Quantitative Estimation of Disposition Index from Postprandial Glucose Data across the Spectrum of Glucose Tolerance

Abstract

Disposition Index (DI), defined as the product of insulin sensitivity and beta-cell responsivity, is the best measure of beta-cell function. This is usually assessed from plasma glucose and insulin, and sometimes C-peptide, data either from surrogate indices or model-based methods. However, the recent advent of continuous glucose monitoring (CGM) systems in non-insulin-treated individuals, raises the possibility of its quantification in outpatients. As a first step, we propose a method to assess DI from glucose concentration data only and validated it against the oral minimal model (OMM).

To do so, we used data from two clinical mixed meal tolerance test (MTT) studies in non-insulin-treated individuals: the first consisted of 14 individuals with type 2 diabetes studied twice, either after receiving a DPP-4 inhibitor or a placebo before the meal, while the second consisted of 62 individuals, with and without pre- or type 2 diabetes. A third, simulated, dataset consisted of 100 virtual subjects from the Padova Type 2 Diabetes Simulator was used for additional tests. Plasma glucose, insulin and C-peptide concentrations were used to estimate the reference DI from the OMM (${\text{DI}}^{\mathrm{MM}}$), while glucose data only were used to calculate the proposed DI (${\text{DI}}^{\mathrm{G}}$).

${\text{DI}}^{\mathrm{G}}$ was well correlated with ${\text{DI}}^{\mathrm{MM}}$ in both the clinical and simulated datasets (R between 0.88 and 0.79, p<0.001), and exhibited the same between-visit or between-group pattern.

${\text{DI}}^{\mathrm{G}}$ can be used to assess therapy effectiveness and degree of glucose tolerance using glucose data only, paving the way to potentially assess beta-cell function in real-life conditions using CGM.

NEW & NOTEWORTHY

The advent of continuous glucose monitoring (CGM) in non-insulin-treated individuals raises the possibility of quantifying Disposition Index (DI), a key metric of beta-cell function usually assessed in research settings, in outpatients. A method for DI estimation from postprandial glucose data only (${\text{DI}}^{\mathrm{G}}$) was developed and validated against reference. ${\text{DI}}^{\mathrm{G}}$ can be used to assess therapy effectiveness and degree of glucose tolerance in non-insulin-treated individuals, paving the way for its quantification in real-life conditions from CGM devices.

Graphical Abstract:

In the last decades the ready availability and use of continuous glucose monitoring (CGM) systems has improved glycemic control in both individuals with type 1 diabetes [1][2] and insulin-treated type 2 diabetes [3][4]. However, only a few studies focusing in individuals with non-insulin-treated type 2 diabetes (T2D) [5][6][7][8], while none in individuals with pre-diabetes (PD), have assessed the role of CGM systems in disease management, demonstrating HbA1c reduction or improvement in standard CGM-based metrics [9], but possibly not leveraging all the information provided by these devices [10]. In addition, recent evidence has also highlighted the potential role of CGM and CGM-based metrics in the screening and management in individuals with prediabetes [11], presymptomatic type 1 diabetes [12] and cystic fibrosis related diabetes [13].

Recently, our group developed a method to quantify an index of insulin sensitivity in individuals with type 1 diabetes (T1D) from modern minimally invasive diabetes technologies, like CGM and insulin pump [14][15]. This method allowed, for the first time, the assessment of this index of glucose-insulin homeostasis in outpatients overcoming the need for specialized testing (see [16] for review). In addition, this index was also successfully used both in silico [17] and in vivo [18][19] to adjust therapy parameters, like the insulin to carbohydrate ratio, in individuals with T1D in real life conditions.

However, when dealing with non-insulin-treated individuals, like people with PD and early-stage T2D, it is an accepted notion that an accurate evaluation of glucose-insulin homeostasis must consider both the ability of insulin to lower glucose concentrations (insulin sensitivity) and the ability of the β-cells to secrete insulin in response to glucose increase (beta-cell responsivity). To date, the disposition index (DI), defined as the product of insulin sensitivity and beta-cell responsivity, is one of the most widely used approaches to do so [20]. DI was firstly introduced by Bergman et al. [20], defined as the product between the minimal model-derived insulin sensitivity [21] and the second-phase post-hepatic insulin secretion after an intravenous glucose tolerance test (IVGTT). Thereafter, other definitions/methods have been proposed to calculate it both from IVGTT, and/or oral/meal glucose tolerance test (OGTT/MTT) experiments [22][23][24][25][26][27]. Of note, all the above methods require the frequent measurement of plasma glucose, insulin, and sometime C-peptide concentrations, therefore limiting its usability to clinical settings. However, a method to quantify DI in everyday life from minimally-invasive technologies, like continuous glucose monitoring (CGM) devices, in noninsulin-treated individuals like those with T2D and PD, may help to assess therapeutic efficacy and/or disease progression. As a first step in this direction, we propose a method to quantify DI from postprandial glucose data only (${\text{DI}}^{\mathrm{G}}$) in non-insulin-treated individuals. To do so, two real (in vivo) datasets were used to develop the proposed ${\text{DI}}^{\mathrm{G}}$ and to validate it against the Oral Minimal Model DI (${\text{DI}}^{\mathrm{MM}}$) [28], which requires plasma glucose, insulin and C-peptide data. Secondly, as a proof of concept, we used a third, simulated (in silico), dataset, which also includes CGM data in addition to plasma glucose and hormones concentrations, to test if the method provided an accurate estimate of ${\text{DI}}^{\mathrm{G}}$ when CGM instead of plasma glucose data were used.

Research Design and Methods

Databases

Two in vivo datasets were used to develop and validate the proposed tool for the quantification of DI from plasma glucose data only. A third, in silico, dataset was also used to both quantify some parameters, not available in the population of interest, and, as a proof of concept, to test if the method could be made to work in real life scenarios using CGM data, since in the first two datasets CGM data were lacking. Moreover, the power of simulation was exploited to assess the robustness of the method to variation in the amount of ingested carbohydrates, which is also a crucial point when moving to a non-standardized outpatient setting.

In all datasets, the new tool was validated against the Oral Minimal Model method [28], which provided a quantification of DI based on plasma glucose, insulin and C-peptide data.

Dataset 1

The first dataset was obtained from a prior study (NCT00351507) [29] and composed of 14 subjects with T2D (5 males, mean±SD: age = 53±8 years, BW = 94±14 kg, HbA1c = 6.1±0.7 %, BMI=33.9±5.6 kg/m²) who underwent a randomized, double-blind, placebo-controlled crossover study receiving either 50 mg of the DPP-4 inhibitor vildagliptin (treatment) or placebo before breakfast and dinner over a 10-day period with a 2-week washout. As per protocol, all agents used for the treatment of diabetes were discontinued three weeks before the study. On day 9, subjects consumed a mixed meal containing 75 g of glucose 30 minutes after the morning dose. Blood measurement of glucose, insulin and C-peptide concentrations were frequently collected for 300 min after meal ingestion and were used to quantify the ${\text{DI}}^{\mathrm{MM}}$. Of note, one subject was monitored, in both visits, just for 240 min and, therefore, was removed from the analysis. Fig. 1 (left panels) shows measured plasma glucose, insulin and C-peptide concentrations for both placebo and treatment visit. More details about the original study are available elsewhere [29].

Figure 1.

Left panels: median and interquartile ranges of plasma glucose (top), insulin (middle), C-peptide (bottom) data during the placebo (continuous line with black triangles) and treatment (dotted line with grey squares) visits in dataset 1. Middle panels: median and interquartile ranges of plasma glucose (top), insulin (middle), C-peptide (bottom) data for individuals with type 2 diabetes (T2D, dotted line with white squares), pre-diabetes (PD, dashed line with grey circles) and healthy control (HC, continuous line with black triangles) in dataset 2. Right panels: median and interquartile ranges of plasma glucose (PG) and continuous glucose monitoring (CGM) data (top, dashed and continuous line respectively), plasma insulin (middle) and C-peptide (bottom) in dataset 3 (simulated) after a meal dose of 75g of glucose.

