Arterial Macrocirculatory Hemodynamics

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1 Arterial Macrocirculatory Hemodynamics 莊漢聲助理教授 Prof. Han Sheng Chuang 9/20/2012 1

2 Arterial Macrocirculatory Hemodynamics Terminology: Hemodynamics, meaning literally "blood movement" is the study of blood flow or the circulation. The circulatory system is a connected series of tubes, which includes the heart, the arteries, the microcirculation, and the veins. It passes nutrients (such as amino acids, electrolytes and lymph), gases, hormones, blood cells, etc. to and from cells in the body to help fight diseases and help stabilize body temperature and ph to maintain homeostasis Arteries are blood vessels that carry blood away from the heart. This blood is normally oxygenated, exceptions made for the pulmonary and umbilical arteries 2

3 Model of Vascular Dimension 3

4 Blood Pressure and Volume Distribution of the Systemic Circulation Blood is pumped through Pulmonary and systemic circulations at a rate of about Rushmer,

5 Fluid Mechanics and Rheology For an incompressible fluid with constant viscosity, the equations governing the fluid motion are continuity and Navier Stokes equation: 0 θ 5

6 A Simple Solution for Viscous, Incompressible Fluids In terms of cylindrical polar coordinates, the Navier Stokes equation can be written as

7 A Simple Solution for Viscous, Incompressible Fluids Assumptions: 1. Steady state. 2. The flow is fully developed in the z direction (independent of z). 3. No body force. 4. The velocity is axisymmetric (independent of θ)

8 A Simple Solution for Viscous, Incompressible Fluids Boundary Conditions: 1. r=0, u z =u 0 or u max. 2. r=r, u z =0 (No slip boundary) One seventh power law 8

9 Laminar and Turbulent Flow Reynolds number Re=

10 Velocity Profiles In the arterial system, the velocity profile is relatively flat or blunt at the entrance, or at the ascending aorta and becomes progressively parabolic when approaching smaller arteries. 10

11 Entrance Length Entrance length is defined as the distance from the entrance or inlet of a vessel at which point the flow is fully developed. However, under pulsatile flow conditions, flow does not fully develop and hence the accuracy in the determination of entrance length becomes an issue. This is normally circumvented by assuming that the flow is almost fully developed in analyzing flow in blood vessels. A common criterion that is used assumes that the centerline velocity is within 1or2%of the centerline velocity according to Poiseuille s flow, i.e. 98% or 99%. A general rule of thumb used follows the following formulation: Laminar 4.4/ Turbulent A lower Reynolds number, such as those occurring at smaller vessels, with much smaller lumen diameters, the requirement of the entrance length becomes much less stringent. 11

12 Boundary Conditions The no slip condition refers to the assumption that concerns the fluid solid interface or blood vessel wall interface. No slip boundary condition refers to the condition when the flow velocity at the tube wall is the same as the wall velocity, such that there is no jump or a step change in velocity to cause discontinuity. The general assumption is that the fluid in contact with the wall does not move at all. This assumption is generally true for the large vessels. In small vessels, plasma dominates as fluid and this no slip condition generally applies to the plasma in contact with the wall, rather than red blood cells or the formed elements in blood. 12

13 Poiseuille Equation Poiseuille equation Flow Rate Flow Resistance L 13

14 Bernoulli Equation and Narrow Vessel h g Along the same stream line. The change rate of total energy of a system is:

15 Bernoulli Equation and Narrow Vessel From the conservation of energy, equating these 2 equations, we have For the case when gravity is ignored or when h l =h 2, we have the simplified Bernoulli equation 1 2 The commonly known phrase that the faster the flow velocity, the lower the pressure, i.e. v 2 >v 1, then p 2 <p 1. 15

16 Newtonian & Non Newtonian Fluids Shear Thickening: cornstarch, quick sand How about blood?? Shear Thinning: cookie dough Bingham Plastic: toothpaste, mayonnaise Blood has plasma, blood cells and other formed elements. In most common analysis of blood flow in vessels, the assumption of blood as a Newtonian fluid seems to work well. Except in the case of very small vessels, such as the small arterioles capillaries where red blood cell size actually approaches that of the vessel lumen diameter. 16

17 Womersley s Number (α) Womersley s number can be viewed as the ratio of oscillatory flow to steady flow. 2r 17

18 The values in following table are typical for individuals at rest. During exercise, cardiac output and hence Reynolds numbers can increase several fold. α w Re α w Re 18

19 Wave Propagation For a pressure pulse wave propagating along a uniform artery without the influence of wave reflections, the pressures measured simultaneously at any two sites along the vessel are related by Where γ is the propagation constant and z is along the longitudinal axis of the artery in the direction of pulse propagation. Pulse wave velocity has been popularly approximated by the so called foot to foot velocity. Here, one simply estimates the pulse wave velocity from the transit time delay ( t) of the onset or the foot between two pressure pulses measured at two different sites along an artery or the pulse propagation path. This requires again, the simultaneous measurements of two pressures separated by a finite distance, z, normally 4 6cmapart. 19

20 Wave Propagation As an example, referring to the above figure, the distance between the two pressure measurement sites is 5cm and the calculated time delay, t, is 60 msec, then the foot to foot velocity is / 20

