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ISRN Applied Mathematics
Volume 2012 (2012), Article ID 470143, 16 pages
http://dx.doi.org/10.5402/2012/470143
Research Article

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

Department of Applied Mathematics, National University of Rwanda, B. P 117 Butare, Rwanda

Received 10 November 2011; Accepted 14 December 2011

Academic Editors: H. Akçay, F. Lamnabhi-Lagarrigue, and C. Lu

Copyright © 2012 Jean Marie Ntaganda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [14]. 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.

470143.fig.001
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 ̇𝑉𝐴. 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 (̇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: 𝑑𝑃as𝑑𝑡=𝑘1𝑃as+𝑃vp𝑎1𝑓1̇𝑉𝐻,𝐴,𝑑𝑃vs𝑑𝑡=𝑘2𝑃vs+𝑃as𝑎2𝑓2̇𝑉𝐻,𝐴,𝑑𝑃𝑑𝑡vp=𝑘3𝑃vp+𝑃vs𝑎3𝑓3̇𝑉𝐻,𝐴,(2.1) where 𝑎1,𝑎2, and 𝑎3 are model constants and 𝑓1,𝑓2, and 𝑓3 denote model functions to be identified. The functions 𝑃as, 𝑃vs, and 𝑃vp 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 𝑃𝑋=as,𝑃vs,𝑃vp𝑇,𝑘𝑞=1,𝑘2,𝑘3,𝑎1,𝑎2,𝑎3𝑇,(2.2)𝐺(𝑋;𝑞)=𝑘1𝑃as+𝑃vp𝑎1𝑓1̇𝑉𝐻,𝐴,𝑘2𝑃vs+𝑃as𝑎2𝑓2̇𝑉𝐻,𝐴,𝑘3𝑃vp+𝑃vs𝑎3𝑓3̇𝑉𝐻,𝐴𝑇.(2.3)

The system (2.1) can be written as follows: ̇𝑋=𝐺(𝑋(𝑡);𝑞(𝑡)).(2.4) We have the following results.

Theorem 2.1. For initial state 𝑋0=(𝑃0as,𝑃0vs,𝑃0vp,)𝑇, it exists 𝑡10 such that it passes a 𝐶1 unique maximal solution of (2.1): 𝑋0,𝑡13(2.5) that satisfies the following condition: 𝑋(0)=𝑋0.(2.6)

Let us take 𝐻𝑒, ̇𝑉𝐴𝑒, 𝑃as𝑒, 𝑃vs𝑒, and 𝑃vp𝑒 the equilibrium parameters of the system (2.1). Hence, we have 𝑘1𝑃as𝑒+𝑃vp𝑒𝑎1𝑓𝑒1=0,𝑘2𝑃vs𝑒+𝑃as𝑒𝑎2𝑓𝑒2=0,𝑘3𝑃vp𝑒+𝑃vs𝑒𝑎3𝑓𝑒3=0,(2.7) where we have set 𝑓𝑒𝑖=𝑓𝑖𝐻𝑒,̇𝑉𝐴𝑒𝑖=1,2,3.(2.8) Since the pressures 𝑃as,𝑃as, and 𝑃vs are positive, by solving the system (2.7) we get the equilibrium state given as follows:𝑃as𝑒=1𝑘1𝑘𝑎3𝑎12𝑘𝑎13𝑓𝑎3𝑎1𝑒2𝑓𝑎1𝑒3𝑓𝑒11/(1𝑎3𝑎2𝑎1),𝑃vs𝑒=1𝑘𝑎21𝑘𝑎1𝑎23𝑘2𝑓𝑎1𝑎2𝑒3𝑓𝑎2𝑒1𝑓𝑒21/(1𝑎3𝑎2𝑎1),𝑃vp𝑒=1𝑘𝑎2𝑎31𝑘𝑎32𝑘3𝑓𝑎2𝑎3𝑒1𝑓𝑎3𝑒2𝑓𝑒31/(1𝑎3𝑎2𝑎1).𝑎3𝑎2𝑎11(2.9) The Jacobian matrix of the system becomes 𝒥𝑒=𝜕𝐺𝑋𝜕𝑋𝑒;𝜆𝑒=𝑘10𝐵1𝐵2𝑘200𝐵3𝑘3,(2.10) where 𝐵1=𝑎1𝑓𝑎3𝑒2𝑓𝑒3𝑓𝑎2𝑎3𝑒1(𝑎11)/(1𝑎3𝑎2𝑎1)𝑓𝑒1,𝐵2=𝑎2𝑓𝑎2𝑒3𝑓𝑎1𝑎3𝑒2𝑓𝑒1(𝑎21)/(1𝑎3𝑎2𝑎1)𝑓𝑒2,𝐵3=𝑎3𝑎2𝑓𝑎1𝑎2𝑒3𝑓𝑒2𝑓𝑎2𝑒1(𝑎31)/(1𝑎3𝑎2𝑎1)𝑓𝑒3.(2.11) The eigenvalues of this matrix are solution of the following characteristic equation: 𝜆3+𝐶1𝜆2+𝐶2𝜆+𝐶3=0,(2.12) where 𝐶1=3𝑖=1𝑘𝑖,𝐶2𝑘=1𝑘2+𝑘2𝑘3+𝑘1𝑘3,𝐶3=3𝑖=1𝑘𝑖+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 𝑑𝑃as𝑑𝑡=𝑃as+𝑃vp𝑎1𝑓1̇𝑉𝐻,𝐴,𝑑𝑃vs𝑑𝑡=𝑃vs+𝑃as𝑎2𝑓2̇𝑉𝐻,𝐴,𝑑𝑃𝑑𝑡vp=𝑃vp+𝑃vs𝑎3𝑓3̇𝑉𝐻,𝐴.(3.1) Let us take 𝑡𝑘𝑢=𝑘Δ𝑡,𝑘=0,,𝑁,withΔ𝑡axedstepoftime,=𝑎1,𝑎2,𝑎3,𝑎4,𝑓1,𝑓2,𝑓3𝑇,(3.2) where 𝑓1=𝑓1𝑡0,,𝑓1𝑡𝑁𝑇,𝑓2=𝑓2𝑡0,,𝑓2𝑡𝑁𝑇,𝑓3=𝑓3𝑡0,,𝑓3𝑡𝑁𝑇.(3.3) Let (𝑃as𝑢,𝑃vs𝑢,𝑃vp𝑢) be the solution of (3.1) that corresponds to parameter vector 𝑢. The identification problem can be formulated as follows.

