A POLYNOMIAL LINEAR REGRESSION APPROACH TO ESTIMATE SENSITIVE PARAMETERS IN THE NOVEL DOUBLE DIABETES MODEL

Sensitivity analysis characterizes the changes in the model outputs due to the changes in the model parameters. In this article, we estimate the most sensitive parameters in the Novel Double Diabetes Model (NDDM) through the polynomial linear regression approach; this way we develop a direct relation between the sensitivity analysis and the paramter estimation. The NDDM has more than seventeen parameters, and estimating them simultanously is difficult. We select the most commonly used five parameters in the glucose-insulin dynamics for the sensitivity analysis. The model outputs-glucose concentrations in the plasma and the subcutaneous compartments are sensitive to the selected parameters whereas the insulin concentrations in the plasma and the subcutaneous compartment are sensitive only to the insulin transfer rate from the subcutaneous to the plasma compartment. System sensitivity of the model for the selected parameters is also in agreement with the individual sensitivities of the parameters. Consequently, we estimate the parameters which are more sensitive by the polynomial linear regression approach. Received July 19th, 2020; accepted August 10th, 2020; published September 3rd, 2020. 2010 Mathematics Subject Classification. 92C45, 34A34.


Introduction
Mathematical Modeling possesses a pivotal position in understanding the different physiological processes [12]. One of the most important biological processes is the glucose-insulin regulatory system [5], [1]. In order to understand the physiological behavior of this system, various mathematical models have been proposed by different scientists, and are available in the literature. Some of these models are very helpful for the development of a closed-loop algorithm in order to control the blood-glucose concentration in the body [18]. Bergman et al. proposed the four compartmental mathematical model of glucose-insulin kinetic on the basis of the Intravenous Glucose Tolerance Test (IVGTT) [3] in the late seventies. This model is commonly used in the mathematical theory and in the clinical research to understand the glucose-insulin system. A detailed study and analysis of this model is found in [13]. In literature, more than 500 studies based on minimal model have been found [13]. Hovarka et al. [6], Sorenson [16] and Puckett [15] modified minimal model and focused on Type-1 diabetic patients [18]. Topp et al., Roy et al., introduced mathematical models [4], which deal with Type-2 diabetes, however, these models ignore the effect of delays. Inclusion of the delay factors are useful for the better understanding of the oscillatory behavior of the insulin and glucose regulatory system.

Researchers suggested two delays, Hepatic Glucose Production(HGP) delay and Insulin Secretion(IS)
delay to be relevant to the oscillations of glucose and insulin. HGP delay was considered as the time taken for "remote insulin" to stimulate HGP to secrete glucose. IS delay is defined as the time for the elevated plasma glucose to the change the insulin secretion. Sturis et al., [17] proposed compartmentel model similar to Bergman minimal model and introduced HGP delay to study oscillating behavior of glucose and insulin.
Li et al., in [9], and [10] proposed one-compartment models, considering IS and HGP delays explicitly. The combined effect of the two delays was considered by using the sum of the two delays and proposed that the combined effect of the two delays influenced the dynamics of the glucose-insulin feedback mechanism, but not each individual delay.
The novel double diabetes model studied in [5] considered the effect of two delays with subcutaneously injected insulin and glucose measurements. Insulin administered via subcutaneous has the advantage to make the management of diabetes more effectively than intravenous. Through sensitivity analysis of the novel double diabetes model, it is possible to investigate the behavior of the system which provides rich information for the diabetes control.
In the present study, we present a novel diabetes model in Section 2. The sensitivity analysis is briefly described in Section 3 and the sensitivity studies of the NDDM model is made in Subsection 3.1. The detail of numerical implementation for the model is given in Section 4 and the results are discussed in Section 5.
The general solution of the model is presented in Section 6 and the selected parameters are estimated in Section 7. The conclusion of the whole analysis is given in Section 8.

