BME 419/519 Hernandez 2002

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1 Vascular Biology 2 - Hemodynamics A. Flow relationships : some basic definitions Q v = A v = velocity, Q = flow rate A = cross sectional area Ohm s Law for fluids: Flow is driven by a pressure gradient P Q = R P = pressure gradient, R = resistance thus, cardiac output: Q = MAP-MVP / total peripheral Resistance. (note about pressure units: 1 mm Hg = 1.36 cm H 2 O = 1330 dynes/cm 2, 1 Newton = 10 5 dynes = 0.22 lb) B. Elastic Properties of Vessels. 1. Elasticity. the vessel walls are elastic and deform if there is a pressure gradient across them. a. Hook s Law. As you apply force, the vessel deforms, storing energy like a spring. F = kx F = force, x = displacement b. Young s elastic modulus: consider a rod with a specific cross sectional area. The Y.M. is the specific stress (Force/Area) needed to double the initial length of the rod. In the case of the vessels, we look at the increase in radius.

2 Material Young s Elastic Mod. dynes/cm 2 Rubber 4x10 7 Steel 2x10 12 VSM 10 6 Elastin 6x10 6 Collagen Compliance : How much the vessel s volume changes as the intraluminal pressure changes (at equilibrium). V C = P C = compliance, V = change in blood volume due to P = change in blood pressure. 3. Distensibility: compliance relative to some initial state (at equilibrium). V D = PV i D = distensibility, Vi = initial blood volume 4. Windkessel Effect. The previous relationships are true for equilibrium conditions. However, the vessels take some time to distend. Relationship between the rate of pressure build/up and the concomitant rate of volume change. dv dp = C dt dt simple example: aortic pressure following diastole: dv = Qin ( t) Qout ( t) dt P dp = Qin ( t) = C R dt dp 1 = P( t) dt CR behaves like a discharging capacitor (see below)

3 Note the analogy between fluid mechanics and circuits: Q= flow I = current P= Pressure Drop V = voltage drop C=compliance C = capacitance V= volume Q = charge R = resistance R = resistance You can use the same techniques on both! C. Blood s viscosity and flow : Poiseuille s equation 1. Viscosity: mechanical property of fluids that slows down their flow due to internal forces. Newton s definition: shear stress τ F / A η = = = shear rate du / dy U / Y non-newtonian fluid is one that doesn t behave like this (non-constant relationship between shear stress and shear rate) 2. Poiseuille s Equation: determines the resistance to flow of a vessel given the viscoelastic properties of the fluid under the following assumptions: -Laminar flow -Newtonian fluid -Straight, rigid pipe -Constant flow and therefore, 8ηL R = π 4 r P Q = = R R = resistance, η = viscosity (function of hematocrit primarily) L = length (won t usually change) r = radius : this is the most critical. Arterioles can essentially shunt flow because of this property. 4 Pπr 8ηL P = pressure drop through a segment of length L

4 3. Considerations: a. Combined resistance : this works just like circuits do i. Series ii. Parallel b. The real world: Non-Newtonian Behavior (??) i. Plug flow happens near the inlet of a tube, before laminar floe is fully developed. Capillaries can also show plug flow. ii. Distortion of erythrocytes. Greater hematocrit greater viscosity

5 c. Different types of flow exist: i. Plug flow: all molecules move at the same speed. Happens only at very small diameters, and slow flows. ii. Laminar Flow. Due to friction against vessel walls, the blood near the center of the tube flows faster than that on the periphery. Infinitesimally thin concentric cylinders sliding past each other. The velocity profile is shaped like a parabola. iii. Turbulent flow. Chaotic, random. Occurs when the Reynolds number for a fluid is exceeded. 2rvρ Re = η d. shear stress (force/area) : the viscous drag of the blood creates a shear force on the intraluminal side of the vessel walls. Using Poiseuille s eq.

6 F Pr 4ηQ τ w = = = 3 A 2L πr τ w = wall shear stress this can cause tears inside the lumen (dissecting aneurysm). High velocity in the aorta more likely place to happen : bad news! D. Pressure inside capillaries: Law of LaPlace Sources of pressure: a. Hydrostatic pressure: pressure due to gravity function of body part, height, position,.etc. P hs = ρ h g ρ = fluid density, h =vertical distance to a reference ( phlebostatic )level g = gravitational force constant b. Static (intraluminal or transluminal) pressure : Pressure in the vessels without the hydrostatic pressure. I.e. measured at the reference level: patient is supine and all organs are at the same level as the heart. -- Law of Laplace: T = rp T = tension in vessel wall, P = intraluminal pressure r = radius of vessel Implication thin walled capillaries can stand high internal pressures, because of their small lumen

7 . Stress : force per unit area on the vessel wall. Strain is the resultant deformation. Stress in vessel wall σ = rp w σ = vessel wall stress w = wall thickness BUT: As the vessel gets stretched out, the wall gets thinner, more fragile (ie greater stress with the same pressure), less compliant. (notice table above: the capillaries and the aorta withstand similar (ratio ~ 10) pressure, but there is a lot less tension in capillaries (ratio~10 9 ). This radius dependence keeps the capillaries from rupturing. E. Bernoulli s Relationships Under the following conditions: i.constant flow ii.non-viscous fluid iii.incompressible fluid. the total pressure in a section of a vessel is constant and can be divided into a static and a dynamic component. Bernoulli s Law: 2 P + ρ v = constant 2 2 v P d = ρ 2 P d = dynamic component to pressure ρ = density of the fluid v = velocity of flow (analogous to conservation of energy: P.E. + K.E. = constant)

8 as a fluid moves faster, it exerts a smaller radial pressure on the vessel. (note: the L shaped tube measures the total pressure. The straight tube measures the static radial pressure) Physiological Examples: Stenosis, Aneurysm: Consider a long continuous tube. Flow must be the same throughout the whole length (conservation of mass). If we reduce the cross-sectional area of a segment (stenosis), then the flow velocity must increase proportionally to maintain flow constant. The static pressure is reduced. An aneurysm is exactly the opposite effect.

9 In light of the fluid mechanics principles we have seen, and what we know about the geometry the above picture should make more sense.

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