A glucose-insulin-glucagon coupled model of the isoglycemic intravenous glucose infusion experiment

Type 2 diabetes (T2D) is a pathophysiology that is characterized by insulin resistance, beta- and alpha-cell dysfunction. Mathematical models of various glucose challenge experiments have been developed to quantify the contribution of insulin and beta-cell dysfunction to the pathophysiology of T2D. There is a need for effective extended models that also capture the impact of alpha-cell dysregulation on T2D. In this paper a delay differential equation-based model is developed to describe the coupled glucose-insulin-glucagon dynamics in the isoglycemic intravenous glucose infusion (IIGI) experiment. As the glucose profile in IIGI is tailored to match that of a corresponding oral glucose tolerance test (OGTT), it provides a perfect method for studying hormone responses that are in the normal physiological domain and without the confounding effect of incretins and other gut mediated factors. The model was fit to IIGI data from individuals with and without T2D. Parameters related to glucagon action, suppression, and secretion as well as measures of insulin sensitivity, and glucose stimulated response were determined simultaneously. Significant impairment in glucose dependent glucagon suppression was observed in patients with T2D (duration of T2D: 8 (6–36) months) relative to weight matched control subjects (CS) without diabetes (k1 (mM)−1: 0.16 ± 0.015 (T2D, n = 7); 0.26 ± 0.047 (CS, n = 7)). Insulin action was significantly lower in patients with T2D (a1 (10 pM min)−1: 0.000084 ± 0.0000075 (T2D); 0.00052 ± 0.00015 (CS)) and the Hill coefficient in the equation for glucose dependent insulin response was found to be significantly different in T2D patients relative to CS (h: 1.4 ± 0.15; 1.9 ± 0.14). Trends in parameters with respect to fasting plasma glucose, HbA1c and 2-h glucose values are also presented. Significantly, a negative linear relationship is observed between the glucagon suppression parameter, k1, and the three markers for diabetes and is thus indicative of the role of glucagon in exacerbating the pathophysiology of diabetes (Spearman Rank Correlation: (n = 12; (−0.79, 0.002), (−0.73,.007), (−0.86,.0003)) respectively).


Validation of the Glucose-Insulin-Glucagon coupled Model
There are different methods of validating ODE based biological models.
(1) One method would be to fit data obtained under one set of conditions and validating the model by simulating a second data set obtained for the same individual under different conditions. As data from IIGI experiments for the cohort studied matched to three different OGTT glucose loads was available, validation using this method was possible. The model developed in this paper was validated by simulating the glucoseinsulin-glucagon dynamics for IIGI experiments matched to the 25g OGTT glucose profiles of patients with T2D and IIGI experiments matched to 125g OGTT glucose profiles of CS. The simulation was carried out as follows: The glucose infusion, RIIGI, from the 25 g or 125 g equivalent IIGI experiment was used as input and the coupled Equations (1)-(3) were solved using the parameters obtained by fitting the IIGI data matched to the 75 g OGTT as presented in the paper for each patient with T2D and CS respectively.
In the CS simulations, all fit parameters from the 75g equivalent IIGI were kept fixed. The simulated glucose, insulin and glucagon profiles visually match the data from the 125 g IIGI experiment as seen in Supplementary Figure 1.
In the case of patients with T2D, all parameters obtained in the fit of the data from the 75g equivalent IIGI were kept fixed in patients 1 and 7. In the other patients, the glucagon secretion parameter 2 and the time delays  and 1 were adjusted manually (Supplementary Table 1). The baseline glucagon levels were significantly different on the two different experimental days (OGTT vs IIGI) for most patients, which could explain why the parameter 2 needed to be adjusted to get good fits. The simulated glucose, insulin and glucagon profiles visually match the data from the 25g IIGI experiment as seen in supplementary Figure 2. It is interesting to note that despite discrepancies in baseline glucagon values, good fits were obtained with most of the parameters kept fixed including the glucagon suppression parameter. Only the glucagon secretion parameter had to be adjusted. It is unclear how and why the delays in RIIGI and insulin secretion vary as a function of the glucose load.

Supplementary Figures and Tables
Supplementary Figure 1. IIGI data corresponding to the 125 g OGTT and simulations using parameters from the fit of the IIGI matched to 75 g OGTT in CS (panels A-F). The visual match between data (black dots) and simulation (blue line) in most individuals is good. Figure 2. Simulation of the IIGI corresponding to the 25 g OGTT using parameters from the fit of the IIGI matched to 75 g OGTT in patients with T2D. The parameters 2,  and  were manually adjusted for a better fit. The visual match in most patients with T2D is good. Table 1   Table S1. To obtain a good description of the 25 g OGTT matched IIGI data in T2D patients as seen in Supplementary Figure 2, the following parameters were adjusted: the glucagon secretion parameter, 2, and the time delays  and 1. No adjustments were necessary in two patients (1 and 7) where baseline glucose and hormone levels matched on both experimental days. The insulin secretion parameter, 1, had to be adjusted for patients 6 and 8.  Figure 3. Plots of the fits obtained with hysteresis Model 2 for patients with T2D in panels A-G. The visual fit of the recovery arm of the insulin secretion profile is not as good as Model 1 particularly in patients 6 and 8 (panels E and G) where the recovery is incomplete. The adjusted R 2 of the overall fits is comparable to model 1.