Fuzzy Logic for Solving an Optimal Control Problem of Hypoxemic Hypoxia Tissue Blood Carbon Dioxide Exchange during Physical Activity

This paper aims at using of an approach integrating the fuzzy logic strategy for hypoxemic hypoxia tissue blood carbon dioxide human optimal control problem. To test the efficiency of this strategy, the authors propose a numerical comparison with the direct method by taking the values of determinant parameters of cardiovascular-respiratory system for a 30 years old woman in jogging as her regular physical activity. The results are in good agreement with experimental data.


Introduction
Hypoxia, or hypoxiation, is a pathological condition related to adequate oxygen supply in human body.The derived adequate oxygen supply can be whole body (generalized hypoxia) or its region (tissue hypoxia).Generalized hypoxia occurs in healthy people when they ascend to high altitude, where it causes altitude sickness leading to potentially fatal complications: high altitude pulmonary edema (HAPE) and high altitude cerebral edema

Methods
First of all we focus on the models equations as developed by Guillermo Gutierrez [2].The diagram for a two compartmental model is illustrated in the Figure 1 where mass transport model of tissue 2 CO exchange is de- veloped to examine the relative contributions of blood flow and cellular hypoxia (dysoxia) to increases in tissue and venous blood 2 CO concentration.From the diagram presented in the Figure 1, the equations of the model can be formulated as follows.CO v are function of time we prefer to denote also t.
It is known that the human respiratory control system varies the ventilation rate A V  in response to the levels of 2 CO and 2 O in the body and the control mechanisms of cardiovascular system influences global control in the blood vessels as well as heart rate H for changing blood flow Q  [3].Generally, during physical activity in altitude and particular in the hypoxia case, the control mechanism of these two system plays a crucial role.
Consequently, the control of cardiovascular and respiratory system is described via the following two ordinary differential equations respectively.( ) ( ) where the functions ( ) u t and ( ) v t are determined by an optimality criterion.Now let be interested in writing the arterial blood 2 PaO and 2 PaCO .First of all the venous concentration of 2 O in the vascular compartment is calculated from Fick's equation that allows to determine the rate at which oxygen is being used during physical activity as where [ ] where [ ] Hb is the blood hemoglobin concentration in g/L By considering the calculation done in [2], the arterial pressure of 2 O [4] is where pH is taken as constant 7.40 and PaCO is calculated on the basis of the Henderson-Hasselbach equation [5]   ( ) where B is 2 CO content of plasma defined by Douglas [6] as follows subject to the system (1)-( 2) and ( 3)-( 4).
In the relation (7), the positive scalar coefficients q , u q and v q determine how much weight is attached to each cost component term in the integrand while max T denotes the maximum time that the physical activity can take.
Let us consider N V the vector space that is span of a base of linear B-splines functions { } , 1, , on a regular grid max , 0, , .
The functions verify the following relation ( ) where δ denotes Kronecker symbol.The discretisation of the optimal problem (7) is done by setting the state vector ( ) PaCO ,PaO x = and the desired final vector ( ) P aCO ,P aO such that it can be written as follows. ( u v k q q q q q λ = = = with i x , f i x , i λ , i k and i q respectively the th i component of the vectors x , f x , λ , k and q .
We are looking for , an approximated solution of (10) in the set ( ) Therefore the cost function (10) becomes , with where ( 12) is determined using rectangular method such that the discretisation is done on a regular grid M Ω .Finally, the discrete formulation of optimal problem (7) subject to (1)-( 2) and (3)-( 4) is written as follows.
where M λ is a matrix ( ) denotes the two first components of solution of the system (1)-( 2) and ( 3)-( 4) associated to N λ λ = , R and B are matrix defined by 1 1

