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In this paper, we aim to solve a two compartmental mathematical model of ordinary differential equations for cardiovascular-respiratory system using a new recent method: Perturbation Iteration method. The description of this method for different order derivatives in the Taylor Series expansion is discussed. This method provides the solution in the form of an infinite series for ordinary differential equation. The efficiency of the method used is investigated by a comparison of Euler method and Runge Kutta. Numerical simulations of all these three methods are implemented in Matlab. The validation has been carried out by taking the values of determinant parameters of cardiovascular-respiratory system for a 30 years old woman who is supposed to make practice of three regular physical activities: Walking, Jogging and Running fast. The results are in good agreement with experimental data.

The most important mechanisms leading to following chronic diseases among them there are non-communicable diseases. The development of cardiovascular and metabolic disorders depends on the complex interplay of multiple anatomic and physiologic factors but the mechanism behind these factors and their impacts on the type or degree of the disorder, experimental observations or on the treatment responses remain poorly understood. Therefore, most patients today are being treated with general therapies regardless of the cause of dysfunction. We believe that combining experimental measurements with mathematical modeling will provide important information on the individual key dysfunction, making it possible to start developing personalized therapies. Similarly as observations from data can inspire new theoretical models, the models can translate the measurements first into new ideas, then into testable hypotheses and finally into medical knowledge. Such model-based investigations can therefore provide systematic strategies toward better understanding, predicting or even preventing the disorders.

Naturally human bodies are made in such way that the level of glucose concentration must be maintained in the range of 70 - 120 mg/dl or 3.9 - 6.7 mmol/l. This means that when an individual’s glucose concentration level is found out of that normal range, the said person is judged to have the plasma glucose problems which should be classified in two categories: 1) it can happen Hyperglycemia, this is when the level is greater than 140 mg/dl or 7.8 mmol/l after an Oral Glucose Tolerance Test, or greater than 100 mg/dl or 5.5 mmol/l after a Fasting Glucose Tolerance Test. 2) Alternatively it can happen Hypoglycemia, that is when the level is less than 40 mg/dl or 2.2 mmol/l. On one hand, a prolonged hyperglycemia that is seen to be the major long-term effect of diabetes can cause complications, which may lead to kidney disease, blindness, loss of limbs, and so on. On the other hand, the hypoglycemia can lead to dizziness, coma, or even death. Human body has two main organs that play an important role in regulating blood glycose measurements; those organs are pancreas and liver. During the process of controlling blood glycose, insulin and glucagon are hormones that are mainly involved in that process of controlling glucose. In the pancreas, there are clusters of endocrine cells glucose and insulin. These are the α-cells and the β-cells. The α-cells produce glucagon and the β-cells produce insulin. The pancreas secretes these antagonistic hormones into the extracellular fluid, which then enters the circulatory system and regulates the concentration of glucose in the blood. For biologists, this is known as a simple endocrine pathway. Diabetes Mellitus is an endocrine disorder caused by a deficiency of insulin (Type I Diabetes) or a decreased response to insulin in target tissues (Type II Diabetes) [

Worldwide diabetes disease and human glucose-insulin balance motivated researchers in various directions such as studies on 1) glucose-insulin endocrine metabolic regulatory system [

The current study is framed in six sections. Section one introduces to the audience the motivation of model-based investigation, background on previous research in the domain of dynamical systems related to glucose and insulin and the structure of the paper. In section two, we present the mathematical model of ordinary differential equations. The section three deals with the basic idea of the perturbation-iteration method. The numerical simulation is presented in section four where the comparison is done using Euler method and Runge Kutta methods to test efficiency of perturbation-iteration method. The section five focuses on discussion while section six rounds up and deals with the concluding remarks.

