1. BLOOD FLOW Barbara Grobelnik Advisor: dr. Igor Serša

2. January 2008Blood Flow2 Introduction The study of blood flow behavior: Improving the design of implants (heart valves, artificial heart) and extra-corporeal flow devices (blood oxygenators, dialysis machines) Understanding the connection between flow characteristics and the development of cardiovascular diseases (atherosclerosis, thrombosis) CONTENTS  Cardiovascular physiology  Physical properties of blood  V iscosity  Steady blood flow  Poiseuille ’s equation  Entrance effects  Bernoulli ’s equation  Oscillatory blood flow  Windkessel model  Wommersley equations

3. January 2008Blood Flow3 Cardiovascular Physiology MAIN FUNCTIONS: to deliver oxygen and nutrients to the cells to remove cellular wastes and carbon dioxide to maintain organs at a constant temperature and pH HEART: atrium, ventricles BLOOD VESSELS: aorta, arteries, arterioles, capillaries, veinules, veins left ventricle  aorta  organs and tissues  right atrium right ventricle  lungs  left atrium mean diameter [mm] number of vessels aorta19 - 4.51 arteries4 – 0.15110.000 arterioles0.052.7 ∙10 6 capillaries0.0082.8 ∙10 9

4. January 2008Blood Flow4 Poiseuille flow Steady flow in a rigid cylindrical tube –Pressure gradient –Viscous force rr L r p1p1 p2p2 2r v The forces are equal and opposite: volume flow average velocity v(r=R)=0 v(r=0)≠∞

5. January 2008Blood Flow5 Poiseuille flow - assumptions Newtonian fluid – in large blood vessels (at high shear rates) Laminar flow – Reynold’s numbers below the critical value of about 2000 No slip at the vascular wall – endothelial cells Steady flow – pulsatile flow in arteries Cylindrical shape – elliptical shape (veins, pulmonary arteries), taper Rigid wall – visco-elastic arterial walls Fully developed flow – entrance length; branching points, curved sections    x x x x

6. January 2008Blood Flow6 Physical properties of blood BLOOD = plasma + blood cells ( 55%) (45%) PLASMAWHOLE BLOOD density1035 kg/m 3 1056 kg/m 3 viscosity1.3×10 -3 Pa s3.5 × 10 -3 Pa s Reference values electrolyte solution containing 8% of proteins Red blood cells (95%) White blood cells (0.13%) Platelets (4.9%) RBC: 8 μm 1 μm

7. January 2008Blood Flow7 Viscosity Viscosity varies with samples – variations in species – variations in proteins and RBC Temperature dependent – decrease with increasing T Blood – a non-Newtonian fluid at low shear rates (the agreggates of RBC) – a Newtonian fluid above shear rates of 50 s -1 – Casson’s equation In small tubes the blood viscosity has a very low value because of a cell-free zone near the wall. Fahraeus-Lindqvist effect

8. January 2008Blood Flow8 Fahraeus-Lindqvist Effect Cell-free marginal layer model  Core region μ c, v c, 0  r  R-   Cell-free plasma μ p, v p, R-  r  R The Sigma effect theory · velocity profile is not continuous · small tubes (N red blood cells move abreast) · the volume flow is rewritten region near the wall  the volume flow 1/μ rr μ c, v c μ p, v p R · N concentric laminae, each of thickness ε 1/μ

9. January 2008Blood Flow9 Entrance length The flow of fluid from a reservoir to a pipe –flat velocity profile at the entrance point –the fluid in contact with the wall has zero velocity (‘no slip’) –retardation due to shearing adjacent to the wall –boundary layer (where the viscous effects are present) –acceleration in the core region to maintain the same volume of flow –parabolic velocity profile  FULLY DEVELOPED FLOW viscous force  - boundary layer thickness at z U - free stream velocity inertial force * a=U/t=U/(z/U)

