A Mathematical Modeling Approach to Estimate Blood Glucose Behavior in Individuals with Prediabetes

Abstract

Background: Glucose homeostasis is a crucial physiological process, and its disruption is closely linked to the onset of Type 2 Diabetes Mellitus (T2DM), a major global health issue. Objective: This study presents a novel mathematical model to describe glucose dynamics in both healthy individuals and those with prediabetic risk factors. Methods: We analyzed 311 days of continuous glucose monitoring data from 43 participants (14 healthy and 29 at risk, aged 25–55), using a Dual Extended Kalman Filter to estimate parameters and unmeasurable variables, while accounting for parametric variability. We applied the Levenberg–Marquardt algorithm to minimize estimation error. Results: Based on average parameter values and standardized inputs, 311 simulations were conducted, showing strong agreement with experimental data (r = 0.98, p < 0.01). Conclusions: The model provides an accurate representation of glucose regulation and serves as a valuable in-silico tool for advancing preventive strategies against T2DM, marking one of the first models specifically tailored to individuals with prediabetes.

1. Introduction

Glucose homeostasis is a vital self-regulatory process that maintains blood glucose concentrations within a narrow physiological range, thereby preventing hyperglycemia and hypoglycemia [1]. This balance is primarily regulated by two counter-regulatory hormones: insulin, secreted in response to elevated glucose levels to promote its decrease [2], and glucagon, released when concentrations fall below basal levels to induce an increase in glucose [3]. The main exogenous alteration of this system is dietary intake, particularly carbohydrates, which causes acute elevations in blood glucose [4,5]. Other factors, including physical activity [6], psychological stress [7], and hormonal factors [8], can lead to sustained alterations in glycemic dynamics. Persistent dysregulation of this balance, driven by these exogenous factors, may compromise the homeostatic system, resulting in a wide spectrum of complications [9,10] and an increased risk of non-transmissible chronic diseases [11,12].

Type 2 Diabetes Mellitus (T2DM) is a chronic degenerative metabolic disease characterized by impaired glucose homeostasis and persistently elevated blood glucose levels [13,14]. Its progression is associated with multiple comorbidities that diminish life quality and require ongoing treatment. This impacts the economic burden of the disease, with an estimated annual individual cost of between 1583 and 2842 USD [15], which strains public health systems worldwide. In 2019, regions with the highest disease prevalence reported healthcare expenditures ranging from 52.3 to 294.6 billion USD [16,17]. These costs are expected to increase in parallel with the global rise in diabetes prevalence, which reached 536.6 million cases in 2021 and continues to grow [18]. Therefore, developing effective prevention strategies is essential to mitigate both the health and economic impacts of this disease.

Within diabetes prevention, various strategies have been developed to reduce the prevalence of the disease in society. These include the promotion of healthy eating habits [19], lifestyle modifications [20], and, in some countries, the implementation of front-of-package labeling of processed foods containing excessive amounts of compounds harmful to metabolic health [21]. Mathematical models have long been used to explore glucose-insulin dynamics [22], from population-based simulations predicting disease incidence to individual-level models supporting insulin therapy [23]. However, few models have focused on capturing glucose homeostasis in individuals at the prediabetes stage a critical window for prevention integrating real interstitial glucose data into parameter estimation frameworks. For instance, Ha et al. [24] developed a model that simulates the deterioration of metabolic function leading to T2DM, demonstrating how deficient insulin secretion accelerates disease progression. Using intelligent techniques, Omana et al. [25] proposed a predictive model to estimate the risk of developing T2DM, intending to support early prevention and improve community health outcomes. Similarly, Alonso-Bastida et al. [26] built virtual patients based on glucose homeostasis models of healthy individuals to generate glycemic profiles for the Mexican population, considering dietary habits and consumption patterns. They also evaluated the feasibility of implementing sustainable and healthy diets among at-risk groups in Mexico [27].

Mathematical models of glucose homeostasis are also used as tools to support glycemic control in patients dependent on exogenous insulin. These models provide information on the interaction between food intake and insulin infusion, allowing for the calculation of appropriate insulin doses to prevent hypoglycemic episodes [28]. The study compared directly measured HbA1c with calculated HbA1c, obtained using the formula of Misquith et al. [29]: HbA1c = 2.6 + 0.03 × FBS (mg/dL). Data distribution was assessed using the Kolmogorov–Smirnov test, which indicated non-normality and justified the use of non-parametric methods. Spearman analysis showed strong correlations between calculated HbA1c, calculated average glucose (AG), and estimated average glucose with their measured counterparts. The Wilcoxon signed-rank test revealed a significant difference between calculated and measured HbA1c (Z = −9.49; p < 0.0001). Furthermore, Bablok regression demonstrated significant deviation from linearity, suggesting systematic disagreement between the two methods. In this context, Orozco-López et al. [30] proposed a methodology for generating virtual patients with type 1 diabetes mellitus (T1DM), enhancing individual variability and offering an alternative approach to optimize insulin dosage and improve treatment efficiency. Mughal et al. [31] compared several control strategies based on Hovorka’s glucose homeostasis model in individuals with T1DM [32], evaluating the robustness of glycemic regulation under different exogenous insulin dosing protocols. All these differences highlight the need for health sectors to develop effective strategies for the prevention and management of T2DM.

