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What Are the Basic Properties of the Blood Flow in the Human Body?

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Topic of today: Blood flow 1. Design of cardiac array 31545 Medical Imaging systems 2. Basic observations about flow in the human body Conservation of mass Lecture 5: Blood flow in the human body • Conservation of energy • Viscosity • Turbulence • Jørgen Arendt Jensen 3. Pulsating flow and its modeling Department of Health Technology Section for Ultrasound and Biomechanics 4. Pulse propagation and influence of geometric changes in vessels Technical University of Denmark September 13, 2021 5. Questions on Exercise 2 Reading material: JAJ, ch. 3, pages 45-61 1 2 Design an array for cardiac imaging Clean kitchen :) Penetration depth 15 cm and 300 λ Assume distance between ribs is maximum 3 cm The elevation focus should be at 8 cm 1. What is the element pitch? 2. What is the maximum number of elements in the array? 3. What is the lateral resolution at 7 cm? 4. What is the F-number for the elevation focus? 3 4 Human circulatory system Pulmonary circulation through the lungs What are the basic properties of the blood flow Systemic circulation to the organs in the human body? Type Diameter [cm] Arteries 0.2 – 2.4 Arteriole 0.001 – 0.008 Capillaries 0.0004 – 0.0008 Veins 0.6 – 1.5 5 6 Arteries and veins in the body Physical dimensions of arteries and veins Internal Wall Young’s diameter thickness Length modulus Vessel cm cm cm N/m2 105 Ascending aorta 1.0 – 2.4 0.05 – 0.08 5 3 –· 6 Descending aorta 0.8 – 1.8 0.05 – 0.08 20 3 – 6 Abdominal aorta 0.5 – 1.2 0.04 – 0.06 15 9 – 11 Femoral artery 0.2 – 0.8 0.02 – 0.06 10 9 – 12 Carotid artery 0.2 – 0.8 0.02 – 0.04 10 –20 7 – 11 Arteriole 0.001 – 0.008 0.002 0.1 – 0.2 Capillary 0.0004 – 0.0008 0.0001 0.02 – 0.1 Inferior vena cava 0.6 – 1.5 0.01 – 0.02 20 – 40 0.4 – 1.0 Data taken from Caro et al. (1974) 7 8 Velocity parameters in arteries and veins Characteristics of blood flow in humans Peak Mean Reynolds Pulse propaga- velocity velocity number tion velocity Pulsating flow, repetition from 1 to 3 beats/sec Vessel cm/s cm/s (peak) cm/s • Ascending aorta 20 – 290 10 – 40 4500 400 – 600 Not necessarily laminar flow Descending aorta 25 – 250 10 – 40 3400 400 – 600 • Abdominal aorta 50 – 60 8 – 20 1250 700 – 600 Femoral artery 100 – 120 10 – 15 1000 800 – 1030 Short entrance lengths • Carotid artery 50 – 150 20 – 30 600 – 1100 Arteriole 0.5 – 1.0 0.09 Branching Capillary 0.02 – 0.17 0.001 • Inferior vena cava 15 – 40 700 100 – 700 Reynolds numbers usually below 2500, non-turbulent flow • Data taken from Caro et al. (1974) Very complicated flow patterns • Reynolds number: 2Rρ Re = ¯v µ We will start from simple observations and then develop the theory • Indicates turbulence if Re > 2500 (approximately). R - Vessel radius, ρ - Density of blood, µ - Viscosity, ¯v - Mean cross-sectional velocity 9 10 Conservation of mass Stationary flow properties Mass at both ends must be equal: Height h2 Pressure p2 Laminar flow. Particles flow in par- Area A2 Velocity v A1v1ρ1∆t = A2v2ρ2∆t, allel layers with streamlines that do 2 not intersect. Incompressible fluid: ρ1 = ρ2: A Height h 1 1 v = v Pressure p 2 1 No acceleration at a fixed position: 1 A2 Area A1 Velocity v ∂v 1 A2 > A1: Velocity decreases = 0 A2 < A1: Velocity increases ∂t x,y,z Methods for generating a flow: Note, however, that it is possible that v = 0 at changes in geometry Difference in pressure ∇ 6 • Difference in height • 11 12 Relations for a pressure difference Equations: ρ 2 2 Benoulli’s law for conservation of energy p1 p2 = (v v ) − 2 2 − 1 A1 v2 = v1 A2 v2 v2 Combining gives: p + 1 ρ + gh ρ + U = p + 2 ρ + gh ρ + U 1 2 1 1 1 10 2 2 2 2 2 20 2 ρ A1 2 2 p2 = p1 v v ↑ ↑ ↑ ↑ − 2 A 1 − 1 pressure kinetic potential Internal (heat, chemical, friction) 2 ! 