Find 𝑢=(𝑎1,𝑎2,𝑎3,𝑎4,𝑓1,𝑓2,𝑓3)𝑇 solution of a least square problem 𝐽𝑢=min𝑢𝐽𝑢,(3.4) where 𝐽𝑢=𝑁𝑘=0𝑃as𝑢𝑡𝑘𝑃𝛿asobs𝑡𝑘2+𝑃vs𝑢𝑡𝑘𝑃𝛿vsobs𝑡𝑘2+𝑃vp𝑢𝑡𝑘𝑃𝛿vpobs𝑡𝑘2,(3.5) and where (𝑃as𝑢(𝑡𝑘),𝑃vs𝑢(𝑡𝑘),𝑃vp𝑢(𝑡𝑘)) is solution at the time 𝑡𝑘 of the system (2.1) corresponding to the parameter vector 𝑢, and where 𝑃𝛿asobs𝑡𝑘,𝑃𝛿vsobs𝑡𝑘,𝑃𝛿vpobs𝑡𝑘(3.6) are observed values of variables 𝑃as(𝑡),𝑃vs(𝑡),𝑃vp(𝑡) at time 𝑡𝑘. The measures of those observed values are obtained by taking the the order error 𝛿 such that 𝑃𝛿asobs=𝑃𝛿asobs𝑡0,,𝑃𝛿asobs𝑡𝑁𝑇,𝑃𝛿vsobs=𝑃𝛿vsobs𝑡0,,𝑃𝛿vsobs𝑡𝑁𝑇,𝑃𝛿vpobs=𝑃𝛿vpobs𝑡0,,𝑃𝛿vpobs𝑡𝑁𝑇,(3.7) with 𝑃𝛿asobs𝑡0=𝑃as0,𝑃𝛿vsobs𝑡0=𝑃vs0,𝑃𝛿vpobs𝑡0=𝑃vp0.(3.8) For solving the system (2.1), we approximate it by assuming the following system: ̇𝑃as=𝑃as+𝑃vp𝑎1𝑓𝑁1,̇𝑃vs=𝑃vs+𝑃vp𝑎2𝑓𝑁2,̇𝑃vp=𝑃vp+𝑃vs𝑎3𝑓𝑁3,(3.9) where 𝑓𝑁𝑗(𝑡)=𝑁𝑙=0𝑓𝑗𝑙𝜓𝑁𝑙(𝑡),𝑗=1,2,3,(3.10) with {𝜓𝑁𝑖}𝑁𝑖=0 the set of B-splines linear functions defined on the interval [0,𝑇] and that satisfy 𝜓𝑁𝑖𝑡𝑘=𝛿𝑖𝑘,(3.11) and where we have set 𝑓𝑗𝑘=𝑓𝑗𝐻𝑡𝑘,̇𝑉𝐴𝑡𝑘,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 𝑓1=𝑓1𝑡0,,𝑓1𝑡𝑁𝑇,𝑓2=𝑓2𝑡0,,𝑓2𝑡𝑁𝑇,𝑓3=𝑓3𝑡0,,𝑓3𝑡𝑁𝑇.(3.13) The minimization problem can be formulated as follows.