Dataset 2

The second dataset was also obtained from a prior study [30] composed of 9 subjects with T2D (T2D: 3 males, age = 57±8 years, BW = 87±13 kg, BMI=31.7±4.1 kg/m²), 35 subjects with pre-diabetes, both with impaired fasting and/or tolerance to glucose (PD: 17 males, age = 53±8 years, BW = 91±20 kg, BMI=30.1±5.0 kg/m²), and 18 healthy controls (HC: 6 males, age = 50±8 years, BW = 78±9 kg, BMI=27.7±3.1 kg/m²) who underwent a mixed meal test containing 75 g of glucose. At the time of the study, no individuals were on diabetes-related medications. Blood measurement of glucose, insulin and C-peptide concentrations were frequently collected for 360 min after meal ingestion and were used to quantify the ${\text{DI}}^{\mathrm{MM}}$. Of note, one subject in the HC group was removed from the analysis due to issues with insulin measurement as a consequence of hemolysis. Fig. 1 (middle panels) shows measured plasma glucose, insulin and C-peptide concentrations for individuals with T2D, PD and HC. More details about the original study are available elsewhere [30].

Dataset 3 (simulated)

The Padova Type 2 Diabetes simulator [31] was used to generate the simulated dataset. This consists of a model of glucose–insulin–C-peptide dynamics and a population of 100 in silico adults with T2D (mean±SD: age = 63±29 years, body weight = 93±18 kg). Each in silico subject underwent a mixed meal test containing 75 g of glucose where plasma glucose, insulin, C-peptide concentrations and CGM traces were simulated for 360 min after meal ingestion. In addition, to assess the robustness of the method to variation in the amount of ingested carbohydrates, subjects also underwent a mixed meal with 50 and 100 g of glucose, respectively. Finally, to get realistic data, we superimposed an independent, Gaussian noise, with zero mean and coefficient of variation (CV) equal to 2% to plasma glucose [32]; an independent, Gaussian noise, with zero mean and known variance to plasma C-peptide and insulin [33] and a colored noise to CGM data as described in [34]. In Fig. 1 (right panels) simulated plasma and sensor glucose, insulin, C-peptide concentrations in case of a mixed meal with 75 g of glucose are reported.

Calculations

Assessment of Disposition Index from Glucose Data only (${\text{DI}}^{\mathrm{G}}$)

The proposed method for the assessment of Disposition Index from glucose data only (${\text{DI}}^{\mathrm{G}}$) is derived from the equations of the Oral Minimal Model (OMM) method [28]. To do so, we integrated the differential equations of the Oral Glucose Minimal Model (OGMM) [32], which interprets plasma glucose concentration in relation to the observed changes in insulin concentration during a meal, together with the ones of the Oral Insulin and C-peptide Minimal Model (OICMM) [33][35], which interprets plasma insulin and C-peptide concentration in relation to the observed changes in glucose concentration also measured during a meal.

In particular, by assuming that, especially in individuals with T2D [39] and PD, the pancreatic insulin secretion rate (ISR) is mainly controlled by glucose concentration (static phase of ISR) rather than the rate of glucose increase (dynamic phase of ISR) and stable glucose concentrations at meal time, the proposed disposition index can be estimated as (more details about the derivation of the proposed method are reported in the Appendix):

$${DI}^G=\frac{\frac{D·f\left(t_{end}\right)}{BW}-GE·\underset{t_{meal}}{\overset{t_{end}}{\int }}\mathrm{\Delta }G\left(t\right)dt-V_G\left[G\left(t_{end}\right)-G\left(t_{meal}\right)\right]}{p_2·\frac{V_{Cp}}{V_I}·\alpha ·\underset{t_{meal}}{\overset{t_{end}}{\int }}\left\{\left[e^{-p_2·t}\mathrm{*}{e}^{-n·t}\mathrm{*}\left[\left(1-{HE}_b+a_G·\mathrm{\Delta }G\left(t\right)\right)·\left(e^{-\alpha ·t}\mathrm{*}\mathrm{\Delta }G\left(t\right)\right)\right]\right]·G\left(t\right)\right\}dt}$$ (1)

where $*$ represents the convolution operator, $G\left(t\right)$ is the glucose concentration measured during the experiment while $\mathrm{\Delta }G\left(t\right)$ denotes its value above the pre-meal one $G\left(t_{meal}\right)$ [mg/dL], $D$ is the amount of carbohydrates ingested during the meal [mg], $f(t_{end})$ is the fraction of ingested dose [dimensionless] which has reached plasma at the end of the study $\left(t_{end}\right)$ [14], $t_{meal}$ is the time of meal ingestion, $BW$ is the subject’s body weight [kg]. Parameters in Eq. 1 have to be fixed to their population values. Specifically, for the parameters of the glucose subsystem one has: $GE$ is the glucose effectiveness, $V_G$ is the volume of glucose distribution and $p_2$ the rate constant of insulin action respectively fixed to 0.019 dL/kg/min, 1.45 dL/kg and to 0.0121 min⁻¹ respectively, according to [29][32]. Similarly, for the parameters of the insulin and C-peptide kinetics one has: $V_{Cp}$ is the volume of C-peptide distribution calculated using the Van Cauter population model [37], $n$ [min⁻¹] and $V_I[\mathrm{L}$] are the fractional rate of insulin clearance and the volume of insulin distribution both derived from the Campioni model [38], and $H{E}_b$ is the basal hepatic insulin extraction fixed to 0.6 according to [38]. In addition, since no population values were available for parameters $\alpha$ and $a_G$, representing the rate constant for the provision of new releasable insulin [33] and the effectiveness of glucose in reducing hepatic insulin extraction [36], respectively, they were fixed to 33 min and 0.0026 dL/mg, respectively, according to the results obtained from the OGMM and OICMM identification on the simulated T2D dataset. Of note, when dealing with different populations, like individuals with and without PD and others, some of these fixed parameters may need to be tuned accordingly.

As it is, ${\text{DI}}^{\mathrm{G}}$ can be directly obtained assuming plasma glucose (PG) data as $G\left(t\right)$. Nevertheless, ${\text{DI}}^{\mathrm{G}}$ can be easily extended to work also with CGM data by deriving the plasma glucose signal $G\left(t\right)$ from the CGM sensor signal $CGM\left(t\right)$ and the first order differential equation:

$$\left\{\begin{array}{l}\dot{IG}\left(t\right)=-\frac{1}{\tau }·IG\left(t\right)+\frac{1}{\tau }·G\left(t\right)IG\left(0\right)=G_b\\ CGM\left(t\right)=IG\left(t\right)\end{array}\right.$$ (2)

where $\tau$ is the glucose equilibration time constant [min] between the intravascular to interstitial fluid compartment [40][41]. Assuming that the CGM is calibrated, one can calculate $G\left(t\right)$ from $CGM\left(t\right)$ by inverting Eq. 2 as

$$G\left(t\right)=\tau ·\dot{IG}\left(t\right)+IG\left(t\right)$$ (3)

which can then be used in Eq. 1.

