BLOOD FLOWBarbara Grobelnik Advisor: dr. Igor SeraNovember 20071Blood FlowIntroductionThe 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 Viscosity Steady blood flow Poiseuilles equation Entrance effects Bernoullis equation Oscillatory blood flow Windkessel model Wommersley equationsNovember 20072Blood FlowCardiovascular PhysiologyMAIN 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, veinsleft ventricle aorta organs and tissues right atriumright ventricle lungs left atriummean diameter mmnumber of vessels aorta19 - 4.51arteries4 0.15110.000arterioles0.052.7 106capillaries0.0082.8 109November 20073Blood FlowPoiseuille flow Steady flow in a rigid cylindrical tube Pressure gradient Viscous force rLrp1p22rvThe forces are equal and opposite:volume flowaverage velocityv(r=R)=0v(r=0)November 20074Blood FlowPoiseuille flow - assumptions Newtonian fluid in large blood vessels (at high shear rates) Laminar flow Reynolds 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 sectionsxxxxNovember 20075Blood FlowPhysical properties of bloodBLOOD = plasma + blood cells(55%) (45%)PLASMAWHOLE BLOODdensity 1035 kg/m31056 kg/m3viscosity 1.310-3 Pa s3.5 10-3 Pa sReference valueselectrolyte solution containing 8% of proteinsRed blood cells (95%)White blood cells (0.13%)Platelets (4.9%)RBC:8 m1 mNovember 20076Blood FlowViscosity 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 Cassons equationIn small tubes the blood viscosity has a very low value because of a cell-free zone near the wall.Fahraeus-Lindqvist effectNovember 20077Blood FlowFahraeus-Lindqvist EffectCell-free marginal layer model Core region c , vc , 0rR- Cell-free plasma p , vp , R-r RThe Sigma effect theory velocity profile is not continuous small tubes (N red blood cells move abreast) the volume flow is rewrittenregion near the wall the volume flow1/r c , vcp , vpR N concentric laminae, each of thickness 1/ November 20078Blood FlowEntrance 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 FLOWviscous force - boundary layer thickness at z U - free stream velocityinertial force* a=U/t=U/(z/U)November 20079Blood FlowEntrance 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 fluctuatesNovember 200710Blood FlowApplication of Bernoulli Equation Flow trough stenosis v2 v1 p2 p1 : expansion and bursting of the vessel caused by the weakening of the arterial wallBernoulli equationp2, v2, A2p1 v1A1p2, v2, A2A1 p1 v1A1v1 = A2v2November 200711Blood FlowVacular 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:Mean pressure values mmHg:- arteries 100 - capillaries 30-34 at arterial end, 12-15 at venous endn 2 average d=1.26velocity decreases, pressure gradient increasesNovember 200712Blood FlowTurbulent Flow Reynolds number Flow in the circulatory system is normally laminar Flow in the aorta can destabilize during the deceleration phase of late