Mathematical modelling of a glucose-insulin system for type 2 diabetic patients in Chad

In this paper, we focus on modelling the glucose-insulin system for the purpose of estimating glucagon, insulin, and glucose in the liver in the internal organs of the human body. A three-compartmental mathematical model is proposed. The model parameters are estimated using a nonlinear inverse optimization problem and data collected in Chad. In order to identify insulin and glucose in the liver for type 2 diabetic patients, the Sampling Importance Resampling (SIR) particle filtering algorithm is used and implemented through discretization of the developed mathematical model. The proposed mathematical model allows further investigation of the dynamic behavior of hepatic glucose, insulin, and glucagon in internal organs for type 2 diabetic patients. During periods of hyperglycemia (i.e., after meal ingestion), whereas insulin secretion is increased, glucagon secretion is reduced. The results are in agreement with empirical and clinical data and they are clinically consistent with physiological responses


Introduction
Diabetes is a disease that cannot be cured (chronic) but can be treated.It is due to an abnormal increase in glucose [1].This disease has become a major world health problem, particularly in Chad.Like all developing countries, Chad is paying heavy consequences due to this disease.Chad, the second largest country and the third most overcrowded country in Central Africa, is in the midst of a transition from this disease [2].According to the report of the International Diabetes Federation, 425 million people worldwide were living with the disease in 2015 and this number may increase to an estimated 622 million in 2040 [3].In 2013, 231290 Chadians had diabetes [4], this number may be significant in the future.All these results make diabetes a real public health problem.There are two main types of diabetes: type 1 diabetes, which affects 10 percent of the affected population and type 2 diabetes, which affects the remaining percent [5,6].Type 2 diabetes is therefore important in terms of severity.Consequently, many researchers are trying to find methods to diagnose and treat this disease.Generally, disease dynamics is investigated throughout the application of the mathematical models [7,8,9].One approach is to find a mathematical model that can describe ® Received: 15.11.2022 ® Revised: 24.12.2022® Accepted: 27.12.2022® Published: 30.12.2022 the dynamics of the glucose-insulin system in order to analyze, interpret and predict the results.One of the approaches used is to find a mathematical model capable of describing the dynamics of the glucose-insulin system in order to predict, analyze and interpret the results.In literature, much effort has been made recently to analyze [10] and to develop mathematical models of type 2 diabetes [11,12,13,14].The objective of this paper is to propose a mathematical model of the glucose-insulin system suitable for type 2 diabetic patients in Chad.The glucagon, insulin, and hepatic glucose in internal human organs are estimated using Sampling Importance Resampling (SIR) algorithm.The mathematical model parameters are estimated using a nonlinear inverse optimization problem and data collected in Chad.This paper is structured as follows: Materials and methods are presented in Section 2. It deals with data sources and mathematical model equations.Section 3 focuses on the estimation of the parameters of the model.The numerical tests are given in Section 4 while concluding remarks are presented in Section 5.

Data set
The data are collected in the laboratory of the hematology department at regional Abéché hospital in Chad for a period of one month from 23, January 2019 to end 23, February 2019.Moreover, 900 exams should be done, but we faced some challenge including lack of material such as glucose meter, lower number of enumerators and financial means to organize the transport of participants who live so far from the hospital to come early morning in a fasting state.Consequently, the data have been collected only for 96 participants and 75 among them have type 2 diabetes.

