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Identifying glucose thresholds for incident diabetes by
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physiological analysis: a mathematical solution.
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Ele Ferrannini,1 and Maria Laura Manca,2,3
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CNR Institute of Clinical Physiology, Pisa, Italy, 2 Department of Clinical &
Experimental Medicine, and 3 Department of Mathematics, University of Pisa, Italy
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Short title: Optimal plasma glucose levels by physiological analysis
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Corresponding author: Ele Ferrannini, MD Department of Clinical & Experimental Medicine Via Roma, 67 56126 Pisa, Italy e-mail:
[email protected] phone: +39 050 553272 fax: +39 050 553235
Copyright © 2014 by the American Physiological Society.
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Abstract
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Plasma glucose thresholds for diagnosis of type 2 diabetes are currently based on outcome
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data (risk of retinopathy), an inherently ill-conditioned approach. A radically different approach
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is to consider the mechanisms that control plasma glucose rather than its relation to an outcome.
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We developed a constraint optimization algorithm to find the minimal glucose levels associated
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with the maximized combination of insulin sensitivity and ß-cell function, the two main
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mechanisms of glucose homeostasis. We used a training cohort of 1,474 subjects (22%
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prediabetic, 7.7% diabetic) in whom insulin sensitivity was measured by the clamp technique
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and ß-cell function by mathematical modeling of an OGTT. Optimized fasting glucoses were
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≤87 mg/dL and ≤89 mg/dL in women and men under age 45, and ≤92 and ≤95 mg/dL in women
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and men over age 45; the corresponding optimized 2-hour glycemias were ≤96, ≤98, ≤103, and
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≤105 mg/dL. These thresholds were validated in three prospective cohorts of nondiabetic
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subjects (RISC, Botnia, and Mexico City Diabetes Study) with baseline and follow-up OGTT.
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Altogether, 452 of 5,593 participants progressed to diabetes. Similarly in the three cohorts,
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subjects with glucose values above the estimated thresholds had an odds ratio of 3.74 [95%CI:
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2.64-5.48] of progressing, substantially higher than the risk carried by baseline conventionally
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defined prediabetes (2.32 [CI: 1.91-2.81]. The concept is proven that, optimization of glucose
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concentrations by using direct measures of insulin sensitivity and ß-cell function identifies
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gender- and age-specific thresholds that bear on disease progression in a physiologically sound,
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quantifiable manner.
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Key words: glucose thresholds, ß-cell function, insulin sensitivity, incident diabetes
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Introduction
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Type 2 diabetes (T2DM) is diagnosed based on fasting (≥126 mg/dL) and/or 2-hour plasma
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glucose concentration (≥200 mg/dL) during a standard oral glucose tolerance test (OGTT)
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(1,31). Prediabetes – encompassing impaired fasting glycemia (IFG) and impaired glucose
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tolerance (IGT) – also is diagnosed on the basis of fasting and/or 2-hour plasma glucose
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concentrations (1,31). These threshold values hinge upon epidemiological evidence of risk for
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presence or future development of retinopathy (1,31) as well as some evidence for a bimodal
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distribution of plasma glucose values (25). However, using outcomes to set diagnostic cutoff
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points is inherently ill-conditioned as the choice strongly depends on quality of outcome
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ascertainment, sample size, duration of observation, and identification of point(s) of inflection.
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For example, a recent analysis of three large population-based, cross-sectional studies yielded no
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evidence for a clear glycemic threshold for prevalent or incident retinopathy (30).
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In longitudinal surveys, plasma glucose levels as continuous variables are consistent
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predictors of incident T2DM in nondiabetic individuals along with other risk factors (e.g.,
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positive family history of diabetes, obesity, etc.) (24,26,29). A common approach to improving
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diabetes prediction is to include additional biomarkers – such as HbA1c (24) or multiple other
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biomarkers (13,17) – in the prediction model.
