Open Access e-Journal Cardiometry - No.14 May 2019

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REPORT Submitted: 29.6.2018; Accepted: 13.8.2018; Published online: 21.11.2018Motion of blood formed elementsin a pulsatile hemodynamic flowVladimir I. Timoshenko 1 , Nikolay N. Chernov 1*1Southern Federal UniversityRussia, 347900, Taganrog, Shevchenko st. 2*Corresponding author:phone: +7 (928) 908-11-77, e-mail: nnchernov@sfedu.ruAbstractThis paper offers some results obtained in our studies on themotion of individual formed elements in blood with due regardof a pulsatile flow of the plasma in the blood vessels. An analysisof the process of the motion of the blood particles having differentshapes in a pressure wave variable field of the pressure wavehas been completed herein.KeywordsBlood formed elements, Motion, Pressure wave variable field,Human circulatory system, Physical models of motion of formedelementsImprintVladimir I. Timoshenko, Nikolay N. Chernov. Motion of bloodformed elements in a pulsatile hemodynamic flow. Cardiometry;Issue 13; November 2018; p.22-26; DOI: 10.12710/cardiometry.2018.13.2226;Available from: http://www.cardiometry.net/issues/no13-november-2018/pulsatile-hemodynamic-flowIntroductionCirculating blood represents a suspension comprisedof the plasma as the blood liquid componentand formed elements suspended therein, which are asfollows: erythrocytes, leukocytes and thrombocytes,the sizes of which range up to 10 micrometers. Hemodynamicprocesses, which occur in different phaseswithin a cardiac cycle, determine the respective bloodflow volumes, which are considered to be one of themajor criteria for an assessment of the cardiovascularsystem performance. Electrical potential traced asan electrocardiogram controls the blood flow in theblood vessels in order to provide all required parametersof hemodynamics. From an analysis of an electrocardiogram,which can be interpreted as a sequence ofsimple sinusoidal signals [1], it may be deduced thatthe blood flow is managed under smoothly changingenergetic processes in each cardiac cycle, which showtheir maxima and minima, from their growing tilltheir decaying. An actual frequency of these processesis dictated by the heart rate and cannot exceed 5 Hzthat refers to 300 heart beats per minute. A knowledgeof biophysics processes of blood circulating flows,considering motion of the formed elements in a pulsatileblood flow, is of crucial importance.A set of the respective blood volumes and associateddynamic fluctuations thereof are considered tobe a valuable diagnostics parameter for assessing ofhemodynamic processes is. A simulation of the hemodynamicprocess on the basis of empirical equationsby G.M.Poyedintsev – O.K.Voronova [2] accordingto their concept of elevated fluidity regime,existing in the circulatory system, has made it possibleto bring an analysis of volumetric parameters of acardiovascular cycle to a new higher level [3]. Howeverit should be noted that many factors of biophysics,responsible for the motion of blood as a suspension,comprising the liquid plasma and the formedelements suspended therein, are still not clearly understood.There is still a poor understanding of howthe blood laminar flow process is organized: erythrocytes,leukocytes and thrombocytes, circulating inblood vessels, are non-spherical in their shape. So, itis supposed that the blood elevated fluidity phenomenon,determined by pulsations by the blood vesselwalls, should be accompanied by oscillations of theblood formed elements. Considering the above, anattempt is made in our paper to analyze the processof the motion of the blood elements in a variable fieldof a pressure wave.Materials and methodsThe construction of our fresh theory of the motionof the blood formed elements is based on a solution ofa number of physics-related problems. Following thisway, it should be stressed that the key problem withinthe context is an investigation of the behavior of individualparticles and the system of these smallest particlesin a variable field of pressure. The solution of thisproblem allows discovering applicable laws and regularitiesof the major physical mechanism of the laminarflow of the elements in a variable pressure field.The formed elements, which are suspended in the liquidmedium, under the action of the oscillations, are16 | Cardiometry | Issue 14. May 2019
