A New C-Peptide Minimal Model to Assess $\beta$-Cell Responsiveness to Glucagon in Individuals With and Without Type 2 Diabetes

Abstract

Glucagon regulates its own secretion indirectly by stimulating $\beta$-cells to secrete insulin. This may serve as a compensatory mechanism to enhance $\beta$-cell function in individuals with, or predisposed to, type 2 diabetes. However, tools to quantify glucagon-induced C-peptide secretion (SR) are lacking.

To bridge this gap, we developed a novel model-based method to provide quantitative indices of $\beta$-cell function in response to a glucagon bolus in individuals without and with type 2 diabetes (T2D). Eight individuals without diabetes (3M, age=55±9 yrs, BMI=32±4 kg/m²) and 6 with T2D (1M, age=59±5 yrs, BMI=35±6 kg/m²) underwent a 210-min hyperglycemic clamp (~9 mmol/L). After 180 min, a 1 mg bolus of glucagon was administered over 1 min and plasma glucagon and C-peptide concentrations were frequently measured over 30 min.

We tested a battery of mathematical models and selected the best one based on standard criteria. The optimal model assumes that C-peptide SR is made up of two components, one proportional to the above basal glucagon, through parameter ${\text{Γ}}_{\text{s}}$ (static), and one proportional to glucagon rate of change, through parameter ${\text{Γ}}_{\text{d}}$ (dynamic responsivity to the hormone). An index of total $\beta$-cell responsivity to glucagon, $\text{Γ}$, was also derived from ${\text{Γ}}_{\text{s}}$ and ${\text{Γ}}_{\text{d}}$. Model estimated ${\text{Γ}}_{\text{s}}$ and $\text{Γ}$ were significantly higher in individuals without diabetes compared to T2D (p<0.05), while ${\text{Γ}}_{\text{d}}$ was not.

Our findings reveal notable differences in both static and total insulin secretory response to glucagon in people with diabetes as compared to those without diabetes.

New & Noteworthy:

In this study, we propose a new mathematical model able to quantify C-peptide secretion and $\beta$-cell responsivity in response to a glucagon bolus in individuals without and with type 2 diabetes. We show that individuals with type 2 diabetes exhibit reduced $\beta$-cell responsivity to glucagon.

Graphical Abstract

Introduction

Type 2 Diabetes (T2D) is a metabolic disease characterized by dysfunctional and delayed insulin secretion combined with impaired glucagon suppression, leading to chronically elevated glucose concentrations [1, 2]. The maintenance of glucose homeostasis requires the synchronized secretion and action of several hormones, with the primary ones being insulin, secreted from pancreatic $\beta$-cells, and glucagon, secreted from $\alpha$-cells. Despite the critical role of this hormonal control system, research in diabetic pathophysiology has traditionally centered on insulin.

Since its discovery, glucagon was primarily recognized for its counter-regulatory effect on insulin, and its role as the “anti-insulin” hormone. However, recent research has highlighted that glucagon is more than a hyperglycemic agent, pointing out the importance of completely understanding its metabolic role in the prevention and treatment of T2D [3].

Glucagon concentrations in T2D have been shown to be inappropriate for the prevailing fasting [4] and postprandial glucose concentrations [5]. These abnormalities have been correlated with hepatic fat but also with other factors such as impaired insulin action [6].

Interestingly, glucagon regulates its own secretion indirectly via stimulating $\beta$-cells to release insulin that lowers glucose levels. Glucagon is also a known insulin secretagogue and the $\beta$-cell response to a 1mg glucagon bolus has been used to test the integrity of the $\beta$-cell secretory apparatus [3, 7, 8, 9, 10, 11]. This has led to the hypothesis that increased glucagon secretion is a compensatory mechanism to enhance $\beta$-cell function in individuals with or predisposed to T2D [3]. However, to the best of our knowledge, tools for the quantification of insulin secretion related to a glucagon stimulus are still lacking, although various approaches are available to measure glucose-induced insulin secretion and quantify $\beta$-cell functionality related to glucose [12, 13, 14, 15]. The availability of a mathematical model able to relate the secretion of insulin to glucagon concentration, thus enabling the estimation of quantitative indices to define the $\beta$-cell functionality related to glucagon, may bridge this gap.