Description of Fuzzy Logic Strategy Approach
Let us consider the following problem.Find subject to , where R and Q are positive defined matrices.The problems ( 15) and ( 16) can be solved by the dynamic programming method.This method has a fast convergence, its convergence rate is quadratic and the optimal solution is often represented as a state of control feedback [7].However, the solution determined by this method depends on the choice of the initial trajectory and in some cases this solution is not optimal.It is for this reason that the integration of the fuzzy logic [8] can permit to determine quickly the optimal solution.We develop a linearization strategy of the subject system by an approach based on the fuzzy logic.This approach had been developed by Takagi-Sugeno [9] [10].The model that has been introduced in 1985 by Takagi-Sugeno permits to get some fuzzy linearization regions in the state space [11].While taking these fuzzy regions as basis, non linear system is decomposed in a structure multi models which is composed of several independent linear models [12].The linearization is made around an operating point contained in these regions.
Let's consider the set of operating point i X 1, , i S =  .Different fuzzy approximations of the nonlinear term ( ) NL x can be considered.The approximation of order zero gives: Using the first order of Taylor expansion series we obtain: To improve this approximation, we introduce the factor of the consequence for fuzzy Takagi-Sugeno system.This factor permits to minimize the error between the non linear function and the fuzzy approximation.If ε designates this factor, the approximation (18) can be formulated as the following form: If one replaces the term NL by its value approached in (16), the linearization around i x leads to 1 , , , , 1, , ; 0, , 1 where , i k A and , i k B are square matrix which has N N × order and , i k C matrix with 1 N × order.Therefore, the optimal control problems ( 15) and ( 16) become a linear quadratic problem which the feedback control is given by the following expression [13] [14]: , , 1, , ; 0, , 1.
where ( ) is the feedback gain matrix and i E discreet Riccati equation solution of the following form ( ) It is obvious that the linearization around every operating point gives the system for which the equations have the form (20).Because there are S operating points, we have S systems which have this form.Therefore, according to the relation (21) S controls are determined.The defuzzyfication method [10] permits to deter- mine only one system and only one control k U .Then, this transformation gives the following equation: , , and x ω designates membership degree partner to the operating point i x .

Fuzzy Strategy
To approximate the optimal control problems ( 7), ( 1)-( 2) and ( 3)-( 4), we propose to use the explicit Euler scheme.The stability of this scheme constitutes an advantage to approach some ordinary differential equations.The discretisation of the constraints (1)-( 2) and ( 3)-( 4) is done using the first order explicit Euler method.From the Equations (1)-( 2) and ( 3)-(4) and taking Applying the first order explicit Euler's method, the system (27) is transformed as follow where Let us set the following variable change ( ) PaCO P aCO , PaO P aO The system (27) can be formulated using the relations ( 6) and ( 30) but here we prefer to keep this form.The use of these relations is taken into account in numerical simulation.Therefore, the approximation of objective function ( 7) is made using the rectangular method and it becomes ( ) ( ) where and where R , B and h are the same as taken in subsection 2.1.Finally, the optimal control problems ( 7), ( 1)-( 2) and ( 3)-( 4) becomes the following a linear quadratic (LQ) problem. Find

Direct Approach
To approximate the system (1)-( 4), let us consider a linear B-splines basis functions on the uniform grid max , 0, , Let us introduce the vector space N W whose the basis is .
and let us take the interpolation operator : We verify easily that 0 Therefore, the system (1)-( 4) can be approached by the following form ,0 ,0 0.
We must determine , It is necessary to note that we can write Therefore, we can approximate the objective function by , where max T t N ∆ = .The convergence of the discreet objective function (48) toward the continuous objective func- tion given by the problem (46) has been shown in [15].
Finally, the optimal control problems ( 7), ( 1)-( 2) and ( 3)-( 4) are minimisation problems with constraint.The discreet formulation of such problem can be written as follows. Find where M λ is a matrix ( ) B and Y is the matrix such that the ( ) , is the solution of (50) associated to N λ λ = .

