Glucose is essential for human life and its concentration in the blood is strictly regulated. Alterations in blood glucose are a sign of numerous diseases and several clinical tests are used for diagnosis.

The IVGTT test is a clinical exam that analyzes two important parameters such as the efficacy of glucose and insulin sensitivity. Using a mathematical model, such as the minimum glucose model, it is possible to undertake both prediction and identification studies.

Through the prediction it is possible to identify, starting from the knowledge of the parameters, the temporal evolution of the blood sugar. Once the accuracy of the model has been verified, it is also possible to start from the data of an IVGTT test and trace the estimate of the glucose effectiveness and insulin sensitivity parameters.

The following report is an extract from a project carried out for the course of Modeling and Simulation of Physiological Systems, held for Medical Engineering at the University of Rome Tor Vergata.

## Introduction

Glucose is essential for the life of human beings. It turns out to be the main energy source for most living organisms providing up to 285 kilocalories per mole. The human body ensures a tight regulation of glucose homeostasis by keeping blood sugar levels as constant as possible. These values ​​are altered by various stimuli and the integration of numerous control mechanisms leads to the reestablishment of physiological conditions.

Of these systems the most important are regulated by the pancreas and liver. The liver has a rather limited ability to store glucose and any excess carbohydrates are converted into fat and deposited in the adipose tissue. Starting from the intestinal walls, after ingestion, glucose is absorbed and routed to the central nervous system and then to the liver and muscles, where it is accumulated as a glycogoneum.

The interaction of the endocrine part of the pancreas with the liver and other systems provides for a tight regulation of blood glucose and this process can be described through various mathematical models. Following is an analysis of the minimum glucose model and the various possibilities it offers such as the prediction of blood glucose evolution over time, for example in an IVGGT test, or the identification of parameters of clinical interest such as glucose effectiveness or insulin sensitivity.

## Background

Physiological regulation of glucose is very complex. The body integrates different control systems and pathways to maintain blood glucose levels, away from meals, at a physiological value of 80 ÷ 90 mg/dl. Blood sugar is the measure of the concentration of glucose in the blood. It is strongly regulated by the body and is one of the most important parameters of the body’s homeostasis.

Normally, glucose is stored within the muscles and in the liver in the form of glycogen. This deposit therefore allows to keep the glycaemia constant through different control pathways. Furthermore, the blood glucose value can vary considerably in the presence of strong stimuli, for example after a rich meal it can grow up to 200 ÷ 250 mg/dl.

The growth of plasma glucose levels leads to the activation of numerous mechanisms. The most important mechanism, for restoring rest values, follow the release of insulin into the blood by the beta cells of the pancreas. Several cascade mechanisms are triggered that lead to two main actions:

• Increase in the metabolic consumption of glucose by the body’s cells
• Stimulation of glucose uptake in the liver with consequent accumulation in the form of glycogen

The presence of the opposite stimulus, such as the reduction of blood sugar, instead leads to the activation of the pancreatic cells which in turn release glucagon. This hormone in turn stimulates the liver to release glucose and there will be an increase in the plasma level of glucose [2]. Overall, there are different catabolic hormones such as glucagon, cortisol and catecholamines which lead to an increase in blood sugar but only one anabolic hormone, insulin [3].

### Pathophysiology

These mechanisms are altered in some pathologies. Hyperglycemia is when the blood glucose level remains too high. Maintaining hyperglycemic conditions for a long time can lead to various health problems such as heart disease, cancer, eye and kidney problems.

Glucose levels above 300 mg/dl lead to fatal damage due to ketoacidosis. A characteristic pathology is diabetes mellitus where there is an alteration in different points: in type I it is the beta cells that are unable to start the release of insulin, in type II, however, insulin is not effective.

Even low blood glucose levels can lead to different problems such as lethargy and alteration of some neurological peculiarities such as irritability, weakness and loss of consciousness. We talk about hypoglycemia when the level is below 40 mg/dl.

### Clinic

In this regard, in the clinic, a specific patient test is used, the IVGTT [4]. This test, under the name of intra venous glucose tolerance test, allows to estimate two parameters of interest such as:

• glucose effectiveness, glucose itself has an effect on blood sugar reduction
• insulin sensitivity, measures the effectiveness of insulin in reducing plasma glucose levels

It contrasts with faster tests such as the OGTT oral test and has significantly greater precision for the first phase (acute phase) of insulin release. Glucose is injected by vein and this allows to avoid the digestive system and therefore to have a greater sensitivity.