Currently, the state-of-the-art estimate of insulin secretion is performed by deconvolution of C-peptide plasma concentration employing an established C-peptide kinetics model [16]. In fact, plasma C-peptide concentration is a better marker of insulin secretion than insulin itself, since insulin and C-peptide are secreted equimolarly, but whereas insulin is extracted by the liver, C-peptide is not. Therefore, in principle, a model for glucagon-induced insulin secretion can be developed from C-peptide and glucagon measurements only, similarly to what was previously done in [12, 13], assuming that all other hormones and substrates are in an acceptable steady state.

The aims of this work were, first, to develop a mathematical model able to describe C-peptide plasma concentration and quantify $\beta$-cell function in response to an injection of a glucagon bolus, and, second, to compare $\beta$-cell responsivity to glucagon in healthy individuals (ND) and individuals with T2D to uncover potential differences indicative of $\beta$-cell dysfunction in T2D.

Materials and Methods

Subjects and Protocol

The study was approved by the Mayo Clinic Institutional Review Board (Mayo Clinic College of Medicine, Rochester, MN). To be eligible, healthy subjects had no history of chronic illness or upper gastrointestinal surgery. Additionally, they were not taking medications that could affect glucose metabolism. Subjects with type 2 diabetes had no history of microvascular or macrovascular complications and were treated with lifestyle modification alone or in combination with metformin. Eight individuals without diabetes (ND, 1/7 M/F, age = 59±5 years, BMI = 35±6 kg/m²) and 6 with type 2 diabetes (T2DM, 3/3 M/F, age = 55±9 years, BMI = 32±4 kg/m²) participated to the study.

This data was generated as part of a previously published study [12] {Welch, 2023 #17700}. After written informed consent was obtained, participants were studied twice, at least 2 weeks apart in random order. On one study day, at t = −120 min participants received an infusion of exendin 9–39, on the other study day, saline was infused instead. However, in this work, only data from the saline day are employed. At t = 0 min glucose was infused to raise peripheral concentrations to ~ 160mg/dL and at t = 180 min, 1mg of glucagon was administered intravenously.

Blood samples were collected at t = [–30 0 2 4 6 8 10 20 30 60 90 120 150 180 182 184 186 188 190 200 210] min after the start of the experiment for measurement of plasma glucose, glucagon, and C-peptide concentrations.

Plasma glucose concentrations were measured using a Yellow Springs glucose analyzer. Glucagon was measured using a 2-site ELISA (Mercodia, Winston Salem, NC, USA), which has a Lower Limit of Detectability (LLoD) of 1.7 pmol/L. Plasma C-peptide was measured using a 2-site immunoenzymatic sandwich assay (Roche Diagnostics, Indianapolis, IN, USA). Time courses of plasma glucose, glucagon, and C-peptide concentrations measured after the glucagon bolus administration are reported in Fig. 1. More detailed information about the experimental protocol is available in [7].

Fig. 1.

Average time course of plasma glucose (left), glucagon (middle), and C-peptide concentrations (right) measured from 180 minutes (time of glucagon bolus administration) in individuals without diabetes (ND, continuous line with circles) and with type 2 diabetes (T2D, dashed line with squares). Vertical bars represent ± standard error.

Models

In this work, we tested six models providing different functional descriptions of C-peptide Secretion Rate (SR). They all share the two-compartment description of C-peptide kinetics proposed by Eaton et al. [17]:

$$\left\{\begin{array}{cc}\dot{{CP}_1}\left(t\right)=-\left(k_{01}+k_{21}\right){CP}_1\left(t\right)+k_{12}{CP}_2\left(t\right)+SR\left(t\right)& {CP}_1\left(0\right)={CP}_0\\ \dot{{CP}_2}\left(t\right)=k_{21}{CP}_1\left(t\right)-k_{12}{CP}_2\left(t\right)& {CP}_2\left(0\right)=\frac{k_{21}}{k_{12}}{CP}_0\end{array}\right.$$ (1)

where ${CP}_1$ and ${CP}_2$ (pmol/L) are the C-peptide concentration in the accessible and peripheral compartments, respectively; ${CP}_0$ (pmol/L) is the initial plasma C-peptide concentration. $SR$ (pmol/L · min⁻¹) is the $\beta$-cell C-peptide secretion rate, and $k_{21}$, $k_{12}$, $k_{01}$ (min⁻¹) are transfer rate parameters, fixed to standard population values, estimated using the Van Cauter et al. model [16].