In the work [

To clarify the mathematical model of our study, reference is made to [_{A}) and venous pressure (P_{V}), our mathematical model is made up in such way that they are take into account. In addition blood flows between lungs and heart due to left (Q_{l}) and right (Q_{r}) cardiac output. Consequently, arterial pressure leads the tissues to receive the blood from cardiovascular respiratory system whereas the blood comes to cardiovascular respiratory system from tissues due to the arterial pressure. However, the respiratory control system varies the ventilation rate in response to the levels of dioxide CO_{2} and oxygen O_{2} gases. Consequently, ventilation rate and cardiac output influence each other mutually. Therefore exchanges between LC and PC are controlled by heart rate and alveolar ventilation functions. This mechanism of controlling is not straightforward rather it is represented by outflow functions between systemic arterial and venous compartments that depend on heart rate and alveolar ventilation (

Therefore a nonlinear compartment analysis leads on the following new global model

where the functions

Here physical activity comes in to play the role of maintaining glucose concentration level in a narrow range and to improve insulin sensitivity for the case of Type II diabetes. The numerical simulation is carried on three types of regular physical activity (physical exercise of 30 minutes per day) for a 30 years old woman: Walking, Jogging and Running Fast. We consider the following identified functions f and g as presented in [

Constants δ and σ are set as in [

Perturbation-iteration method has been developed recently by Aksoy and Pakdemirli [

First of all, we discuss the

a vector state. The system first order of K nonlinear ordinary differential equations can be written as follows

where ε is the perturbation parameter and t denotes the independent variable. That is the system

Taking an approximate solution of the system (3) as

where subscript n represents the n’th iteration over this approximate solution, we have a solution with one correction term in the perturbation expansion. The system can be approximated with a Taylor series expansion in the neighborhood of ε = 0

where

is defined for the (n + 1)’th iterative equation

Substituting (6) into (5), we obtain an iteration equation

which is a first order differential equation and can be solved for the correction terms

Note that for a more general algorithm, n correction terms instead of one can be taken in expansion (4) which would then be a

After the discussion of

We consider the general cause of first order of differential equation as

where

where n denotes the n’th iteration over this approximate solution such that for the perturbation parameter ε the expression

Reorganizing Equation (10), we have

where

as integrating factor, our equation is now transformed into the form

so that

that is

Substitution of (12) into (9) and constructing the iteration scheme yields

We consider the rest values of unhealthy woman. Therefore we take 180 mg/dl and 2.1 μU/dl for plasma glucose and insulin respectively. The equilibrium values (values for healthy person) of plasma glucose and insulin are taken as 100 mg/dl and 3.5 μU/dl respectively.

To show the effectiveness of

Taking

where

The system (1)-(2) is solved by using

Setting

with one correction term in the perturbation expansion such that the relation (7) is satisfied for

Walking case | |

Jogging case | |

Running fast case | |

For (15), Equation (7) reduces to

with the solution

and

where

Using the values from

After substituting initial guess in (17) and with the help of Equation (16), the first following approximation has been obtained in applying the iteration formula (18) and (19).

・ Walking case:

・ Jogging case:

・ Running fast case:

To test the efficiency of perturbation iteration algorithm

Exercise intensity | Rest | Walking | Jogging | Running Fast |
---|---|---|---|---|

Ventilation (L/min) | 6 | 8.5 | 15 | 25 |

Heart rate (Beats/min) | 70 | 85 | 140 | 180 |

The heart rate and the alveolar ventilation are two controls of the cardiovascular-respi- ratory system. Consequently they have a great influence in controlling other parameters that flow through this system such as plasma glucose and insulin. Furthermore, the stability of each of those controls at the equilibrium value allows plasma glucose to reach a stabilized state and insulin to be around its sensitivity value. The response of these controls to the plasma glucose and insulin are represented in

We have investigated in this work a new numerical method for solving a system of ordinary differential equations: Perturbation Iteration method. The efficiency of this method is tested using two other convergent methods that are Euler method and Runge- Kutta method. Those all methods are implemented using Matlab packages. The numerical simulations illustrate the responses of plasma glucose and insulin due to the control of heart rate and alveolar ventilation of cardiovascular-respiratory system. The numeri- cal results confirmed the analytical analysis for a 30 years old woman during three different physical activities: Walking, Jogging and Running fast.

Gahamanyi, M., Ntaganda, J.M. and Haggar, M.S.D. (2016) Perturbation-Iteration Method for Solving Mathematical Model of Glucose and Insulin in Diabetic Human during Physical Activity. Open Journal of Applied Sciences, 6, 826- 838. http://dx.doi.org/10.4236/ojapps.2016.612072