BME 501 Advanced Topics in Biomedical Systems Spring 2014 Dr. KayBME 501 Lecture Notes – Mar 31 Hemodynamics • Navier-Stokes Equations & Womersley Number • Damping pulsatile flow • Monitoring hemodynamics • Blood pressure waveformsNavier-Stokes Equations • Consist of – One time-dependent continuity equation for conservation of mass – Three time-dependent conservation of momentum equations – One time-dependent conservation of energy equation • Four independent variables in the problem – Spatial coordinates of chosen domain, x, y, and z – Time, t • Six dependent variables – Pressure, p – Density, ρ – Temperature, T (which is contained in the energy equation through the total energy ET) – Three components of the velocity vector: u component in x direction, v component in y direction, and w component in z direction • All dependent variables are functions of all four independent variables • The differential equations are therefore partial differential equations rather than ordinary differential equations • Equations describe how velocity, pressure, temperature, and density of a moving fluid are related • Equations derived independently by G.G. Stokes (in England) and M. Navier (in France) in early 1800’sNavier-Stokes Equations 3-D unsteady flowNavier-Stokes Equations 3-D unsteady flow where u = velocity vector field (3D) ε = thermodynamic internal energy p = pressure T = temperature ρ = density μ = viscosity (a.k.a., η) KH = heat conduction coefficient F = external force per unit massNavier-Stokes Equation: Newtonian fluids of constant density and viscosity Goes to 0 if at steady flow Goes to 0 if an ideal fluid (μ= 0) Goes to 0 if at hydrostatic equilibrium For pulsatile flow, compare the transient inertial force term with the viscous force term: Transient Inertial ForceShear or Viscous Force=rwVmVR-2=wR2mr-1=wR2n=a2w: angular frequency of oscillations V: stream velocitym: viscosity (a.k.a., h) R: radius of piper: densityn: kinematic viscosity = m/ra= NW= Rwv=D2wv= Womersley NumberWomersley Number: NW • In cardiovascular system: frequency, density, viscosity usually same throughout network • NW depends on changes in tube radii • NW large in large/wide vessels (e.g., aorta, proximal arteries, veins = macrocirculation) and small in small/narrow vessels (e.g., arterioles, capillaries, venules = microcirculation) • Important for keeping dynamic similarity when scaling an experiment – α in aorta of human ≈ 20, dog ≈14, cat ≈ 8, rat ≈ 3 – Sometimes scale up vasculature to model hemodynamics (need to keep NW in mind) • Important for determining boundary layer thickness for entrance effects: d= Rnew/a Rnew: new pipe radiusremember : a=Transient Inertial ForceShear or Viscous Force= RwvWomersley Number: NW • Common values of NW in humans: – Aorta: 15-20 – Femoral artery: 4 – Arterioles: 0.04 – Capillaries: 0.005 – Venules: 0.035 – Inferior vena cava: 10-15 • When α < 1 – Viscous forces dominate – Frequency of pulsations sufficiently low for parabolic velocity profile to develop during each cycle – Flow very nearly in phase with ΔP (Poiseuille’s Law applies) • When α > 10 – Transient inertial forces dominate – Frequency pulsations sufficiently large that velocity profile relatively flat – Flow lags pressure gradient by ~90°Velocity Profiles of Pulsatile Flow For Different Values of α α: 3.34 4.72 5.78 6.67 • Velocity profiles for first four harmonics resulting from pressure gradient cos(ωt) • Increases in α cause flattening of central region, reduction of amplitude and reversal of flow near vessel wallLarge Arteries Act as Hydraulic Filter • Aorta, pulmonary artery, and their major branches have substantial volume and distensibility • Act as hydraulic filter similar to resistance-capacitance filters of electrical circuits • Converts intermittent output of heart to steady flow through capillaries • Capillary flow does not stop while heart is fillingLarge Arteries Act as Hydraulic FilterLarge Arteries Act as Hydraulic Filter Minimize Cardiac Workload Steady flow less work than intermittent flow for a given flow rate (Q = V/t) W = P ×dVt1t2òWhen flow is steady:W = P ×VIf average flow is 100 mL/sec, and peripheral resistance remains constant, work for a 1 second cycle equals A. P*V = 100 mmHg * 100 mL = 104 mmHg-mL B. P*V = 200 mmHg * 100 mL = 2*104 mmHg-mL C. P*V = 100 mmHg * 100 mL = 104 mmHg-mLAortic Grafts • Aortic aneurisms (abnormal widenings) require support/replacement • Grafts have nearly no compliance • Heart must work harder (consumes more oxygen) to pump blood • After graft procedure, common to see hypertrophy (thickening) of left ventricleAge Affects Aortic Compliance • Slope of curve (dV/dP) represents aortic compliance • In young subjects, compliance is – Smallest at very high and low pressures – Greatest at intermediate pressures • Older subjects have less compliant aortas (greater rigidity: arteriosclerosis)Why Are Hemodynamics Important? • Understanding how blood flow behaves in normal and abnormal subjects aids in diagnosis of cardiovascular illness – Insufficient cardiac output? – Excessive blood volume? – Stiffness of vessels? • Assessment of hemodynamics allows evaluation of treatment regimenMonitoring Hemodynamics • Heart Rate (HR): – Cardiac output is a function of both the stroke volume (SV: volume of blood ejected during each contraction) and HR – Changes in HR can indicate changes in underlying hydraulic system • Arterial blood pressure (BP): – Systolic, diastolic and mean pressures – Affected by physical and physiological factors • Flow in aorta – Blood velocity, stroke distance – Changes can indicate damaged heartHeart Rate (HR) • Normal resting HR: 60 –100 beats per minute (bpm) • Can be measured by listening with stethoscope at point of maximum impulse (between 5th and 6th ribs on left side of chest): apex beat • Can be measured at wrist or other “pulse point” (less accurate by about 10 bpm) • Cardiac output (CO) = SV * HR = Q • Compensates for changes in cardiovascular system or in needs of tissues • Tachycardia (rapid heart rate) most common – Low stroke volume – Increased demand for oxygen in tissues • Bradycardia (slow heart rate) – Large stroke volume (e.g., athletes) – Dysfunctions in