involved into different types of the motion. And theviscosity of the medium plays a significant role therein.The laws, regularities and general rules of the motionof a single spherical fluid element in a viscousmedium are quite sufficiently treated in papers by G.G. Stokes, L.D. Landau, E.M. Lifshits [4] and other researchers.In monographs by N.A. Fuks [5], a theoryof the behavior of an individual particle in a viscousmedium under the action of different forces is comprehensivelystudied. Based on the key results of thistheory, the description of the behavior of individualparticles and their system in a variable pressure fieldwill be discussed herein below.The formed particles, residing in a field of periodicalmixing in a viscous medium, are subjected toexternal forces, resistance forces and forces of interactionsbetween the particles. As a result of the collectiveaction of the above forces, a motion of the particlerelative to the medium is initiated. By the externalforces we usually mean gravity, electrostatic forces etc.Those forces, acting on a particle, exerted by a restingmedium (in case of the particle motion) or a movingmedium, are usually not ranked as the external forces.They are referred to as the forces of the resistance. Incases, when the particles are involved in an oscillatorymovement in a variable pressure field, the resistanceforces should be regarded as the acting ones, so thatthey may be taken into account as the external forces.The force, exerted by the medium, which acts ona single particle, can be computed from a solution tothe Navier-Stokes equation. For a spherical solid particlehaving radius R, the solution to the Navier-Stokesequation:∂ V 1+ ( V ∇ ) V = − grad p + ν∆ V∂tρfor a viscous incompressible medium, in case of anarbitrary function of time for medium velocity U c(t),leads to formulation of an expression for time-variableforce F , acting on the particle, as given below:4 2 dV3 3F = 6πηRV + πR ρ g + πRρ +cc3 3 dt2+ 6Rπρ ηct∫0cdVdtidtit − ti(1)(2)where V = (U c– U p) is velocity of flow past a particle,U pis velocity of the particle, η is a coefficient of thedynamic viscosity of the medium; ν = η/ρ сis a coefficientof the kinematic viscosity, ρ сis density of themedium; g is gravitational acceleration; t iis typicaltime at the inverse Fourier transform of the functionV(t); p pressure.The formula (2) holds for small Reynold numbersRe = 2R|V|/ν. The first term in the right side expressesthe Stoke’s force of the resistance at a constant velocityof the motion, equal to the instantaneous value of flowpast at a given instant. The second term represents theArchimedes force, or the buoyant force. The third andthe fourth terms are the forces of the resistance, whichare associated with an energy consumption requiredto initiate the motion of the medium itself. In this case,the third term corresponds to the force of inertia for apotential flow past a medium. It can be interpreted asa mass added to the particle, that is equal to one halfliquid displaced by the particle itself. The fourth termis associated with an energy input to initiate the motionof the medium areas, adjacent to the particle, dueto the mechanism of the internal friction (viscosity).Under the oscillatory motion of the medium, thevelocity of the medium U p, and, consequently, the flowpast velocity are functions of time. However temporalvariations in the velocity will not alter the pattern ofthe viscous motion of the particle, provided that thecondition of the quasi-stationarity [6] is met:2∂V2 ωRν∇V ≈ = 1(3)∂tνwhere ω = 2πf is a circle frequency.The condition of the quasi-stationarity is very crucialfor studies on issues associated with effects madeby oscillations on liquid dispersed media, and it is fulfilledfor the heart pulse frequencies.In addition, we should mention that there maybe some other factors and properties of the particlescapable of influencing the nature of the motion ofthese smallest components, the sizes of which are inthe range of few microns, as follows: their non-sphericity,fluctuating lability, etc. The actions and effectsthereof can be taken into account by introducing therespective correction coefficients. But it is often thecase, when they are ignored to a rough approximation.The motion of the blood formed elements in avariable pressure field can be defined by a system ofparameters derived from the solution of the equationof the motion.Issue 14. May 2019 | Cardiometry | 17
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Page 5 and 6: 64 Prospects for the application of
Page 7 and 8: Prof. Mohammad AleemCairo Universit
Page 9 and 10: Olga K. Voronova, the Russian Mathe
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Open Access e-Journal Cardiometry - No.14 May 2019

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