Looking at the data of Fig. 1, C-peptide secretion increases after glucagon concentration rises. Therefore, it is reasonable to assume a positive relationship between SR and glucagon concentration. As a first attempt to assess SR following a bolus of glucagon, we tested two models, structurally similar to those proposed by Toffolo et al. [12], and Breda et al. [13] (Model 1 and Model 2, in the following), but using glucagon instead of glucose concentration as the primary driver. As discussed in the Results section, the performance of these models was not satisfactory. Therefore, we formulated and tested four new models (Models 3–6), which accounted for either a glucose contribution to C-peptide secretion or a modulation of glucose on glucagon-stimulated C-peptide secretion.

Model 1

This model, similar to that developed by Toffolo et al. [12] to describe C-peptide secretion during an Intravenous Glucose Tolerance Test (IVGTT), is based on the packet storage hypothesis of insulin/C-peptide secretion. In particular, the rise in glucagon above the threshold level $h_{Gn}$ (pmol/L) causes C-peptide secretion, proportional, through the constant rate $m$ (min⁻¹), to the C-peptide readily available in $\beta$-cells for the release, $X$ (pmol/L):

$$SR\left(t\right)=\left\{\begin{array}{cc}mX\left(t\right)& \text{if}Gn\left(t\right)\ge h_{Gn}\\ 0& \text{if}Gn\left(t\right)<h_{Gn}\end{array}\right.$$ (2)

where $Gn$ is glucagon concentration and the $h_{Gn}$ was fixed to $Gn$ basal value, calculated as the mean of the samples drawn at time 150 and 180 min.

$X$ is refilled via provision of new C-peptide, $Y$:

$$\dot{X}\left(t\right)=-SR\left(t\right)+Y\left(t\right)X\left(0\right)=X_0+X_{0b}$$ (3)

where $X_0$ (pmol/L) is the C-peptide made available for secretion right after the glucagon stimulus, and $X_{0b}$ (pmol/L) is the C-peptide stored in the system before the glucagon stimulus, calculated from the steady-state constraint as:

$$X_{0b}=C{P}_0\frac{k_{01}}{m}$$ (4)

Finally, $Y\left(t\right)$ (pmol/L · min⁻¹) is the provision of new C-peptide, proportional through parameter ${\beta }_{Gn}$ (min⁻¹) to the glucagon concentration above a certain threshold ($h_{Gn}$) and delayed by the rate constant $\alpha$ (min⁻¹):

$$\dot{Y}\left(t\right)=-\alpha \left[Y\left(t\right)-{\beta }_{Gn}\left(Gn\left(t\right)-h_{Gn}\right)-S{R}_b\right]Y\left(0\right)=k_{01}C{P}_0$$ (5)

where, $S{R}_b$, can be obtained from the steady-state constraint of Eq. 1:

$$S{R}_b=k_{01}C{P}_0$$ (6)

Model 2

This model is similar to that developed by Breda et al. [13] to describe C-peptide secretion during an oral glucose tolerance test or a mixed meal tolerance test. The model assumes that SR is the sum of a static ($S{R}_s$) and a dynamic component ($S{R}_d$):

$$SR\left(t\right)=S{R}_s\left(t\right)+S{R}_d\left(t\right)$$ (7)

$S{R}_s$ and $S{R}_d$ are controlled by delayed glucagon concentration (static glucagon control) and by its rate of change (dynamic glucagon control), respectively. $S{R}_s$ is assumed to be equal to the provision of new insulin to the $\beta$-cells $Y$, described in Eq. 5:

$$S{R}_s\left(t\right)=Y\left(t\right)$$ (8)

