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

Authors: Jean Marie Ntaganda
Journal: ISRN Applied Mathematics
 (2012)

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## 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.


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## Body

## 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. Section2 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 (Figure1): 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 V̇A. 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 (V̇A) 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:
(2.1)dPasdt=-k1Pas+(Pvp)a1f1(H,V̇A),dPvsdt=-k2Pvs+(Pas)a2f2(H,V̇A),ddtPvp=-k3Pvp+(Pvs)a3f3(H,V̇A),
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(2.2)X=(Pas,Pvs,Pvp)T,q=(k1,k2,k3,a1,a2,a3)T,(2.3)G(X;q)=(-k1Pas+(Pvp)a1f1(H,V̇A),-k2Pvs+(Pas)a2f2(H,V̇A),-k3Pvp+(Pvs)a3f3(H,V̇A))T.The system (2.1) can be written as follows:
(2.4)Ẋ=G(X(t);q(t)).
We have the following results.Theorem 2.1.
For initial stateX0=(Pas0,Pvs0,Pvp0,)T, it exists t1≥0 such that it passes a C1 unique maximal solution of (2.1):
(2.5)X:[0,t1]⟶R3
that satisfies the following condition:
(2.6)X(0)=X0.Let us takeHe, V̇Ae, Pase, Pvse, and Pvpe the equilibrium parameters of the system (2.1). Hence, we have
(2.7)-k1Pase+(Pvpe)a1fe1=0,-k2Pvse+(Pase)a2fe2=0,-k3Pvpe+(Pvse)a3fe3=0,
where we have set
(2.8)fei=fi(He,V̇Ae)i=1,2,3.
Since the pressures Pas,Pas, and Pvs are positive, by solving the system (2.7) we get the equilibrium state given as follows:(2.9)Pase=(1k1k2a3a1k3a1(fe2a3a1fe3a1fe1))1/(1-a3a2a1),Pvse=(1k1a2k3a1a2k2(fe3a1a2fe1a2fe2))1/(1-a3a2a1),Pvpe=(1k1a2a3k2a3k3(fe1a2a3fe2a3fe3))1/(1-a3a2a1).(a3a2a1≠1)
The Jacobian matrix of the system becomes
(2.10)Je=∂G∂X(Xe;λe)=[-k10B1B2-k200B3-k3],
where
(2.11)B1=a1(fe2a3fe3fe1a2a3)(a1-1)/(1-a3a2a1)fe1,B2=a2f(e3a2fe2a1a3fe1)(a2-1)/(1-a3a2a1)fe2,B3=a3a2(fe3a1a2fe2fe1a2)(a3-1)/(1-a3a2a1)fe3.
The eigenvalues of this matrix are solution of the following characteristic equation:
(2.12)λ3+C1λ2+C2λ+C3=0,
where
(2.13)C1=-∑i=13ki,C2=-(k1k2+k2k3+k1k3),C3=-∏i=13ki+a1a2a3.
Thereafter we set
(2.14)k1=k2=k3=1.
Therefore, (2.12) becomes
(2.15)-(λ+1)3+a1a2a3=0.
Consequently we have the following result.Proposition 2.2.
If(2.16)0<a1a2a3<1
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.