Numerical Simulation
Let us consider a hypoxic patient (a 30 year old woman) practicing jogging as physical activity for a period of max 10 T = minutes.The values of determinant parameters of a 30 years old woman in this physical activity are given in [15].Setting up 100 N = we have 0.1 h = .We consider a universe of discourse X which has four linguistic variables: tissue total CO concentration, heart rate and ventilation rate) vary such that we can consider a universe of discourse X where the labels are centered at 25, 30 and 35 (respectively 15 , 22.5 and 30; 50 , 115 and 180 and 4, 14.5 and 25 ).It is obvious that these points take the corresponding values in the labels centers of a universe of discourse X [8].
The operating points associated to those linguistic variables are given in the Table 1, membership functions associated to this labeling are represented in the Figure 2 and Figure 3 and Table 2 shows the obtained degrees of membership of each linguistic variable.
Using parameters values from Table 3 we get 0.028 α = and 7.068 and the following matrices     The next step of the fuzzy logic strategy at this point is the defuzzification.The formulas used in the defuzzification are illustrated in (26).Now considering our Equation (29), the matrices s A and s B remain unchanged.However, as matrices 1 2 , C C and 3 C contain a variable A V  , calculation is made after replacing A V  with its operating points presented in Table 1.With parameters from Table 3, we get 0.995 0.005 0.0028 0 0 0 0 0 0 0.005 0.395 0 0 0 0 0 0 0.2437 , , and .0 0 1 0 0.1 0 0 0 0 0 0 0 1 0 0.1 0 0 0 Since there are three linear state systems, the solution leads to three feedback controls of the form , , 1, 2, 3, where i K is a gain feedback.The implementation can be made in several platforms.Here we use MATLAB package where we use the built-in function dare for solving discrete Riccati Equation ( 23 After calculation, we obtain 0.0104 0.0013 3.3354 0 0 0 0 2.2468 .0.1880 0.0125 0.0110 0 0.0296 0.0401 0.0031 0 For solving the optimal control problem (7) subject to the system (1)-( 2) and ( 3)-( 4), we take The numerical simulation gives the graphical results.The Figure 4 illustrates both the variation of the heart rate and ventilation rate.Figure 5 presents the impact of physical activity due to two controls of cardiovascular-respiratory system in the variation of tissue and vascular carbon dioxide while Figure 6 shows the responses of the partial pressure of carbon dioxide and oxygen.The controls variation of the cardiovascular respiratory system are represented in Figure 4 which shows the increase of both the heart rate and the alveolar ventilation until they reach a stabilized state.It is a perfect representation of the importance of physical activity in the regulation of the cardiovascular respiratory system; in order to avoid or even heal non severe Hypoxemic-Hypoxia.In the case of Hypoxemic-Hypoxia, there is a perfect deficit of oxygen in the body.The ventilation rate plays an important role in the gas supply and regulation through the body.An increase in heart rate and ventilation rate results in an adequate and regular supply of both oxygen and carbon dioxide in the body.The Figure 5 shows a decrease of tissue and venous carbon dioxide concentration.This results from the brut increase of ventilation during the initial stage of the physical activity which is followed by a gradual increase of ventilation.The absence of a perfect ventilation leads to an increase (decrease) of carbon dioxide ( ) PaCO and oxygen ( ) PaO resulting from an accumulation of lactic acid.For a 30 years old woman during jogging as her physical activity, the Figure 6 shows the arterial partial pressure of carbon dioxide (resp.oxygen) decreases (resp.increases) in the time of physical activity until the stabilization at normal value.The results obtained in this work are rather satisfactory.In particular, the reaction of the cardiovascular and respiratory system to physical activity can be modeled and a feedback can be approximated by the solution of a linear quadratic problem.Physical activity reduces the risk of Hypoxemic-Hypoxia or contacting any cardiovascular-respiratory disease.Physical activity induces important changes in the stabilization of cardiac, vascular and blood

Concluding Remarks
In this work, two numerical approaches have been compared to determine the optimal trajectories of arterial pressures of of carbon dioxide and oxygen as response to controls of cardiovascular-respiratory system subjected to a physical activity.The finding results show that those two used methods are satisfactory and closed.Consequently, the approach integrating the fuzzy logic strategy is very important for the resolution of the optimal control problem.In particular, it gives the optimal trajectories of cardiovascular-respiratory system in the same way it ensures their performance.

Figure 1 .
Figure 1.Diagram for the tissue

λ
are components of the function N j λ in the set N B and Y represents the matrix with ( ) the th i components of the vector 0 , , f y y y and j λ and f j λ denote th j components of the vector λ and .f λ Therefore, the problems (1) and (2) can take the following compact form

Figure 3 .
Figure 3. Membership function of H (a) and A V  (b).

Figure 4 .
Figure 4. Variation of heart rate (a) and Ventilation rate (b) for a 30 years old woman during jogging as her physical activity.The curves in dotted line represent the parameter for the direct approach.The curve dashed line show the parameter for the approach integrating the fuzzy logic strategy.

Figure 5 .Figure 6 .
Figure 5. Variation of carbon dioxide in tissue (a) and in vascular (b) for a 30 years old woman during jogging as her physical activity.The curves in dotted line represent the parameter for the the direct approach.The curve dashed line show the parameter for the approach integrating the fuzzy logic strategy.

Table 1 .
Variables and their operating points.

Table 2 .
Variables and their corresponding degrees of membership.

Table 3 .
Value of used parameters.