On the other hand, $S{R}_d$ represents the secretion of C-peptide stored in the $\beta$-cells in a promptly releasable form (labile hormone) and it is proportional to the glucagon rate of increase, ($\dot{Gn}$) (pmol/L· min⁻¹), through the parameter $k_d$ (dimensionless):

$$S{R}_d\left(t\right)=\left\{\begin{array}{cc}{k}_d\cdot \dot{Gn}\left(t\right)& \text{if}\dot{Gn}\left(t\right)>0\\ 0& \text{if}\dot{Gn}\left(t\right)\le 0\end{array}\right.$$ (9)

Model 3

This and the following models are based on Model 2 and share with it the model structure, the assumption that SR, is made up of two components (Eq. 7), and the description of $S{R}_d$ (Eq. 9). However, they differ in the description of $S{R}_s$, Eq. 8. In particular, with this model, we tested the hypothesis that glucagon concentration above a threshold level $h_{Gn}$ (pmol/L) acts with no delay on the static glucagon control through the parameter ${\beta }_{Gn}$ (min⁻¹):

$$S{R}_s\left(t\right)={\beta }_{Gn}\left(Gn\left(t\right)-h_{Gn}\right)+S{R}_b$$ (10)

Model 4

Based on the assumption that, during a clamp, glucose concentration is maintained almost constant, Models 1, 2, and 3 do not include any glucose effect on SR. However, in a nonnegligible percentage of subjects, fluctuations in glucose concentration were observed experimentally. Therefore, to account for this confounding factor, in Model 4, we added the action of plasma glucose concentration to the static C-peptide secretion rate and assumed that the static secretion is dependent on both glucagon above a threshold level $h_{Gn}$ (pmol/L), through the parameter ${\beta }_{Gn}$ (min⁻¹), and glucose concentration above a threshold level $h_{Gs}$ (mmol/L), through the parameter ${\beta }_{Gn}$ (10⁻⁹ min⁻¹):

$$S{R}_s\left(t\right)={\beta }_{Gn}\left(Gn\left(t\right)-h_{Gn}\right)+{\beta }_{Gs}\left(Gs\left(t\right)-h_{Gs}\right)+S{R}_b$$ (11)

where $h_{Gs}$ has been fixed to the value of glucose concentration at t = 180 min.

Model 5

At variance with Models 3 and 4, which assume that plasma glucagon has a direct effect on SR, in Model 5, we tested the hypothesis that glucose concentration modulates glucagon-stimulated C-peptide secretion. Hence, the static secretion becomes:

$$S{R}_s\left(t\right)=\left[{\beta }_{Gn}+{\beta }_{Gs}\left(Gs\left(t\right)-h_{Gs}\right)\right]\left(Gn\left(t\right)-h_{Gn}\right)+S{R}_b$$ (12)

Model 6

Model 6 was proposed to test whether glucagon and glucose concentration exert a bilinear control on the static SR through the parameter $\beta$ (L/mmol min⁻¹):

$$S{R}_s\left(t\right)=\beta \left(Gn\left(t\right)-h_{Gn}\right)\left(Gs\left(t\right)-h_{Gs}\right)+S{R}_b$$ (13)

Indices of Beta-cell Responsivity to Glucagon

Models 2, 3, 4 and 5 share the assumption that SR consists of a dynamic and a static component (Eq. 7). While the dynamic component remains the same across these models, the static component differs. This allows the estimation of three indices of $\beta$-cell function related to glucagon. The static responsivity to glucagon, ${\text{Γ}}_s$ (min⁻¹), measures the effect of plasma glucagon concentration on $\beta$-cell secretion:

$${\text{Γ}}_s={\beta }_{Gn}$$ (14)

The dynamic sensitivity to glucagon, ${\text{Γ}}_d$ (dimensionless), quantifies the stimulatory effect of the glucagon rate of change on the secretion of stored C-peptide.

$${\text{Γ}}_d=k_d$$ (15)