10. January 2008Blood Flow10 Entrance length equating the viscous and inertial force k – proportionality constant derived from experiments, approximately 0.06 the boundary layer thickness the entrance length (when  =D/2 the flow becomes fully established) The above derivation is valid only for the flow originating from a very large reservoir, where the velocity profile at the entrance point is relatively flat. In other cases, the entrance length is shorter. Pulsatile flow – the entrance length fluctuates

11. January 2008Blood Flow11 Application of Bernoulli Equation Flow trough stenosis – v 2 > v 1 – p 2 < p 1 : caving or closing of the vessel – decrease in v 2 – reopening of the vessel – fluttering Flow in aneurysms – v 2 < v 1 – p 2 > p 1 : expansion and bursting of the vessel – caused by the weakening of the arterial wall Bernoulli equation p 2, v 2, A 2 p1p1 v1v1 A1A1 A1p1v1A1p1v1 A 1 v 1 = A 2 v 2

12. January 2008Blood Flow12 Vacular resistance and branching Vascular resistance – for Poiseuille flow – major drop in the mean pressure in arterioles (60 mmHg) autonomic nervous system controls muscle tension arterioles distend or contract Succesive branching: – Increase in the total cross- section area – dA1=nA2:– dA1=nA2: Mean pressure values [mmHg]: - arteries 100 - capillaries 30-34 at arterial end, 12-15 at venous end n ≥ 2 average d=1.26 velocity decreases, pressure gradient increases

13. January 2008Blood Flow13 Turbulent Flow Reynolds number Flow in the circulatory system is normally laminar Flow in the aorta can destabilize during the deceleration phase of late systole – too short time period for the flow to become fully turbulent Diseased conditions can result in turbulent blood flow – vessel narrowing at atherosclerosis, defective heart valves – weakening of the wall, progression of the disease critical value R e > 2000 for flow in rigid straight cylindrical pipes

14. January 2008Blood Flow14 Unsteady flow models The pressure pulse: –generated by the contraction of the left ventricle –travels with a finite speed through the arterial wall –change in a shape due to interaction with reflected waves Windkessel model –the arteries: a system of interconnected tubes with a storage capacity –distensibility D i = dV/dp –Inflow – Outflow = Rate of Storage SYSTOLEDIASTOLE Q=Q 0, 0  t  t s Q=0, t s  t  T p(t)  b-(b-p 0 )e -t/a p(t)  e (T-t)/a A typical pressure pulse curve. p0p0 tsts T b diastolesystole

15. January 2008Blood Flow15 Wommersley equations The equation for the motion of a viscous liquid in a cylindrical tube (general form): Arterial pulse = periodic function  the sum of harmonics The solution: – J 0 (xi 3/2 ) is a Bessel function of the first kind of order zero and complex argument – y=r/R The flow velocity pulse and the arterial pressure pulse (femoral artery of a dog). – Wommersley number 

16. January 2008Blood Flow16 The role of Wommersley number  - unsteady inertial forces vs. viscous forces (viscous forces dominate when   1) 10 -3    18 capillaries aorta  : 3.34 4.72 5.78 6.67 The velocity profiles for the first four harmonics resulting from the pressure gradient cos ωt  Parabolic profile is not formed · The laminae near the wall move first · Solid mass in the centre  Increase in  : flattening of the central region, reduction of amplitude and reversal of flow at the wall

17. January 2008Blood Flow17 The sum of harmonics y=r/R The first four harmonics summed together with a parabola (representing the steady forward flow). The time dependence of velocity at different distances y. · Parabolic shape in the fast systolic rush · Phase lag between the pressure gradient and the movement of the liquid  The reversal begins in the peripheral laminae (the point of flow reversal: 25° after the pressure  Back flow: harmonics are out of phase and the profile is flattened gradient)  The peak forward and backward velocities: 165 cm/s at 75° 35 cm/s at 165°

18. January 2008Blood Flow18 Conclusion What have we learned? - basic equations of blood flow Why am I interested in blood flow? future experiment: dissolving blood clots under physiological conditions PULSATILE FLOW Artificial heart.

19. January 2008Blood Flow Non-Newtonian fluid behavior  Bingham plastic  Power law fluid  Cassons fluid Velocity profiles in a round rigid tube.