Model equations
More multiple pancreatic hormones are involved in glucose homeostasis [15], but the potent hormone regulators of both glucose appearance and disappearance in the circulation are insulin and glucagon.Indeed, insulin is the key regulatory hormone of glucose disappearance, and glucagon is a major regulator of glucose appearance.Consequently, the deficiency of these hormones is the main cause of type 2 diabetes.During the fed state, the rate of gastric emptying is the major determinant source of circulating glucose and other sources of how glucose appears in the circulation are derived from hepatic processes.Glucagon plays a major role in sustaining plasma glucose during fasting conditions by stimulating hepatic glucose production.During the first 8-12 hours of fasting, glucagon facilitates this process and thus promotes glucose appearance in the circulation throughout the glycogenolysis mechanism [16].Over longer periods of fasting, glucose, produced by gluconeogenesis, glucose is released from the liver which is the sole source of endogenous glucose production.Most tissues have the ability to hydrolyze glycogen and glucose removal into skeletal muscle and adipose tissue is driven mainly by insulin in the immediate post-feeding state [16].In addition, insulin contributes to augmenting glucose uptake in peripheral tissues and in the liver by affecting the activity of different enzymes.The peripheral insulin resistance and relative insulin deficiency in type 2 diabetic patients have resulted in low glucose uptake rates by muscle cells and adipose tissue cells [17].The studies show that insulin-induced stimulation effects on hepatic glucose uptake and hepatic glucose production rate are impaired in type 2 diabetic patients.This leads to reduced hepatic glucose uptake [18].A type 2 diabetic patient is experienced with postprandial β-cell action that becomes abnormal due to the loss of immediate insulin response to a meal [19].Therefore, hyperglycemia in type 2 diabetic patients is caused by the resistance of peripheral insulin resistance coupled with progressive β-cell failure and decreased availability of insulin and other hormones that is amylin (a neuroendocrine hormone coexpressed and consecrated with insulin by pancreatic β-cells in response to nutrient stimuli, and GLP-1 (more potent incretin hormone secreted in greater concentrations and is more physiologically relevant in humans) [20].Due to high glucose level that can induce vascular endothelial cell dysfunction and affect blood viscosity and arterial wall tension, type 2 diabetes patients are at higher risk for the development of vascular complications than non-diabetic persons [21].Due to such a high demand for understanding the blood flow characteristics in type 2 diabetic patients, in this study, we develop the mathematical model of type 2 diabetes patients.We propose a three-compartmental mathematical model where the compartments are the vascular and tissues compartment, liver compartment and pancreas compartment.The state variables are are glucagon (Γ ), insulin (I p ), hepatic glucose (G L ), glucose (G).Taking into consideration the diagram model illustrated in Figure 1.The model equations are as follows: where n i denotes the rate of blood coming indirectly in compartment i (i = Γ , I p , G L G) from vascular circulation.α, β, γ and δ are the constants to be estimated and refer to the non-linearity of the corresponding determinant variable.The parameters and variables are described in Table 1.

Estimation of parameters
We consider that the volumes are obtained from literature and they are presented in Table 2 [14].
Table 2. Parameters used from the literature [14] To estimate other parameters, let be a vector of parameters to be estimated.We fix a positive parameter and we consider the following discretisation of the interval 0, T max , where T max denotes a positive time parameter.For a positive integer parameter N and G σ (t k ) corresponding data measurement of real values G(t k ) at time t k we take The superscript σ is the perturbation parameter due to some imprecision on measured data [22].The identification of parameters can be done by formulating nonlinear inverse problem which is solved using regularization techniques [23].Let G be vector solutions in R N at time grid points of the system (1) depending of the parameter vector u.We want to solve the coefficient identification problem where G σ is given by (2).
This inverse problem can be formulated as follows.Find subject to (1).We should notify that the (3) is not linear inverse problem and it is ill-posed.This means that little perturbation on data produces a solution that is very different of the original ones.Furthermore, the solution µ does not depend continuously on the data.To find the regularization of the problem (3), we use Tikhonov techniques (See [23] for details).Therefore, this problem becomes Taking subject to (1) where we have set for good θ such that µ θ converges to the solution µ as θ → 0.Here L is an operator used for stabilization (i.e., L is the identity, a differentiation operator, etc.).The solution of (4) subject to (1) is obtained using the Least square method and the data collected in Chad.The numerical simulations are carried out using Matlab built-in function fmincon which allows solving constrained optimization problems.The parameters estimated are shown in Table 3.