In addition to ${\text{Γ}}_s$ and ${\text{Γ}}_d$, it is also useful to derive an index of total $\beta$-cell sensitivity to glucagon ($\text{Γ}$ [min⁻¹]), which effectively integrates both the static (${\text{Γ}}_s$) and dynamic (${\text{Γ}}_d$) control indices. This index can be calculated as the average increase above basal of pancreatic secretion, Eq. 7, over the average glucagon stimulus above the threshold level $h_{Gn}$:

$$\mathrm{\Gamma }=\frac{{\int }_0^{\mathrm{\infty }}SR\left(t\right)-S{R}_{b}dt}{{\int }_0^{\mathrm{\infty }}[Gn\left(t\right)-h_{Gn}]dt}$$ (16)

In practice, $\text{Γ}$ can be calculated from the other two responsivity indices, and the area under the curve of glucagon above the threshold level $h_{Gn}$:

$$\mathrm{\Gamma }={\mathrm{\Gamma }}_s+\frac{{\mathrm{\Gamma }}_d\cdot \left(G{n}_{max}-h_{Gn}\right)}{{\int }_0^{\mathrm{\infty }}[Gn\left(t\right)-h_{Gn}]dt}$$ (17)

Indices of Beta-cell Responsivity to Glucose

Models 4 and 5 also allow to estimate the static responsivity to glucose, ${\varphi }_s$ (10⁻⁹ min⁻¹), which measures the effect of plasma glucose concentration on $\beta$-cell secretion, as reported in [13]:

$${\varphi }_s={\beta }_{Gs}$$ (18)

Model Identification

All the proposed SR models are a priori uniquely identifiable. The identifiability of the models was tested using the transfer function method [18]. Model parameters were estimated from plasma C-peptide concentrations using nonlinear weighted least squares method [18] implemented in Matlab (version R2023b, RRID:SCR_001622) [19]. Measurement error on C-peptide (CP) concentration was assumed to be independent, Gaussian, with zero mean and with variance modeled as proposed in [14]: $Var\left(CP\right)=2000+0.001\times {CP}^2$ [pmol²/L²].

Model Assessment

Subject data were used for model development and to select the most parsimonious one. Model performances were compared based on standard criteria [18]: ability to describe the data by looking at the Weighted Residual Sum of Squares (WRSS) and the distribution and randomness of the weighted residuals, precision of parameter estimates expressed as percentage Coefficient of Variation (CV), and physiological plausibility of model estimates. Among the models that fitted the data well and provided plausible estimates with precision, the best one was selected using the Akaike Information Criterion (AIC) [20]:

$$AIC=WRSS+2P$$ (19)

where $P$ is the number of model parameters.

Statistical Analysis

Data and results are presented as mean ± standard deviation unless otherwise specified. Statistical analysis was performed using Matlab [19]. Normality of sample distributions was assessed using the Lilliefors’ test [21]. Randomness of the residuals was assessed by visual inspection since the Run test cannot be applied to time series shorter than 20 samples. Comparison of distributions of independent samples was done by unpaired Student’s T-test, for normally distributed variables, or Wilcoxon’s rank-sum test, otherwise, with significance level set to 0.05 in both cases. Areas Under the Curve (AUC) were calculated using the trapezoidal rule.

Results

Model Assessment

When fitted to C-peptide data, all the tested models provided weighted residuals that were satisfactorily distributed, with the standard deviation bars that always crossed the zero line, and, by visual inspection, quite random (Fig. 2). Results are summarized in Table 1. Models 1, 2, 5 and 6 did not allow precise estimates of the parameters in a significant percentage of the subjects. Conversely, Model 3 provided precise parameter estimates in all subjects apart from two. Model 4 provided precise parameter estimate of ${\text{Γ}}_s$ and ${\text{Γ}}_d$ in all subjects. However, in 7 subjects, where glucose concentrations were approximately constant, ${\varphi }_s$ was estimated close to zero and, consequently, with poor precision (high CV). In these cases, ${\varphi }_s$ was fixed to 0, making Model 4 collapsing into Model 3. Ultimately, the lowest AIC index was provided by Model 4 (AIC = 62.36 ± 50.98). These results suggest selecting Model 4 as the optimal model, provided that ${\varphi }_s$ is fixed at zero when glucose concentration do not vary over time. Fig. 3 shows both the average data vs model prediction and data vs model prediction in representative individuals from those subjects with and without diabetes, obtained with the selected model.

