### Abstract

- Top of page
- Abstract
- Introduction
- Methods and Procedures
- Results
- Discussion
- Acknowledgment
- Disclosure
- REFERENCES

The utility of the disposition index as a measure of β-cell compensatory capacity rests on the established hyperbolic relationship between its component insulin secretion and sensitivity measures as derived from the intravenous glucose tolerance test (IVGTT). If one is to derive an analogous measure of β-cell compensation from the oral glucose tolerance test (OGTT), it is thus necessary to first establish the existence of this hyperbolic relationship between OGTT-based measures of insulin secretion and insulin sensitivity. In this context, we tested five OGTT-based measures of secretion (insulinogenic index, Stumvoll first phase, Stumvoll second phase, ratio of total area-under-the-insulin-curve to area-under-the-glucose-curve (AUC_{ins/gluc}), and incremental AUC_{ins/gluc}) with two measures of sensitivity (Matsuda index and 1/Homeostasis Model of Assessment for insulin resistance (HOMA-IR)). Using a model of log(secretion measure) = constant + β × log(sensitivity measure), a hyperbolic relationship can be established if β is approximately equal to −1, with 95% confidence interval (CI) excluding 0. In 277 women with normal glucose tolerance (NGT), the pairing of total AUC_{ins/gluc} and Matsuda index was the only combination that satisfied these criteria (β = −0.99, 95% CI (−1.66, −0.33)). This pairing also satisfied hyperbolic criteria in 53 women with impaired glucose tolerance (IGT) (β = −1.02, (−1.72, −0.32)). In a separate data set, this pairing yielded distinct hyperbolae for NGT (*n* = 245) (β = −0.99, (−1.67, −0.32)), IGT (*n* = 116) (β = −1.18, (−1.84, −0.53)), and diabetes (*n* = 43) (β = −1.37, (−2.46, −0.29)). Moreover, the product of AUC_{ins/gluc} and Matsuda index progressively decreased from NGT (212) to IGT (193) to diabetes (104) (*P* < 0.001), consistent with declining β-cell function. In summary, a hyperbolic relationship can be demonstrated between OGTT-derived AUC_{ins/gluc} and Matsuda index across a range of glucose tolerance. Based on these findings, the product of these two indices emerges as a potential OGTT-based measure of β-cell function.

### Introduction

- Top of page
- Abstract
- Introduction
- Methods and Procedures
- Results
- Discussion
- Acknowledgment
- Disclosure
- REFERENCES

Insulin secretion and insulin sensitivity are linked through a negative feedback loop, whereby pancreatic β-cells compensate for changes in whole-body insulin sensitivity through a proportionate and reciprocal change in insulin secretion (1,2). Accordingly, in a classic paper from 1981, Bergman *et al.* first postulated that the relationship between insulin secretion and insulin sensitivity would be best characterized by a rectangular hyperbolic function (i.e., *y* = constant/*x*) (3). In 1993, Kahn *et al.* confirmed the existence of this hyperbolic relationship in human subjects using the acute-insulin-response-to-glucose (AIR_{g}) and the insulin sensitivity index (S_{I}), measures of insulin secretion and sensitivity, respectively, obtained from the intravenous glucose tolerance test (IVGTT) (4). This hyperbolic relationship implies that the product of AIR_{g} and S_{I}, termed the “disposition index,” should yield a constant for a given degree of glucose tolerance (3,4). Indeed, in the progression from normal glucose tolerance (NGT) to impaired glucose tolerance (IGT) to diabetes, the disposition index typically decreases, which reflects the concomitant deterioration in β-cell compensation for ambient insulin resistance (5,6). Thus, by evaluating insulin secretion in the context of prevailing insulin sensitivity, the disposition index has emerged as an important integrated measure of β-cell compensation *in vivo*. Furthermore, because it is a heritable feature in subjects at risk of diabetes, the disposition index can also provide a marker to identify such high-risk individuals (5,7).

