Modelling Blood and Pulmonary Pressure for Solving a Performance Optimal Problem for Sportsmen

This paper aims at designing a three-compartmental mathematical model for determining the impact and response of blood pressures on cardiovascular and respiratory parameters. Three nonlinear ordinary differential equations are derived from three compartments. Stability conditions are established and inverse techniques are proposed for identifying model parameters. To test the efficiency of the found model, a validation is achieved based on an existing mathematical model through a comparative study.


Introduction
An important problem of human health is the control of cardiovascular and respiratory system.The goodness knowledge of this control is very useful for improving the diagnostics, treatment of this system.This knowledge by professional people allows them to provide necessary advices to patients.Among those advices, the regular physical activity is given to prevent some diseases of cardiovascular and respiratory system.The determination of heart rate and alveolar ventilation is the question that often arises for controlling the cardiovascular and respiratory system to prevent chronic diseases.For a healthy people, it is well known that the variation of heart rate and alveolar ventilation venous of determining heart rate and alveolar ventilation for controlling systemic arterial and venous blood pressures to prevent cardiac accidents.For a healthy subject, it is well known that the coaching sessions allow the behavior variation of heart rate and alveolar ventilation.
Since 1950 mathematical models for cardiovascular and respiratory systems have been proposed through compartmental theory 1-4 .Recently, a two compartmental model with stable equilibrium states is designed in 5 .A global mathematical model is proposed in 3 .

Arterial systemic compartment
Lungs V However, the equilibrium states of this model are unstable.Again this model does not allow the knowledge of a long term of cardiovascular respiratory system in some cases of physical activity such as aerobic.By using this existing global model, we design a new mathematical model based on inverse techniques to stabilize blood pressures which reach their desired values.This paper is organised as follows.Section 2 presents a three-compartmental model and inverse techniques to determine the unknown constants and functions of model.Section 3 focuses on the estimation of parameters.In Section 4 we present the numerical result for a healthy woman subject who is 30 years old.The concluding remarks are presented in Section 5.

Outline of the Model
Let us consider the system of three compartments Figure 1 : the arterial systemic compartment, the venous systemic compartment, and pulmonary venous compartment.
The exchanges between compartments are controlled by heart rate H and alveolar ventilation VA functions.This control mechanism is not direct and can be represented by outflow functions between two compartments which depend on heart rate alveolar ventilation Figure 1 .Therefore, a nonlinear compartment analysis leads to the following new global model: where a 1 , a 2 , and a 3 are model constants and f 1 , f 2 , and f 3 denote model functions to be identified.The functions P as , P vs , and P vp denote, respectively, mean blood pressures in systemic arterial region, in systemic venous region, and in venous pulmonary region.Equation 2.1 arises from straightforward development of mass balance between three compartments.They are obtained by using Fick's law, Boyle's law relating the concentration of the gas in the solution to the partial pressure.

Stability Analysis
Let us set The system 2.1 can be written as follows: Ẋ G X t ; q t .

2.4
We have the following results.
Theorem 2.1.For initial state X 0 P 0 as , P 0 vs , P 0 vp , T , it exists t 1 ≥ 0 such that it passes a C 1 unique maximal solution of 2.1 : that satisfies the following condition: Since the pressures P as , P as , and P vs are positive, by solving the system 2.7 we get the equilibrium state given as follows: The Jacobian matrix of the system becomes where

2.11
The eigenvalues of this matrix are solution of the following characteristic equation: where

2.13
Thereafter we set Therefore, 2.12 becomes Consequently we have the following result.
then the mathematical model 2.1 has only one equilibrium state that is asymptotically stable.
We can refer to 6, 7 , for the proof of Proposition 2.2.It is should be mentioned that the bifurcation analysis technique may predict the existence of the Hopf bifurcation at parameter values where the equilibrium loses its stability and stable periodicals solutions exist if the value of parameter increases.In this work we mainly focus our attention on the identification of the model parameters leading to asymptotically stable solutions.

