Topic of today: flow

1. Design of cardiac array 31545 Medical Imaging systems 2. Basic observations about flow in the 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

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Design an array for 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

Pulmonary circulation through the

What are the basic properties of the blood flow Systemic circulation to the organs in the human body?

Type Diameter [cm] 0.2 – 2.4 0.001 – 0.008 0.0004 – 0.0008 0.6 – 1.5

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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 1.0 – 2.4 0.05 – 0.08 5 3 –· 6 0.8 – 1.8 0.05 – 0.08 20 3 – 6 0.5 – 1.2 0.04 – 0.06 15 9 – 11 Femoral 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 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 • 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 – 600 100 – 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

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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 = (v2 v1) Height h2 − 2 − Pressure p2 Area A 2 A1 Velocity v2 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 Height h1 ρ A1 2 2 Pressure p p2 = p1 v v ↑ ↑ ↑ ↑ 1 − 2 A 1 − 1 pressure kinetic potential Internal (heat, chemical, friction) Area A1 2 ! Velocity v   1 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)

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Viscosity Viscosity properties v Area A F Newtonian fluids have a linear relation between shear stress and v/d. Shear stress: •

F v0 For non-Newtonian fluids µ is dependent on 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 Value for water is 1 cP. • Shear stress: Blood 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, R v(r) Stationary flow: no pul- • R v(r) parabolic flow: r sation r

vo v Laminar flow o ∆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 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. 0.3 T − Z0 The spatial average velocity is then: 0.2 0.1 Velocity [m/s] ∞ ¯v(t) = v0 + Vm cos(mωt φm) 0 | | − 0 50 100 150 200 250 300 350 m=1 Phase [deg.] -1 0 1 -1 0 1 = X Relative radius Relative radius φm 6 Vm. Spatial mean velocities from the common femoral (top) and carotid arteries Profiles at time zero are shown at the bottom of the figure and time is increased toward (bottom). the top. One whole is covered and the dotted lines indicate zero velocity.

27 28 Modeling pulsatile flow: II Modeling pulsatile flow: III Newtonian fluid makes it possible to add Fourier components 250 Find spatial and temporal distribution from Womersley-Evans theory: 14 200 Sinusoidal pressure difference: r 2 ∞

10 12 150 10 v(t, r/R) = 2v0 1 + Vm ψm(r/R, τm) cos(mωt φm + χm(r/R, τm)) /M’ /M’ 2 − R | || | − 2

α =1 p(t) = ∆p cos(ωt φ) 100 α   m − 10   X 50 8 Volume flow using Womersley’s formula: 0 0 5 10 15 0 1 2 3 α α vm(t, r/R) = Vm ψm(r/R, τm) cos(ωmt φm + χm) 8 M100 | || | r − Q(t) = ∆P sin(ωt φ + 100 ) τ J (τ ) τ J ( τ ) R α2 − 100 m 0 m m 0 R m f ψm(r/R, τm) = − ρ 80 τmJ0(τm) 2J1(τm) α = R ω - Womersley’s number 60 − 10 ’ µ ε χm(r/R, τm) = ψ(r/R, τm) r 40 6 8µL ρ Rf = 20 3/2 πR4 τm = j R ωm, 0 µ 0 2 4 6 8 10 12 14 s α α < 1: Pressure and flow rate in phase 2 R - Vessel radius, r - Radial position in vessel, ρ - Density of blood, α > 1: Pressure and flow rate out-of phase The Womersley’s parameters α /M100 and 100 as a function of α. µ - Viscosity, m - Harmonic number

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Fourier components for flow velocity Pulse propagation velocity