Fig. 2.

Average weighted residuals (continuous line with circles) provided by the different models. Vertical bars represent ± standard deviation.

Table 1:

Models performance in nondiabetic (ND) and individuals with type 2 diabetes (T2D).

Model	WRSS	Average CV%	${\text{N}}_{\text{CV}>100\%}$	AIC
1	80.85 ± 120.80	>100	7	88.85 ± 120.80
2	63.34 ± 81.89	>100	5	69.34 ± 81.89
3	80.29 ± 89.12	52.47	2	84.29 ± 89.12
4	57.36 ± 50.84	23.64	0	62.36 ± 50.98
5	260.13 ± 430.94	>100	10	266.13 ± 430.94
6	168.81± 228.25	>100	4	172.81± 228.25

${\text{N}}_{CV>100\%}$ indicates the number of subjects in which at least one parameter is estimated with CV > 100%. WRSS, weighted residual sum of squares; CV, coefficient of variation; AIC, Akaike information criterion.

Fig. 3.

Upper panel: average C-peptide concentrations (filled circles) vs model prediction (continuous line). Vertical bars represent ± standard error. Middle and lower panels: C-peptide concentrations (filled circles) vs model prediction (continuous line) in a representative individual without (ND, middle) and with type 2 diabetes (T2D, lower panel).

ND vs T2D

Notable differences were found in both static, and total insulin secretory response between individuals without and with T2D, as shown in Fig. 4. In particular, ${\text{Γ}}_s$ was significantly higher in ND than T2D (0.05 ± 0.03 vs 0.01 ± 0.01 min⁻¹; p-value = 0.009). ${\text{Γ}}_d$ was not significantly different in ND than in T2D (0.21 ± 0.16 vs 0.11 ± 0.13; p-value = 0.14). The calculated $\text{Γ}$ was significantly higher in ND than T2D (0.08 ± 0.05 vs 0.03 ± 0.01 min⁻¹; p-value = 0.022). No significant differences in ${\varphi }_s$ were detected between ND and T2D (26.30 ± 25.57 vs 13.66 ± 26.80 10⁻⁹ min⁻¹; p-value = 0.48). The time courses of SR, estimated with Model 4, are shown in Fig. 5. Finally, in ND individuals, the AUC of SR was significantly higher compared to that of T2D individuals (1.06·10⁴ ± 3.21·10³ vs 5.97·10³ ± 2.04·10³ pmol/L, p-value = 0.006).

Fig. 4.

Indices of $\beta$-cell function in individuals without (ND, light grey) and with type 2 diabetes (T2D, dark grey). Bars represent mean values, whereas vertical lines represent the standard errors. Asterisks indicate statistically significant differences according to an unpaired Student T-test or Wilcoxon’s rank-sum test (p-value < 0.05).

Fig. 5.

SR(t) estimated with Model 4 in individuals without (ND, continuous line) and with type 2 diabetes (T2D, dashed line). Solid lines represent medians, whereas, shaded areas represent the 25th-75th interquartile ranges.

Discussion

Increased (and abnormal) glucagon secretion for a given glucose concentration is observed in people with type 2 diabetes. This $\alpha$-cell dysfunction may represent an effort to stimulate impaired $\beta$-cell function in these individuals. A detailed understanding of glucagon kinetics and its interactions with insulin is crucial for assessing the complex mechanisms underlying glucagon-insulin-glucose regulation. Given the intricate relationship among these factors, mathematical modeling can provide a powerful tool to quantitatively describe the dynamics of such interactions.

The aim of this work was the development of a mathematical model to evaluate insulin secretion after a glucagon bolus and use it in individuals with and without T2D to assess possible differences in $\beta$-cell sensitivity to glucagon. However, insulin is extracted by the liver right after its secretion. For this reason, models developed as part of this work employ measurements of C-peptide concentration. In fact, C-peptide is secreted in a 1:1 ratio with insulin from the pancreatic $\beta$-cells, but unlike insulin, it is not extracted by the liver.