2 ρ A1 2 = p1 + 1 v1 2 − A2 ! Assume same height h1 = h2, incompressible fluid ρ1 = ρ2 = ρ, and same internal energy gives Examples: ρ 2 2 p1 p2 = (v2 v1) A < A p < p and v > v − 2 − 2 1 ⇒ 2 1 2 1 A > A p > p and v < v 2 1 ⇒ 2 1 2 1 (Incompressible fluid and same height) 13 14 Viscosity Viscosity properties v Area A F Newtonian fluids have a linear relationHeight between h shear stress and v/d. Shear stress: 2 • Pressure p2 Area A F v 2 0 For non-Newtonian fluids µ is dependentVelocity on v2 shear stress Sxy. d = µ • A d Unit of viscosity is kg/[m s] or centipoise cP (g/[cm s]) µ - Viscosity of fluid • v = 0 Fluid Height h Value1 for water is 1 cP. •Pressure p1 Area A1 Velocity v Shear stress: Blood1 is a Newtonian fluid in major vessels For small fluid element: • ∆F ∆v = µ The viscosity in blood depends on hematocrit (fraction of red blood cell volume), ∆A ∆y • temperature, and pressure. ∆F v+∆v ∆y In the limit: o ∆A v In major vessel the viscosity is usually around 4 cP at 37 and a hematocrit of 45%. y, u ∂vx ∂vy • Sxy = µ + ∂x ∂y ! Viscosity will give rise to parabolic velocity profiles for a laminar, stationary flow. x, v • for a Newtonian fluid. 15 16 Flow models: Stationary parabolic flow Poiseuille flow Pressure difference for a laminar, v(r) Stationary flow: no pul- R • parabolic flow: r sation v o Laminar flow ∆p = Rf Q • Long entrance length • Rf - Viscose resistance Q - Volume flow velocity Profile: 2 r Q = A¯v = v(x, y)dxdy [m3/s] v(r) = 1 v0, − R2! ZZxy Spatial mean velocity over cross section: 8µl = v Rf 4 ¯v = 0 πR 2 l - Distance for pressure drop v0 - Peak velocity at center of vessel, r - Radial position in vessel, - Radius of vessel R - Radius of vessel R 17 18 Example Straight vessel with a diameter of 2 cm, µ = 4 cP rises 20 cm straight up. Mean velocity is 30 cm/s and density is 1060 kg/m3. Giraffe assignment Pressure difference between the ends of the tube? Neglect viscosity, the pressure difference is then given by Bernoulli’s equation What is the change in ρ 2 2 ∆P = (v v ) + ρg(h2 h1). pressure in a Giraffe’s 2 2 − 1 − Straight vessel gives v1 = v2: neck, when it moves it’s head from drink- ∆P = ρg(h2 h1) = 1060 9.82 0.2 = 2082 Pa. − · · Pressure drop due to viscosity is ing to eating leaves in a tree? 8µl 2 8µl 8 0.004 0.2 ∆P = Rf Q = ¯vπR = ¯v = · · 0.3 = 19.2 Pa. πR4 R2 0.012 For a vessel with a diameter of 0.2 cm and a mean velocity of 10 cm/s, the pressure The height of a giraffe drop due to viscosity is is up to 6 m. 8 0.004 0.2 ∆P = · · 0.1 = 640 Pa. 0.0012 There is, thus, a considerable difference in the pressure relationships between a standing person and a person lying flat on a bed. 19 20 Why doesn’t the Giraffe explode? Turbulent flow in tubes Photo of dye injected into a tube with flowing water. The images are taken of the original apparatus as used by Reynold in 1883. Velocity is increasing from the top tube to the bottom one. Image courtesy of Hans Nygaard, Aarhus University. Courtesy of C. Lowe, Manchester School of Engineering. 21 22 Turbulent flow in tubes Disturbed flow in the body Normally non-turbulent flow exists in the body, but it can appear at constrictions, directional changes, and bifurcations: 0.8 Determined by Reynolds number: 0.7 5 2Rρ Re = ¯v 0.6 µ 10 Laminar flow usually found if: 0.5 0.4 Re < 2000 15 Axial distance [mm] 0.3 Turbulent flow found if : 20 0.2 Re > 2500 0.1 Usually non-turbulent flow is found 25 0 in most vessels −5 0 5 [m/s] Lateral distance [mm] Display of velocity direction and magnitude in the carotid bifurcation right after peak systole for a normal volunteer (Courtesy of Jesper Udesen). 23 24 Entrance effects Theoretical velocity profiles 1 0.8 Profiles: p0 0.6 r v(r) = 1 v0 − R 0.4 0.2 p = 32 o 0 p = 8 Development of a steady-state parabolic velocity profile at the entrance to a tube. o Spatial mean velocity −0.2 p = 4 o over cross section: Entrance length: Normalized radial position −0.4 p = 2 o 2 RR ¯v = v0 1 e −0.6 − p + 2 Ze 0 ! ≈ 15 −0.8 −1 Example (aorta): 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 v0 - Peak velocity at center of Velocity [m/s] vessel 0.01 1600 r - Radial position in vessel Ze · 1m Velocity profiles for different values of p0. ≈ 15 ≈ R - Radius of vessel The spatial mean velocity is the same for all profiles. 25 26 Velocity profiles for femoral and carotid artery Modeling pulsatile flow: I Profiles for the femoral artery Profiles for the carotid artery 1 Periodic waveform: 0.5 0 Fourier decomposition of spatial average ve- Velocity [m/s] -0.5 locity: 0 50 100 150 200 250 300 350 Phase [deg.] 1 T Velocity Velocity Vm = ¯v(t) exp( jmωt)dt.
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