Instead of the IVGTT that is required for calculation of the disposition index, large clinical and epidemiological studies typically use the oral glucose tolerance test (OGTT). Dalla Man, Cobelli and colleagues have introduced minimal model assessment of insulin sensitivity on a multisample OGTT (seven samples over 2 h) and have extensively studied the hyperbolic relationship between insulin secretion and insulin sensitivity using this multisampling protocol (8,9,10). In contrast to this seven-sample protocol, however, the OGTT methodology in large epidemiological studies typically involves sampling only at fasting and 120 min, with possible additional samples at 30 or 60 min. As such, several simple OGTT-based indices that can be obtained from these 2–4 samples have been derived for the estimation of insulin secretion (e.g., insulinogenic index), insulin sensitivity (e.g., Matsuda index), and insulin resistance (e.g., Homeostasis Model of Assessment for insulin resistance (HOMA-IR)), respectively (11,12,13,14). To assess β-cell function, investigators have applied these indices to measure insulin secretion in the context of ambient insulin sensitivity/resistance (e.g., insulinogenic index divided by HOMA-IR) (1,15,16,17). Importantly, however, to our knowledge, a hyperbolic relationship between these simple OGTT-based indices of insulin secretion and insulin sensitivity has never been formally established, a necessary first step for the derivation of an OGTT-based measure of β-cell compensation analogous to the disposition index that can be used in large clinical studies. Thus, our objective in this analysis was to confirm the existence of a hyperbolic relationship between specific simple OGTT-based indices of insulin secretion and insulin sensitivity, using the same methodology first used by Kahn *et al.* in demonstrating this relationship for IVGTT-based measures (4).

### Methods and Procedures

- Top of page
- Abstract
- Introduction
- Methods and Procedures
- Results
- Discussion
- Acknowledgment
- Disclosure
- REFERENCES

This analysis was conducted in three stages, using 2-h 75-g OGTT data from three studies undertaken at our institution (Studies A, B, and C). All participants in these studies gave written informed consent and the study protocols were approved by institutional research ethics committees.

Study A is an ongoing observational study prospectively evaluating determinants of glucose tolerance in young women over the first year postpartum. Healthy women attending outpatient obstetrics clinics are recruited during glucose tolerance screening in late pregnancy and then return to the clinical investigational unit for OGTT at 3-months postpartum. This analysis used data from the OGTT at 3-months postpartum, so far completed by 339 women.

Study B was designed to assess the reproducibility of the OGTT at different dilutions of oral glucose (18). In this study, 35 healthy subjects completed three replicate OGTTs at each of three dilutions of 75 g of glucose (300, 600, and 900 ml) in random order. This analysis was restricted to the data from the OGTTs that used the standard 300-ml dilution, which were conducted in the 30 subjects with NGT (i.e., yielding data from 3 × 30 = 90 OGTTs).

Study C is an ongoing clinical trial designed to determine whether the combination of metformin and rosiglitazone, in conjunction with lifestyle intervention, can prevent the development of type 2 diabetes in individuals with IGT (19). This analysis used data from the screening OGTTs that determined baseline glucose tolerance status (and hence eligibility for the trial) in 404 individuals.

In all studies, glucose tolerance status was determined on the OGTT. NGT was defined by fasting glucose <6.1 mmol/l and 2-h glucose <7.8 mmol/l. IGT was defined by fasting glucose <7.0 mmol/l and 2-h glucose between 7.8 and 11.0 mmol/l inclusive. Diabetes was defined by fasting glucose ≥7.0 mmol/l or 2-h glucose ≥11.1 mmol/l.

#### Laboratory assays

In Study A, venous blood samples for measurement of glucose and insulin were drawn at 0, 30, 60, and 120 min during the OGTT. Glucose and insulin measurements from the same time points from Study B were used for this analysis. In Study C, glucose and insulin were measured at 0, 30, and 120 min. All assays for glucose and insulin were performed at the Banting and Best Diabetes Centre Core Laboratory, as previously described (18,19).