We tested six models of different complexity, starting from two established in the literature [12, 13], as per good modeling practice.

The rationale for starting from these models is based on the fact that, although glucose, as a nutrient, and glucagon, as a hormone, utilize distinct initial signaling mechanisms (glucose primarily stimulates insulin secretion via calcium influx triggered by its metabolism, while glucagon acts by binding to its cognate receptor and the GLP-1 receptor on $\text{β}$-cells) they ultimately converge on the activation of the SNARE complex, which mediates insulin granule exocytosis [8].

Model 1 [12] assumes that the rise of glucagon above a certain level causes a delayed C-peptide secretion proportional to the C-peptide stored in the $\beta$-cells. Whereas, Model 2 [13] assumes that glucagon stimulates C-peptide secretion by exerting both a static, dependent on plasma glucagon concentration, and a dynamic control, dependent on glucagon rate of change. While these models fitted the data satisfactorily, model parameters were estimated with poor precision in 63% of the ND and 33% of the T2D, with Model 1, and in 13% of the ND and 67% of the T2D individuals, with Model 2, suggesting that both Model 1 and Model 2 were too complex for the available data. In Model 3, we assumed that glucagon acts with no delay, since the hormone is rapidly injected, sharply increasing its concentration to levels that may have a direct stimulatory effect on $\beta$-cells.

These three models ignored the glucose contribution to C-peptide secretion since, by design, the experiment was performed during a hyperglycemic clamp. However, data analysis (Fig. 1) revealed modest but nonnegligible fluctuations in glucose concentration in most of the participants. In Model 4, we added a term accounting for the glucose-induced C-peptide secretion, proportional through the parameter ${\beta }_{Gs}$ to the above basal glucose concentration. However, this last could not be precisely estimated in all individuals. Specifically, in those where the glucose clamps did not present significant glucose fluctuations, the glucose effect on SR could not be detected, and therefore, ${\beta }_{Gs}$ was estimated close to zero, with a high coefficient of variation. Therefore, for subjects with almost constant glucose levels (50% of the total), the parameter ${\beta }_{Gs}$ was fixed to 0.

At variance within the model proposed in the reference work [13], the dynamic component of SR in this study omits the contribution of glucose rate of change. This decision was taken because, in our study, glucose was clamped, and therefore glucose derivative were much lower than those observed postprandially in 204 healthy individuals [22] (0.046 ± 0.030 vs 0.135 ± 0.034 mmol/L/min, p < 0.05). Notwithstanding, we also tested a model that included the glucose derivative in the dynamic component of the SR (not shown). The index of dynamic responsivity to glucose, however, was estimated very close to zero, indicating no contribution from glucose derivative in this experimental condition. This finding supported our decision to exclude glucose rate of change from the drivers of the dynamic component of SR. Nevertheless, we acknowledge that, if we had observed higher glucose derivatives, we would have to adjust the model accordingly.

Finally, in Model 5 and Model 6, we tested the hypothesis that glucose modulates glucagon concentrations and that glucose and glucagon exert a bilinear control on C-peptide secretion, respectively.

The systematic model comparison suggested that Model 4 was the model of choice since it provided a satisfactory fit of the data (Fig. 2 and 3) with precise parameter estimates, being also the most parsimonious one (Table 1). Thus, we analyzed the results of Model 4 to assess if, and to what extent, the $\beta$-cell responsivity differed in ND and T2D. Model-based results suggested that the static responsivity of $\beta$-cell to glucagon was significantly higher in ND than in T2D (p-value = 0.009). However, the same cannot be said for the dynamic responsivity ${\text{Γ}}_d$, which did not show a statistically significant difference among ND and T2D (p-value = 0.14). This was somewhat unexpected, since the prompt $\beta$-cell response is the first to vanish in T2D onset [18]. We hypothesized that this was likely due to the limited number of individuals included in the study and that a higher sample size could have reveal a significant difference. On the other hand, the total responsivity index $\text{Γ}$, which is the combination of the static and dynamic responsivity (Eq. 17), was significantly higher in ND than in T2D (p-value = 0.014) (Fig. 4). In addition, by looking at the estimated SR curves, we observed that ND individuals have a faster dynamic with a higher peak compared to those of T2D, (Fig. 5). This was also confirmed by the statistical comparison of the AUC that showed significantly higher AUC for ND than T2DM individuals. Finally, regarding ${\varphi }_s$, it was not possible to detect any significant differences between the two populations (p-value = 0.48), likely because this parameter was set to 0 in a significant percentage of the subjects, those in which glucose was well clamped.