Find 𝑢=𝑎1,𝑎2,𝑎3,𝑓1,𝑓2,𝑓3𝑇(3.14) solution of least square problem 𝐽𝜆𝑢=min𝑢𝐽𝜆𝑢,(3.15) where 𝐽𝜆𝑢𝑢=𝐽𝑢+𝜆𝑢3(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 𝑢̂𝜆 solution of the following minimization problem: min0<𝑢𝜆<𝐽𝜃(𝜆)=log𝑁𝑘=0𝑃as𝑢𝜆𝑡𝑘𝑃as𝑡𝑘2+𝑃vs𝑢𝜆𝑡𝑘𝑃vs(𝑡𝑘)2+𝑃vp𝑢𝜆𝑡𝑘𝑃vp𝑡𝑘2𝑢+log𝜆𝑢23(𝑁+2),(3.17) subject to (3.9). Let us set 𝜇=𝑁𝑘=0𝑃as𝑢̂𝜆𝑡𝑘𝑃as𝑡𝑘+𝑃vs𝑢̂𝜆𝑡𝑘𝑃vs𝑡𝑘+𝑃vs𝑢̂𝜆𝑡𝑘𝑃vs𝑡𝑘.(3.18) The value of the Tikhonov regularization is given by ̂𝜇𝜆=𝑢̂𝜆𝑢3(𝑁+2),(3.19) which is calculated from the following iterative scheme: 𝜆𝑘+1=𝜇𝑘𝑢𝜆𝑘𝑢𝜆𝑘3(𝑁+2),𝑘=0,,(3.20) where 𝜇𝑘=𝑁𝑘=0𝑃as𝑢𝜆𝑘𝑡𝑘𝑃as𝑡𝑘+𝑃vs𝑢𝜆𝑘𝑡𝑘𝑃𝑣𝑠𝑡𝑘+𝑃vp𝑢𝜆𝑘𝑡𝑘𝑃vp𝑡𝑘.(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].

tab1
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.

fig2
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.

fig3
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 (𝜆𝑘) given in the relation (3.20) we have the curve shown in Figure 4.

470143.fig.004
Figure 4: Curve in L form where F denotes the numerator of the relation (3.20) and 𝐿 represents its denominator.

The value of ̂𝜆 is given by the quotient of abscissa by ordinate for the point 𝑃𝑐 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 𝑓1,𝑓2, and 𝑓3 to be identified according to the observed values of heart rate 𝐻(𝑡𝑘) and alveolar ventilation ̇𝑉𝐴(𝑡𝑘) at each time 𝑡𝑘.

tab2
Table 2: Table of constants got from the Tikhonov regularization.
fig5
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: 𝑓1̇𝑉𝐻,𝐴̇𝑉55.1166𝐴0.01382̇𝑉+8.7797𝐻×𝐴𝑓2259.4624,2̇𝑉𝐻,𝐴0.0046(𝐻7.6765)2+̇𝑉𝐴3.2718𝑓21.5285,3̇𝑉𝐻,𝐴̇𝑉0.00152𝐴×(𝐻+6.0647)+25.1458.(3.22)(2)Jogging case: 𝑓1̇𝑉𝐻,𝐴̇𝑉74.0790𝐴+0.02012̇𝑉+9.1872𝐻×𝐴𝑓2457.4610,2̇𝑉𝐻,𝐴0.0011(𝐻3.3702)2+̇𝑉𝐴0.5162𝑓15.1199,3̇𝑉𝐻,𝐴̇𝑉0.00102𝐴×(𝐻+6.0980)+11.5553.(3.23)(3)Running case: 𝑓1̇𝑉𝐻,𝐴̇𝑉62.3280𝐴0.01182̇𝑉+9.1371𝐻×𝐴𝑓1959.5886,2̇𝑉𝐻,𝐴0.0009(𝐻3.7713)2+̇𝑉𝐴0.5019𝑓23.3508,3̇𝑉𝐻,𝐴̇𝑉0.00042𝐴×(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: 𝑃minvs𝑃vs𝑒2+𝑃as𝑃as𝑒2+𝑃vp𝑃vp𝑒+𝐻𝐻𝑒2+̇𝑉𝐴̇𝑉𝐴𝑒2(4.1) subject to the system (2.1) with initial values that correspond to the rest state. Here 𝑃vs𝑒, 𝑃as𝑒,𝑃vp𝑒,𝐻𝑒, and ̇𝑉𝐴𝑒 are means values given in Table 1.

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

fig6
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.
fig7
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.
fig8
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.
fig9
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.
fig10
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.
fig11
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.

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