dr. David A. Ehrmann (Department of Medicine, University of Chicago) for having kindly provided unpublished data.