Numerical tests
The measurement of glucose and insulin from different parts of the body needs to take blood samples to determine their concentration.However, to take blood samples from many internal body organs for instance heart and the liver is clinically impossible.We solve this problem, we can estimate concentrations of these parameters using measurements from peripheral tissues along with a mathematical model and a filtering algorithm.Sampling Importance Resampling (SIR) algorithm is one of the methods that can give the estimated solution.For more details about the SIR algorithm, we refer the readers to [24,25].The data collected in Chad is the glucose measured in the tissues of patients.In our simulation we estimate glucose in the liver, insulin and glucagon in the pancreas by taking this measured glucose as measurements.We assume that noisy measurements of glucose in tissues and the heart are available measured data.To apply the SIR algorithm, we consider a fixed-step backward difference approximation by where and If M denotes the total number of discretization intervals we have which refers to the dynamic equations with Using ( 8) our state dynamic model ( 1) is estimated recursively by the following compact form where we set Y k = G k that is measurement of glucose in tissues, v k-1 and e k-1 are the stochastic process and measurement noise, respectively and they are independent and identically distributed (i.i.d.).P denotes the matrix from the equations modelling the sensors referred to as the measurement model given as P = (0, 0, 0, 1) t .
In numerical simulation, we consider M = 100, T max = 300 minutes, N = 1000 particles and we assume that the state X 0 and state measurement noises have a Gaussian probability density function that is X 0 ∼ N (0, 5), v k ∼ N (0, 20) and e k ∼ N (0, 1) where N means normal distribution.The numerical results are illustrated in the Figure 2, 3 and 4.
Figure 2 shows the concentration of glucagon in the pancreas using SIR implemented on the mathematical model.There is not a significant variation of this parameter that plays a crucial role in the regulation of blood glucose.This means that blood glucose is not regulated for a type 2 diabetic patient.The pancreatic insulin decreases as shown in the Figure 3. Therefore, there is no role of both insulin and glucagon as regulators of blood glucose.During periods of hyperglycemia (i.e., after meal ingestion), whereas insulin secretion is increased, glucagon secretion is reduced.All those results are justified by the increase of glucose in the liver (See Figure 4).Hence, the liver affects insulin concentrations, since about 50% of insulin is extracted at first passage, and this fraction may be reduced in the insulin-resistant liver, leading to hyperinsulinemia [26].The elevated rate of hepatic glucose production (HGP) is a major cause of fasting hyperglycemia and a lack of suppression of HGP may contribute to postprandial hyperglycemia.Through an increased secretion of triglyceride, the liver may also contribute to peripheral insulin resistance and thereby further increase postprandial hyperglycemia [26].

Concluding remarks
The mechanical behaviour of the glucose-insulin system depends on the type of mathematical model that describes its dynamics.The simple mathematical model should describe accurately this system vis-a-vis the diseases that affect it.In this work, we have proposed a three-compartmental mathematical model for type 2 diabetic patients that describes the variation of glucose and insulin in Chad context based on data collected at Abéché hospital.To estimate the dynamics of glucagon, insulin and hepatic glucose in internal organs of the human body, the Sampling Importance Resampling (SIR) particle filtering algorithm is used and implemented using the developed mathematical model.Numerical results show that the new glucose-insulin system model

Figure 1 .
Figure 1.Diagram of the mathematical model.The dash lines notify that a certain quantity of mass flows non-linearly from one compartment/sub-compartment to another.The dashed dot line between the heart-tissues compartment means that a small quantity of glucose is transmitted to the liver compartment.

Figure 2 .Figure 3 .
Figure 2. Variation of glucagon in the pancreas

Figure 4 .
Figure 4. Variation of glucose in the liver

Table 1 .
Description of variables and parameters of the mathematical model

Table 3 .
Estimated model parameters