The precision of ${\text{DI}}^{\mathrm{G}}$ was assessed by propagating the measurement error of glucose data on ${\text{DI}}^{\mathrm{G}}$ and quantified as coefficient of variation (CV, %). In particular, in case of PG data, measurement error was assumed an independent, Gaussian noise, with zero mean and CV equal to 2% [32]; while, in case of CGM data (simulated dataset only), a mean absolute relative deviation (MARD) equal to 9% [42] was assumed. As a result, similarly to what done in [15], ${\text{DI}}^{\mathrm{G}}$ cannot be reliably quantified if it is poorly estimated, as defined by its CV higher than 100%, or in case it results in unphysiological, e.g. negative, values. As a matter of fact, the latter may occur in case of very high glucose excursion (see the numerator of Eq. 1), which is a clear marker of low glucose tolerance.

Validation against the Disposition Index from the Oral Minimal Model (${\text{DI}}^{\mathrm{MM}}$)

${\text{DI}}^{\mathrm{G}}$ was validated against the disposition index obtained from the OMM method (${\text{DI}}^{\mathrm{MM}}$). The latter, as by definition, was calculated as the product of insulin sensitivity (S_I), estimated from the Oral Glucose Minimal Model (OGMM), and total beta-cell responsivity ($\varphi$), estimated from the Oral C-peptide Minimal Model (OCMM), respectively.

In particular, from one side, the OGMM [32] is described by

$$\left\{\begin{array}{ll}\dot{G}\left(t\right)=-\frac{GE}{V_G}·\left(G\left(t\right)-G_b\right)-\frac{S_I}{V_G}·X\left(t\right)·G(t)+\frac{{Ra}_G\left(t\right)}{V_G}& G\left(0\right)=G_b\\ \dot{X}\left(t\right)=-p_2·X\left(t\right)+p_2·I\left(t\right)& X\left(0\right)=0\end{array}\right.$$ (4)

where $G$ is the plasma glucose concentration [mg/dL], with $G_b$ denoting its basal value, $X$ is the remote insulin compartment from which insulin exerts its action [μU/mL], $I$ is the above basal plasma insulin concentration [μU/mL], $GE$ is the glucose effectiveness [dL/kg/min], $V_G$ is the glucose distribution volume [dL/kg], $p_2$ is the speed of rise and decay of insulin action [min⁻¹], $S_I$ is the insulin sensitivity [dL/kg/min per μU/mL] and $R{a}_{\mathrm{G}}$ is the rate of glucose absorption from the meal [mg/kg/min] described by a piecewise linear function with known break points $t_i$ and unknown amplitude ${\alpha }_i$ as

$${Ra}_G\left(t\right)=\left\{\begin{array}{cc}{\alpha }_{i-1}+\frac{{\alpha }_i-{\alpha }_{i-1}}{t_i-t_{i-1}}·\left(t_i-t_{i-1}\right)& for{t}_i\le t\le t_i,i=1,\dots ,7\\ 0& otherwise\end{array}\right.$$ (5)

From the other side, the OCMM [43] is described by

$$\left\{\begin{array}{ll}\dot{{Cp}_1}\left(t\right)=-\left(k_{01}+k_{21}\right)·{Cp}_1\left(t\right)+k_{12}·{Cp}_2\left(t\right)+ISR\left(t\right)& \hfill {Cp}_1\left(0\right)=0\\ \dot{{Cp}_2}\left(t\right)=-k_{12}·{Cp}_2\left(t\right)+k_{21}·{Cp}_1\left(t\right)& \hfill {Cp}_2\left(0\right)=0\\ ISR\left(t\right)=Y\left(t\right)+{\varphi }_d·{\dot{G}\left(t\right)}^{+}& ISR\left(0\right)=0\\ \dot{Y}\left(t\right)=-\alpha ·\left[Y\left(t\right)-{\varphi }_s·\left(G\left(t\right)-h\right)\right]& \hfill Y\left(0\right)=0\end{array}\right.$$ (6)

where $C_{P1}$ and $C_{P2}$ are the above basal C-peptide concentration in the accessible and remote compartments, with $k_{01},k_{12},k_{21}$ the C-peptide kinetic parameters [min⁻¹], $ISR$ is the above basal (C-peptide) insulin secretion rate which is modeled as the sum of two components: a “static” one, controlled by glucose concentration, and a “dynamic” one, controlled by its rate of increase ${\dot{G}\left(t\right)}^{+}$. In particular, the “static” component represents the provision of new releasable insulin $Y(t)$, which tends with a time constant $1/\mathrm{\alpha }$ [min] towards a steady-state value that is linearly dependent to glucose concentration above a threshold level $h$ (mg/dL) through a proportionality constant ${\varphi }_s$ [min⁻¹] (static beta-cell responsivity). The “dynamic” component represents the secretion of promptly releasable insulin and is dependent to rate of glucose increase ${\dot{G}\left(t\right)}^{+}$ through a proportionality constant ${\varphi }_d$ [dimensionless] (dynamic beta-cell responsivity). Finally, as done in [43], from ${\varphi }_s$ and ${\varphi }_d$, an index of total beta-cell responsivity $\varphi$ is calculated as

$$\varphi ={\varphi }_s+\frac{{\varphi }_d\cdot \left(G_{max}-G_b\right)}{{\int }_0^{\mathrm{\infty }}\left[G\left(t\right)-h\right]dt}$$ (7)

The two model were separately identified. In particular, the OGMM was identified using plasma insulin as input while glucose data as output, using a Bayesian Maximum a Posteriori (MAP) estimator and fixing $GE$ and $f(t_{\mathit{end}})$ to their population values [29][32], while the OCMM was identified from plasma glucose as input while C-peptide data as output, using a Bayesian Maximum a Posteriori (MAP) estimator and fixing the C-peptide kinetics parameters to those predicted by the Van Cauter population model [37]. Measurement error on plasma glucose concentration was assumed to be independent, Gaussian, with zero mean and known standard deviation (CV = 2%) [32], while measurement error on plasma C-peptide was assumed to be independent, Gaussian, with zero mean and known variance [33]. In addition, for both models, precision of model parameters was obtained from the Fisher Information matrix [44].

Finally, as a result, the disposition index from the OMM method ($\mathrm{D}{\mathrm{I}}^{\mathrm{M}\mathrm{M}}$) was obtained as

$$D{I}^{MM}=S_I\cdot \varphi$$ (8)

Robustness tests of ${\text{DI}}^{\mathrm{G}}$

A battery of robustness tests was conducted on ${\mathrm{D}\mathrm{I}}^{\mathrm{G}}$, exploiting the simulated dataset while using CGM data as input, to assess the impact of: i) fixing some parameters to population values; ii) duration of the experiment; iii) meal glucose dose.

In particular, to assess the impact of fixing some parameters to population values for the calculation of the proposed $\mathrm{D}\mathrm{I}$, a sensitivity analysis was performed. This has been done focusing on those parameters which were fixed for the calculation of ${\mathrm{D}\mathrm{I}}^{\mathrm{G}}$, while either estimated from the data ($\alpha$ and $p_2$) or even not used ($a_G$ and $H{E}_b$) for the calculation of ${\mathrm{D}\mathrm{I}}^{\mathrm{M}\mathrm{M}}$. In fact, ${\mathrm{D}\mathrm{I}}^{\mathrm{M}\mathrm{M}}$ is simply the product of insulin sensitivity $\left({\mathrm{S}}_{\mathrm{I}}\right)$, derived from the OGMM [32] using plasma insulin as input and glucose as output, and the total beta-cell responsivity $\left(\mathrm{\varphi }\right)$, derived from the OCMM [43] using plasma glucose as input and C-peptide as output, and these models are separately identified. The availability of plasma insulin measurements avoids the need to describe the insulin extraction and kinetics and thus the knowledge of $a_G$ and $H{E}_b$ for the calculation of ${\mathrm{D}\mathrm{I}}^{\mathrm{M}\mathrm{M}}$. For each parameter: a) a random generation of 100 values, extracted from a uniform distribution ranging between ±50% from its nominal value was generated; b) these generated values were used to re-calculate the ${\mathrm{D}\mathrm{I}}^{\mathrm{G}}({{\mathrm{D}\mathrm{I}}^{\mathrm{G}}}_{\text{gen}})$, by randomly assigning these values to the subjects instead of the nominal one; c) ${{\mathrm{D}\mathrm{I}}^{\mathrm{G}}}_{\text{gen}}$ was compared with the ${\text{DI}}^{\mathrm{G}}$ calculated with the nominal parameter values (${{\mathrm{D}\mathrm{I}}^{\mathrm{G}}}_{\text{gen}}$). Of note, the selected range of variability was, on average, in line with the actual range of variability observed in the virtual and real population with type 2 diabetes: 61% for ${\mathrm{a}}_{\mathrm{G}}$, 63% for $\mathrm{\alpha }$, 27% for ${\mathrm{H}\mathrm{E}}_{\mathrm{b}}$ and 28% for p₂.