## 2.1. Outline of the Model

Let us consider the system of three compartments (Figure1): 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 V̇A. 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 (V̇A) 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:
(2.1)dPasdt=-k1Pas+(Pvp)a1f1(H,V̇A),dPvsdt=-k2Pvs+(Pas)a2f2(H,V̇A),ddtPvp=-k3Pvp+(Pvs)a3f3(H,V̇A),
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(2.2)X=(Pas,Pvs,Pvp)T,q=(k1,k2,k3,a1,a2,a3)T,(2.3)G(X;q)=(-k1Pas+(Pvp)a1f1(H,V̇A),-k2Pvs+(Pas)a2f2(H,V̇A),-k3Pvp+(Pvs)a3f3(H,V̇A))T.The system (2.1) can be written as follows:
(2.4)Ẋ=G(X(t);q(t)).
We have the following results.Theorem 2.1.
For initial stateX0=(Pas0,Pvs0,Pvp0,)T, it exists t1≥0 such that it passes a C1 unique maximal solution of (2.1):
(2.5)X:[0,t1]⟶R3
that satisfies the following condition:
(2.6)X(0)=X0.Let us takeHe, V̇Ae, Pase, Pvse, and Pvpe the equilibrium parameters of the system (2.1). Hence, we have
(2.7)-k1Pase+(Pvpe)a1fe1=0,-k2Pvse+(Pase)a2fe2=0,-k3Pvpe+(Pvse)a3fe3=0,
where we have set
(2.8)fei=fi(He,V̇Ae)i=1,2,3.
Since the pressures Pas,Pas, and Pvs are positive, by solving the system (2.7) we get the equilibrium state given as follows:(2.9)Pase=(1k1k2a3a1k3a1(fe2a3a1fe3a1fe1))1/(1-a3a2a1),Pvse=(1k1a2k3a1a2k2(fe3a1a2fe1a2fe2))1/(1-a3a2a1),Pvpe=(1k1a2a3k2a3k3(fe1a2a3fe2a3fe3))1/(1-a3a2a1).(a3a2a1≠1)
The Jacobian matrix of the system becomes
(2.10)Je=∂G∂X(Xe;λe)=[-k10B1B2-k200B3-k3],
where
(2.11)B1=a1(fe2a3fe3fe1a2a3)(a1-1)/(1-a3a2a1)fe1,B2=a2f(e3a2fe2a1a3fe1)(a2-1)/(1-a3a2a1)fe2,B3=a3a2(fe3a1a2fe2fe1a2)(a3-1)/(1-a3a2a1)fe3.
The eigenvalues of this matrix are solution of the following characteristic equation:
(2.12)λ3+C1λ2+C2λ+C3=0,
where
(2.13)C1=-∑i=13ki,C2=-(k1k2+k2k3+k1k3),C3=-∏i=13ki+a1a2a3.
Thereafter we set
(2.14)k1=k2=k3=1.
Therefore, (2.12) becomes
(2.15)-(λ+1)3+a1a2a3=0.
Consequently we have the following result.Proposition 2.2.
If(2.16)0<a1a2a3<1
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
(3.1)dPasdt=-Pas+(Pvp)a1f1(H,V̇A),dPvsdt=-Pvs+(Pas)a2f2(H,V̇A),ddtPvp=-Pvp+(Pvs)a3f3(H,V̇A).
Let us take
(3.2)tk=kΔt,k=0,…,N,withΔtafixedstepoftime,u̲=(a1,a2,a3,a4,f1,f2,f3)T,
where
(3.3)f1=(f1(t0),…,f1(tN))T,f2=(f2(t0),…,f2(tN))T,f3=(f3(t0),…,f3(tN))T.
Let (Pasu̲,Pvsu̲,Pvpu̲) be the solution of (3.1) that corresponds to parameter vector u̲. The identification problem can be formulated as follows.Findu̲*=(a1*,a2*,a3*,a4*,f1*,f2*,f3*)T solution of a least square problem
(3.4)J(u̲*)=min⁡u̲J(u̲),
where
(3.5)J(u̲)=∑k=0N[(Pasu̲(tk)-Pasobsδ(tk))2+(Pvsu̲(tk)-Pvsobsδ(tk))2+(Pvpu̲(tk)-Pvpobsδ(tk))2],
and where (Pasu̲(tk),Pvsu̲(tk),Pvpu̲(tk)) is solution at the time tk of the system (2.1) corresponding to the parameter vector u̲, and where
(3.6)Pasobsδ(tk),Pvsobsδ(tk),Pvpobsδ(tk)
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
(3.7)Pasobsδ=(Pasobsδ(t0),…,Pasobsδ(tN))T,Pvsobsδ=(Pvsobsδ(t0),…,Pvsobsδ(tN))T,Pvpobsδ=(Pvpobsδ(t0),…,Pvpobsδ(tN))T,
with
(3.8)Pasobsδ(t0)=(Pas)0,Pvsobsδ(t0)=(Pvs)0,Pvpobsδ(t0)=(Pvp)0.
For solving the system (2.1), we approximate it by assuming the following system:
(3.9)Ṗas=-Pas+(Pvp)a1f1N,Ṗvs=-Pvs+(Pvp)a2f2N,Ṗvp=-Pvp+(Pvs)a3f3N,
where
(3.10)fjN(t)=∑l=0NfjlψlN(t),j=1,2,3,
with {ψiN}i=0N the set of B-splines linear functions defined on the interval [0,T] and that satisfy
(3.11)ψiN(tk)=δik,
and where we have set
(3.12)fjk=fj(H(tk),V̇A(tk)),j=1,2;k=0,…,N.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)f1*=(f1*(t0),…,f1*(tN))T,f2*=(f2*(t0),…,f2*(tN))T,f3*=(f3*(t0),…,f3*(tN))T.
The minimization problem can be formulated as follows.Find(3.14)u̲*=(a1*,a2*,a3*,f1*,f2*,f3*)T
solution of least square problem
(3.15)Jλ(u̲*)=min⁡u̲Jλ(u̲),
where
(3.16)Jλ(u̲)=J(u̲)+λ‖u̲-u†‖R3(N+2),