Computing Model Parameters
By considering the assumption 2.14 made on k 1 , k 2 , and k 3 the ordinary differential system 2.1 becomes

3.1
Let us take where

3.3
Let P as u , P vs u , P vp u be the solution of 3.1 that corresponds to parameter vector u.The identification problem can be formulated as follows.
Find For solving the system 2.1 , we approximate it by assuming the following system:

3.9
where with {ψ N i } N i 0 the set of B-splines linear functions defined on the interval 0, T and that satisfy 11 and where we have set

3.12
It is important to point out to the problem 3.4 that is ill-posed.Therefore, we need the regularization methods to make it wee posed.One method we can use is the Tikhonov regularization 8 .Therefore, let us consider

3.13
The minimization problem can be formulated as follows.Find 3.14 solution of least square problem where and where J u is the relation 3.5 , λ denotes the regularization parameter to be chosen while u † represents a additional information on u * .The choice technique of λ can be found in 8 .We are only focusing on how to use this technique for determining it.For that let us consider u λ solution of the following minimization problem:

3.18
The value of the Tikhonov regularization is given by which is calculated from the following iterative scheme: Now, we are interested in coefficients and functions identification of mathematical model.After that we focus on validation of identified model and stabilization of parameter around their equilibrium states.

3.19
For a healthy woman in physical activity the parameters of cardiovascular respiratory system can reach the values given by the Table 1 5 .
The results obtained from the mathematical model of Kappel et al. 2 show that for an untrained woman in physical activity such as jogging, the heart rate and alveolar ventilation slightly vary 5 .We consider the parameters illustrated in the Figure 2 as observed values.
By using these parameters we have the observed solutions of the model 2.1 shown in the Figure 3.
By considering the sequence λ k given in the relation 3.20 we have the curve shown in Figure 4.
The value of λ is given by the quotient of abscissa by ordinate for the point P c in the Figure 4.It yields that λ 0.0964.
By using the parameter an considering a sequence of data in rough and perturbed data of order δ 0.01, the problem 3.15 has one unique solution.Table 2 presents the constants.Figure 5 shows the functions f 1 ,f 2 , and f 3 to be identified according to the observed values of heart rate H t k and alveolar ventilation VA t k at each time t k .
The technique of smoothing gives the following explicit forms for the functions f 1 , f 2 , and f 3 .1 Walking case: A × H 6.0647 25.1458.

Test Results
To test our models we consider the acute respiratory response for a healthy trained woman who is 30 years old whose mean values are given in Table 1.The autoregulation process states that the cardiovascular and respiratory systems evolves in the optimal way toward these values.This allows us to solve the following optimal control problem:      The Figures 6, 8, and 10 represent the variation of heart rate and alveolar ventilation for a woman in physical activity in the case of walking, jogging and running respectively.
One minute after the beginning of the physical activity, those variations reach the wished value and they are subjected to the small variations around this value.This leads to cardiovascular respiratory system to fit to physical activity such that they supply them with the energy and oxygen and eliminate the off cuts that are results of contracting lactates and dioxide of carbon .
The heart rate and alveolar ventilation play a crucial role in controlling the cardiovascular respiratory system.Their variations have a general influence on other parameters and in particular on arterial systemic pressure, venous systemic pressure, and pulmonary venous pressure.The response of this influence on these parameters is illustrated respectively in the Figures 7, 9, and 11.
In the case of walking and running, one minute after the beginning of physical activity, the arterial systemic pressure is above the wished value.After this time, it is stabilized on this value Figures 7 and 11 a .In the jogging case, this parameter reaches the wished value one minute after the beginning of physical activity.After this time it oscillates around this value Figure 9 a .
Regarding the venous systemic pressure in the walking case, it is stabilised to the wished value at sixth minute Figure 7 b .The decreasing of this parameter appears also in the case of jogging Figure 9 b where it reaches 2.8 mmHg pressure which is below the wished value 3.28 mmHg .Between 2 and 3 minutes it is constant.After this time it increases and it is stabilised to wished value.The previous situation does not appear in the running case Figure 11 b because at the beginning of physical activity, the pulmonary venous pressure increases to reach 3.8 mmHg which is maximum value in 2.5 minutes.After this time it waits for the seventh minute to decrease slightly and to be stabilised to wished value.
In the jogging case, the pulmonary venous pressure decreases but one minutes after the beginning of physical activity it increases to stabilise to wished value Figure 9 c .In the walking case, the pulmonary venous pressure before stabilizing to the wished value between the sixth and eighth minute, after the beginning of physical activity Figure 7 c .This behaviour of pulmonary venous pressure at the beginning of physical activity does not appear in the running case.Therefore, between the third and seventh minutes it oscillates around 9 mmHg.After this time it is stabilised to the wished value.