To assess the impact of ${\mathrm{D}\mathrm{I}}^{\mathrm{G}}$ calculation with respect to the duration of the experiment $\left(t_{\mathit{end}}\right)$, the estimates of ${\mathrm{D}\mathrm{I}}^{\mathrm{G}}$ obtained using data collected in the 360 min following meal ingestion were compared to those obtained from shorter integration intervals (up to 120 min).

Finally, to assess the impact of meal dose in ${\mathrm{D}\mathrm{I}}^{\mathrm{G}}$ calculation, in addition to a meal with 75 g of glucose, in silico subjects also underwent mixed meals with 50 and 100 g of glucose and both ${\mathrm{D}\mathrm{I}}^{\mathrm{M}\mathrm{M}}$ and ${\mathrm{D}\mathrm{I}}^{\mathrm{G}}$ were calculated and compared.

Statistical Analysis

Data are presented as median and interquartile range, unless otherwise specified. Two-sample comparisons were done by paired Student’s t-test, for normally distributed variables, or Wilcoxon signed rank test, otherwise. Similarly, for normally distributed variables, one-way ANOVA was used to assess differences between different groups (T2D vs. PD vs HC), and post-hoc analysis was performed using the Tukey-Kramer test correction for multiple comparisons; Kruskal-Wallis and Dunn-Sidak tests were used in case of non-normally distributed variables, instead. Normality of the distributions was assessed by the Lilliefors test. Pearson’s correlation was used to evaluate univariate linear correlation and [95% C.I.] were calculated exploiting Fisher transformation. A p-value of less than 0.05 was considered statistically significant.

Results

Dataset 1

${\text{DI}}^{\mathrm{G}}$ was reliably calculated from plasma glucose data in all the subjects/visits. The OMM always fitted the data well and parameters were estimated with precision in all the subjects (not shown). ${\text{DI}}^{\mathrm{G}}$ was 69.4 [52.6, 193.5] and 197.8 [79.5, 290.9] 10⁻¹⁴ dL/kg/min² per pmol/L in the placebo and treatment visit, respectively (median CV=13%), while ${\text{DI}}^{\mathrm{MM}}$ was 175.5 [108.9, 238.7] and 231.2 [132.7, 401.1] 10⁻¹⁴ dL/kg/min² per pmol/L in the two visits (median CV=11%). Both ${\text{DI}}^{\mathrm{G}}$ and ${\text{DI}}^{\mathrm{MM}}$ were significantly lower in the placebo than treatment visit (p<0.001 and p=0.04, respectively) (Fig. 2, left bottom panel). The overall correlation [95% C.I.] between the two indices was high (R=0.88 [0.76, 0.95], p<0.001) (Fig. 2, left top panel).

Figure 2.

Scatter plot (top panels) and boxplot (bottom panels) of the disposition index (DI) obtained with the Oral Minimal Model (${\text{DI}}^{\mathrm{MM}}$), exploiting plasma glucose, insulin and C-peptide concentration data, vs. the proposed method (${\text{DI}}^{\mathrm{G}}$), exploiting plasma glucose data only, in dataset 1 (left panels), in the placebo (black triangles) and treatment (grey squares) visits, and in dataset 2, in individuals with the type 2 diabetes (T2D, white squares), pre-diabetes (PD, grey circles) and healthy controls (HC, black triangles). For ${\text{DI}}^{\mathrm{G}}$ calculation model parameters are fixed to T2D population values. Dashed black line represents the diagonal line.

Dataset 2

${\text{DI}}^{\mathrm{G}}$ was reliably calculated from plasma glucose data in 99% the subjects, while the remaining was removed from the subsequent analysis (CV>100%). Similarly, the OMM fitted the data well in almost all (99%) of the subjects (not shown). ${\text{DI}}^{\mathrm{G}}$ was 251.0 [241.1, 260.8], 1333.9 [656.1, 2011.8] and 1803.7 [1749.9, 1857.5] 10⁻¹⁴ dL/kg/min² per pmol/L in the T2D, PD and HC group (median CV=9%) respectively, while ${\text{DI}}^{\mathrm{MM}}$ was 149.3 [132.1, 166.5], 481.2 [326.3, 636.1] and 810.8 [804.8, 816.7] 10⁻¹⁴ dL/kg/min² per pmol/L (median CV = 4%), respectively. Both ${\text{DI}}^{\mathrm{G}}$ and ${\text{DI}}^{\mathrm{MM}}$ differed significantly between groups (p<0.001), with post-hoc analysis highlighting both ${\text{DI}}^{\mathrm{G}}$ and ${\text{DI}}^{\mathrm{MM}}$ to be significantly lower in T2D vs. PD (p=0.007 and p=0.024, in ${\text{DI}}^{\mathrm{G}}$ and ${\text{DI}}^{\mathrm{MM}}$, respectively) and HC (p<0.001) as well as a significantly lower DI in PD vs. HC (p=0.018 and p=0.01, in ${\text{DI}}^{\mathrm{G}}$ and ${\text{DI}}^{\mathrm{MM}}$, respectively), (Fig. 2, right bottom panel). The correlation [95% C.I.] between the two indices (Fig. 2, right top panel) was good (R=0.79 [0.67, 0.87], p<0.001). Correlation [95% C.I.] was also calculated for each subpopulation, resulting in R=0.40 [−0.36,0.84] (p=NS), 0.72 [0.50,0.85] (p<0.001) and 0.73 [0.39,0.89] (p<0.001) for T2D, PD and HC individuals respectively. Of note, correlation between ${\text{DI}}^{\mathrm{G}}$ and ${\text{DI}}^{\mathrm{MM}}$ in T2D was not statistically significant, possibly due to the low number of subject (N=9). Finally, no statistically significant difference between ${\text{DI}}^{\mathrm{G}}$ and ${\text{DI}}^{\mathrm{MM}}$ association among subpopulations was found.

Of note, since some parameters ($a_G,H{E}_b,\alpha$ and $p_2$) are usually unknown, we fixed these values to the ones estimated in the simulated T2D dataset (dataset 3) with the OMM. Nevertheless, we also assessed the effect of using population-specific parameters in the calculation of ${\text{DI}}^{\mathrm{G}}$, i.e. by using the median of population-specific values estimated by the OMM on the same dataset. By doing so, ${\text{DI}}^{\mathrm{G}}$ was numerically more similar to ${\text{DI}}^{\mathrm{MM}}$ than in the previous case, i.e. 230.0 [221.8, 238.1], 939.5 [442.3, 1436.7] and 1309.2 [1289.7, 1328.6] 10⁻¹⁴ dL/kg/min² per pmol/L in the T2D, PD and HC group, especially for PD and HC populations. Moreover, similar statistically significant differences were observed among groups in ${\text{DI}}^{\mathrm{G}}$. Specifically, T2D was significantly lower than PD and HC (p=0.045 and p<0.001, respectively) and PD was significantly lower than HC (p=0.006). ${\text{DI}}^{\mathrm{G}}$ was also well correlated with ${\text{DI}}^{\mathrm{MM}}$ (R=0.79, p<0.001) (Supplemental Fig. S1).

