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

Abstract

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.

1. 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]. 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.

2. Model Design

2.1. 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.

Figure 1 

Diagram of three-compartmental model of human cardiovascular respiratory system. 𝑓1,𝑓2, and 𝑓3 are functions depend on heart rate 𝐻 and ventilation alveolar 𝑉𝐴̇. The states variables are arterial systemic pressure (𝑃as), venous systemic pressure (𝑃vs), and venous pulmonary pressure (𝑃vp).

The exchanges between compartments are controlled by heart rate (𝐻) and alveolar ventilation (𝑉_𝐴̇) 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: 𝑑𝑃_as𝑑𝑡=−𝑘¹𝑑𝑃_vs𝑃_as+𝑃_vp^𝑎1𝑓₁𝐻,𝑉_𝐴̇,𝑑𝑡=−𝑘²𝑑𝑃_vs+𝑃_as^𝑎2𝑓₂𝐻,𝑉_𝐴̇,𝑑𝑡𝑃^vp=−𝑘₃𝑃_vp+𝑃_vs^𝑎3𝑓₃𝐻,𝑉_𝐴̇,(2.1) where 𝑎₁,𝑎₂, and 𝑎₃ are model constants and 𝑓₁,𝑓₂, and 𝑓₃ 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.

2.2. Stability Analysis

Let us set 𝑞=𝑋𝑘=₁𝑃_as,𝑃_vs,𝑃_vp^𝑇,,𝑘₂,𝑘₃,𝑎₁,𝑎₂,𝑎₃^𝑇,(2.2)𝐺(𝑋;𝑞)=−−𝑘𝑘₁₃𝑃_as+𝑃_vp^𝑎1𝑓₁𝐻,𝑉_𝐴̇,−𝑘₂𝑃_vs+𝑃_as^𝑎2𝑓₂𝐻,𝑉_𝐴̇,𝑃_vp+𝑃_vs^𝑎3𝑓₃𝐻,𝑉_𝐴̇^𝑇.(2.3)

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 𝑋₀=(𝑃⁰_as,𝑃⁰_vs,𝑃⁰_vp,)^𝑇, it exists 𝑡₁≥0 such that it passes a 𝐶¹ unique maximal solution of (2.1): 𝑋∶0,𝑡₁⟶ℝ³(2.5) that satisfies the following condition: 𝑋(0)=𝑋₀.(2.6)

Let us take 𝐻_𝑒, 𝑉̇_𝐴𝑒, 𝑃_as𝑒, 𝑃_vs𝑒, and 𝑃_vp𝑒 the equilibrium parameters of the system (2.1). Hence, we have −−𝑘𝑘₁₂𝑃_as𝑒+𝑃_vp𝑒^𝑎1𝑓_𝑒1=0,−𝑘₃𝑃_vs𝑒+𝑃_as𝑒^𝑎2𝑓_𝑒2=0,𝑃_vp𝑒+𝑃_vs𝑒^𝑎3𝑓_𝑒3=0,(2.7) where we have set 𝑓_𝑒𝑖=𝑓_𝑖𝐻_𝑒,𝑉̇_𝐴𝑒𝑖=1,2,3.(2.8) 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:𝑃_as𝑒=1𝑘₁𝑘^𝑎3𝑎1₂𝑘^𝑎1₃𝑃_vs𝑒𝑓^𝑎3𝑎1_𝑒2𝑓^𝑎1_𝑒3𝑓_𝑒1^{1/(1−𝑎3𝑎2𝑎1)},=1𝑘^𝑎2₁𝑘^𝑎1𝑎2₃𝑘₂𝑃_vp𝑒𝑓^𝑎1𝑎2_𝑒3𝑓^𝑎2_𝑒1𝑓_𝑒2^{1/(1−𝑎3𝑎2𝑎1)},=1𝑘^𝑎2𝑎3₁𝑘^𝑎3₂𝑘₃𝑓^𝑎2𝑎3_𝑒1𝑓^𝑎3_𝑒2𝑓_𝑒3^{1/(1−𝑎3𝑎2𝑎1)}.𝑎³𝑎₂𝑎₁≠1(2.9) The Jacobian matrix of the system becomes 𝒥_𝑒=𝜕𝐺𝜕𝑋𝑋^𝑒;𝜆_𝑒=⎡⎢⎢⎢⎢⎣−𝑘¹0𝐵₁𝐵0₂−𝐵𝑘₂₃0−𝑘₃⎤⎥⎥⎥⎥⎦,(2.10) where 𝐵𝐵₁₂=𝑎₁𝑓^𝑎3_𝑒2𝑓_𝑒3𝑓^𝑎2𝑎3_𝑒1^{(𝑎1−1)/(1−𝑎3𝑎2𝑎1)}𝑓_𝑒1,𝐵₃=𝑎₂𝑓^𝑎2_𝑒3𝑓^𝑎1𝑎3_𝑒2𝑓_𝑒1^{(𝑎2−1)/(1−𝑎3𝑎2𝑎1)}𝑓_𝑒2,=𝑎₃𝑎₂𝑓^𝑎1𝑎2_𝑒3𝑓_𝑒2𝑓^𝑎2_𝑒1^{(𝑎3−1)/(1−𝑎3𝑎2𝑎1)}𝑓_𝑒3.(2.11) The eigenvalues of this matrix are solution of the following characteristic equation: 𝜆³+𝐶₁𝜆²+𝐶₂𝜆+𝐶₃=0,(2.12) where 𝐶₁=−³_𝑖=1𝑘_𝑖,𝐶₂=−𝑘₁𝑘₂+𝑘₂𝑘₃+𝑘₁𝑘₃,𝐶₃=−³_𝑖=1𝑘_𝑖+𝑎₁𝑎₂𝑎₃.(2.13) Thereafter we set 𝑘₁=𝑘₂=𝑘₃=1.(2.14) Therefore, (2.12) becomes −(𝜆+1)³+𝑎₁𝑎₂𝑎₃=0.(2.15) Consequently we have the following result.