Novel Double Diabetes Model
Novel double diabetes model consists of a set of six ordinary differential equations and seven subfunctions [18]. This model has seventeen parameters. The description of all state variables, sub-functions and parameters of the model are described in Table( 1), Table( 2), and Table( 3) respectively [18].
The seven sub-functions used in the model are described below: (1) HGP (I P ) = 160 S(G p ) = 210 The mechanism of delays plays a vital role in physiological system [18]. In the model, two delay parameters τ 1 and τ 2 are used which enhance the accuracy of the glucose-insulin system under examination.
Glucose moves in arteries and peripherals. Brain and Central Nervous System(CNS) use glucose without insulin dependency, this mechanism is known as insulin independent glucose utilization U ii . Liver, muscles and adipose tissue use glucose depending upon insulin; this mechanism is known as insulin dependent glucose utilization U id .

Sensitivity Analysis
Since novel model is a compartmental model with six outputs G p , G i , Q 1a , Q 1b and Q 2 , we consider a multiple-output system with the measurable outputs [8], [14] modeled by where T > 0 is fixed. The open set, θ ∈ U ⊂ R p , is the set of admissible parameters. We assume that the output model given by Equation , is defined by.
Here assume that f k and θ n are non-zero k = 1, · · · , , and n = 1, · · · , r. The sensitivity function describes the behavior of model function output by changing the parameters' values [11]. It quantifies to which parameter is the model output is the most or less sensitive. The sensitivity functions are used in the parameter identification problems [7].
The system sensitivity with respect to a model parameter θ n over time interval [0,T] describes the sensitivity of all model outputs f k , k = 1, · · · , with respect to θ n , and is defined by the following relation [13].
3.1 Sensitivity Analysis of the NDDM. In order to check the sensitivity of the parameters (k 1 , k 2 , k a1 , k a2 , k e ), we use all parametric values except (k 1 , k 2 , k a1 , k a2 , k e ) in the model. Then, the model equations becomė (3.4)

Numerical Implementation
We take θ = (k 1 , k 2 , k a1 , k a2 , k e ), then a system of sensitivity equations is obtained using Equation

Results
In first subsection, we find the sensitivities of the model outputs with respect to the model parameters, and in the second subsection we find the system senstivity of the whole model output.
5.1 Sensitivites. Sensitivity analysis identifies the parameters to which the model output is the most or the least sensitive [2]. The sensitivities of the model outputs with respect to the selected parameters are given in the following parts.
(1) Sensitivity of G p with respect to Parameters k 1 , k 2 , k a1 , k a2 , k e : Sensitivity of the plasma glucose, G p , with respect to the parameters k 1 , k 2 , k a1 , k a2 and k e is presented in Figure (1). It shows that the plasma glucose, G p , is sensitive to all five parameter upto 10 minutes. This means the changes in the true value of these parameters change G p output.
The Parameter k 1 , k 2 are the glucose transfer rates modeling the material exchange between the plasma and the subcutaneous region. Since the output, G p , is sensitive to k 1 and k 2 , these parameters affect the evaluation of the glucose concentration in the plasma glucose region. If k 1 is greater in amount, then the glucose transfers from the plasma to the subcutaneous compartment and extra glucose moves back from subcutaneous to the plasma compartment through k 2 , then the glucose level tends to normal range. If k 1 is smaller in amount, then less amount of glucose transfers from the plasma to the subcutaneous compartment through k 2 and less glucose moves from the subcutaneous to the plasma compartment, then the glucose level increases from normal range that leads to diabetes.
The model output, G p , is sensitive to the paramters k a1 , k a2 . The glucose-insulin regulatory system shows large variation due to small changes in these parameters. If the transfer rates, k a1 , k a2 , are greater, than more insulin moves from the subcutaneous tissues to the plasma, and maintain the blood glucose level.
Plasma glucose, G p , is also sensitive to the insulin clearance rate, namely k e . When k e decreases, its results is shown in higher plasma insulin that maintains the blood glucose level. When it increases, the results is shown in terms of low plasma insulin that leads to diabetes.
The combined sensitivity of the output, G p , for the selected parameter is presented in Figure (3) (upper panel) which shows that G p is the most sensitive to the parameters k 1 , k e , k a1 and less sensitive to the parameters k 2 , and k a2 .
(2) Sensitivity of G i with respect to Parameters k 1 , k 2 , k a1 , k a2 , k e : Sensitivity of the subcutaneous glucose, G i , with respect to the selected parameters is presented in Figure (2). The combined sensitivity of the output, G i , for the selected parameters is presented in Figure (3) (lower panel) which shows that G i is the most sensitive to the parameter k 1 than the other four parameters. (3) Sensitivity of Q 1a , Q 1b , Q 2 and I p with respect to the Parameters k 1 , k 2 , k a1 , k a2 , k e : The combined sensitivities of the outputs Q 1a , Q 1b , Q 2 and I p w.r.t the parameters k 1 , k 2 , k a1 , k a2 , k e are presented in Figure (4).
It is observed that Q 1a , Q 2 and I p outputs are not sensitive to the four parameters k 1 , k 2 , k a2 and k e upto 10 minutes. This means the changes in the true value of the parameters k 1 , k 2 , k a2 and k e do not bring changes in Q 1a , Q 2 and I p output. These output are only sensitive to k a1 for the 1 minutes. This implies that changes in parameter, k a1 , will change in the outputs Q 1a , Q 2 and I p in the beginning and not afterwards. Moreover, Q 1b is not sensitive to all the five parameters k 1 , k 2 , k a1 , k a2 and k e upto 10 minutes. This implies the changes in the true value of the parameters k 1 , k 2 , k a1 , k a2 and k e do not bring change in the output, Q 1b . The sensitivity analysis signifies that the parameters are identified by the availability of data for any