The main limitation of this study is the relatively small sample size of participants which reduced the statistical power of the analysis and potentially limited the detection of the impairment in the glucagon dynamic control in T2D individuals. A second limitation of this study was the impossibility of simultaneously estimating C-peptide secretion and kinetic parameters, forcing us to fix the latter to the values calculated using the Van Cauter et al. model [16]. However, without performing a two-stage experiment to assess both plasma kinetics and secretion, this is what is usually done in practice [12, 13]. Moreover, to make the model identifiable, $h_{Gn}$ and $h_{Gs}$ were fixed to basal glucagon and glucose concentrations, respectively, since we assumed that subjects were in steady-state before the glucagon bolus injection. Whether or not these assumptions are correct will require further study.

Finally, to isolate the specific contribution of glucagon to C-peptide secretion we administered a glucagon bolus during a hyperglycemic clamp. The response to glucagon in the presence of hyperglycemia has been thought to better represent a sub-maximal secretory response compared to the fasting state [23]. While this experimental design does not fully replicate the physiological patterns of glucagon secretion, it ensures we were measuring the response to glucagon. Furthermore, the glucagon stimulation test has been used in clinical practice as a measure of $\text{β}$-cell function previously [10]. Results and conclusions might change if a more physiological protocol was used. Additionally, $\beta$-cell responsivity may be influenced by other factors, however, investigating the role of such players was beyond the scope of this study. Future research could investigate these additional variables using complementary datasets or models.

Overall, the developed model is able to precisely quantify the loss of glucagon sensitivity in the $\beta$-cells of individuals with T2D. This model will be used in the study of the onset and progression of type 2 diabetes, providing clinicians and researchers with a novel tool for the assessment of pancreatic function.

Future work will include testing the model on a larger dataset and with more frequently sampled C-peptide profiles. This will allow the detection of a significant difference between people with and without type 2 diabetes, if any, also in the $\beta$-cell dynamic responsivity index to glucagon. In addition, future developments will concern the integration of this new model in a validated simulator of the glucose and insulin system [24] for the generation of T2D virtual subjects.

However, further studies will be required to assess the validity of this model during a more physiological setup, with glucagon changing more slowly over time.

In conclusion, we developed a novel model to accurately quantify the insulin secretagogue effect of glucagon in subjects with and without type 2 diabetes and used it to assess the $\beta$-cell function in response to a glucagon bolus during a hyperglycemic clamp. The model provides three indices that quantify the static (${\text{Γ}}_s$), the dynamic (${\text{Γ}}_d$), and the total ($\text{Γ}$) $\beta$-cell responsivity to glucagon. Additionally, if the experimental setup allows it, the model can simultaneously provide an index of static $\beta$-cell responsivity to glucose (${\varphi }_s$).

Glossary

$k_{01}$, $k_{12}$, $k_{21}$	transfer rate parameters	$m{in}^{-1}$
$m$	constant rate	$m{in}^{-1}$
$h_{Gn}$	glucagon threshold level	$pmol/L$
$X_0$	C-peptide available after the stimulus	$pmol/L$
$X_{0b}$	C-peptide available before the stimulus	$pmol/L$
$\alpha$	time constant	$m{in}^{-1}$
$h_{Gs}$	glucose threshold level	$mmol/L$
${\mathrm{\Gamma }}_s$	static responsivity to glucagon	$m{in}^{-1}$
${\mathrm{\Gamma }}_d$	dynamic responsivity to glucagon	dimensionless
$\mathrm{\Gamma }$	total responsivity to glucagon	$m{in}^{-1}$
${\varphi }_s$	static responsivity to glucose	${10}^{-9}\cdot m{in}^{-1}$

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request. No applicable resources were generated or analyzed during the current study.