From a mathematical perspective, modeling glucose homeostasis provides a useful framework for exploring the dynamics of metabolic regulation and early dysregulation. In this study, we developed a simplified but physiologically consistent model describing glucose regulation in healthy and prediabetic individuals, using interstitial glucose measurements to better capture early metabolic alterations. The proposed model is intended to support in-silico experimentation and guide public health strategies aimed at preventing Type 2 Diabetes Mellitus. To accomplish this, it incorporates a Dual Extended Kalman Filter (DEKF) for dynamic parameter estimation, improving interpretability and robustness in simulating glucose–insulin interactions compared with previous approaches.

2. Materials and Methods

• Experimental Protocol in Human Subjects: All procedures involving human participants complied with the Declaration of Helsinki, the Belmont Report, and national regulations. The study protocol was reviewed and approved by the Ethics Committee of the Faculty of Medicine, Universidad Autónoma del Estado de Morelos (Approval Code: CONBIOETICA-17-CEI-003-201-81112). Recruitment took place between August 2022 and August 2023. Informed consent was obtained from all participants, and personal data was anonymized and securely stored. Additional details on ethical and methodological procedures are provided in Supplementary Materials, Section S1.
• Experimental tests: Participants were selected and classified according to their fasting capillary glucose levels: those with values below 100 mg/dL were considered normoglycemic, while levels between 100–126 mg/dL indicated prediabetes [33]. Selected participants underwent anthropometric and body composition assessments and were implanted with an interstitial glucose sensor to allow continuous monitoring of glucose levels.
• Mathematical modeling: To start the model, a mathematical representation of glucose homeostasis was generated. Using experimental data, glucose signals were processed, and key model parameters and variables were estimated. Local minima for these parameters were then calculated, and the model was validated using correlation and error analyses, comparing measured values with model approximations.

2.1. Instrumentation

An interstitial continuous glucose monitoring (CGM) system (FreeStyle Libre, Abbott^®; Witney, UK) was used to obtain constant glucose measurements. Body measurements were carried out by using a segmental bioimpedance scale (BC-545 Segmental, Tanita^®, Tokyo, Japan) and a stadiometer. Additionally, for capillary glucose measurement, the Accu-Check Instant system (Roche^®, Mannheim, Germany) was used. Finally, a food intake and physical activity diary was used to count macronutrients and the intensity of physical activity performed.

2.2. Study Population

Participants were classified into two groups based on their glycemic status and associated health indicators. The first group consisted of 14 individuals (5 women and 9 men), aged 25 to 55 years, with no prior diagnosis of diabetes or chronic degenerative diseases, who maintained constant glucose levels below 170 mg/dL throughout the day. These participants were classified as having a healthy weight based on body mass index (BMI) criteria (18.5–24.9 kg/m²). The second group included 29 individuals (16 women and 13 men), also aged 25 to 55 years, who presented metabolic risk factors and daily peak glucose levels below 200 mg/dL [33]. This group was classified as overweight, defined as a BMI of 25.0–29.9 kg/m² for both genders. Group categorization was based on current clinical standards for glucose monitoring and BMI classification. Throughout the study, person-centered and non-stigmatizing language was consistently used when describing participant groups.

2.3. Mathematical Representation

The proposed glucose homeostasis dynamics for individuals with prediabetes and without diagnosed diabetes are based on Bergman’s minimal model and adapted by Kaveh et al. [34]. Our model introduces mathematical and parametric adjustments to improve simplicity and fidelity. The model consists of four components: (1) plasma glucose concentration, (2) insulin-induced glucose reduction, (3) plasma insulin concentration, and (4) interstitial glucose concentration (Figure 1). Parameter definitions are provided in the Nomenclature section.

