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EQUATIONS OF HEMODYNAMICS IN THE QUASI-ONE-DIMENSIONAL APPROXIMATION FOR BLOOD VESSEL
Dilafruz Shukrullaevna Nurjabova
Tashkent University of Information Technologies Karshi branch, independent researcher of department "Software Engineering"
Annotation. This article describes an overview of the mathematical model of the circulatory system for the cardiovascular system and provides a basis for the mathematical representation of aggregate medical parameters, such as the, objects and research methods, goals, and objectives of the thesis, blood volume, scientific novelty, practical significance, self-regulation and influence on the upper and inner parts of the heart.
Key words: Equations of hemodynamics in the quasi-one-dimensional approximation, science, common vascular zone, self-regulation, effect on the upper and working heart, medical parameters.
INTRODUCTION
Equations of hemodynamics in the quasi-one-dimensional approximation represent a system of two differential equations in partial derivatives, which is closed by one algebraic relation.
Equations of continuity and motion: The differential equation (the continuity equation) expressing the law of conservation of mass looks like this:
dS duS dt dt
The law of change in momentum leads to a differential Equation
duS du2 S dpS duS
P —I" P —;--1" —;--P —;— = qf S,
dt dx dx dx 1
which is converted to the form
dy dy 1 dp dt dx p dx p
Here f q is the bulk density of external forces. One variable t is time.
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As a spatial variable x, the length of the arc passes through the centers of the circular cross-sections of the vessel. S(x, t) - area cross-section of the vessel, depending on the coordinate x and time t.
The speed of blood movement is considered to be directed along the axis of the vessel, the same in the entire circular section of the vessel, and denotes(x, t). The pressure inside the liquid (blood) will be denoted p(x, t). Blood density S = S(p) counts are constant t (incompressible fluid). State equation: To close equations (1.1.1) and (1.1.2)
we use the additional relation
Equation (1.1.3) can be considered experimentally established the relationship between the cross-sectional area of the vessel and the pressure inside the vessel. In hemodynamic problems, the equation S = S( p) plays a role similar to the equation of state for a bar tropic gas in gas dynamics, and by analogy, henceforth we will call it that.
In this paper, we restrict ourselves to the closure of the system of equations the position that the cross-sectional area of the vessel depends on the pressure of the blood inside the vessel and we will take into account the elastic-mechanical properties of the vessel according to limiting values of its cross-sectional area, achieved at small and high-pressure values. Characteristic of the main vessels of the arterial part of the vascular human systems, the type of dependence (1.1.3) is shown in Fig.1.1.1.
An important characteristic feature of the dependence of the cross-section on pressure is the increase in the cross-section with increasing pressure, that is, the performance conditions
5 = S(p)
METHODS
S
ST
p
p
The simplest form of the model equation of state is presented in Fig.1.1.2 and is determined by the formula
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Pmin < p < Ртах P ^ Pmin Ртах ^ P
where min max min max S , S , p , p are characteristics of a particular vessel.
Smin
Smax
Pniax
* P
Pmiii
Fig. 1.1.2. General equations of state
In what follows, we will use more general equations of state. Equations (1.1.1), (1.1.2), and (1.1.3) together are a system of equations of hemodynamics in the quasi-one-dimensional approximation.
Properties of hemodynamic equations. Equations derived From hemodynamics (1.1.1), (1.1.2 ), and (1.1.3) have a number of characteristic properties. At, in particular, the type of equations and various forms of their representation is of interest.
Hyperbolic type of equations of hemodynamics. Everywhere below, we will assume that
v
с
,2
S(p)
pS'vip)
Denote
—» rp
Let us introduce the vector Y = (p ,u)1 ,F = ( 0 , qy/ p/)
T
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Then equations (1.1.1), (1.1.2) can be written as
dY dY ^ —r- + A— = F
dt dx
Solving the characteristic equation d et {A — AE) = 0 Find eigenvalues of the matrix A, which are equal to
and
Under condition (1.1.4), these numbers are real and different.
Consequently, the system of equations (1.1.5) is a system of hyperbolic type [127].
Note that the value has meaningful meaning
S(P)
pS'p(p)
c =
M
similar to the speed of sound in gas dynamics [127, 133]. It is significant that for flow in large arterial vessels of the circulatory system, characteristic is the overwhelming predominance of the speed of sound over fluid flow velocity, i.e. M/c « 1
Riemann parameters in hemodynamic equations and characteristic form writing equations. Characteristic relations for equations hemodynamics have the form
1 dp du ( 1 dp du>
du ^ / 1 dp du\ qf dt ~\pc(p)dt dt) p
pc(p) dt dt ~ \ pc(p) dt dt) p Using the function (p {p) = /J ,
where a is an arbitrary fixed number, we can write the last equations in the form
d . d qf
— (±<p + u)+A± —(±(p + it) = — dt dx p
We introduce two new functions (the Riemann parameters [127])
^(u^) = u± (p(p)
As a result, equations (1.1.6) are reduced to two transfer equations for parameters R+ and R- :
dR± , dR± qf dt dx p
Equations (1.1.7) are hemodynamic equations written for the Riemann parameters.