#### Insulin secretion and sensitivity indices

The following five OGTT-based measures of insulin secretion were studied: (i) insulinogenic index, (ii) Stumvoll first phase index, (iii) Stumvoll second phase index, (iv) ratio of the total area-under-the-insulin-curve to the total area-under-the-glucose-curve (total AUC_{ins/gluc}), and (v) ratio of the incremental area-under-the-insulin-curve to the incremental area-under-the-glucose-curve (incremental AUC_{ins/gluc}). Insulinogenic index is calculated as follows: (Ins_{30} − Ins_{0})/(Gluc_{30} − Gluc_{0}), where Ins_{y} and Gluc_{y} represent insulin and glucose values, respectively, at time *y* min during the OGTT (15,16). The Stumvoll first phase index is defined by the following formula: 1194 + 4.724 × Ins_{0} − 117.0 × Gluc_{60} + 1.414 × Ins_{60} (20). The Stumvoll second phase index is defined as follows: 295 + 0.349 × Ins_{60} − 25.72 × Gluc_{60} + 1.107 × Ins_{0} (20). Total AUC_{ins/gluc}, an emerging measure of secretion (15,21), was calculated using the trapezoidal rule applied to the insulin and glucose curves during the OGTT. Incremental AUC_{ins/gluc} was similarly calculated using the trapezoidal rule with determination of the incremental area above baseline insulin and baseline glucose, respectively.

The following two OGTT-based indices of insulin sensitivity were studied: (i) Matsuda index and (ii) 1/HOMA-IR. Matsuda index (13) is defined by the following formula: HOMA-IR (14), a measure of insulin resistance, is defined as follows: (Gluc_{0} × Ins_{0})/22.5. Thus, 1/HOMA-IR, a measure of insulin sensitivity, is defined by the reciprocal of this formula. Both Matsuda index and 1/HOMA-IR have been validated against the euglycemic-hyperinsulinemic clamp (13,22). These two measures of insulin sensitivity were chosen for this analysis because they represent a fasting index (1/HOMA-IR) and an OGTT-stimulated index (Matsuda).

#### Confirmation of hyperbolic relationship

A rectangular hyperbolic relationship between insulin secretion and insulin sensitivity can be characterized by the following equation: insulin secretion = constant/insulin sensitivity (4,23). With log transformation, this equation becomes the following: log(insulin secretion) = constant − log(insulin sensitivity). In this context, we applied regression analysis to combinations of OGTT-based measures of insulin secretion and insulin sensitivity to determine the regression coefficient β for the following model: log(secretion measure) = constant + β × log(sensitivity measure). As noted by Kahn *et al.* (4), a hyperbolic relationship between a measure of secretion and a measure of sensitivity would be represented by β = −1. Thus, with regression analysis applied to this model, a hyperbolic relationship can be established if the following criteria are satisfied: (i) β is approximately equal to −1 and (ii) the 95% confidence interval (CI) of β excludes 0.

With a simple linear regression model (i.e., *y = βx + ɛ*, where ɛ is the sampling error associated with dependent random variable *y*), if all of the assumptions for linear regression are met (e.g., such as that the variance of ɛ is constant) and we assume that neither *x* nor *y* has measurement error, then ordinary least squares regression will give the best linear unbiased estimate of the true parameters. However, if the observed *x* is associated with random measurement error, this random measurement error will result in larger variance for the observed variable and will attenuate the correlation between *y* and *x*, thereby causing a downward bias in the regression coefficient β (the slope). Conversely, if *y* is associated with random measurement error while *x* is perfectly measured, it will not affect the estimation of regression coefficient β, but will lead to a larger standard error of β. Furthermore, in the case of glycemic measures, this bias is amplified in the setting of hyperglycemia (24).