Equation (1) captures the rate of change in plasma glucose $G\left(t\right)$, integrating its return to a set-point $G_{st}\left(t\right)$, insulin-mediated uptake $X\left(t\right)$, and glucose input from carbohydrate ingestion $G_{int}\left(t\right)$. Equation (2) models insulin action $X\left(t\right)$ as a function of its decay and stimulation by deviations in plasma insulin $I\left(t\right)$ from basal levels $I_b$. Equation (3) describes insulin secretion and clearance, driven by glucose levels exceeding a threshold $h$ and modulated by time. Finally, Equation (4) represents the rate of change in the interstitial glucose $G_{sc}\left(t\right)$, which adjusts toward plasma glucose with a delay determined by $k_{sc}$.

$$\dot{G}\left(t\right)=-p_1\left[G\left(t\right)-G_{st}\left(t\right)\right]-X\left(t\right)G\left(t\right)+{p_{4}G}_{int}\left(t\right);$$

(1)

$$\dot{X}\left(t\right)=-p_{2}X\left(t\right)+p_3\left[I\left(t\right)-I_b\right];$$

(2)

$$\dot{I}\left(t\right)=-\eta \left[I\left(t\right)-I_b\right]+\gamma \left[G\left(t\right)-h\right]t;$$

(3)

$${\dot{G}}_{sc}\left(t\right)=k_{sc}\left[G\left(t\right)-G_{sc}\left(t\right)\right].$$

(4)

2.3.1. Steady-State Glucose Variation

Glucose homeostasis in humans is regulated by complex physiological mechanisms that prevent plasma glucose levels from remaining strictly at basal concentrations throughout the day. These fluctuations are influenced by endogenous rhythms, particularly the circadian cycle, which modulates metabolic activity over the course of 24 h. To account for this, we define the steady-state glucose term $G_{st}\left(t\right)$, which incorporates both the basal glucose level $G_b$ and the time-dependent variation induced by circadian influences $G_c\left(t\right)$, as described by Mansell et al. [35]. This relationship is expressed mathematically in Equation (5).

$$G_{st}\left(t\right)=G_b+G_c\left(t\right).$$

(5)

2.3.2. Circadian Cycle Effect

The circadian component $G_c\left(t\right)$ captures the endogenous variation in glucose levels driven by biological rhythms that regulate hormonal secretion, insulin sensitivity, and hepatic glucose production. This effect is modeled using a third-degree polynomial that describes glucose dynamics over a full day (0 ≤ t ≤ 1440 min). For simulations extending beyond 24 h, the time variable $t$ is reset at the start of each new day to preserve periodicity. The circadian variation is defined in Equation (6).

$$G_c\left(t\right)=g_1{t}^3+g_2{t}^2+g_{3}t+g_4;$$

(6)

To evaluate the impact of inter-individual biological variability, a sensitivity analysis of the circadian parameters $g_1$, …, $g_4$ is presented in Supplementary Materials, Section S5, illustrating how their variation within plausible physiological ranges affects glucose dynamics across simulated subjects.

2.3.3. Digestion Dynamics

In glucose–insulin system modeling, nutrient intake is typically represented by carbohydrate absorption, often approximated by an exponential decay function [36,37]. However, the human homeostatic response to feeding involves subtle physiological phenomena, such as variability in gastric emptying, changes in eating-related behaviors, and other factors that disrupt average postprandial glucose patterns, including oscillatory dynamics [38,39,40]. These effects are difficult to capture explicitly, but they can significantly influence glucose regulation. To explain this, we propose a novel formulation of the digestion effect, modeled as an underdamped oscillatory signal modulated by a Gaussian envelope $f_G\left(t\right)$, as shown in Equation (7).

$$G_{int}\left(t\right)=k_{Int}·Intake\left(t\right)·\left({\displaystyle \frac{2t_I\left[sinsin{t}_I^2·coscos\left(1\right)-t_I^{2}coscos{t}_I^2·sinsin\left(1\right)\right]}{t_I^{k_s}-1}}\right)+f_G\left(t\right);$$

(7)

This equation provides an alternative approach to represent unmodeled regulatory effects that influence postprandial glucose variability. Its flexible structure supports digestion-related simulations and sensitivity analyses in intervention studies, where timing, magnitude, and response shape are critical. By adjusting $k_{Int}$, $k_s$, and the form of $f_G\left(t\right)$, we could variability and model robustness. A sensitivity analysis of Equation (7) is included in Supplementary Materials, Section S5, highlighting how parameter variation reflects inter-individual differences in digestive glucose dynamics.

2.3.4. Gaussian Function as Particularity Factor

The Gaussian function ($f_G\left(t\right)$) modulates the digestion signal and can be included or omitted depending on the modeling objective. Its role is to introduce intake-specific variability, reflecting subtle differences in glucose absorption due to circadian timing and exogenous factors. This modulation allows amplification or attenuation of the postprandial glucose effect during the absorption window. The function is defined in Equations (8) and (9).

$$f_G\left(t\right)=k_a{e}^{-\left({\displaystyle \frac{{\left(t-k_b\right)}^2}{2k_c^2}}\right)};$$

(8)

$$t_I=k_t·t_{intake}.$$

(9)

A sensitivity analysis of the Gaussian parameters is presented in Section S5, to explore how variations in $k_a$, $k_b$, and $k_c$ influence glucose responses across different intake scenarios.