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RESULTS
Consider equations (1.1.7) on two families of characteristics in the plane of independent variables (x, t ) given by the equations
Equations (1.1.7) for the functions R ± t, x± (t) ) respectively, and the characteristics, can be represented as X and X:
dR~ qf
From (1.1.8), in particular, it follows that if there are no external forces, then is qy (x, t ) = 0 , then the Riemann parameters R+ and R- are constant on characteristics X andX, respectively.
In the case , taking into account the expressions for the functions ,
q (p), and dependence S =S(p) after transformations, we obtain that the characteristics X andX, respectively, the following characteristic
dx+ (t) _ dx (t)
dt ' dt
dt p
dR qy
dt p
relations have written for differentials.
CONCLUSION
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In this graph shows results of doctors how can use desktop program via using above mathematical model. The described approach to constructing a numerical implementation makes it possible to divide the problem into independent blocks for
The main conclusions after the interview'
■ The user-friendly interface allows patients to work with the command
1,2
0.8 0.6
0,2 0
Mammadova D. Zhaiilov A Ravshanov. S Kaxmansv r Mirzaev.B Ergasheva. 3
Figure.1.3. Main findings after the interview calculating the flow in each vessel and at each point of their docking. The simplicity of the model makes it possible to complicate it and thereby take into account the influence of many factors.
Taskl.Treatment methods in Task 2. If the patient's usual Task 3. In-depth survey Task 4. Collecting cardiac -
this section should be tests are out + then do not consists of calculated data data To process data for S,
derived from conventional send for a deep examination based on mat. models that U, p, T, x, ro, L, T, K and
tests , if on the contrary, the will help you detect the BZ, displaying data as a
patient will be sent for a disease at what stage chart
1 deep examination
2 Success Time Success Success Time Success Success Time Success Success Time Success
3 P1 1 30 1 1 160 1 0 35 1 0 70 1
4 P2 1 45 1 0 90 0 0 50 0 1 100 0
5 P3 1 45 0 1 90 1 1 50 1 1 100 1
6 P4 1 50 1 0 100 0 0 55 1 0 100 0
7 Average 1,0 42,5 0,8 0,5 110,0 0,5 0,3 47,5 0,8 0,5 92,5 0,5
8 Number of reps 4 4 4 4 4 4 4 4 4 4 4 4
Figure.1.4. Metric of Mathematical modeling
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Scientific Journal Impact Factor (SJIF 2022=4.63) Passport: http://sjifactor.com/passport.php?id=22230
3. Lishchuk V.A. Mathematical theory of blood circulation. - M.: Medicine, 1991. - 256 p.
4. Pedley T. Hydrodynamics of large blood vessels. - M.: Mir, 1983. - 400 p.
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13. Popel A.S., Regier A.S. On the basic equations of blood hydrodynamics. / Scientific. tr. Institute of Mechanics, Moscow State University. - 1970. No. 1. - With. 3-
14. Young D.F., Shih C.C. Some experiments on the effect of isolated proturbberances on flow through tubes // Experimental Mechanics. - 1969. Vol. 9, No.5. - P. 225-229.
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18. Dilafruz Shukrullaevna Nurjabova, Ravshan Narzullaevich Abdullaev, JournalNX ,ISSN (E): 2581-4230 Journal Impact Factor: 7.232 Editor In- Chief Dr. Rajinder Singh Sodhi ,Volume 7, Issue 9 | September, 2021, USING NUMERICAL SOLUTION OF
20.
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THE NAVIER-STOKES EQUATIONS AND LINEARIZED NAVIER-STOKES EQUATIONS VISCOUS NEWTONIAN FLUID MODEL FOR BLOOD VESSEL WALLS, 175-179 p
19.Dilafruz Shukrullaevna Nurjabova, JournalNX ,ISSN (E): 2581-4230 Journal Impact Factor: 7.232 Editor In- Chief Dr. Rajinder Singh Sodhi ,Volume 7, Issue 9 | September, 2021, USING AND MODELIZATION OF THE LAW THREE-DIMENSIONAL MODEL OF THE FLOW OF AN INCOMPRESSIBLE VISCOUS NEWTONIAN FLUID MODEL FOR BLOOD VESSEL WALLS , 172-174 p
20.Dilafruz Shukrullaevna Nurjabova, Har. Edu.a.sci.rev. 0362-8027 Vol.1. Issue 1 Pages 96-106.10.5281/zenodo.5670030, Using numerical solution of nonlinear navier-stokes equations fluid model for blood vessel walls.
21.Dilafruz Shukrullaevna Nurjabova, Sojida Rayimberdi qizi Ochilova 197-201, USING AND MODELIZATION OF THE LAW OF EHERGY UNDER THE PATHOLOGY OF BLOOD FLOW IN THE CONSTRUCTION OF ELASTIC VARIABLE MODELS FOR BLOOD VESSEL WALLS, A Multidisciplinary Peer Reviewed Journal Volume 7, Issue 5, May, 2021ISSN: 2581-4230 Impact Factor: 7.232.
22.Dilafruz Shukrullaevna Nurjabova , 217-221 MODELING THE INFLUENCE OF PATHOLOGIES ON BLOOD FLOW BY MODIFYING THE ELASTIC MODEL FORVASCULAR WALLS A Multidisciplinary Peer Reviewed Journal Volume 7, Issue , May, 2021 ISSN: 2581-4230 Impact Factor: 7.23.