Proposition 2.2. If 0<𝑎₁𝑎₂𝑎₃<1(2.16) 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.

3. Computing Model Parameters

By considering the assumption (2.14) made on 𝑘₁,𝑘₂, and 𝑘₃ the ordinary differential system (2.1) becomes 𝑑𝑃_as𝑑𝑡=−𝑃^as𝑑𝑃_vs+𝑃_vp^𝑎1𝑓₁𝐻,𝑉_𝐴̇,𝑑𝑡=−𝑃^vs𝑑+𝑃_as^𝑎2𝑓₂𝐻,𝑉_𝐴̇,𝑑𝑡𝑃^vp=−𝑃_vp+𝑃_vs^𝑎3𝑓₃𝐻,𝑉_𝐴̇.(3.1) Let us take 𝑡_𝑘=𝑘Δ𝑡,𝑘𝑢=0,…,𝑁,withΔ𝑡afixedstepoftime,=𝑎₁,𝑎₂,𝑎₃,𝑎₄,𝑓₁,𝑓₂,𝑓₃^𝑇,(3.2) where 𝑓𝑓₁₂=𝑓₁𝑡₀,…,𝑓₁𝑡_𝑁^𝑇,𝑓₃=𝑓₂𝑡₀,…,𝑓₂𝑡_𝑁^𝑇,=𝑓₃𝑡₀,…,𝑓₃𝑡_𝑁^𝑇.(3.3) Let (𝑃_as𝑢,𝑃_vs𝑢,𝑃_vp𝑢) be the solution of (3.1) that corresponds to parameter vector 𝑢. The identification problem can be formulated as follows.