Dataset 3 (simulated)

${\text{DI}}^{\mathrm{G}}$ was reliably calculated from plasma glucose data in 92% of the subjects, while the remaining was removed from the subsequent analysis (${\text{DI}}^{\mathrm{G}}$ was negative in 6 out of 8 subjects while, in the remaining, we obtained a CV>100%). Similarly, the OMM well fitted the data in most (92%) of the subjects (not shown), while the remaining was removed from the subsequent analysis. ${\text{DI}}^{\mathrm{G}}$ was 116.1 [49.3, 250.5] 10⁻¹⁴ dL/kg/min² per pmol/L (median CV=12%), and ${\text{DI}}^{\mathrm{MM}}$ was 109.2 [50.5, 202.0] (median CV = 9%). The correlation [95% C.I.] between ${\text{DI}}^{\mathrm{G}}$ and ${\text{DI}}^{\mathrm{MM}}$ (Fig. 3) was good (R=0.86 [0.80, 0.91], p<0.001). As a proof of concept, the simulated dataset was also used to quantify ${\text{DI}}^{\mathrm{G}}$ from CGM data and test it against ${\text{DI}}^{\mathrm{MM}}$. In particular, ${\text{DI}}^{\mathrm{G}}$ from CGM data was reliably calculated in 83% of the subjects, while the remaining was removed from the subsequent analysis (${\text{DI}}^{\mathrm{G}}$ was negative in 7 out of 17 subjects while, in the remaining, we obtained a CV>100%). ${\text{DI}}^{\mathrm{G}}$ was on median [25^th, 75^th] percentiles equal to 117.7 [64.5, 235.8] 10⁻¹⁴ dL/kg/min² per pmol/L (median CV=45%), and it was also well correlated with ${\text{DI}}^{\mathrm{MM}}$ (R=0.83 [0.75, 0.89], p<0.001).

Figure 3.

Scatter plot of the disposition index (DI) obtained with the Oral Minimal Model (${\text{DI}}^{\mathrm{MM}}$), exploiting plasma glucose, insulin and C-peptide concentration data, vs. the proposed method (${\text{DI}}^{\mathrm{G}}$), exploiting plasma glucose data (PG, black squares) or continuous glucose monitoring (CGM, grey triangles) data, in the simulated database. Dashed black line represents the diagonal line.

Sensitivity analysis on ${\text{DI}}^{\mathrm{G}}$ was performed with respect to model parameters $a_G,H{E}_b,p_2$ and $\alpha$ and assessed by calculating the slope (S) of the regression line between the relative deviation of each model parameters vs. ${\text{DI}}^{\mathrm{G}}$. The sensitivity of ${\text{DI}}^{\mathrm{G}}$ to $H{E}_b$ was high and positive (S = 1.04); this means that, if $H{E}_b$ deviated from its nominal value by a given percentage, we observed a deviation in ${\text{DI}}^{\mathrm{G}}$ of a similar percent magnitude. A much lower and negative sensitivity of ${\text{DI}}^{\mathrm{G}}$ was found with respect to $a_G$ (S = −0.39), $p_2$ (S = −0.41) and $\alpha$ (S = −0.21) (Fig. 4, left panel), meaning that a large deviation of these parameters from their nominal values modestly affects ${\text{DI}}^{\mathrm{G}}$ estimation. Nevertheless, the correlation between the ${{\mathrm{D}\mathrm{I}}^{\mathrm{G}}}_{\text{gen}}$ vs. DI^G_nom was very high (R ≥ 0.98) in case of deviation of $a_G,p_2$ and $\alpha$ from their nominal values, and slightly lower, but still high (R = 0.87), for $H{E}_b$ (Fig. 4, right panel).

Figure 4.

Sensitivity analysis of ${\text{DI}}^{\mathrm{G}}$ to population parameters ($a_G,H{E}_b,p_2$ and $\alpha$) using the in silico population: scatter plot between the percent deviation of each parameter variation from its nominal value (x-axis) vs. the respective percent deviation from its nominal value of ${\text{DI}}^{\mathrm{G}}$ (y-axis, left panel), and correlation analysis between the DI^G_nom vs. ${{\mathrm{D}\mathrm{I}}^{\mathrm{G}}}_{\text{gen}}$ for each parameter variation (right panel).

When looking at the impact of the duration of the experiment on ${\text{DI}}^{\mathrm{G}}$ calculation, we found no significant difference between DI^G_{360 min} and DI^G_{300 min}, while ${\text{DI}}^{\mathrm{G}}$ tended to be significantly overestimated, with respect to DI^G_{360 min}, when the integration interval was equal to or lower than 240 min (Fig. 5, top panel). This was accompanied by a worsening in the correlation between DI^G_300min, DI^G_{240 min}, DI^G_{180 min} and DI^G_{120 min} with the reference DI^G_{360 min}, (R = 0.99, 0.96, 0.86 and 0.67, respectively) (Fig. 5, middle panel), with a median absolute relative deviation of ${\text{DI}}^{\mathrm{G}}$ equal to 7%, 17%, 44% and 164%, for 300, 240, 180 and 120 min respectively (Fig. 5, bottom panel).

Figure 5.

Sensitivity analysis of ${\text{DI}}^{\mathrm{G}}$ to time integration intervals (360, 300, 240, 180 and 120 min) using the in silico population. Boxplots of ${\text{DI}}^{\mathrm{G}}$ (top panel), and correlation indices (middle panel) and absolute relative deviation (bottom panel) from the ${\text{DI}}^{\mathrm{G}}$ calculated at 360 min (DI^G_{360 min}).

In addition, to possibly compensate the effect of reduced integration intervals in ${\text{DI}}^{\mathrm{G}}$ calculation, we also calculated ${\text{DI}}^{\mathrm{G}}$ by assuming that glucose has returned to its premeal level at 360 min, i.e. G(360) = G(0). Hence, using that assumption, we linearly extrapolated glucose traces beyond the last available sample, i.e. G(t_end) with t_end = 300, 240, 180 and 120 min, and calculated the respective ${\text{DI}}^{\mathrm{G}}$. As a result, with this assumption, ${\text{DI}}^{\mathrm{G}}$ estimated from the shorter intervals were better correlated with the reference value obtained from a 360 min experiment (DI^G₃₆₀) (R=0.99, 0.97, 0.94 and 0.83 for t_end = 300, 240, 180 and 120 min, respectively; Supplemental Fig. S2, middle panel), with median absolute relative deviations of ${\text{DI}}^{\mathrm{G}}$ equal to 10%, 17%, 21% and 40%, for 300, 240, 180 and 120 min respectively (Supplemental Fig. S2, bottom panel), although the estimates tended to slightly, but significantly, underestimate ${\text{DI}}^{\mathrm{G}}$ when compared against DI^G₃₆₀ (Supplemental Fig. S2, top panel).

