Dynamics of Blood FlowDynamics of Blood Flow26.3.1226.3.12

Transport SystemTransport System

A closed double-pump system:A closed double-pump system:

SystemicCirculation

LungCirculation

Left side of heart

Right side of heart

Transport SystemTransport System

Branching of blood vesselsBranching of blood vessels– Ateries branch into arterioles, veins into Ateries branch into arterioles, veins into

venulesvenules

ArteriesArterioles

Capillaries

Venules

Veins

Heart

Volume Flow RateVolume Flow Rate

The average flow from the heart is the The average flow from the heart is the stroke volume (the volume of blood stroke volume (the volume of blood ejected in each beat) x number of beats ejected in each beat) x number of beats per second. This is ~ 60 (ml/beat) x 80 per second. This is ~ 60 (ml/beat) x 80 (beats/min) = 4800 ml/min(beats/min) = 4800 ml/min

Hagen–Poiseuille lawHagen–Poiseuille law or or Poiseuille lawPoiseuille law

In fluid dynamics, the In fluid dynamics, the Hagen–Poiseuille equationHagen–Poiseuille equation is a physical law is a physical law that states that for steady laminar flow of a Newtonian fluid through that states that for steady laminar flow of a Newtonian fluid through a cylindrical tube, the flow rate is directly proportional to the a cylindrical tube, the flow rate is directly proportional to the pressure drop, fourth power of radius of tube and inversely to the pressure drop, fourth power of radius of tube and inversely to the length and viscosity of fluidlength and viscosity of fluid

oror

Where ∆P is the pressure drop Where ∆P is the pressure drop

L is the length of pipe, L is the length of pipe,

ƞ is the dynamic viscosity, ƞ is the dynamic viscosity,

Q is the volumetric flow rate and Q is the volumetric flow rate and

r is the radius of the piper is the radius of the pipe L

r

P1 P2

P= P1 - P2

Limitations of Poiseuille lawLimitations of Poiseuille lawThe assumptions of the equation areThe assumptions of the equation are

• A long rigid cylinder with length much greater than the radiusA long rigid cylinder with length much greater than the radius• Fluid has constant viscosity and is incompressible Fluid has constant viscosity and is incompressible • Steady Laminar flow that is not pulsatile and turbulentSteady Laminar flow that is not pulsatile and turbulent• The fluid velocity at the edges of tube is zeroThe fluid velocity at the edges of tube is zero

Poiseuille law has certain limitations when applied to circulating blood Poiseuille law has certain limitations when applied to circulating blood in vivoin vivo

• Blood vessels are not rigid tubes and are quite distensible so that their Blood vessels are not rigid tubes and are quite distensible so that their size depends on the blood pressure within them as size depends on the blood pressure within them as well as upon well as upon the contraction of smooth muscles in the vessel wallsthe contraction of smooth muscles in the vessel walls

• Blood is a Non-Newtonian fluid and fluid viscosity is not Blood is a Non-Newtonian fluid and fluid viscosity is not constantconstant• The flow is not steady but pulsatile in most parts of the vascular bedThe flow is not steady but pulsatile in most parts of the vascular bed

Blood volume flow rate QBlood volume flow rate Q

Liquid flows along the lumen of a rigid tube from a higher Liquid flows along the lumen of a rigid tube from a higher to lower hydrostatic pressureto lower hydrostatic pressure

In the vascular system, the rate of blood flow In the vascular system, the rate of blood flow (volume/unit time) is proportional to the hydrostatic (volume/unit time) is proportional to the hydrostatic pressure gradient (∆P) across the vessel and inversely pressure gradient (∆P) across the vessel and inversely to the resistance (R) offered to its flowto the resistance (R) offered to its flow

Analogous to Ohms law for electrical circuits (I=V/R) we Analogous to Ohms law for electrical circuits (I=V/R) we can writecan write

Blood flow rate Q = ∆P / R = (PBlood flow rate Q = ∆P / R = (Paa-P-Pvv) / R) / R

Vascular ResistanceVascular ResistanceVascular resistanceVascular resistance is a term used to define the is a term used to define the resistance to flow that must be overcome to push blood resistance to flow that must be overcome to push blood through the circulatory system. through the circulatory system.

The resistance offered by the peripheral circulation is The resistance offered by the peripheral circulation is known as the known as the systemic vascular resistancesystemic vascular resistance ( (SVRSVR))

The systemic vascular resistance may also be referred The systemic vascular resistance may also be referred to as the total peripheral resistanceto as the total peripheral resistance

Resistance RResistance RResistance is dependent on the vessel’s dimensions and Resistance is dependent on the vessel’s dimensions and the viscosity of bloodthe viscosity of blood

From From Poiseuille law, Poiseuille law,

The resistance decreases rapidly as r increases The resistance decreases rapidly as r increases R = R = ΔΔP/Q =P/Q = 8 8 L L ηη / / ππ rr44

A narrowing of an artery leads to a large increase in the A narrowing of an artery leads to a large increase in the resistance to blood flow because of 1/ rresistance to blood flow because of 1/ r44 term term

Vasoconstriction (i.e., decrease in blood vessel diameter) Vasoconstriction (i.e., decrease in blood vessel diameter) increases SVR, whereas vasodilation (increase in increases SVR, whereas vasodilation (increase in diameter) decreases SVRdiameter) decreases SVR

Peripheral resistance can be equated to DC resistance in Peripheral resistance can be equated to DC resistance in electrical circuitselectrical circuits

Arrangement of vessels also determines resistance.Arrangement of vessels also determines resistance.