In this study, where both the dependent variable (insulin secretion) and the independent variable (insulin sensitivity) are measured with error, ordinary least squares regression would yield both a biased regression coefficient (underestimated) and a biased standard error of the coefficient estimate (overestimated) (4,25). For this reason, like Kahn *et al.* (4), we used Perpendicular Least Squares Properly Weighted (PW) regression (26), a method that takes account of the error in both the dependent and independent variables, and thereby provides a better coefficient estimate. The coefficient estimate from the PW regression is represented by the following equation:

where *s* _{yy} is the sum of squared deviations from sample mean of independent variables, *s* _{xx} is the sum of squared deviations from sample mean of dependent variables, *s* _{xy} is the sum of products of deviations in independent variables and dependent variables, and λ is the ratio of the variance of the error perturbing the dependent variable to that of the error perturbing the independent variable. Again, like Kahn *et al.* (4), these error estimates were based on the coefficient of variation (CV) for each insulin secretion and insulin sensitivity measure in our lab, as determined from replicate OGTTs in 30 subjects with NGT (18). The CV values were as follows: (i) insulinogenic index: 40.6%; (ii) Stumvoll first phase: 19.2%; (iii) Stumvoll second phase: 15.9%; (iv) total AUC_{ins/gluc}: 24.8%; (v) incremental AUC_{ins/gluc}: 46.3%; (vi) Matsuda index: 21.9%; and (vii) 1/HOMA-IR: 21.1%.

After estimating β using PW regression, it is necessary to determine its 95% CI, to establish whether the hyperbolic criteria have been met. The bootstrap method was used to calculate the standard error of the coefficient estimate of β (27). We drew 100, 500, and 1,000 bootstrap samples from the original data set with replacement. The bootstrap standard error estimates of β were similar for 100, 500, and 1,000 bootstrap samples, indicating that the randomness in the bootstrap standard error from using a finite number of bootstrap sample sizes was negligible when we took over 100 bootstrap samples. Therefore, we determined the 95% CI of β, where β was estimated by PW regression and the standard error of β was obtained using the bootstrap method with 1,000 sets of data randomly selected from the original data set for each analysis.

#### Statistical analysis

All analyses were conducted using the statistical analysis system (SAS, version 9.1; SAS Institute, Cary, NC). Plots were prepared using SigmaPlot Version 10.0 (Systat Software, Richmond, CA). Descriptive data are presented as mean ± s.d., unless otherwise indicated. The analysis consisted of the following three phases:

*Phase 1*. The 10 combinations of five insulin secretion and two insulin sensitivity measures were first tested for the existence of a hyperbolic relationship in women with NGT and IGT, respectively, from Study A (**Table 1**). This stage identified those combinations that satisfied the hyperbolic criteria.

Table 1. Estimated regression coefficient (β) and 95% CI for β, for combinations of insulin secretion, and sensitivity measures regressed using the following model: log(secretion measure) = constant + β × log(sensitivity measure) |

*Phase 2*. As OGTT-based indices of insulin secretion and insulin sensitivity share many of the same measurements of glucose and insulin (i.e., fasting values, 120-min values), the possibility of statistical auto-correlation as a basis for any apparent hyperbolic relationship must be considered (15,28). For this reason, combinations identified in phase 1 were next evaluated using the replicate OGTTs from Study B, with insulin secretion and insulin sensitivity measures taken from different OGTTs in the same individual (**Figure 2**). Because these indices do not share the same insulin and glucose measurements, any observed hyperbolic relationship on this analysis could not be on the basis of statistical auto-correlation.

*Phase 3*. The physiological relationship between insulin secretion and insulin sensitivity would be expected to display distinct hyperbolae for different degrees of glucose tolerance (as seen with the disposition index (5,6)). Thus, the combination identified in phase 2 was next evaluated within each of the three glucose tolerance groups represented in Study C (NGT, IGT, diabetes). Analysis of variance was used to compare the mean product of total AUC_{ins/gluc} and Matsuda index from each of the respective glucose tolerance groups.