and where J(u̲) is the relation (3.5), λ denotes the regularization parameter to be chosen while u† represents a additional information on u*.The choice technique ofλ can be found in [8]. We are only focusing on how to use this technique for determining it. For that let us consider uλ̂ solution of the following minimization problem:
(3.17)min⁡0<uλ<∞Jθ(λ)=log⁡[∑k=0N((Pasuλ(tk)-Pas(tk))2+(Pvsuλ(tk)-Pvs(tk))2+(Pvpuλ(tk)-Pvp(tk))2)+log⁡(‖uλ-u†‖R3(N+2)2)],
subject to (3.9). Let us set
(3.18)μ=∑k=0N[‖Pasuλ̂(tk)-Pas(tk)‖+‖Pvsuλ̂(tk)-Pvs(tk)‖+‖Pvsuλ̂(tk)-Pvs(tk)‖].
The value of the Tikhonov regularization is given by
(3.19)λ̂=μ‖uλ̂-u†‖R3(N+2),
which is calculated from the following iterative scheme:
(3.20)λk+1=μk‖uλk-uλk†‖R3(N+2),k=0,…,
where
(3.21)μk=∑k=0N[‖Pasuλk(tk)-Pas(tk)‖+‖Pvsuλk(tk)-Pvs(tk)‖+‖Pvpuλk(tk)-Pvp(tk)‖].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 Table1 [5].Table 1
Optimal parameters of cardiovascular and et respiratory system for a woman in physical activity.
Sport typeRestWalkingJoggingRunningVentilation (L/min)68.51525Heartrate (beats/min)7085140180Pas(mmHg)104110135170Pvs(mmHg)3.5663.433.283.23Pvp(mmHg)7.6877.98.59.2The 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.The observed values of heart rate (a) and alveolar ventilation (b) during jogging for a woman who is 30 years old.
(a)(b)By using these parameters we have the observed solutions of the model (2.1) shown in the 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).
(a)(b)(c)By considering the sequence(λk) given in the relation (3.20) we have the curve shown in Figure 4.Figure 4
Curve inL form where F denotes the numerator of the relation (3.20) and L represents its denominator.The value ofλ̂ is given by the quotient of abscissa by ordinate for the point Pc 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 orderδ=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 V̇A(tk) at each time tk.Table 2
Table of constants got from the Tikhonov regularization.
ConstantValuea10.1967a2−0.1237a3−0.4541Identified functionsf1,f2, and f3.
(a)(b)(c)The technique of smoothing gives the following explicit forms for the functionsf1, f2, and f3.(1)
Walking case:(3.22)f1(H,V̇A)≈-55.1166(V̇A-0.0138)2+8.7797H×V̇A-2259.4624,f2(H,V̇A)≈0.0046(H-7.6765)2+V̇A-3.2718-21.5285,f3(H,V̇A)≈-0.0015V̇A2×(H+6.0647)+25.1458.(2)
Jogging case:(3.23)f1(H,V̇A)≈-74.0790(V̇A+0.0201)2+9.1872H×V̇A-2457.4610,f2(H,V̇A)≈0.0011(H-3.3702)2+V̇A-0.5162-15.1199,f3(H,V̇A)≈-0.0010V̇A2×(H+6.0980)+11.5553.(3)
Running case:(3.24)f1(H,V̇A)≈-62.3280(V̇A-0.0118)2+9.1371H×V̇A-1959.5886,f2(H,V̇A)≈0.0009(H-3.7713)2+V̇A-0.5019-23.3508,f3(H,V̇A)≈-0.0004V̇A2×(H+0.0612)+17.5554.


## 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 Table1. 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:
(4.1)min⁡‖Pvs-Pvse‖2+‖Pas-Pase‖2+‖Pvp-Pvpe‖+‖H-He‖2+‖V̇A-V̇Ae‖2
subject to the system (2.1) with initial values that correspond to the rest state. Here Pvse, Pase,Pvpe,He, and V̇Ae are means values given in Table 1.The numerical simulation result are illustrated in the Figures6, 7, 8, 9, 10, and 11.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.
(a)(b)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.
(a)(b)(c)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.
(a)(b)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.
(a)(b)(c)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.
(a)(b)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.
(a)(b)(c)The Figures6, 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 Figures7, 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 (Figures7 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 (Figure7(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 (Figure9(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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