2.4. Characterization Methodology

Given the experimental conditions and the complexity of the proposed model, certain parameters and variables are not directly measurable. To address this, we developed a characterization methodology that enables parametric estimation under these constraints:

1. Dynamics of the Effect of Digestion on the Glucose-Insulin System: The digestion process was represented by adapting the model proposed by Kaveh et al. [34], which treats carbohydrate uptake as a perturbation $D\left(t\right)$. Using daily glucose measurements and their oscillation minima, we derived $D\left(t\right)$ and assumed it as the digestion effect $G_{int}\left(t\right)$, capturing both dietary intake and homeostatic influences. This signal was used as input for characterizing our model. A detailed overview of the estimation process is provided in Supplementary Materials, Section S2.

2. Estimation of Non-Measurable Variables and Parameters of the Model: To address uncertainty in signals and parameters, the methodology of Alonso-Bastida et al. [41] was used, employing a Dual Extended Kalman Filter (DEKF) to estimate both system states and non-measurable parameters.

3. Generation of the Local Minimum in the Estimated Parameters: The Levenberg–Marquardt algorithm was used to identify the local minima of the estimated parameters, minimizing the error between model predictions and observed glucose dynamics. This step ensures the convergence of the estimation process and the stability of parameter values.

4. Validation of the Mathematical Model: We validated the model by comparing DEKF-estimated variables with measured glucose values, using error dynamics, performance indices, and Spearman’s rho correlation. The results were contrasted with the local minima of the estimated parameters to assess consistency and reliability.

5. Stability of the Mathematical Model: The stability of the nonlinear system was evaluated by using the Lyapunov theorem combined with Monte Carlo simulations, considering the upper and lower bounds of each model parameter.

This methodological framework constitutes the core contribution of the study. While the DEKF [42,43,44] and Levenberg–Marquardt [45,46,47] algorithms are well established in the literature, their integration within this modeling context provides a novel computational strategy to describe glucose homeostasis in healthy and prediabetic individuals. Full algorithmic details and implementation procedures are provided in Supplementary Materials, Section S3.

3. Results

3.1. Signal and Parameter Estimation

We used the DEKF methodology to estimate states and parameters under parametric uncertainty in nonlinear systems, incorporating digestion effects as an input to the glucose-insulin system. This approach uses a proposed mathematical model of glucose homeostasis to implement DEKF across both glucose profiles. The general DEKF conditions are as follows:

• System Configuration for DEKF Estimation: The estimated states are ${\widehat{x}}^T=\left[GXI{G}_{sc}\right]$, with parametric uncertainty addressed individually for each participant. For healthy individuals, the parameters are ${\widehat{\theta }}_H=\left[p_1{p}_4{p}_2{p}_3\eta \gamma k_{sc}\right]$; for those with prediabetes, ${\widehat{\theta }}_{RF}=\left[p_1{p}_4{p}_2{p}_3\eta \gamma h\right]$. The system input is $u=G_{int}$, and measurable outputs are $Y^T=\left[G00G_{sc}\right]$. Notably, since only interstitial glucose is measured, $G\left(t\right)$ is considered with a 15-min lead relative to $G_{sc}\left(t\right)$, following Stout et al. [48].
• Observability Analysis and Model Conditioning: Perform observability analysis and adapt the nonlinear model to meet DEKF requirements.
• Initialization of Estimation Conditions: Define initial states ($x_o$), parameters (${\theta }_o$), estimates (${\widehat{x}}_o,{\widehat{\theta }}_o$), noise covariance matrices ($v,\omega$), sampling time ($t_s$) and apply a truncated Taylor series expansion.
• Iterative Estimation Procedure: Implement the DEKF update cycle based on the number of available system measurements.

Note: The application of DEKF within the system is described in detail in the Supplementary Materials, Sections S3 and S4.

Partial results from a healthy participant, as an example of the estimation of the glucose process, are presented in Figure 2 and organized into four sections. Section (a) shows the comparison between the measured interstitial glucose (blue line) and its DEKF-based estimation (red dashed line), demonstrating convergence between both signals. Section (b) complements this by illustrating the estimation error dynamics, which fluctuate near zero, further confirming convergence. Sections (c) and (d) display the temporal evolution of the estimated parameters “Insulin-independent constant rate” ($p_1$) and “Ratio of plasma glucose to interstitial glucose” ($k_{sc}$) during DEKF initialization, highlighting their variability over time.

3.2. Model Performance

We evaluated the model performance in two stages for both profiles (healthy and prediabetic). First, we assessed the accuracy of state estimation using DEKF and experimental data. Then, we evaluated how well the proposed model reproduced the system dynamics using the local minimum values of the estimated parameters.