When 𝛽-pancreatic cells release insulin into the portal vein, it passes into the liver and is partially eliminated before entering the circulation. Furthermore, the clearance rate of hepatic insulin can vary due to both physiological and pharmacological mechanisms. Thus, the insulin concentration measured in the peripheral vessels can be markedly different from that released by the pancreas. The IVGTT test requires a mathematical model to estimate parameters and the glucose minimal model is often used to estimate pancreatic response and insulin sensitivity.

It should also be noted that the proper conduct of an IVGTT test requires qualified personnel who are also able to apply mathematical modeling [5]. The test can last several hours and the concentration due to the release of glucose and the concentration of insulin or, alternatively, the C-peptide are collected. The following sections do not analyze the C-peptide models but consider the insulin concentration measured in the blood.

### Glucose minimal models

The minimum glucose model is a mathematical model that allows you to describe the evolution of plasma glucose concentration.

Like any model, a compromise must be found between accuracy, reliability, computational cost and the number of parameters to be estimated in order to use it.

This model, as indicated by the name, uses the minimum number of parameters possible in order to have a sufficiently accurate description to allow the evaluation of the two clinically interesting quantities, such as glucose effectiveness and glucose sensitivity. This model was introduced by Bergman and Cobelli [6] and provides for the subdivision into two compartments (fig. 3), represented by two differential equations:

$$\begin{cases}\frac{d G}{d t}=S_{g}\left(G_{b}-G\right)-X G, & \mathrm{G}(0)=G_{0} \\ \frac{d X}{d t}=k\left[S_{i}\left(I-I_{b}\right)-X\right], & \mathrm{X}(0)=X_{0}\end{cases}$$

#### Compartment of glucose

The first compartment represents the time-varying plasma glucose concentration 𝐺 (𝑡). It is affected by the glucose concentration itself in two different ways. Through the glucose effectiveness 𝑆𝑔 there is a direction of growth proportional to the difference between the basal concentration and the current one, it should be observed that this difference becomes negative so that the more the gap increases, compared to the basal concentration, the more the glucose will tend to decrease. The second term predicts a growth rate inversely proportional to the effective insulin concentration 𝑋 (𝑡).

This quantity is not directly measurable but is related to the second compartment and in turn has a growth rate influenced by two parameters. There is a negative feedback from its own concentration and a feedback proportional to the difference between the current and basal insulin concentration, through a proportionality coefficient 𝑘⋅𝑆𝑖, therefore linked to insulin sensitivity 𝑆𝑖. The growth rate will be positive as the insulin concentration increases.

This second compartment takes into account the fact that the level of insulin in the plasma reaches a plateau around 90 𝜇U/ml and during its evolution the mobility of glucose increases.

Using this model, it is possible to trace the parameters of clinical interest, such as glucose effectiveness and insulin sensitivity, starting from the data of an IVGTT test. Alternatively, if you know the descriptive parameters, it is possible to predict the trend of the model’s concentration. Both possibilities are then analyzed using Matlab and Simulink [7].

## Prediction of glucose levels

For the prediction we considered the parameters in Tab. 1. The test data can be taken from clinical data or generated through Simulink, implementing the model with coefficients known in the literature [8].

### ODE solution

Starting from the insulin values, sampled and present in the data in fig. 4b it is possible to implement and solve the model using a differential equation solver, ode45(). The algorithm is based on the explicit Runge-Kutta method of order 4 and 5 thus requiring only the solution in the previous step, beyond the initial parameters, and solving in a single time step [9].

For implementation, Matlab requires you to pass differential equations and parameters to ode45(). The equation is passed through odefcn() while the parameters are joined within an array (fig. 5) In addition, the initial values ​​and the time vector are passed.

For the time vector it is advisable to pass only the initial and final instant, leaving the algorithm free to choose the step, so as to have a more accurate result. This time vector is part of the experimental data also containing the glucose samples, in fig. 4a, used for comparison, and of insulin, in fig. 4b, used in prediction.

#### Mathematical model

The mathematical model in eq. (1) is then inserted inside odefcn(), in fig. 6, with the foresight to guarantee the insulin value 𝐼 (𝑡) at every moment in time. To do this, a linear interpolation is used on the sampled values, reading the value at the same instant in time used in the solution of the differential equation system. The model, once solved, provides the curve 𝐺 (𝑡) which can be compared with the sampled values (fig. 4a).

