HemodynamicsHemodynamics Michael G. Levitzky, Ph.D. Professor of Physiology LSUHSC (504)

FLUID DYNAMICS PRESSURE = FORCE / UNIT AREA = Dynes / cm 2 FLOW = VOLUME / TIME = cm 3 / sec RESISTANCE : POISEUILLE’S LAW P 1 - P 2 = F x R R = P 1 - P 2 F F Dynes / cm 2 cm 3 / sec cm 5 Dyn sec = = = =

POISEUILLE’S LAW AIR FLOW : P 1 - P 2 = V x R. BLOOD FLOW : P 1 - P 2 = Q x R.

RESISTANCE R = 8L8L 8L8L  r 4  = viscosity of fluid L = Length of the tube r r = Radius of the tube

Constant flow P1P1 P2P2 PLPL  (P 1 – P 2 )r 4 8L8L Poiseuille’s law: Q =.

POISEUILLE’S LAW - ASSUMPTIONS: 1.Newtonian or ideal fluid - viscosity of fluid is independent of force and velocity gradient 1.Newtonian or ideal fluid - viscosity of fluid is independent of force and velocity gradient 2.Laminar flow 3. Lamina in contact with wall doesn’t slip 4. Cylindrical vessels 5. Rigid vessels 6. Steady flow

RESISTANCES IN SERIES : RESISTANCES IN PARALLEL : R T = R 1 + R 2 + R RTRT RTRT R 1 R2R2 R2R2 R3R3 R3R = =

R1R1 R2R2 R3R3 R T = R 1 + R 2 + R 3 R1R1 R2R2 R3R3 1/R T = 1/R 1 + 1/R 2 + 1/R 3

x Boundary layer edge 

LAMINAR FLOW  P  Q x R. TURBULENT FLOW  P  Q 2 x R.

ml / sec Pressure Gradient (cm water)

TURBULENCE REYNOLD’S NUMBER REYNOLD’S NUMBER = = (  ) (Ve) ( D)    =Density of the fluid Ve =Linear velocity of the fluid D = Diameter of the tube  =Viscosity of the fluid

HYDRAULIC ENERGY ENERGY = FORCE x DISTANCE units = dyn cm ENERGY = FORCE x DISTANCE units = dyn cm ENERGY = PRESSURE x VOLUME ENERGY = ( dyn / cm 2 ) x cm 3 = dyn cm ENERGY = PRESSURE x VOLUME ENERGY = ( dyn / cm 2 ) x cm 3 = dyn cm

HYDRAULIC ENERGY THREE KINDS OF ENERGY ASSOCIATED WITH LIQUID FLOW: THREE KINDS OF ENERGY ASSOCIATED WITH LIQUID FLOW: 1.Pressure energy ( “lateral energy”) a. Gravitational pressure energy b.Pressure energy from conversion of kinetic energy c.Viscous flow pressure 1.Pressure energy ( “lateral energy”) a. Gravitational pressure energy b.Pressure energy from conversion of kinetic energy c.Viscous flow pressure 2.Gravitational potential energy 3.Kinetic energy = 1/2 mv 2 = 1/2  Vv 2

T Pi r TT Po Laplace’s Law Transmural pressure = Pi - Po T = Pr

GRAVITATIONAL PRESSURE ENERGY PASCAL’S LAW The pressure at the bottom of a column of liquid is equal to the density of the liquid times gravity times the height of the column. The pressure at the bottom of a column of liquid is equal to the density of the liquid times gravity times the height of the column. P =  x g x h GRAVITATIONAL PRESSURE ENERGY =  x g x h x V

IN A CLOSED SYSTEM OF A LIQUID AT CONSTANT TEMPERATURE THE TOTAL OF GRAVITATIONAL PRESSURE ENERGY AND GRAVITATIONAL POTENTIAL ENERGY IS CONSTANT. IN A CLOSED SYSTEM OF A LIQUID AT CONSTANT TEMPERATURE THE TOTAL OF GRAVITATIONAL PRESSURE ENERGY AND GRAVITATIONAL POTENTIAL ENERGY IS CONSTANT.