Finally, when assessing the impact of meal dose on ${\text{DI}}^{\mathrm{G}}$ calculation, correlation between ${\text{DI}}^{\mathrm{MM}}$ and ${\text{DI}}^{\mathrm{G}}$ was good for all the meal doses (R=0.75, 0.83 and 0.84 for 50, 75 and 100 g respectively; p<10⁻³). ${\text{DI}}^{\mathrm{G}}$ was significantly lower when increasing meal dose, with a median reduction of 13% and 26% when comparing 75g and 100g to 50g respectively, while only a trend was shown by ${\text{DI}}^{\mathrm{MM}}$, with a median reduction by 3% and 4%, respectively (Supplemental Fig. S3, left panels). We speculated that the different reductions between ${\text{DI}}^{\mathrm{MM}}$ and ${\text{DI}}^{\mathrm{G}}$, when varying the meal dose, may be due to variations is some of the parameters estimated with the OMM, like the timing of insulin action ($p_2$) [45]. In particular, this parameter showed a statistically significant reduction when increasing the meal dose (with a median value of 0.012 min⁻¹, 0.011 min⁻¹ and 0.010 min⁻¹, when the meal dose was 50 g, 75 g and 100 g respectively; p<10⁻³), while this was fixed to population value for the calculation of ${\text{DI}}^{\mathrm{G}}$. Hence, we tested the effect of fixing this parameter to population value also when estimating ${\text{DI}}^{\mathrm{MM}}$ which resulted, similarly to ${\text{DI}}^{\mathrm{G}}$, in a statistically significant reduction in ${\text{DI}}^{\mathrm{MM}}$ when increasing the meal dose (by 4% and 12% when comparing 75g and 100g to 50g respectively; Supplemental Fig. S3 right panels).

Discussion

In this work, we describe a method to estimate disposition index in non-insulin-treated individuals from postprandial glucose data only (${\text{DI}}^{\mathrm{G}}$). In particular, the method was tested and validated exploiting plasma glucose data in vivo, in individuals with type 2 diabetes either receiving placebo or a drug improving glucose tolerance (a DPP-4 inhibitor) [29] and in individuals with type 2 diabetes, pre-diabetes and without diabetes [30], since individuals did not wear CGM in the original studies. In addition, an in silico database of virtual subjects with type 2 diabetes [31] was used as a tool for the quantification of fixed model parameters included in the proposed method and, as a proof of concept, to test if the method showed comparable results both using plasma glucose as well as CGM sensor data. In both cases, the method proposed was compared against the reference ${\text{DI}}^{\mathrm{MM}}$ estimated using the Oral Minimal Model method [28] based on plasma glucose, insulin and C-peptide data.

In the in vivo datasets, the results showed a good agreement between ${\text{DI}}^{\mathrm{G}}$ and ${\text{DI}}^{\mathrm{MM}}$ (Fig. 2), with a high correlation between the two indices and a similar pattern both between the placebo vs. treatment visit (dataset 1) as well as between different populations (dataset 2). This also applied in the simulated dataset, both when using plasma glucose or CGM data as input for the ${\text{DI}}^{\mathrm{G}}$ calculation (Fig. 3). Of note, in dataset 2, the method was applied to different groups, e.g. type 2 diabetes, pre-diabetes and healthy individuals, while some parameters of the proposed methodology were originally tuned for type 2 diabetes individuals. Hence, we also assessed the effect of using population-specific parameters in the calculation of ${\text{DI}}^{\mathrm{G}}$ achieving a similar correlation and pattern with respect to ${\text{DI}}^{\mathrm{MM}}$ while also showing that, when using population-specific parameters, ${\text{DI}}^{\mathrm{G}}$ was numerically more similar to ${\text{DI}}^{\mathrm{MM}}$ (Supplemental Fig. S1). As a matter of fact, it would be preferable to use population-specific parameters to properly translate the glucose excursion into the corresponding DI but, if not available, suboptimal population parameters can still be used. In line with this, future work will focus on assessing the use of population models to possibly derive such fixed parameters from subject demographics, like age, BSA, body weight, etc., and/or characteristics, degree of glucose tolerance and/or other disease-related conditions.

Despite the good correlation achieved between ${\text{DI}}^{\mathrm{MM}}$ and ${\text{DI}}^{\mathrm{G}}$, some differences in the numerical values were still observable. This was, at least in part, related to parameters which were fixed to population values in the calculation of ${\text{DI}}^{\mathrm{G}}$ while can be either estimated ($\alpha$ and $p_2$) or even not used (a_G and ${\mathrm{HE}}_{\mathrm{b}}$) for the calculation of ${\text{DI}}^{\mathrm{MM}}$. In line with this, we also performed a sensitivity analysis on those parameters to assess the effect of fixing such parameters to population values (Fig. 4). The results showed that, an increase in the basal hepatic insulin extraction $\left(H{E}_b\right)$, a reduction in the glucose-dependent suppression of hepatic insulin extraction $\left(a_G\right)$, a greater delay in insulin secretion and action (i.e. reduction of $\alpha$ or $p_2$) all led to an increase of ${\text{DI}}^{\mathrm{G}}$. This occurred since, in all those cases, the model predicted a lower insulin exposure, which, with glucose being the same, implies a higher disposition index. Nevertheless, even in the presence of large variations of such parameters, the resulting ${{\mathrm{D}\mathrm{I}}^{\mathrm{G}}}_{\text{gen}}$ was still highly correlated with the nominal one (DI^G_nom), with correlation indices ranging from a minimum of 0.87, in case of variation in $H{E}_b$, to 0.98, in case of variation of the remaining parameters. We also assessed the robustness of ${\text{DI}}^{\mathrm{G}}$ with respect to a reduction in the experiment duration (Fig. 5): ${\text{DI}}^{\mathrm{G}}$ did not differ significantly from, and was highly correlated with, ${\text{DI}}^{\mathrm{G}}$ calculated at 360 min (DI^G_{360 min}) if the integration interval was limited to 300 min. ${\text{DI}}^{\mathrm{G}}$ tended to be progressively overestimated and less correlated for shorter time integration intervals (from 240 min, with R=0.96, up to 120 min, with R=0.67). This implies that ${\text{DI}}^{\mathrm{G}}$ estimated using CGM readings collected in time intervals of different duration are not directly comparable. Despite this, when calculated from CGM data, the median absolute relative deviation of ${\text{DI}}^{\mathrm{G}}$ is within the ${\text{DI}}^{\mathrm{G}}$ estimation error (median CV ~45%, database 3) if the integration interval is higher or equal to 180 min. In addition, if one assumes that glucose has returned to premeal level at 360 min, the median absolute relative deviation of ${\text{DI}}^{\mathrm{G}}$ was lower than the ${\text{DI}}^{\mathrm{G}}$ estimation error even when the experiment has ended at 120 min (Supplemental Fig. S2). On the other hand, when calculated from glucose data, to achieve a median absolute relative deviation around the ${\text{DI}}^{\mathrm{G}}$ estimation error (median CV ~9% and ~13%, database 1 and 2 respectively), the integration interval must be higher or equal to 240 min. In any case, if one aims to properly compare ${\text{DI}}^{\mathrm{G}}$ on different occasions, the best option would be to standardize ${\text{DI}}^{\mathrm{G}}$ calculation by using a fixed integration interval and, possibly, equal to 360 min. This is because, in most of the cases, glucose has returned to its premeal level after 360 min thus allowing proper quantification of the regulation of glucose by insulin.

Finally, leveraging on the simulation framework [31], we also tested the robustness of the method in response to different meal glucose doses (Supplemental Fig. S3). ${\text{DI}}^{\mathrm{G}}$ still showed a good correlation against ${\text{DI}}^{\mathrm{MM}}$, regardless of the meal dose, while a negative trend was observed when increasing the meal dose. This would be in keeping with the observation of saturation mechanisms of glucose utilization in the presence of increasing glucose levels [46].