3.2.1. Performance of State Estimation

Since only $G\left(t\right)$ and $G_{sc}\left(t\right)$ are measurable, we evaluated the performance of their DEKF-based estimates using standard error rates for each profile. The applied metrics were the Integral of Absolute Error (IAE), Integral of Squared Error (ISE), and Mean Squared Error (MSE).

Table 1. Performance rates of methodologies used.

Dual Extended Kalman Filter Estimation with Parameter Variation	System Variables
	$\mathbf{G}\left(\mathbf{t}\right)$, mg/dL		${\mathbf{G}}_{\mathbf{s}\mathbf{c}}\left(\mathbf{t}\right)$, mg/dL
	Healthy		Healthy
	IAE: 883.9 (427.5, 1759.2)		IAE: 682.9 (327.8, 1297.1)
	ISE: 1981.3 (520.3, 6637.5)		ISE: 1257.5 (409.9, 3627.0)
	MSE: 2.1 (0.5, 7.0)		MSE: 1.3 (0.4, 3.8)
	Prediabetes		Prediabetes
	IAE: 967.5 (146.1, 1740.7)		IAE: 793.5 (114.1, 1380.9)
	ISE: 2255.6 (286.0, 5533.9)		ISE: 1583.9 (278.1, 3735.6)
	MSE: 2.3 (0.3, 5.8)		MSE: 1.6 (0.2, 3.9)
Estimation of the proposed model with local minimum parameters	System Variables
	$\mathbf{G}\left(\mathbf{t}\right)$, mg/dL	$\mathbf{X}\left(\mathbf{t}\right)$, 1/min	$\mathbf{I}\left(\mathbf{t}\right)$, mU/dL	${\mathbf{G}}_{\mathbf{s}\mathbf{c}}\left(\mathbf{t}\right)$, mg/dL
	Healthy	Healthy	Healthy	Healthy
	IAE: 2249.8	IAE: 3.6 × ${10}^{-3}$	IAE: 850.4	IAE: 882.8
	(933.9, 4770.3)	(1.3 × ${10}^{-3}$, 7.3 × ${10}^{-3}$)	(312.7, 1534.8)	(423.8, 1666.5)
	ISE: 8997.9	ISE: 2.5 × ${10}^{-8}$	ISE: 1847.1	ISE: 2119.9
	(1882.2, 2.8 × ${10}^4$)	(3.7 × ${10}^{-9},$ 7.8 × ${10}^{-8}$)	(237.4, 5854.3)	(650.7, 7213.4)
	MSE: 9.5	MSE: 2.7 × ${10}^{-11}$	MSE: 1.9	MSE: 2.2
	(2.0, 30.6)	(3.9 × ${10}^{-12},$ 8.3 × ${10}^{-11}$)	(0.2, 6.2)	(0.6, 7.6)
	Prediabetes	Prediabetes	Prediabetes	Prediabetes
	IAE: 5533.4	IAE: 1.6	IAE: 1191.0	IAE: 2391.5
	(1819.0, 10,315.0)	(0.7, 3.7)	(428.4, 3059.8)	(263.3, 4287.9)
	ISE: 6.5 × ${10}^4$	ISE: 5.6 × ${10}^{-3}$	ISE: 3876.5	ISE: 1.3 × ${10}^4$
	(8.9 × ${10}^3$, 2.1 × ${10}^5$)	(1.3 × ${10}^{-3}$, 2.3 × ${10}^{-2}$)	(400.2, 2.0 × ${10}^4$)	(222.8, 4.6 × ${10}^4$)
	MSE: 69.1	MSE: 5.9 × ${10}^{-6}$	MSE: 4.1	MSE: 14.7
	(9.5, 224.3)	(1.3 × ${10}^{-6}$, 2.4 × ${10}^{-5}$)	(0.4, 22.1)	(0.2, 49.3)

$G\left(t\right)$: Plasma glucose concentration; $G_{sc}\left(t\right)$: Interstitial glucose concentration; $X\left(t\right)$: Glucose reduction effect derived from insulin action; $I\left(t\right)$: Plasma insulin concentration.

, located in the section “Dual Extended Kalman Filter Estimation with Parameter Variation” summarizes the results, with MSE values ranging from 1.3 to 2.3 (mg/dL)² for both estimates across profiles. These results indicate that the estimation is appropriate given the experimental conditions.

3.2.2. Model Performance with Local Minimum Values

To ensure good model performance, we accounted for the high parametric variability over time observed in Figure 2. We applied the Levenberg–Marquardt algorithm to identify the local minima of each estimated parameter, yielding optimal values for approximating glycemic behavior using a model with constant parameters. We evaluated model performance by comparing its response—using the locally optimized parameters and the same input signal $G_{int}\left(t\right)$—with the state estimates from DEKF ($\widehat{G},\widehat{X},\widehat{I},{\widehat{G}}_{sc}$) for both profiles.