Find 𝑢^∗=(𝑎^∗1,𝑎^∗2,𝑎^∗3,𝑎^∗4,𝑓^∗1,𝑓^∗2,𝑓^∗3)^𝑇 solution of a least square problem 𝐽𝑢^∗=min_𝑢𝐽𝑢,(3.4) where 𝐽𝑢=^𝑁_𝑘=0𝑃^as𝑢𝑡_𝑘−𝑃^𝛿_asobs𝑡_𝑘²+𝑃_vs𝑢𝑡_𝑘−𝑃^𝛿_vsobs𝑡_𝑘²+𝑃_vp𝑢𝑡_𝑘−𝑃^𝛿_vpobs𝑡_𝑘²,(3.5) and where (𝑃_as𝑢(𝑡_𝑘),𝑃_vs𝑢(𝑡_𝑘),𝑃_vp𝑢(𝑡_𝑘)) is solution at the time 𝑡_𝑘 of the system (2.1) corresponding to the parameter vector 𝑢, and where 𝑃^𝛿_asobs𝑡_𝑘,𝑃^𝛿_vsobs𝑡_𝑘,𝑃^𝛿_vpobs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 𝑃𝑃^𝛿_{asobs𝛿vsobs}=𝑃^𝛿_asobs𝑡₀,…,𝑃^𝛿_asobs𝑡_𝑁^𝑇,𝑃_𝛿vpobs=𝑃^𝛿_vsobs𝑡₀,…,𝑃^𝛿_vsobs𝑡_𝑁^𝑇,=𝑃^𝛿_vpobs𝑡₀,…,𝑃^𝛿_vpobs𝑡_𝑁^𝑇,(3.7) with 𝑃^𝛿_asobs𝑡₀=𝑃_as₀,𝑃^𝛿_vsobs𝑡₀=𝑃_vs₀,𝑃^𝛿_vpobs𝑡₀=𝑃_vp₀.(3.8) For solving the system (2.1), we approximate it by assuming the following system: 𝑃𝑃̇̇_asvs=−𝑃_as+𝑃_vp^𝑎1𝑓^𝑁1,𝑃̇_vp=−𝑃_vs+𝑃_vp^𝑎2𝑓^𝑁2,=−𝑃_vp+𝑃_vs^𝑎3𝑓^𝑁3,(3.9) where 𝑓^𝑁𝑗(𝑡)=^𝑁_𝑙=0𝑓_𝑗𝑙𝜓^𝑁𝑙(𝑡),𝑗=1,2,3,(3.10) with {𝜓^𝑁𝑖}^𝑁𝑖=0 the set of B-splines linear functions defined on the interval [0,𝑇] and that satisfy 𝜓^𝑁𝑖𝑡_𝑘=𝛿_𝑖𝑘,(3.11) and where we have set 𝑓_𝑗𝑘=𝑓_𝑗𝐻𝑡_𝑘,𝑉_𝐴̇𝑡_𝑘,𝑗=1,2;𝑘=0,…,𝑁.(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 𝑓𝑓^∗1_∗2=𝑓^∗1𝑡₀,…,𝑓^∗1𝑡_𝑁^𝑇,𝑓_∗3=𝑓^∗2𝑡₀,…,𝑓^∗2𝑡_𝑁^𝑇,=𝑓^∗3𝑡₀,…,𝑓^∗3𝑡_𝑁^𝑇.(3.13) The minimization problem can be formulated as follows.

Find 𝑢^∗=𝑎^∗1,𝑎^∗2,𝑎^∗3,𝑓^∗1,𝑓^∗2,𝑓^∗3^𝑇(3.14) solution of least square problem 𝐽^𝜆𝑢^∗=min_𝑢𝐽^𝜆𝑢,(3.15) where 𝐽^𝜆𝑢=𝐽𝑢+𝜆‖‖𝑢−𝑢^†‖‖_ℝ3(𝑁+2),(3.16) 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: min_{0<𝑢𝜆<∞}𝐽_𝜃(𝜆)=log^𝑁_𝑘=0𝑃^as𝑢𝜆𝑡_𝑘−𝑃_as𝑡_𝑘²+𝑃_vs𝑢𝜆𝑡_𝑘−𝑃_vs(𝑡_𝑘)²+𝑃_vp𝑢𝜆𝑡_𝑘−𝑃_vp𝑡_𝑘²+log‖‖𝑢^𝜆−𝑢^†‖‖^2ℝ_3(𝑁+2),(3.17) subject to (3.9). Let us set 𝜇=^𝑁_𝑘=0‖‖‖𝑃^as𝑢̂𝜆𝑡_𝑘−𝑃_as𝑡_𝑘‖‖‖+‖‖‖𝑃_vs𝑢̂𝜆𝑡_𝑘−𝑃_vs𝑡_𝑘‖‖‖+‖‖‖𝑃_vs𝑢̂𝜆𝑡_𝑘−𝑃_vs𝑡_𝑘‖‖‖.(3.18) The value of the Tikhonov regularization is given by 𝜆̂=𝜇‖‖𝑢_̂𝜆−𝑢^†‖‖_ℝ3(𝑁+2),(3.19) which is calculated from the following iterative scheme: 𝜆_𝑘+1=𝜇^𝑘‖‖𝑢_𝜆𝑘−𝑢^†_𝜆𝑘‖‖_ℝ3(𝑁+2),𝑘=0,…,(3.20) where 𝜇_𝑘=^𝑁_𝑘=0‖‖‖𝑃^{as𝑢𝜆𝑘}𝑡_𝑘−𝑃_as𝑡_𝑘‖‖‖+‖‖‖𝑃_{vs𝑢𝜆𝑘}𝑡_𝑘−𝑃_𝑣𝑠𝑡_𝑘‖‖‖+‖‖‖𝑃_{vp𝑢𝜆𝑘}𝑡_𝑘−𝑃_vp𝑡_𝑘‖‖‖.(3.21)

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.

