## Introduction

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).