Reference plane E1E1 E2E2 Gravitational pressure E = 0 (atmospheric) Gravitational potential E = X +  gh · V Thermal E = UV h Total E 1 = X +  gh · V + UV Gravitational pressure E =  gh · V Gravitational potential at reference plane E = X Thermal E = UV Total E 2 = X +  gh · V + UV

E = ( P +  gh + 1/2  v 2 ) V Gravitational Potential Kinetic Energy Gravitational and Viscous Flow Pressures TOTAL HYDRAULIC ENERGY (E) TOTAL HYDRAULIC ENERGY (E)

(P 1 +  gh 1 + 1/2  v 1 2 ) V = (P 2 +  gh 2 + 1/2  v 2 2 ) V BERNOULLI’S LAW FOR A NONVISCOUS LIQUID IN STEADY LAMINAR FLOW, THE TOTAL ENERGY PER UNIT VOLUME IS CONSTANT.

Linear Velocity = Flow / Cross-sectional area cm/sec = (cm 3 / sec) / cm 2

Bernoulli’s Law of Gases (or liquids in horizontal plane) [ P 1 + ½  v 1 2 ] V = [ P 2 + ½  v 2 2 ] V lateral pressure kinetic energy

The Bernoulli Principle PLPL PLPL PLPL Constant flow (effects of resistance and viscosity omitted) Increased velocity Increased kinetic energy Decreased lateral pressure

LOSS OF ENERGY AS FRICTIONAL HEAT LOSS OF ENERGY AS FRICTIONAL HEAT U x V

E = (PV) + (±  gh V) + ( 1/2  v 2 V) + (U V) TOTAL ENERGY PER UNIT VOLUME AT ANY POINT PRESSURE ENERGY GRAVITATIONAL POTENTIAL ENERGY KINETIC ENERGY THERMAL ENERGY VISCOUS FLOW PRESSURE GRAVITATIONAL PRESSURE (±  gh) (  QR) TOTAL ENERGY

 UV = Frictional heat (  internal energy) ½  v 2 ·V = Kinetic energy PV = Viscous flow pressure energy E = Total energy h KE +UV E1E1 E2E2 E3E3 Reference plane P1P1 P2P2 P3P3

Reference plane E1E1 E2E2 E3E3 P1P1 P2P2 P3P3 KE + UV

Reference plane E1E1 E2E2 E3E3 E4E4 P1P1 P2P2 P3P3 P4P4 viscous flow P gravitational energy

b a  gh

Reference plane P1P1 P2P2 P3P3 P4P4 P5P5 E1E1 E2E2 E3E3 E4E4 E5E5 KE + UV

Pressure equivalent of KE

ArteriesCapillariesVeins h (cm) P’ (mm Hg) 44 12

ArteriesCapillariesVeins h (cm) P’ (mm Hg) (12) (9)(3) 0 (0) (9)(3) Q = 1.0

h (cm) P’ (mm Hg) (12) (9)(3) 0 (0) (10.7) (8.1) Q = Q = 1.43 Q = 1.0

1210 a 4 b Q -6 c d h (cm) P’ (mm Hg)

(Pa – Pv) (mmHg) Flow (ml/min) Pv (mmHg)

VISCOSITY Internal friction between lamina of a fluid STRESS (S) = FORCE / UNIT AREA S =  dv dx  = S dv dx dv dx Is called the rate of shear; units are sec -1 The viscosity of most fluids increases as temperature decreases

A v1v1 v2v2 === dx

VISCOSITY OF BLOOD 1.Viscosity increases with hematocrit. 2.Viscosity of blood is relatively constant at high shear rates in vessels > 1mm diameter (APPARENT VISCOSITY) 3.At low shear rates apparent viscosity increases (ANOMALOUS VISCOSITY) because erythrocytes tend to form rouleaux at low velocities and because of their deformability. 4.Viscosity decreases at high shear rates in vessels < 1mm diameter (FAHRAEUS-LINDQUIST EFFECT). This is because of “plasma skimming” of blood from outer lamina.