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A radically different approach is to consider the mechanisms that control the level of the
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biomarker rather than its relation to an outcome. If the relationship between mechanism and
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biomarker can be described quantitatively (in the form of a so-called objective function), one can
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determine the best level of control mechanism that optimizes the biomarker, which can then be
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tested as a threshold. In applied mathematics, this approach is known as an optimization
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problem. Optimization programs are ubiquitous in engineering (20) and science (12); to our
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knowledge, they have never been proposed for plasma glucose. In the case of glucose, it is well
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established that both ß-cell function and insulin sensitivity are strong independent determinants
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of plasma glucose concentrations (7,10,15). Therefore, by using measures of insulin secretion or
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insulin action one can find the minimal plasma glucose level, or threshold, above which the risk
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of incident diabetes becomes clinically significant.
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In the present work, we took advantage of a very large database in which both insulin
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sensitivity and ß-cell function were measured by gold-standard methodology across the full
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range of glucose tolerance. These data were used to build a ‘bi-objective’ optimization problem,
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i.e., to estimate the best combination of these two control variables that simultaneously
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minimizes fasting and 2-hour plasma glucose levels. The value of these thresholds was then
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assessed in 3 longitudinal cohorts of incident T2DM.
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Materials and Methods
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Clinical data The training cohort consisted of 1,474 adult subjects (22% prediabetic, 7.7%
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T2DM) in whom insulin sensitivity was measured by the euglycemic clamp technique and ß-cell
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function by mathematical modeling of a standard (2 hours, 75 g) OGTT. This cohort included
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nondiabetic participants of the RISC Study (8) as well as subjects and patients studied at our
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Center as part of different project. As per previously published methods, insulin sensitivity was
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indexed as the steady-state whole-body glucose utilization rate (M) normalized by the steady-
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state plasma insulin concentration (M/I, µmol.min-1.m-2.mM-1) (5), while ß-cell function was
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indexed as ß-cell glucose sensitivity (ßGS, pmol.min-1.m-2.mM-1), i.e., the mean slope of the
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dose-response function of insulin secretion rates (reconstructed by deconvolution of plasma C-
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peptide levels) vs plasma glucose levels during the OGTT (18). The anthropometric and clinical
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characteristics of the training cohort are given in Table 1 grouped by gender and glucose
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tolerance status (normal glucose tolerance (NGT), prediabetes (encompassing IFG and IGT),
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and T2DM according to the criteria of the American Diabetes Association (1).
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Three prospective, observational studies whose participants received a standard OGTT both
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at baseline and follow up (with measurement of fasting and 2-hour plasma glucose levels) were
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used as validation cohorts. (a) In the RISC Study (8), nondiabetic participants were recruited at
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19 centers in 13 countries in Europe, according to the following inclusion criteria: men or
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women, age between 30-60 years (stratified by sex and age), and clinically healthy. Initial
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exclusion criteria were: treatment for obesity, hypertension, lipid disorders or diabetes,
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pregnancy, cardiovascular or chronic lung disease, weight change of ≥5 kg in last 6 months,
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cancer (in last 5 years) and renal failure. Exclusion criteria after screening were: arterial blood
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pressure ≥140/90 mmHg, prediabetes or known diabetes, total serum cholesterol ≥7.8 mmol/l,
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serum triglycerides ≥4.6 mmol/l, and ECG abnormalities. Baseline examinations were
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completed in July 2005; 1,048 subjects (503 women, aged 44±14 years, and 545 men, aged
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47±12 years) participated in the follow-up examination 3 years later. Subjects were classified as
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progressors (n=130) if they stepped up along the sequences NGTprediabetes, NGTT2DM,
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prediabetesT2DM between baseline and follow up.
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(b) The Botnia Study is a family-based, observational study started in 1990 on the West coast
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of Finland aiming at identifying diabetes susceptibility genes (16). The prospective part included
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2,585 nondiabetic family members and/or their spouses (1,186 men, aged 45±13 years, and
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1,399 women, aged 46±14 years) who were followed up for 9.5 years. Progressors (n=152) were
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those nondiabetic participants who had developed T2DM by the end of the study.