Concluding Remarks
In this work we have investigated a mathematical model that describes blood and pulmo nary pressure.We have verified that the optimal functions of cardiovascular and respiratory system depend on stability of its control heart rate and alveolar ventilation .The cardiovascular and respiratory system is comprised of a multitude factors.The increasing requires the meaning of those measurable controls and blood and pulmonary pressure by considering the actions of both physiological and pathological conditions.Consequently, those conditions allow the investigation of a simple models that should be able to describe exactly the mechanical behaviour of cardiovascular respiratory system.A qualitative study leads to determine constraint equations on estimated parameters.The mathematical model is tested by using the determinant constants to get satisfactory results.Consequently, this model can be useful to control and stabilise certain parameters of cardiovascular and respiratory system to ensure their performance.

Figure 2 :
Figure 2: The observed values of heart rate a and alveolar ventilation b during jogging for a woman who is 30 years old.

Figure 3 :
Figure 3: Arterial systemic pressure a , venous systemic pressure b and pulmonary venous pressure c .Dashed line denotes the observed values while solid line illustrates the solution of the model 2.1 .

Figure 4 :
Figure 4: Curve in L form where F denotes the numerator of the relation 3.20 and L represents its denominator.

Figure 6 :
Figure 6: Variation dashed line of optimal heart rate a and optimal alveolar ventilation b for a woman in physical activity and their corresponding wished values solid line in the walking case.

Figure 7 :
Figure 7:The variation dashed line of optimal arterial systemic pressure a , optimal venous systemic pressure b and optimal pulmonary venous pressure c and their corresponding wished values solid line in walking case.

Figure 8 :
Figure 8: Variation dashed line of optimal heart rate of rythm a and optimal alveolar ventilation b for a woman in physical activity and their corresponding wished values solid line in jogging case.

Figure 9 :Figure 10 :
Figure9: The variation dashed line of optimal arterial systemic pressure: a optimal venous systemic pressure, b , optimal pulmonary venous pressure, c and their corresponding wished values solid line in jogging case.

Figure 11 :
Figure11: The variation dashed line of optimal arterial systemic pressure: a optimal venous systemic pressure, b optimal pulmonary venous pressure, c and their corresponding wished values solid line in running case.
Diagram of three-compartmental model of human cardiovascular respiratory system.f 1 , f 2 , and f 3 are functions depend on heart rate H and ventilation alveolar VA .The states variables are arterial systemic pressure P as , venous systemic pressure P vs , and venous pulmonary pressure P vp .

6
Let us take H e , VA e , P as e , P vs e , and P vp e the equilibrium parameters of the system 2.1 .
as u t k , P vs u t k , P vp u t k is solution at the time t k of the system 2.1 corresponding to the parameter vector u, and where P δ as obs t k , P δ vs obs t k , P δ vp obs t k 3.6 are observed values of variables P as t ,P vs t , P vp t at time t k .The measures of those observed values are obtained by taking the the order error δ such that

Table 1 :
Optimal parameters of cardiovascular and et respiratory system for a woman in physical activity.

Table 2 :
Table of constants got from the Tikhonov regularization.