The model allows you to predict values at any time and in particular with an average error, with the exception of the first sample where there is a strong difference of 6.6% compared to the available samples.

parameters =[Sg ,Gb ,k,Ib ,Si];
[t,y] = ode45(@(t,y) odefcn(t,y,insuline ,time ,parameters), [time (1), time(end)],[G0 ,x0]);
function dydt=odefcn(t,y,insuline,time,parameters) Sg=parameters(1);
Gb=parameters(2); k=parameters(3);
Ib=parameters(4); Si=parameters(5);
% y(1) = G(t); % y(2)= X(t)
% dGdt=Sg (Gb-G(t))-X(t)*G(t);
% dXdt=k*(Si*(I(t)-Ib)-X(t))
% The solver need I(t) at every t wtih linear interpolation
I_inter=interp1(time,insuline,t); dydt(1)=Sg*(Gb-y(1))-y(2)*y(1);
dydt(2)=k*(Si*(I_inter-Ib)-y(2)); dydt=dydt ’;
end

The same system of differential equations can be implemented in Simulink. It remains necessary to provide the descriptive function of the time course of insulin and to do this you can use a Matlab function block by doing linear interpolation.

Sample data and parameters are loaded via an .m file. To distinguish the two compartments it is also useful to use masks and sub-systems, grouping the relative components. In other words, we pass from an extended model (fig. 7a) to a compact model (fig. 7b) and observe how the two compartments are linked precisely by the product 𝑋 (𝑡) 𝐺 (𝑡).

## Parametric identification

Reasoning less from a simulation perspective and focusing more on clinical aspects, it is possible to use this model also for diagnoses. That is, it is possible to approach a parametric identification of parameters of clinical interest such as glucose effectiveness and insulin sensitivity.

In particular, the curves 𝐺 (𝑡) and 𝐼 (𝑡) are assumed (for example from the IVGTT test) together with the basal glucose concentration 𝐺𝑏 and insulin 𝐼𝑏 and the initial condition 𝑋0 and identify the four missing parameters:

$$\mathbf{P}=\left\{S_{g}, S_{i}, G_{0}, k\right\}$$

Parametric identification is nothing more than the attempt to cancel the error function, given by the difference between the known values and the prediction of the model with the estimated parameters:

$$\underline{e}(\mathbf{P})=g l u c o s e\left[t_{i}\right]-f\left(t_{i}, \mathbf{P}\right)$$

That is, it is a minimization problem of the objective function defined as the square norm of this error function.

### Minimization

To implement numerical minimization we use lsqnonlin(), an algorithm in Matlab for the solution of nonlinear least squares problems [10]. The algorithm uses the LevenbergMarquardt method, an algorithm that lies in the middle between the Gauss-Newton algorithm and the gradient descent method. The optimization algorithm requires an initial guess and values known in the literature are provided by Pacini and Bergman [8].

The algorithm setting, present in fig. 9, recalls the objective function present within @ 𝚘𝚙𝚝𝚏𝚞𝚗 (). Within this function, the known quantities are recalled and the error function is defined as the difference between the glucose samples and the result of the numerical model calculated by passing the parameters updated each iteration to the ode45 solver.

p_init =[0.399e-1, 0.2e-1,0.4e-4 ,0.287e3];
options = optimoptions(’lsqnonlin’,’Display’,’iter’);
options.Algorithm = ’levenberg -marquardt’;
options.StepTolerance=1e-8;
options.PlotFcn=’optimplotresnorm’;
options.Display=’iter -detailed’;
[x,resnorm ,residual ,exitflag ,output ,lambda ,jacobian] = lsqnonlin(@optfcn ,p_init ,[],[],options );
disp([’Error: ’,num2str (100*( p_true -x)./p_true),’%’])

The results is shown in fig. 8a where the curves are quite close despite the error on the individual parameters can also be high. The numerical results are present in Tab. 2. The results are not convincing.

A slightly better estimate can be obtained by observing that experimentally the first sample of the glucose concentration is considerably lower than the expected 𝐺0. Neglecting this data, a trend closer to the experimental data is obtained (Fig. 8b).

Despite this, the numerical estimate indicated in the table as 𝚂𝚝𝚒𝚖𝚊𝚝𝚒 (𝚋) still remains far from the expected values.

## Conclusions

The implementation of the minimum glucose model allows you to correctly predict the trend in blood glucose concentration.

This model requires the estimation of descriptive parameters such as glucose effectiveness and insulin sensitivity which can be estimated starting from the results of an IVGTT test. This estimate, being a non-linear problem, is strongly affected by the initial guess and the data available for parametric identification.

The estimation of the parameters could lead to a qualitatively representative curve of the samples but the numerical result on the estimation of the parameters is not convincing.

## Code availability

The code is available at the project’s GitHub repository.

## References

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