System
In order to estimate model parameter, we need to find the general solution.

Linearization and General Solution
NDDM is nonlinear system. We linearize it about the equilibrium point. The equilibrium point is obtained by putting the rate of change of each state to zero, and then assigning values to parameters [18], as given in Table 3. We get the feasible equilibrium point as To estimate the parameters, (k 1 , k 2 , k a1 , k a2 , k e ), we use the model Equation (3.4) and linearize it using the Jacobin matrix given by Equation (6.2) about the equilibrium point given in (6.1).
Putting the valuse of the corresponding entries in the Jacobian matrix A, we get the matrices A, A 2 , A 3 , · · · , as follow.
General solution is When we neglect A 3 , and the higher power of A, we get, exp(At) = I + At + A 2 t 2 /2!.

Polynomial Regression Least Square Approach
Polynomial regression least square method is used to estimate the rate parameters k 1 , k 2 , k a1 , k a2 by the collection of data of the model output, G p . Through sensitive analysis of G p with respect to parameters, it t 0 120 90 150 120 300 G p (t) 150 210 240 225 250 215 Table 4. Numerical measurements of G p has been observed that G p is more sensitive with respect to k 1 , k a1 , k e rather than k 2 , k a2 . So, we assign parametric values from Table 3 to k 2 and k a2 , we only estimate the sensitive parameters k 1 , k a1 , and k e are estimated. From Equation (6.5), taking: 386k e − 0.133, β 3 = −9.9 × 10 −6 k 2 + 0.265k a2 + 0.265k a1 .
and similarly adjusting β 4 and β 5 , then G p (t) becomes: The Equation (7.2) yields polynomial regression model We consider the measurement scheme of G p (t) against the values of time T as given in Table( 4) [18].
Then the least square estimatesβ are determined bŷ By using data, Equation (7.4) becomes and,   Table 5. Estimated Values of Sensitive Parameter

Conclusions
(1) Plasma glucose G p and subcutaneous glucose G i are sensitive with respect to all selected parameters k 1 , k 2 , k a1 , k a2 , k e .
(2) Two insulin subcutaneous compartments Q 1a , Q 2 and one plasma insulin compartment I p are sensitive with respect to k a1 .
(3) Insulin subcutaneous compartments Q 1b is not sensitive with respect to all parameters.
(4) The system sensitivity of the model output is also in agreement with the individual sensitivity of the parameters.
(5) The sensitivity analysis signifies that parameters are identified by availability of data for any compartment G p , G i , Q 1a , Q 1b , Q 2 , I p .
(6) Finally, we concluded that model output G p is more sensitive with respect to k 1 , k a1 , k e rather than k 2 , k a2 . Therefore, sensitive parameters k 1 , k a1 , k e are estimated using polynomial regression least square method and found closed to the literature value.
(7) In a future work, we want to implement this method to a more precise and studied model so that the results are more close to the real values.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.