Non-Newtonian behavior of normal human blood Apparent Viscosity  (poise) Rate of Shear (sec -1 )

Hematocrit Relative Viscosity Effects of Hematocrit on Human Blood Viscosity / sec 212 / sec

PULSATILE FLOW 1. The less distensible the vessel wall, the greater the pressure and flow wave velocities, and the smaller the differential pressure. 2. The smaller the differential pressure in a given vessel, the smaller the flow pulsations. 3.Larger arteries are generally more distensible than smaller ones. A.More distal vessels are less distensible. B.Pulse wave velocity increases as waves move more distally. 4.As pulse waves move through the cardiovascular system they are modified by viscous energy losses and reflected waves. 5.Most reflections occur at branch points and at arterioles.

Definitions (Mostly from Milnor) Elasticity: Can be elongated or deformed by stress and completely recovers original dimensions when stress is removed. Strain: Degree of deformation. Change in length/Original length. ΔL/Lo Extensibility: ΔL/Stress (≈ Compliance = ΔV/ΔP) Viscoelastic: Strain changes with time. Elasticity: Expressed by Young’s Modulus. E = ΔF/A = Stress ΔL/Lo Strain Elastance: Inverse of compliance. Distensibility: Virtually synonymous with compliance, but used more broadly. Stiffness: Virtually synonymous with elastance. ΔF/ΔL

Distance from the Arch m / sec Carotid Arch Thoracic Aorta Diaphragm Inguinal ligament Knee Tibial Femoral Illiac Abdominal Aorta Ascending Aorta Bifid 2.5 Progressive increase in wave front velocity of the pressure wave with increasing distance from the heart. Mean pressures were 97 – 120 mmHg. (Average of 3 dogs)

V (cm/sec) P (mmHg) AscendingThoracicAbdominalFemoralSaphenous Aorta

1. Ascending aorta 2. Aortic arch 3. Descending thoracic aorta 4. Abdominal aorta 5. Abdominal aorta 6. Femoral Artery 7. Saphenous artery Pressure waves recorded at various points in the aorta and arteries of the dog, showing the change in shape and time delay as the wave is propagated.

Flow (ml / sec) Pressure (mmHg) Pressure Flow Experimental records of pressure and flow in the canine ascending aorta, scaled so that the heights of the curves are approximately the same. If no reflected waves are present, the pressure wave would follow the contour of the flow wave, as indicated by the dotted line. Sustained pressure during ejection and diastole are presumably due to reflected waves returning from the periphery. Sloping dashed line is an estimate of flow out of the ascending aorta during the same period of time.

Pulmonary Artery flow Pulmonary Artery Pressure kPa / mmHg Aortic Pressure kPa / mmHg 2.5 kPa 20mmHg 5 kPa 40mmHg 100 mls -1 Aortic flow

CAPACITANCE(COMPLIANCE) Ca =  V V V V  P P P P

During pulsatile flow, additional energy is needed to overcome the elastic recoil of the larger arteries, wave reflections, and the inertia of the blood. The total energy per unit volume at any point equals : E = (PV) + (±  gh V) + ( 1/2  v V 2 ) + (U V) TOTAL ENERGY PRESSURE ENERGY GRAVITATIONAL POTENTIAL ENERGY KINETIC ENERGY THERMAL ENERGY VISCOUS FLOW PRESSURE GRAVITATIONAL PRESSURE STEADY FLOW COMPONENT PULSATILE FLOW COMPONENT STEADY FLOW COMPONENT (±  gh) PULSATILE FLOW COMPONENT “MEAN VELOCITY” “INSTANTANEOUS VELOCITY” *(  V/C)(  QR) (POTENTIAL ENERGY IN WALLS OF VESSELS) ( 1/2  v 2 V)

ReferencesReferences Badeer, Henry.S., Elementary Hemodynamic Principles Based on Modified Bernoulli’s Equation. The Physiologist, Vol 28, No. 1, Milnor, W.R., Hemodynamics Williams and Wilkins, 1982.