### Discussion

- Top of page
- Abstract
- Introduction
- Methods and Procedures
- Results
- Discussion
- Acknowledgment
- Disclosure
- REFERENCES

In this report, we confirm that a hyperbolic relationship can be consistently demonstrated between OGTT-based total AUC_{ins/gluc} and the Matsuda index in three different data sets. Furthermore, this pairing yields distinct hyperbolae for different degrees of glucose tolerance, ranging from NGT to IGT to diabetes. Based on these findings, the product of these two indices emerges as a potential OGTT-based measure of β-cell function, analogous to the disposition index.

In order to maintain glucose homeostasis, pancreatic β-cells compensate for changes in whole-body insulin sensitivity by a proportionate and reciprocal change in insulin secretion (1,2). The resultant rectangular hyperbolic relationship between insulin secretion and insulin sensitivity has been shown to exist between AIR_{g} and S_{I}, as measured on IVGTT, thereby giving rise to the concept of the disposition index (the product of AIR_{g} and S_{I}) as an integrated measure of β-cell function (3,4). As large clinical studies typically use the OGTT (rather than the intravenous test), the development of an OGTT-based measure of β-cell compensation, analogous to the disposition index, would be of interest. To derive such a measure, however, a necessary first step is the confirmation of a hyperbolic relationship between the component insulin secretion and insulin sensitivity indices.

To our knowledge, a hyperbolic relationship has not been formally demonstrated between simple OGTT-based measures of insulin secretion and insulin sensitivity (such as insulinogenic index and 1/HOMA-IR, respectively). Specifically, although this relationship has been studied in the context of a multisample minimal model OGTT (10), it has not been demonstrated in the setting of the standard OGTT protocol that is typically used in large clinical and epidemiological studies. Instead, the relationship between standard OGTT-derived insulin secretion and sensitivity indices has been simply described as hyperbolic in appearance, albeit without mathematical confirmation (29,30). In other studies, mathematical confirmation has been sought through the application of statistical regression to characterize the relationship between secretion and sensitivity (31,32,33). In these studies, however, an analytical issue that has not been considered is that both the secretion and sensitivity indices are measured with error. As standard regression assumes that error is present only in the dependent variable (and not in the independent variable), its usage in the setting of error in both variables will lead to an underestimation of the regression coefficient that relates them (4,25). Indeed, for this reason, in demonstrating the hyperbolic relationship between AIR_{g} and S_{I}, Kahn *et al.* used a regression method (PW regression) that corrects this bias by incorporating a factor that accounts for the error in both variables (4). Accordingly, in this report, we have applied the same PW regression methodology to confirm formally the existence of a hyperbolic relationship between OGTT-derived AUC_{ins/gluc} and Matsuda index.

It is well recognized that statistical auto-correlation can underlie an apparent hyperbolic relationship between measures of insulin secretion and insulin sensitivity. For example, an earlier study noted above (31) that had the same objective as this current analysis (but that did not use PW regression) reported a hyperbolic relationship between the fasting glucose-to-insulin ratio (G/I) and the Homeostasis Model of Assessment of β-cell function (HOMA-β = 20 × I/(G-3.5)). As was subsequently noted (34), however, these two indices have nearly reciprocal formulae (i.e., G/I vs. I/G) and hence will automatically exhibit a hyperbolic relationship on that basis alone (i.e., rather than reflecting a physiological association). In this analysis, we feel that auto-correlation is not driving the hyperbolic relationship between OGTT-derived AUC_{ins/gluc} and Matsuda index for three reasons. First, the formulae for these two indices (provided in the Methods and Procedures section) are not reciprocals. Second, as shown in phase 3 of the Results section, the product of these two indices is distinct for different degrees of glucose tolerance (whereas it would not be expected to differ between glucose tolerance groups if the two component indices were reciprocals, as the product should then be the same in all cases, regardless of glucose tolerance status). Third, as shown in **Figure 2**, total AUC_{ins/gluc} and Matsuda index measured on separate days in the same individual still maintain a hyperbolic association, despite being derived from distinct measurements. Thus, in ruling out the possibility of auto-correlation, this analysis suggests that the observed hyperbolic relationship indeed reflects the physiological feedback loop relating insulin secretion and insulin sensitivity within the individual.