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(c) The Mexico City Diabetes Study (MCDS) is a population-based cohort participating in a
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longitudinal survey of incident diabetes and cardiovascular risk factors (6). Among the 15,532
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inhabitants of low-income neighborhoods in Mexico City, 2,282 subjects randomly selected
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were examined between 1990-1992 and invited to return for two follow-up examinations, at
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3.25 and 7 years. Among the1,960 nondiabetic subjects (1,146 men, aged 46±8 years, and 814
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women, aged 47±8 years) who were seen at the 7-year follow up, progressors (n=170) were
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those found to have T2DM.
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The studies of the three validation cohorts (RISC, Botnia, MCDS) had been approved by the
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respective Ethics Committees (6,8,16). The present analysis of data from those cohorts has been
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approved by the Ethics Committee of the University of Pisa.
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Optimization modeling Optimization refers to a branch of applied mathematics concerned
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with the maximization or minimization of certain functions (objective functions), possibly under
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constraints (11,21). The present study, based on an optimization problem with constraints,
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aimed at maximizing both insulin sensitivity and glucose sensitivity, expressed as simultaneous
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functions of fasting (FPG) and postglucose plasma glucose concentrations (2-hrG). In other
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words, in the training dataset we searched for the values of the independent variables, FPG and
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2-hrG, that correspond to maximum levels of ßGS and M/I. The first step was the construction
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of the two objective functions, using the IBM SPSS20® software. The training dataset was
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divided into four subgroups based on gender and age (≤ or > 45 years). From a preliminary
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investigation of the data, nonlinear interactions emerged between the dependent variables (M/I,
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ßGS) and the independent variables (FPG, 2-hrG); therefore, these variables were all
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transformed into their natural logarithms for use in linear modeling. After each transform, we
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used an automatic data preparation in order to maximize the predictive power of the model, by
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excluding values of variables lying outside 3 standard deviations from the mean (n=21). Next,
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the bestsubsets method was applied. This method checks all possible models, or at least a larger
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subset of the possible models than the forward stepwise method, to choose the best model as
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judged by the adjusted R2.
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The mathematical procedure for determining Pareto optima and their range by sensitivity
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analysis, detailed in the Appendix and Supplemental Table 1, yields an upper boundary for
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FPG and 2-hrG in the 4 subsets of individuals, and the ‘minimal’ value of the associated control
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variables (M/I and ßGS). All subjects with both FPG and 2-hrG values below the optimized
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threshold (Opt) could then be compared to those non-optimized glucose values (non-Opt) in the
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validation cohorts.
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Statistical analysis Data are presented as mean±SD. Group comparisons were carried out
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using Mann Whitney or Kruskall-Wallis test. Adjustment for covariates was done by multiple
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regression analysis. Univariate and multivariate logistic analysis was used to relate endpoints –
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i.e., incident dysglycemia (RISC) or overt diabetes (BOTNIA, MCDS) – to baseline predictors;
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results are given as odds ratio (OR) and 95% confidence interval (CI). Relative risk (and 95%
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CI) and population-attributable risk were calculated by standard formulae.
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Results
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In the training dataset, prevalence of prediabetes and T2DM were typical of a European
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population (28). While body mass index (BMI) and fasting and 2-hr plasma glucose levels
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increased, insulin sensitivity and ß-cell glucose sensitivity decreased across glucose tolerance
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status, in women as well as men (Table 1). Of note, both M/I and ßGS were inversely related to
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age (p 201
> 136
123 [101]
106 [93]
105 [71]
87 [68]
> 142
> 124
> 131
> 120
487 488 489 490 491
Table 3 – Clinical and metabolic phenotype of nondiabetic subjects with
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optimal (Opt) or non-optimal (non-Opt) plasma glucose concentrations in the 3
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validation cohorts.