Table 1, under the section “Estimation of the proposed model with local minimum values of the parameters” presents the results. As expected, the model with static parameters performs slightly worse than DEKF, with MSE differences of 9.5 (mg/dL)² for G(t) in healthy individuals and 69.1 (mg/dL)² in those with prediabetes. Nonetheless, these values are highly acceptable and adequately capture the system’s dynamics.

3.2.3. Comparative Analysis Between Measured Glucose

To compare the model with real-life behavior, we used experimental data and the average parameter values calculated for each profile to simulate the system. Each simulation scenario incorporated the same intake conditions (previously determined, see Supplementary Materials, Section S2) and the corresponding average parameter values. The simulated results were compared with daily measurements, using the average glucose level for each day as a reference. Figure 3 presents the outcomes, highlighting three key aspects of the comparative analysis.

• The first aspect highlights the real-time comparison between daily glucose measurements and model-based estimations. Panels “a” and “d” show the measured glucose (blue line) and the estimated signal (orange dashed line), selected based on median performance indices. These examples demonstrate the model’s fidelity, with only minor underestimation.
• The second aspect, shown in panels “b” and “e,” illustrates the error dynamics between measurement and estimation. These results confirm the model’s predictive capacity, with the largest deviations—typically triggered by intake events—ranging from 5 to 20 mg/dL across both profiles.
• The third aspect, shown in panels “c” and “f,” evaluates the model’s effectiveness across measurement days by comparing average glucose values from the measurements and estimates. This approach assesses the cumulative estimation error and reveals a strong positive correlation: Spearman’s r = 0.9839, p < 0.01 for healthy individuals, and r = 0.9851, p < 0.01 for those with prediabetes. We calculated correlations using MATLAB’s (version R2023b) “corr (‘Type’, ’Spearman’)” function.

3.3. Local Minima of Model Parameters

We applied the Levenberg–Marquardt algorithm to obtain the local minimum values for each parameter in the proposed model, ensuring system stability. For parameters related to feeding effects and circadian rhythms, we used MATLAB’s “Curve Fitting” tool to optimize each value.

Table 2. Parameters of the proposed model.