Some limitations must be acknowledged. The first is that, by design, the method does not allow separate assessment of insulin sensitivity, i.e. the ability of insulin to lower plasma glucose concentrations, from the β-cell responsivity, i.e. the ability of glucose to stimulate insulin secretion, but only their product, i.e. the disposition index. In fact, to separately assess the two components, measurements of plasma insulin and C-peptide concentrations would be needed. In addition, the method is based on dynamic data, e.g. data collected after meal/glucose ingestion, and therefore the method cannot be used to assess glucose tolerance in non-stimulated conditions, e.g. overnight. A second limitation of the method is that, unlike ${\text{DI}}^{\mathrm{MM}}$, the calculation of ${\text{DI}}^{\mathrm{G}}$ assumes that β-cell responsivity is mainly controlled by glucose level (static phase ISR) rather than also its rate of increase (dynamic phase ISR). This is usually the case in individuals with type 2 diabetes. However, in health and in the early stages of prediabetes the contribution of the dynamic phase ISR is usually not negligible. Despite these limitations, the method still provided an estimate of DI that correlated well with its MM counterpart and was able to capture differences among populations with different levels of glucose tolerance, as well as quantify the effect of treatment in noninsulin-treated individuals with type 2 diabetes. Future work will aim to extend the methodology to quantify the overall contribution, i.e. static and dynamic, of ISR on DI. A third limitation is that estimation of the carbohydrate content of a meal is prone to error, for individuals living with T2D. Nevertheless, the method can still be used in two ways: either to run physiology studies to track metabolic status in the outpatient setting. If a standardized dose of carbohydrate is used, this overcomes the uncertainty regarding the carbohydrate content of the meal challenge. In the second case, if an individual ingests two meals with same carbohydrate content a week or a month apart any error in estimating the actual amount of carbohydrate ingested cancels itself; the method would still be able to detect a between-meal change in glucose tolerance. Finally, as a proof of concept, simulated data were used to test the ability of the method to quantify a DI from CGM data in individuals with type 2 diabetes. However, future studies are needed for independent validation of the methodology proposed in subjects wearing a CGM sensor while also measuring plasma glucose and hormone data.

We conclude that the method described enables assessment of the disposition index in noninsulin-treated individuals from glucose data. This method was validated in vivo using plasma glucose data and, potentially, can be used to quantify the effect of therapeutic treatments as well as defects in glucose tolerance. In addition, as a proof of concept, the method was tested in silico to assess if it allows a robust estimation of DI from CGM traces. Future studies are needed for its validation in a real population of non-insulin-treated individuals, e.g. people with type 2 diabetes and/or pre-diabetes, wearing a CGM sensor and thus allowing to quantify intra- and inter-day variability of this index in daily life conditions.

Supplementary Material

Supplemental Fig. S1-S2-S3: https://doi.org/10.6084/m9.figshare.29517992. The code running the proposed methodology will be shared upon reasonable request.

Appendix

Derivation of the Disposition Index from Glucose Data only (${\text{DI}}^{\mathrm{G}}$)

The method for the estimation of a Disposition Index based on glucose data only (${\text{DI}}^{\mathrm{G}}$) presented in this work is derived from the equations of the Oral Minimal Model (OMM) method [28]. In particular, the OMM describes two key processes in glucose-insulin regulation during a meal: one interpreting plasma glucose concentration (output) in relation to the observed changes in insulin concentration (input), i.e. the Oral Glucose Minimal Model (OGMM) [32], and the other one interpreting plasma C-peptide concentrations (outputs) in relation to the observed changes in glucose concentration (input), i.e. the Oral C-peptide Minimal Model (OCMM) [43]. Here, in particular, to complete the picture of the feedback mechanism of glucose-insulin regulation, we exploited the Oral Insulin and C-peptide Minimal Model (OICMM), instead of the OCMM, allowing also to interpret plasma insulin concentration data, in addition to C-peptide, by means of a model of hepatic insulin extraction and kinetics [33]. Here below the description of the two models and the derivation of the proposed method is reported in detail.

The OGMM equations [32] are:

$$\left\{\begin{array}{ll}\dot{G}\left(t\right)=-\frac{GE}{V_G}·\left(G\left(t\right)-G_b\right)-\frac{S_I}{V_G}·X\left(t\right)·G(t)+\frac{{Ra}_G\left(t\right)}{V_G}& G\left(0\right)=G_b\\ \dot{X}\left(t\right)=-p_2·X\left(t\right)+p_2·I\left(t\right)& X\left(0\right)=0\end{array}\right.$$ (S1)

where $G$ is the plasma glucose concentration [mg/dL], with $G_b$ denoting its basal value, $X$ is the remote insulin compartment from which insulin exerts its action [μU/mL], $I$ is the above basal plasma insulin concentration [μU/mL], $GE$ is the glucose effectiveness [dL/kg/min], $V_G$ is the glucose distribution volume [dL/kg], $p_2$ is the speed of rise and decay of insulin action [min⁻¹], $S_I$ is the insulin sensitivity [dL/kg/min per μU/mL] and ${Ra}_G$ is the rate of glucose absorption from the meal [mg/kg/min].

The OICMM consists of two sub-models, i.e. a model of C-peptide secretion [43] and kinetics [35] (OCMM):

$$\left\{\begin{array}{ll}\dot{{Cp}_1}\left(t\right)=-\left(k_{01}+k_{21}\right)·{Cp}_1\left(t\right)+k_{12}·{Cp}_2\left(t\right)+ISR\left(t\right)& \hfill {Cp}_1\left(0\right)=0\\ \dot{{Cp}_2}\left(t\right)=-k_{12}·{Cp}_2\left(t\right)+k_{21}·{Cp}_1\left(t\right)& \hfill {Cp}_2\left(0\right)=0\\ ISR\left(t\right)=Y\left(t\right)+{\varphi }_d·{\dot{G}\left(t\right)}^{+}& \hfill ISR\left(0\right)=0\\ \dot{Y}\left(t\right)=-\alpha ·\left[Y\left(t\right)-{\varphi }_s·\left(G\left(t\right)-h\right)\right]& \hfill Y\left(0\right)=0\end{array}\right.$$ (S2.1)

and a model of hepatic insulin extraction (here we used that proposed by Piccinini et al. [36]) and post-hepatic insulin kinetics [38]:

$$\left\{\begin{array}{ll}HE\left(t\right)={HE}_b-a_G·\mathrm{\Delta }G\left(t\right)& \hfill HE\left(0\right)={HE}_b\hfill \\ \dot{I}\left(t\right)=-n·I\left(t\right)+\left(1-HE\left(t\right)\right)·\frac{V_{Cp}}{V_I}·ISR\left(t\right)& \hfill I\left(0\right)=0\hfill \end{array}\right.$$ (S2.2)

In Eq. S2.1 $C_{P1}$ and $C_{P2}$ are the above basal C-peptide concentration in the accessible and remote compartments, with $k_{01},k_{12},k_{21}$ and $V_{Cp}$ the C-peptide kinetic parameters [min⁻¹] and distribution volume in the accessible compartment [L] derived from the Van Cauter population model [37]. ISR is the above basal (C-peptide) insulin secretion rate normalized by $V_{Cp}$. which is modeled as the sum of two components: a “static” one, controlled by glucose concentration, and a “dynamic” one, controlled by its rate of increase ${\dot{G}\left(t\right)}^{+}$. In particular, the “static” component represents the provision of new releasable insulin $Y(t)$, which tends with a time constant $1/\mathrm{\alpha }$ [min] towards a steady-state value that is linearly dependent to glucose concentration above a threshold level $h$ (mg/dL), here fixed to G_b [47][48], through a proportionality constant ${\varphi }_s$ [min⁻¹] (static beta-cell responsivity). The “dynamic” component represents the secretion of promptly releasable insulin and is dependent to rate of glucose increase ${\dot{G}\left(t\right)}^{+}$ through a proportionality constant ${\varphi }_d$ [dimensionless] (dynamic beta-cell responsivity).