21 Pulse Wave Velocity Pulse wave velocity recorded as foot to foot velocity measured in different arteries. Higher wave velocity in smaller arteries are seen. 21

22 Modeling Aspects of the Arterial System There are three equations generally thought to be sufficient to characterize the propagation of the pulse wave. The first of these equations describes fluid motion, 1 The second is the equation of continuity to describe the incompressibility of the fluid: 1 2 In a cylindrical blood vessel, the cross sectional area A, is related to its inner lumen radius r. The third equation is the equation of state to describe the elastic properties of the vessel wall where k is a constant. 22

Modeling Aspects of the Arterial System 1 1 1 1 1 2 2 2 Take the 2 nd

23 Modeling Aspects of the Arterial System Take the 2 nd derivative of u z w.r.t. time 3 rd Eq. to 1 st Eq. 2 nd Eq. to 1 st Eq. Wave Equation 23

24 Modeling Aspects of the Arterial System (Cont.) Define Young s modulus of elasticity as: where and 2 which gives the well known Moens Korteweg formula for pulse wave velocity as 24

25 Pressure and Flow Relations and 25

26 Vascular Impedance to Blood Flow Vascular impedance has both a magnitude and a phase for each harmonic. Since pressure and flow are generally not in phase, the impedance possesses a phase angle within 90. This is attributed to the time delayed arrival between the pressure pulse and the flow pulse. Pulsatile blood pressure waveform can be considered an oscillatory part with sinusoidal components oscillating at different harmonic frequencies, nω, and phase, φ n, superimposed on a DC component or mean blood pressure: 26

27 Vascular Impedance to Blood Flow Since pressure and flow are generally not in phase, the impedance possesses a phase angle within 90. When the particular pressure harmonic leads the flow harmonic, then the phase angle between them is. Conversely, when the pressure harmonic lags behind the corresponding flow harmonic, then the phase is. the impedance measured at the ascending aorta or input impedance of the systemic arterial tree in five normal adults. 27

28 Wave Reflection Phenomena The amplification of pressure pulses has been attributed to the in phase summation of reflected waves arising from structural and geometric nonuniformities. The microvascular beds have been recognized as the principal reflection sites. Thus, pulsatile pressure and flow waveforms contain information about the heart as well as the vascular system. Measured pressure (P) and flow (Q) waveforms measured at any site in the vascular system can be considered as the summation of a forward, or antegrade, traveling wave and a reflected, or retrograde, traveling wave: /2 /2 For an infinitely long straight tube with constant properties, input impedance will be independent of position in the tube and dependent only on vessel and fluid properties. The corresponding value of input impedance is called the characteristic impedance Z 0 given by where ρ is the density of blood (1.06 g/cm3), C is pulse wave velocity, πr 2 is the crosssectional area of the artery. 28

29 Wave Reflection Phenomena Similarly, resolution of flow into its forward and reflected components can be obtained from a set of two equations: 29

30 Wave Reflection Phenomena From the above equations, it can be seen that wave reflection has opposite effects on pressure and flow. An increase in wave reflection increases the pressure amplitude, but decreases the flow amplitude. Vasoconstriction Vasodilation 30

31 Reflection Coefficients Reflection coefficient can be defined in terms of vascular impedances. For a vessel with characteristic impedance Z 0, and terminated with vascular load impedance Z, the reflection coefficient Γ is given by: Γ The reflection coefficient so obtained is therefore a complex quantity with modulus Γ and phase φ Γ varying with frequency. Wave reflection sites exist all over the systemic arterial tree, duetogeometricand elastic nonuniformities, branching, and impedance mismatching at arterial terminations. Therefore, reflections cannot originate from one site only. Indeed, there is no agreement on the location of reflecting sites. Although there is no major agreement on the reflecting sites, the arterioles are recognized as being the principal sites for wave reflection, and the reflection coefficient as being high. 31

32 Impedance Matching and Wave Reflection A local reflection coefficient can be defined for vascular branching only: Γ For instance, for a bifurcation with daughter branch characteristic impedances of Z 01,andZ 02, we have: we obtain for the local wave reflection coefficient due to equi vascular branching (Z 01 =Z 02 ) only: Branching daughter vessels Γ

33 Velocity Profile in a Curved Pipe The interaction of the effects of curvature and helicity results in complex flow patterns. It has been argued that the presence of helical, swirling flows in the heart and major arteries confers advantages in terms of flow stability and reduced energy dissipation. 33

34 Velocity Profile in a Symmetric bifurcation 34

35 Velocity Profile in an Asymmetric Bifurcation 35

36 Fluid Mechanics Associated with Atherosclerosis and Stenosis The susceptibility of vascular branches to atherosclerosis is believed due in part to the unusual fluid dynamic environments that the vessel wall experiences in the regions. Fluid mechanical studies have shown that atherosclerosis may occur at branching points where the geometry is complex, a large Reynolds number and a lower than average wall shear stress. In general, the complex flow pattern is associated with a spatially nonuniform shear stress and wall curvature. The rate of change of shear stress and shear rate have been shown to be important, as well as local turbulence and unsteady flow. In addition local disturbed flow patterns, recirculation zones, long particle residence times have been suggested to play significant roles in the onset and development of atherosclerosis. 36

37 Remember to download the course materials from the website prior to the class next week 37

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