Process	Parameter, (Units)	Healthy 1	Prediabetes 2
Glucose dynamics	$p_1$, (${\displaystyle \frac{1}{\mathrm{m}\mathrm{i}\mathrm{n}}}$)	0.0542 (0.0478, 0.0597)	0.0545 (0.0469, 0.0638)
	$p_4$, (${\displaystyle \frac{1}{\mathrm{m}\mathrm{i}\mathrm{n}}}$)	1.4980 (1.2999, 1.6796)	1.7023 (1.4402, 1.9112)
	$G_b$, (${\displaystyle \frac{\mathrm{m}\mathrm{g}}{\mathrm{d}\mathrm{L}}}$)	70 (65, 75)	80 (75, 85)
	$g_1$, (${\displaystyle \frac{\mathrm{m}\mathrm{g}}{(\mathrm{d}\mathrm{L}·\mathrm{m}\mathrm{i}{\mathrm{n}}^3)}}$)	$P_{5th}$: −2.125 × ${10}^{-8}$	$P_{5th}$: 8.953 × ${10}^{-9}$
		$P_{7.5th}$: −2.645 × ${10}^{-8}$	$P_{7.5th}$: 6.725 × ${10}^{-9}$
		$P_{10th}$: −2.398 × ${10}^{-8}$	$P_{10th}$: 5.727 × ${10}^{-9}$
	$g_2$, (${\displaystyle \frac{\mathrm{m}\mathrm{g}}{(\mathrm{d}\mathrm{L}·\mathrm{m}\mathrm{i}{\mathrm{n}}^2)}}$)	$P_{5th}$: 4.008 × ${10}^{-5}$	$P_{5th}$: −1.576 × ${10}^{-5}$
		$P_{7.5th}$: 5.055 × ${10}^{-5}$	$P_{7.5th}$: −1.025 × ${10}^{-5}$
		$P_{10th}$: 4.426 × ${10}^{-5}$	$P_{10th}$: −6.536 × ${10}^{-5}$
	$g_3$, (${\displaystyle \frac{\mathrm{m}\mathrm{g}}{(\mathrm{d}\mathrm{L}·\mathrm{m}\mathrm{i}\mathrm{n})}}$)	$P_{5th}$: −0.0112	$P_{5th}$: 0.0057
		$P_{7.5th}$: −0.0162	$P_{7.5th}$: 0.0019
		$P_{10th}$: −0.0118	$P_{10th}$: −0.0020
	$g_4$, (${\displaystyle \frac{\mathrm{m}\mathrm{g}}{\mathrm{d}\mathrm{L}}}$)	$P_{5th}$: 6.206	$P_{5th}$: 4.813
		$P_{7.5th}$: 8.626	$P_{7.5th}$: 7.605
		$P_{10th}$: 10.75	$P_{10th}$: 11.67
	$k_{sc}$, (${\displaystyle \frac{1}{\mathrm{m}\mathrm{i}\mathrm{n}}}$)	0.1085 (0.0965, 0.1240)	0.1085 (0.0965, 0.1240)
Dynamics of the effect of insulin on glucose concentration reduction	$p_2$, (${\displaystyle \frac{1}{\mathrm{m}\mathrm{i}\mathrm{n}}}$)	0.1228 (0.1010, 0.1410)	0.1224 (0.1059, 0.1408)
	$p_3$,$\left({\displaystyle \frac{\left(\frac{\mathsf{\mu }\mathrm{U}}{\mathrm{m}\mathrm{L}}\right)}{{\mathrm{m}\mathrm{i}\mathrm{n}}^2}}\right)$	$9.21\times {10}^{-8}$	$9.56\times {10}^{-8}$
		$\left(8.95\times {10}^{-8},9.44\times {10}^{-8}\right)$	$\left(9.09\times {10}^{-8},1.05\times {10}^{-7}\right)$
Insulin dynamics	$\eta$, (${\displaystyle \frac{1}{\mathrm{m}\mathrm{i}\mathrm{n}}}$)	0.2465 (0.2129, 0.2758)	0.2463 (0.2154, 0.2735)
	$\gamma$, $\left({\displaystyle \frac{\left(\frac{\mathsf{\mu }\mathrm{U}}{\mathrm{m}\mathrm{L}}\right)}{{\mathrm{m}\mathrm{i}\mathrm{n}}^2\left(\frac{\mathrm{m}\mathrm{g}}{\mathrm{d}\mathrm{L}}\right)}}\right)$	5.63 × ${10}^{-5}$	5.61 × ${10}^{-5}$
		(4.60 × ${10}^{-5}$, 6.33 × ${10}^{-5}$)	(4.72 × ${10}^{-5}$, 6.56 × ${10}^{-5}$)
	$h$, (${\displaystyle \frac{\mathrm{m}\mathrm{g}}{\mathrm{d}\mathrm{L}}}$)	79.03 (70.68, 122.28) [28,49]	94.857 (76.482, 109.785)
Digestion dynamics	$k_{Int}$, ($1/\mathrm{d}\mathrm{L}$)	0.0039 (−0.0333, 0.0333)	0.0039 (−0.0333, 0.0333)
	$k_s$, (-)	4.203 (2.755, 6.188)	4.203 (2.755, 6.188)
	$k_t$, ($1/\mathrm{m}\mathrm{i}\mathrm{n}$)	0.0167 (0.0144, 0.0275)	0.0167 (0.0144, 0.0275)
	$k_a$, (${\displaystyle \frac{\mathrm{m}\mathrm{g}}{\mathrm{d}\mathrm{L}}}$)	0.1 (−4.9140, 5.0480)	0.1 (−4.9140, 5.0480)
	$k_b$, ($\mathrm{m}\mathrm{i}\mathrm{n}$)	96.781 (20.74, 189.60)	96.781 (20.74, 189.60)
	$k_c$, ($\mathrm{m}\mathrm{i}\mathrm{n}$)	20.081 (−19.34, 115.90)	20.081 (−19.34, 115.90)

¹ Healthy: People without a diabetes diagnosis (people apparently healthy). ² Prediabetes: People at risk of developing type 2 Diabetes Mellitus.

summarizes all model parameters, including their average, minimum, and maximum values across samples. Columns 3 and 4 present the numerical results for healthy individuals and those with prediabetes, respectively.

3.4. Stability of the Mathematical Model

In this work, we use the Levenberg-Marquardt method to compute the local minimum values of the model parameters. These optimized values ensure the stability of the nonlinear system, and we validated this stability using the Lyapunov theorem for nonlinear dynamics.

Theorem 1

(Indirect Method, [48]). Let the origin $x=0$ be an equilibrium point of the nonlinear system $\dot{x}=f\left(x\right)$, where $f:D\subset R^n$is a continuously differentiable function and $D\subset R^n$ is a neighborhood of the origin. Let

$$A={{\displaystyle \frac{\partial f\left(x\right)}{\partial x}}|}_{x=0}$$

Then, the origin is asymptotically stable if all the eigenvalues of A have a negative real part. The origin is UNSTABLE if one or more eigenvalues of A have a positive real part.

We followed the formal proof outlined by Hassan [49] and implemented a Monte Carlo methodology [50] to evaluate the system’s stability under parametric uncertainty. Using normally distributed variations bounded by the minimum and maximum values of each parameter, we generated 100,000 scenarios per profile. In all cases, the system’s eigenvalues had negative real parts (ranging from −0.2271 to −0.0470), confirming asymptotic stability across all simulations.