494 RISC
Cohort
Category (n)
BOTNIA Opt
non-Opt
(n=316)
(n=732)
Opt
MCDS non-Opt (n=2,319)
Opt
non-Opt
(n=593)
(n=1,367)
(n=266) Gender (% women)
48*
55
49
55
33*
70
Familial T2DM (%)
20*
32
83
83
-
-
50 ± 12*
44 ± 14
47 ± 14
45 ± 14
44 ± 8*
47 ± 8
Age (years)
Body mass index (kg.m- 24.0 ± 3.6* 2
25.9 ± 4.1 26.5 ± 3.5* 28.6 ± 4.4 26.1 ± 3.9
24.5 ± 3.4*
0.88 ± 0.11
-
)
Waist-to-hip ratio (cm/cm)
0.83±
-
0.08*
0.98 ± 0.07
0.07*
Fasting glucose (mg/dL)
84 ± 7*
95 ± 9
85 ± 5*
101 ± 10
77 ± 9*
88 ± 12
2-hr glucose (mg/dL)
83 ± 13*
113 ± 27
82 ± 14*
115 ± 27
77 ± 16*
118 ± 29
Fasting insulin (pmol/L) 27 ± 14*
37 ± 20
40 ± 21*
50 ± 31
69 ± 52*
104 ± 93
2-hr insulin (pmol/L)
227 ± 206* 194 ± 184
Total cholesterol
4.69 ±
(mmol/L)
0.90*
173 ± 132*
Triglycerides (mmol/L)
277 ± 236 327 ± 271* 679 ± 520
4.94 ± 0.84
-
1.17 ± 0.95
-
0.93 ±
495 496 497
0.96 ±
4.69 ± 1.04 4.92 ± 1.10
-
2.43 ± 1.65
0.60*
HDL-cholesterol
1.50 ±
(mmol/L)
0.37*
2.30 ± 1.72*
1.40 ± 0.38
0.83 ±
-
0.85 ± 0.23 0.23*
Table 4 – Coefficients of constrained linear programming [2].
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Women Age Coefficients of Single Objective Function
≤ 45 y
> 45 y
- 0.76, -1.86, - 1.94, - 1.73,
Men ≤ 45 y
> 45 y
- 0.66, - 1.20,
- 1.54, - 1.75,
14.08
15.97
12.47
14.84
- 1.38, -2.98
- 2.15, -4.05
- 0.90, - 2.20
- 1.69, - 3.99
0.99, 2.78
1.28, 2.80
1.13, 2.89
1.28, 2.89
0.92, 3.33
0.79, 3.28
0.83, 3.11
0.99, 3.30
Right-hand side range of 1st and 2nd constr.
Right-hand side range of ln[FPG] constr. (d1, e1) Right-hand side range of ln[2-hrG] constr. 500 501 502
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Table 5 – Optimized glucose values obtained by using only one independent
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variable or only one objective function.
505 Women
Men
≤ 45 y
> 45 y
≤ 45 y
> 45 y
≤89
≤92
≤90
≤94
2-hr glucose (mg/dL)
≤97
≤102
≤103
≤105
f (ln[FPG], ln[2-
Fasting glucose
≤290
≤295
≤324
≤325
hrG)]
(mg/dL)
f (ln[FPG], ln[2-
2-hr glucose (mg/dL)
≤501
≤477
≤403
≤488
ln[ßGS] & ln[M/I] f (ln[FPG])
f (ln[2-hrG])
Fasting glucose (mg/dL)
ln[ßGS] or ln[M/I]
hrG)] 506 507
RISC(uni)
RISC(multi) BOTNIA(uni)
BOTNIA(multi) MCDS(uni)
MCDS(multi) 0.5
1
3
5
10
15
Odds ratio (95% C.I.) Figure 1
20
Opt
Relative risk
10
NGT 1
0.4
Population-attributable risk
60 65 70 75 80 85 90 95 100 105 110 115 120 125
0.9
Opt
0.8 0.7 0.6 0.5 0.4
NGT
0.3 0.2 0.1 0 60 65 70 75 80 85 90 95 100 105 110 115 120 125
Fasting plasma glucose (mg/dL)
Figure 2
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Relative risk
10
Opt NGT 1
0.4 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145
Population-attributable risk
0.9 0.8 0.7 0.6 0.5
Opt
0.4 0.3 0.2 0.1
NGT
0 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145
2-hour plasma glucose (mg/dL)
Figure 3