In this report, the hyperbolic relationship between AUC_{ins/gluc} and Matsuda index is shown to exist in subjects with NGT, IGT, and diabetes, respectively, as evidenced by three distinct hyperbolae. Similar to that which is observed with disposition index curves (5,6), these hyperbolae exhibit a shift toward the origin as glucose tolerance worsens (**Figure 3**). Mathematically, this feature is represented by a progressive decline in the product of total AUC_{ins/gluc} and Matsuda index, when moving from NGT to IGT to diabetes. Thus, analogous to the disposition index, this product emerges as a novel OGTT-based measure capable of identifying the differences in β-cell function between these glucose tolerance groups. In light of growing recognition of the importance of β-cell dysfunction in the early pathophysiology of type 2 diabetes (1), the development of such a measure that can be derived from a standard OGTT could prove to be very useful for clinical and epidemiological studies. Further study of the potential application of the product of total AUC_{ins/gluc} and Matsuda index as a measure of β-cell function is warranted.

A limitation of this analysis in subjects with IGT and diabetes is that the CVs for the insulin secretion and insulin sensitivity measures (needed for PW regression) were derived from a set of glucose-tolerant subjects who underwent multiple OGTTs. These CVs may be different in individuals with glucose intolerance. Nevertheless, when we determined the CVs using data from four subjects with IGT who underwent repeat OGTTs and then applied these values to the regression analysis of the 53 subjects with IGT from Study A, the pairing of total AUC_{ins/gluc} and Matsuda index again met hyperbolic criteria (β = −1.05, (−1.85, −0.25)). Second, validation of the analyses reported herein against IVGTT data would be desirable but was not possible, as IVGTTs were not performed in the three study populations. Finally, another issue to note in this study is that insulin levels during the OGTT may be affected by other factors apart from β-cell function. Specifically, circulating insulin levels are affected by incretin hormones and by hepatic extraction, both of which may limit the degree to which insulin measurements during the OGTT can reflect β-cell function.

It is important to recognize that β-cell function can be assessed on OGTT by several different means, including (i) mathematical modeling of dynamic parameters of β-cell function (10,35), (ii) measurement of relative proinsulin secretion (36), and (iii) evaluation of insulin secretion adjusted for ambient insulin sensitivity (e.g., insulinogenic index/HOMA-IR) (1). The purpose of this analysis was to determine whether a hyperbolic relationship can be established between simple OGTT-based indices of insulin secretion and insulin sensitivity, as a necessary first step toward the development of an OGTT-based measure of β-cell compensation analogous to the disposition index. Nevertheless, the failure to demonstrate a hyperbolic relationship between specific combinations of insulin secretion and insulin sensitivity measures in this study should not be interpreted as an argument against the use of these measures in the evaluation of β-cell function. In the case of the insulinogenic index, the variability inherent in this measure, as evidenced by both its CV and the wide CIs for β in its pairings with sensitivity measures, precluded the satisfaction of hyperbolic criteria. Indeed, marked within-patient variability of this measure, as seen in this study, has been previously reported (37). Thus, it remains possible that a hyperbolic relationship exists between insulinogenic index and either 1/HOMA-IR or Matsuda index, but could not be established in this analysis. A similar situation exists with incremental AUC_{ins/gluc}, which was also a noisy measure (in this study and in ref. 37). The relative success of the closely related total AUC_{ins/gluc} in meeting hyperbolic criteria (with Matsuda index) is likely in part due to the lesser degree of variability with this measure, compared with its incremental counterpart.

In conclusion, a hyperbolic relationship can be consistently demonstrated between OGTT-based total AUC_{ins/gluc} and Matsuda index. Moreover, this pairing of secretion and sensitivity measures yields distinct hyperbolae for different degrees of glucose tolerance, ranging from NGT to IGT to diabetes. Based on these findings, the product of these two indices emerges as a potential OGTT-based measure of β-cell function.