3.5. Comparative Analysis Between Measured Glucose Profiles and Model Simulation

We conducted a graphical comparison between glucose measurements from healthy and prediabetic individuals, arranging the simulation results of the proposed model into three panels (Figure 4): the first two graphs show measured glucose levels (gray lines) for each group, in contrast to the model simulation (black dashed line), which assumes three standardized intakes at hours 7.5, 13, and 21, each with 250 g of carbohydrates. We modeled the dietary effect using Equations (7)–(9) and illustrated in the lower panel. In this part, the model evaluates the performance under average population conditions, defined by the 5th and 95th percentile glucose levels (healthy: 77.09–125.59 mg/dL; prediabetic: 81.97–140.65 mg/dL). The simulation remains within these bounds most of the time, except during brief drops caused by intake effects, indicating acceptable performance within the expected variability of real-life measurements.

4. Discussion

The increasing number of T2DM cases worldwide places a significant economic burden on public health systems, highlighting the need to understand glucose homeostasis for effective prevention and intervention. In this work, we propose a mathematical model that estimates the average glucose–insulin dynamics in healthy and prediabetic individuals. Inspired by Bergman’s formulation [51], our model integrates the effects of circadian rhythms and digestion on blood glucose regulation.

The proposed model shows strong versatility in representing glycemic profiles of both healthy and prediabetic individuals, due to its parametric flexibility. This represents a significant advance in glucose homeostasis modeling, as it includes prediabetic dynamics for the first time. Similar to Bergman’s minimal model [51], it can now estimate glycemic variation in healthy, prediabetic, and T1DM populations, demonstrating that a relatively simple structure can generate accurate predictions. This is supported by an average Spearman correlation of r = 0.98, interpreted cautiously due to the limited sample size.

The proposed model maintains a high degree of mathematical simplicity, consisting of four differential equations and three algebraic relations to account for food intake. In contrast, analogous models in the literature are more complex: Dalla et al. [52] use 12 equations for healthy individuals; López-Palau et al. [53] and Visentin et al. [54] present models for T2DM with 28 and 14 equations, respectively; and for T1DM, Hovorka et al. [32] and Visentin et al. [55], used 9 and 16 differential equations. While more equations allow for richer dynamics, glucose and insulin concentrations remain the essential variables for guiding decisions related to glucose homeostasis. This underscores the value of simpler mathematical alternatives that enable rapid and efficient decision-making.

The robustness of the proposed model stems from key additions to classical formulations. We modeled the circadian cycle effect, previously introduced by Mansell et al. [35] using triangular functions, with a polynomial based on minimum glucose values at each time point—offering a simpler yet effective representation of daily glucose settling. Additionally, the incorporation of food intake dynamics significantly enhances model robustness. In healthy and prediabetic individuals, glucose exhibits underdamped oscillatory behavior due to autoregulatory processes not captured in prior models. This pattern is aligns with previously reported observations [40].

However, the model has limitations. We developed the model using a small population sample, which may not fully capture the heterogeneity of prediabetes phenotypes [56,57]. We estimated glucose concentrations from interstitial glucose measurements, which inherently present a temporal delay of approximately 13.5 min relative to blood glucose levels [58]. This delay introduces uncertainty in the characterization of rapid glycemic changes, particularly postprandial responses [59]. Furthermore, the estimation and characterization methodology relies on average dynamics and parametric fitting, which, while effective for population-level modeling, may not reflect individual variability or transient physiological adaptations.

Regarding the analysis performed, we demonstrate that the proposed model offers a high degree of specificity compared to existing literature, due to its mathematical structure and applicability in various glucose-related contexts. The model presents several distinctive contributions to the study of glucose homeostasis:

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Robust extension of the classical Bergman representation

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Mathematical simplicity that enables fast and efficient decision-making

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Novel integration of dietary intake effects on the glucose–insulin system

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Flexibility to generate individualized dynamics for virtual patients and populations

5. Conclusions

This study presents a new mathematical model that delineates glucose homeostasis in both healthy and prediabetic individuals, a group often underrepresented in existing modeling approaches. The model demonstrated a strong fit with experimental data (r = 0.98, p < 0.01), supporting its potential to capture key aspects of glucose regulation. Nonetheless, the results should be interpreted with caution given the limited sample size and the use of interstitial glucose data, which inherently involves an approximate 15-min delay relative to plasma glucose levels.

A major strength of this work lies in the integration of a Dual Extended Kalman Filter (DEKF), enabling the model to estimate parameters and minimize prediction error in real time. This feature improves the model’s ability to represent physiological glucose dynamics and offers a practical platform for in-silico experimentation. Beyond its computational contribution, this approach can inform predictive and preventive strategies against Type 2 Diabetes Mellitus. Further research with larger cohorts will help validate and extend its clinical applicability.