Common femoral Common carotid How fast a pressure disturbance travels in a vessel Diameter = 8.4 mm Diameter = 6.0 mm rate = 62 bpm Heart rate = 62 bpm Moens–Korteweg equation: Viscosity = 0.004 kg/[m s] Viscosity = 0.004 kg/[m s] · · Eh cp = , m f α Vm /v0 φm m f α Vm /v0 φm s2ρR 0 - -| 1.00| - 0 - -| 1.00| - 1 1.03 5.5 1.89 32 1 1.03 3.9 0.33 74 Assumes thin wall. 2 2.05 7.7 2.49 85 2 2.05 5.5 0.24 79 3 3.08 9.5 1.28 156 3 3.08 6.8 0.24 121 A more precise equation by Nichols and O’Rourke (1990): 4 4.10 10.9 0.32 193 4 4.10 7.8 0.12 146 5 5.13 12.2 0.27 133 5 5.13 8.7 0.11 147 Eh 2 6 6.15 13.4 0.32 155 6 6.15 9.6 0.13 179 cp = (1 σ ), 7 7.18 14.5 0.28 195 7 7.18 10.3 0.06 233 s2ρR − 8 8.21 15.5 0.01 310 8 8.21 12.4 0.04 218 σ = 0.5 decreases propagation velocity by 3/4. q Data from Evans et al (1989) Normally the propagation velocity is 5 to 10 m/s. h - wall thickness, ρ - density of the wall, R - vessel radius E - Young’s modulus for elasticity of vessel wall Computer simulation: flow demo.m σ - ratio of transverse to longitudinal strain called Poisson ratio (often assumed 0.5) 31 32 Vessel branching Velocity parameters in arteries and veins Cross-sectional area of arterial • tree is gradually increased n Peak Mean Reynolds Pulse propaga- 2 R velocity velocity number tion velocity The ratio between the total ar- Vessel cm/s cm/s (peak) cm/s • eas of the branching vessels in Ascending aorta 20 – 290 10 – 40 4500 400 – 600 2 Rn v Descending aorta 25 – 250 10 – 40 3400 400 – 600 n the human body is on the order v Abdominal aorta 50 – 60 8 – 20 1250 700 – 600 v n 2R 0 0 v of 1.26 Femoral artery 100 – 120 10 – 15 1000 800 – 1030 n v Carotid artery 50 – 150 20 – 30 600 – 1100 n Increase in pressure gradient for 2 R n • Arteriole 0.5 – 1.0 0.09 the bifurcation is on the order of Capillary 0.02 – 0.17 0.001 2 Inferior vena cava 15 – 40 700 100 – 700 2/1.26 = 1.26. 2 R n Decrease in velocity is by a factor Data taken from Caro et al. (1974) • of 1.26, thus v2 = 0.8v0.

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Properties of blood flow in the human body

Influence of geometric changes Spatially variant • Time variant (pulsating flow) • 0.8 Curving of vessel change the pro- Different geometric dimensions • • file. 0.7 Vessels curves and branches re- 5 • A parabolic profile will be skewed peatedly • 0.6 out from the center of the vessel Can at times be turbulent 10 • 0.5 due to centrifugal forces. Flow in all directions • Results in a profile with larger 0.4 • 15 velocities closer to the center of A velocity estimation system Axial distance [mm] 0.3 curvature than at the far wall. • should be able to measure with a 20 Vessels gradually narrow and 0.2 • high resolution in time and space Reynold’s number is successively 0.1 25 decreased and the flow is stabi- 0 lized. The topic for the next lectures: −5 0 5 [m/s] • Lateral distance [mm] Chapter 4 and 6, Pages 63-79 and 113-129 35 36 Discussion for next time Exercise 2 in generating ultrasound images

You should determine the demands on a blood velocity estimation system Basic model: based on the temporal and spatial velocity span in the human body for the carotid and femoral artery. r(z, x) = p(z, x) s(z, x) ∗ ∗

Base your assessment on slide 27 and the flow demo. r(z, x) - Received voltage signal (time converted to depth using the speed of sound) 1. What are the largest positive and negative velocities in the vessels? p(z, x) - 2D pulsed ultrasound field

2. Assume we can accept a 10% variation in velocity for one measure- - 2D convolution ment. What is the longest time for obtaining one estimate? ∗∗

s(z, x) - Scatterer amplitudes (white, random) 3. What must the spatial resolution be to have 10 independent velocity estimates across the vessel? z - Depth, x - Lateral distance

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Signal processing

Hint 1. Find 2D ultrasound field (load from file) Hint to make the scatterer map:

2. Make scatterers with cyst hole % Make the scattere image

3. Make 2D convolution Nz=round(40/1000/dz); Nx=round(40/1000/dx); R=5/1000; 4. Find compressed envelope data e=randn (Nz, Nx); x=ones(Nz,1)*(-Nx/2:Nx/2-1)*dx; 5. Display the image z=(-Nz/2:Nz/2-1)’*ones(1,Nx)*dz; outside = sqrt(z.^2 + x.^2) > R; e=e.*outside; 6. Compare with another pulsed field

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