BME 501 Lecture Notes – Mar 26 Hemodynamics • Determinants of fluid behavior • Types of flow • Shear stress and blood vessels • Bernoulli’s equationFlow Rate, Velocity & Area • Flow rate or volume flow (Q): volume of fluid moving per unit time (e.g., cm3/sec) • Velocity or linear velocity (v): rate of displacement with respect to time (e.g., cm/sec) • Area (A): cross-sectional area of conduit (e.g., cm2) v =QA or Q = v× AFlow Rate, Velocity & Area Q = v1× A1= v2×A2= v3×A3v1v2=A2A1 v1v3=A3A1 v2v3=A3A2Pressure Gradient, Conduit & Viscosity • Flow rate (Q) is proportional to the difference in pressure between two regions (ΔP) • Relationship depends on characteristics of conduit: Q is – Inversely proportional to l (lower case “L”), the length of the conduit – Proportional to r4, the radius of the conduit raised to the fourth power Q µ DP Q µ1l Q µ r4Pressure Gradient, Conduit & Viscosity • For a given pressure difference, for a cylindrical tube of given dimensions, flow depends on viscosity (η) • Viscosity: – Measure of fluid’s resistance to gradual deformation by shear or tensile stress (relates to notion of “thickness”) – Ratio of shear stress to shear rate • For Newtonian fluids, flow is inversely related to viscosity (commonly, dynessec/cm2 = poise)Shear Stress Force tending to cause deformation of material by slippage along plane or planes parallel to imposed stressPressure Gradient, Conduit & Viscosity Q µ DPQ µ1lQ µ r4Q µ1hPressure Gradient, Conduit & Viscosity Poiseuille’s Law • For steady, laminar flow of a Newtonian fluid through a cylindrical tube Q =DP ×p×r48×h×l where p8 is constant of proportionalityResistance to Flow • Encapsulates all components of hydraulic system that hinder flow • As with electrical systems, where , relationship between Q and change in pressure can be summarized using hydraulic resistance equation for a given conduit segment: DP = Pin- Pout= Q × RR =DPQ=8×h×lp× r4DV = I × R or DVI= RResistance to Flow • Resistance in series rememeber : R =DPQResistance to Flow • Resistance in parallel rememeber : R =DPQLaminar Flow • Sometimes called “streamlined” • All elements of fluid move in streams/lines parallel to axis of tube • Velocity profile is paraboloid • Thin layer in contact with wall is motionless • Pressure drop is proportional to first power of flow rate (i.e., ΔP Q) µTurbulent Flow • Irregular motion of fluid • Fluid elements do not remain confined to definite laminae (rapid, radial mixing occurs) • Much greater pressure required to force a given flow through same tube when flow is turbulent (vs. laminar) • Pressure drop approximately proportionate to square of flow rate (i.e., ΔP Q2) µReynolds Number: Re or NR • Allows prediction of whether flow will be laminar or turbulent • Dimensionless value • Represents ratio of inertial to viscous forces: Re =inertiaviscous=rv2DH2hvDH=rvDHhr: fluid densityv : mean fluid velocityDH: hydraulic diameter of pipe (a.k.a., L : traveled length)h: fluid (dynamic) viscosityReynolds Number: Re or NR • Flow is laminar when viscous forces prevail (i.e., when Reynolds number is “small”) • Flow is turbulent when inertial forces prevail (i.e., when Reynolds number is “large”) • “small” and “large” Re values: – Re < ~2,300: laminar flow – ~2,300 < Re < ~4,000: transitional flow – Re > ~4,000: turbulent flow • Turbulent flow more likely if: – Fluid density, mean velocity or pipe diameter large – Fluid viscosity low remember : inertiaviscous=rvDHhReynolds Number: Re or NR • Behavior of fluid around an obstacle depends on Re • Reynolds number in – Aorta: ~2,000 to 3,500 – Arteries: ~500 – Capillaries: ~0.001 to 0.01 – Veins: ~140 – Vena cava: ~3300Turbulent Flow in Cardiovascular System • Laminar flow is normal condition for blood flow through most of cardiovascular system • Turbulent flow occurs – In large arteries at branch points – At points where vessels narrow (rapidly increasing velocity) – When blood passes over rough surfacesTurbulent Flow in Cardiovascular System • Turbulence usually accompanied by audible vibrations: detected as murmur • Heart must pump harder to push turbulent flow of blood • Thrombi (blood clots) much more likely to develop in turbulent flow (vs. laminar)Turbulence in AortaBlood Viscosity • Normal blood viscosity: 3 to 4 centipoise at body temperature (37 degrees Celsius) • Viscosity of blood does change with shear rates (undergoes shear thinning): non-Newtonian fluid • Factors increasing blood viscosity – Increased hematocrit – Stiff red blood cells – Cell aggregation – Hypothermia (low body temperature)Cardiac Murmurs • Turbulence in vessels/heart detectable as murmur • With severe anemia – Have low hematocrit • Decreases oxygen carrying capacity of blood • Decreases viscosity of blood – Heart must pump more to provide oxygen to body, leading to high flow velocities • With narrowed (stenotic) cardiac valve or abnormal widening (aneurysm) of large arteries – Cross sectional area of blood stream suddenly changes – Velocity suddenly changesShear Stress and Blood Vessels • If re-arrange equation for viscosity to solve for shear stress: • As velocity in vessel increases, shear stress increases on vessel walls • Viscous drag: – Rapidly flowing blood in large arteries tends to pull endothelial lining of artery along with it – • Normal/physiologic shear stress – Arteries: 10 to 70 dynes/cm2 – Veins: 1 to 6 dynes/cm2 h=tdu /dy ® t=hdudyt= 4hQ /pr3Shear Stress and Blood Vessels • At high shear stresses, conformational changes in endothelial cells cause synthesis and release of nitric oxide (NO) and prostaglandin I2 (PGI2) • NO and PGI2 cause relaxation of smooth muscle in vessel wall, resulting in blood vessel dilation • Dilation of blood vessel is same as increasing radius, thereby reducing shear stress (negative feedback loop) remember : t= 4hQ /pr3Shear Stress and Blood Vessels • Laminar flow with high shear can protect against atherosclerosis • High shear upregulates athero-protective genes and downregulates atherogenic genes • Turbulent flow patterns disrupt shear stress on walls, likely cause upregulation of genes promoting inflammationDissecting Aneurysm • Hypertension tends to cause sub-endothelial