For a healthy woman in physical activity the parameters of cardiovascular respiratory system can reach the values given by the Table 1 [5].

Table 1 

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

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.

(a)

(b)

Figure 2 

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

By using these parameters we have the observed solutions of the model (2.1) shown in the Figure 3.

(a)

(b)

(c)

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).

By considering the sequence (𝜆_𝑘) given in the relation (3.20) we have the curve shown in Figure 4.

Figure 4 

Curve in L form where F denotes the numerator of the relation (3.20) and 𝐿 represents its denominator.

The value of 𝜆̂ is given by the quotient of abscissa by ordinate for the point 𝑃_𝑐 in the Figure 4. It yields that 𝜆̂=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 𝑓₁,𝑓₂, and 𝑓₃ to be identified according to the observed values of heart rate 𝐻(𝑡_𝑘) and alveolar ventilation 𝑉_𝐴̇(𝑡_𝑘) at each time 𝑡_𝑘.

Table 2 

Table of constants got from the Tikhonov regularization.

(a)

(b)

(c)

Figure 5 

Identified functions 𝑓1,𝑓2, and 𝑓3.

The technique of smoothing gives the following explicit forms for the functions 𝑓₁, 𝑓₂, and 𝑓₃.(1)Walking case: 𝑓₁𝐻,𝑓𝑉_𝐴̇≈−55.1166𝑉_𝐴̇−0.0138²+8.7797𝐻×𝑉_𝐴̇−2259.4624,₂𝑓𝐻,𝑉_𝐴̇≈0.0046(𝐻−7.6765)²+𝑉̇^{𝐴−3.2718}−21.5285,₃𝐻,𝑉_𝐴̇≈−0.0015𝑉^2𝐴̇×(𝐻+6.0647)+25.1458.(3.22)(2)Jogging case: 𝑓₁𝐻,𝑓𝑉_𝐴̇≈−74.0790𝑉_𝐴̇+0.0201²+9.1872𝐻×𝑉_𝐴̇−2457.4610,₂𝑓𝐻,𝑉_𝐴̇≈0.0011(𝐻−3.3702)²+𝑉̇^{𝐴−0.5162}−15.1199,₃𝐻,𝑉_𝐴̇≈−0.0010𝑉^2𝐴̇×(𝐻+6.0980)+11.5553.(3.23)(3)Running case: 𝑓₁𝐻,𝑓𝑉_𝐴̇≈−62.3280𝑉_𝐴̇−0.0118²+9.1371𝐻×𝑉_𝐴̇−1959.5886,₂𝑓𝐻,𝑉_𝐴̇≈0.0009(𝐻−3.7713)²+𝑉̇^{𝐴−0.5019}−23.3508,₃𝐻,𝑉_𝐴̇≈−0.0004𝑉^2𝐴̇×(𝐻+0.0612)+17.5554.(3.24)

4. 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: min‖‖𝑃_vs−𝑃_vs𝑒‖‖²+‖‖𝑃_as−𝑃_as𝑒‖‖²+‖‖𝑃_vp−𝑃_vp𝑒‖‖+‖‖𝐻−𝐻_𝑒‖‖²+‖‖𝑉_𝐴̇−𝑉̇_𝐴𝑒‖‖²(4.1) subject to the system (2.1) with initial values that correspond to the rest state. Here 𝑃_vs𝑒, 𝑃_as𝑒,𝑃_vp𝑒,𝐻_𝑒, and 𝑉̇_𝐴𝑒 are means values given in Table 1.

The numerical simulation result are illustrated in the Figures 6, 7, 8, 9, 10, and 11.

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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.

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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.

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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.

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Figure 9 

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.

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Figure 10 

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 running case.

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Figure 11 

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.

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.

5. Concluding Remarks

In this work we have investigated a mathematical model that describes blood and pulmonary 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.