In Eq. S2.2, HE is the hepatic insulin extraction, with ${\mathit{HE}}_b$ denoting its basal value, ${\mathrm{a}}_{\mathrm{G}}$ the effectiveness of glucose in reducing hepatic insulin extraction, while $n$ and ${\mathit{V}}_I$ are the fractional insulin clearance [min⁻¹] and distribution volume [L] derived from the Campioni population model [38].

By integrating Eq. S1 one can calculate $S_I$ as:

$$S_I=\frac{\underset{t_{meal}}{\overset{t_{end}}{\int }}{Ra}_G\left(t\right)dt-GE·\underset{t_{meal}}{\overset{t_{end}}{\int }}\mathrm{\Delta }G\left(t\right)dt-V_G\left[G\left(t_{end}\right)-G\left(t_{meal}\right)\right]}{\underset{t_{meal}}{\overset{t_{end}}{\int }}\left(X^{\prime }\left(t\right)·G\left(t\right)\right)dt}$$ (S3)

where $t_{meal}$ and $t_{end}$ are the time of meal ingestion and the end of the study, respectively, and $\mathrm{\Delta }G$ is the above basal glucose concentration. The first integral in the numerator, representing the amount of carbohydrates absorbed during the study, can be calculated as

$$\underset{t_{meal}}{\overset{t_{end}}{\int }}{Ra}_G\left(t\right)dt=\frac{D·f\left(t_{end}\right)}{BW}$$ (S4)

assuming $D$ to represent the amount of carbohydrates ingested during the meal [mg], $f(t_{end})$ the fraction of ingested carbohydrate which has reached plasma at $t=t_{end}$ [dimensionless] as described in [14][15], and $BW$ the subject’s body weight [kg]. In particular, a meal is assumed to be completely absorbed at 6 hours after meal ingestion ($t_{end}$).

Similarly, by integrating Eq. S2 one can calculate the insulin concentration above its basal value $It)$ as:

$$I\left(t\right)=e^{-n·t}\mathrm{*}\left[\left(1-{HE}_b+a_G·\mathrm{\Delta }G\left(t\right)\right)·\frac{V_{Cp}}{V_I}·\left(\alpha ·{\varphi }_s·\left[e^{-\alpha ·t}\mathrm{*}\mathrm{\Delta }G\left(t\right)\right]+{\varphi }_d·{\dot{G}\left(t\right)}^{+}\right)\right]$$ (S5)

where * represents the convolution operator. In particular, since here the signals are discrete-time and supported between the time of meal ingestion ($t_{\mathit{meal}}$) and the end of the experiment ($t_{\mathit{end}}$), this can be approximated by the discrete convolution operator assuming signals to be sampled with a sufficiently fine grid within the interval (here a 1-min linear interpolation was used).

Following the same reasoning, one can calculate the insulin action $X^{’}(t)$, by substituting Eq. S5 into S1 as:

$$\begin{gathered} X^{\prime }(t)=\left(p_2\cdot \frac{V_{Cp}}{V_I}\cdot \alpha \cdot {\varphi }_s\right)\cdot e^{-p_2\cdot t}\mathrm{*}{e}^{-n\cdot t}\mathrm{*}\left[\left(1-H{E}_b+a_G\cdot \mathrm{\Delta }G(t)\right)\cdot \left(e^{-\alpha \cdot t}\mathrm{*}\mathrm{\Delta }G(t)\right)\right] \\ +\left(p_2\cdot \frac{V_{Cp}}{V_I}\cdot {\varphi }_d\right)\cdot e^{-p_2\cdot t}\mathrm{*}{e}^{-n\cdot t}\mathrm{*}\left[\left(1-H{E}_b+a_G\cdot \mathrm{\Delta }G(t)\right)\cdot \dot{G}{\left(t\right)}^{+}\right] \end{gathered}$$ (S6)

Now, assuming the contribution of promptly releasable insulin on pancreatic secretion is mainly controlled by glucose level rather than the rate of glucose increase in individuals with type 2 diabetes [39] and pre-diabetes, we assumed ${\varphi }_d$ equal to zero in this population. This allows to simplify Eq. S6 as:

$$X^{\prime }\left(t\right)=\left(p_2·\frac{V_{Cp}}{V_I}·\alpha ·{\varphi }_s\right)·e^{-p_2·t}\mathrm{*}{e}^{-n·t}\mathrm{*}\left[\left(1-{HE}_b+a_G·\mathrm{\Delta }G\left(t\right)\right)·\left(e^{-\alpha ·t}\mathrm{*}\mathrm{\Delta }G\left(t\right)\right)\right]$$ (S7)

Finally, by substituting Eq. S7 into S3 and defining ${\mathrm{D}\mathrm{I}}^{\mathrm{G}}$ as the product between insulin sensitivity ($S_I$) and static beta-cell responsivity to glucose (${\varphi }_s$), one obtains:

$${DI}^G=S_I·{\varphi }_s=\frac{\frac{D·f\left(t_{end}\right)}{BW}-GE·\underset{t_{meal}}{\overset{t_{end}}{\int }}\mathrm{\Delta }G\left(t\right)dt-V_G\left[G\left(t_{end}\right)-G\left(t_{meal}\right)\right]}{p_2·\frac{V_{Cp}}{V_I}·\alpha ·\underset{t_{meal}}{\overset{t_{end}}{\int }}\left\{\left[e^{-p_2·t}\mathrm{*}{e}^{-n·t}\mathrm{*}\left[\left(1-{HE}_b+a_G·\mathrm{\Delta }G\left(t\right)\right)·\left(e^{-\alpha ·t}\mathrm{*}\mathrm{\Delta }G\left(t\right)\right)\right]\right]·G\left(t\right)\right\}dt}$$ (S8)

Last but not least, the plasma glucose signal $G(t)$ used in the previous equations can be derived from CGM sensor by the following first order differential equation as:

$$\left\{\begin{array}{l}\dot{IG}\left(t\right)=-\frac{1}{\tau }·IG\left(t\right)+\frac{1}{\tau }·G\left(t\right)IG\left(0\right)=G_b\\ CGM\left(t\right)=IG\left(t\right)\end{array}\right.$$ (S9)

where $\tau$ is the glucose equilibration time constant [min] between the intravascular to interstitial fluid compartment [40][41]. Assuming that the CGM is calibrated, one can calculate $G(t)$ from $CGM(t)$ by inverting Eq. S9 as

$$G\left(t\right)=\tau ·\dot{IG}\left(t\right)+IG\left(t\right)$$ (S10)

which can be used in Eq. S8.

As a matter of fact, in order to use the proposed methodology, some parameters need to be fixed to population values. Specifically, for the parameters of the glucose subsystem one has: $GE$ is the glucose effectiveness, $V_G$ is the volume of glucose distribution and $p_2$ the rate constant of insulin action respectively fixed to 0.019 dL/kg/min, 1.45 dL/kg and to 0.0121 min⁻¹ respectively, according to [29][32]. Similarly, for the parameters of the insulin and C-peptide kinetics one has: $V_{Cp}$ is the volume of C-peptide distribution calculated using the Van Cauter population model [37], $n$ [min⁻¹] and $V_I$ [L] are the fractional rate of insulin clearance and the volume of insulin distribution both derived from the Campioni model [38], and $H{E}_b$ is the basal hepatic insulin extraction fixed to 0.6 according to [38]. In addition, since no population values were available for parameters $\alpha$ and $a_G$, representing the rate constant for the provision of new releasable insulin and the effectiveness of glucose in reducing hepatic insulin extraction, respectively, they were fixed to 33 min and 0.0026 dL/mg, respectively, according to the results obtained from the OGMM and OICMM identification on the simulated T2D dataset [31]. Of note, when dealing with different populations, like individuals with and without PD and others, some of these fixed parameters may need to be tuned accordingly.