When the vessels are arranged in series, the total When the vessels are arranged in series, the total resistance to flow through all the vessels is the sum of resistance to flow through all the vessels is the sum of individual resistances, whereas when they are arranged individual resistances, whereas when they are arranged in parallel the reciprocal of the total resistance is the sum in parallel the reciprocal of the total resistance is the sum of all the reciprocals of the individual resistanceof all the reciprocals of the individual resistance

Less resistance is offered to blood flow when vessels are Less resistance is offered to blood flow when vessels are arranged in parallel rather than in seriesarranged in parallel rather than in series

Volume Flow RateVolume Flow Rate

Often convenient to define a Often convenient to define a resistance, Rresistance, R to flow, such that to flow, such that P=QRP=QR

P1 P2 P3

R1 R2 R3

P= P1 + P2 + P3

=QR1+QR2+QR3

=QRR=R1+R2+R3

Series Parallel

R1,Q1

R2,Q2

Q=Q1+Q2

=P/R1+P/R2

=P/RR=1/R1+1/R2

Resistances in series add Resistances in series add directly while resistances in directly while resistances in parallel add in reciprocals parallel add in reciprocals

Arteries, arterioles, capillaries, Arteries, arterioles, capillaries, venules and veins are in general venules and veins are in general arranged in series with respect arranged in series with respect to each other. However, the to each other. However, the vascular supply to the various vascular supply to the various organs and the vessels e.g. organs and the vessels e.g. capillaries within an organ are capillaries within an organ are arranged in parallel arranged in parallel

Right and left sides of the heart which are connected in series. Also seen are the various systemic organs receiving blood through parallel arrangement of vessels

Rate of blood flowRate of blood flow

Blood leaves heart at ~ 30 cm/sBlood leaves heart at ~ 30 cm/s

In capillaries, flow slows to ~ 1mm/sIn capillaries, flow slows to ~ 1mm/s– Surprising - continuity should imply higher Surprising - continuity should imply higher

flowflow

Equation of continuity, Bernoulli Equation of continuity, Bernoulli effecteffect

aa11 and a and a22 are areas of cross are areas of cross

section and vsection and v11 and v and v22 are are

velocitiesvelocities

If cross sectional area is large, If cross sectional area is large, velocity is low and pressure is velocity is low and pressure is highhigh

If cross sectional area of pipe is If cross sectional area of pipe is small, velocity is high and small, velocity is high and pressure is lowpressure is low

Cross sectional area of various Cross sectional area of various blood vesselsblood vessels

Linear velocity of blood (cm/s) Linear velocity of blood (cm/s)

With cross sectional area of 2.5 cmWith cross sectional area of 2.5 cm2 2 ,linear velocity of ,linear velocity of blood in aorta is 22.5cm/sblood in aorta is 22.5cm/s

On the other hand, in capillaries with cross sectional On the other hand, in capillaries with cross sectional area of 2500 cmarea of 2500 cm22, linear velocity of blood is simply , linear velocity of blood is simply 0.05cm/s0.05cm/s

Hence aorta has smallest Hence aorta has smallest cross sectional area but the cross sectional area but the mean flow velocity is highestmean flow velocity is highest

Each capillary is tiny, but Each capillary is tiny, but since the overall capillary since the overall capillary bed contains many billions bed contains many billions of vessel, it has total cross of vessel, it has total cross sectional area several sectional area several hundred times that of the hundred times that of the aortaand hence the mean aortaand hence the mean blood flow velocity falls blood flow velocity falls

several foldsseveral folds

Linear velocity of blood (cm/s) Linear velocity of blood (cm/s)

Vessel cross sectional area versus velocity of blood flowVessel cross sectional area versus velocity of blood flow

Vessel cross sectional area vs Vessel cross sectional area vs velocity of blood flowvelocity of blood flow

To understand the effect of cross sectional area on flow To understand the effect of cross sectional area on flow velocity, a mechanical model has been suggestedvelocity, a mechanical model has been suggested

Here a series of 1cm diameter balls are depicted as Here a series of 1cm diameter balls are depicted as being pushed down a single tube. The tube branches being pushed down a single tube. The tube branches into narrower tubes. Each tributary tube has a area of into narrower tubes. Each tributary tube has a area of cross section much smaller than that of the wider tube cross section much smaller than that of the wider tube

Suppose in wide tube each ball moves at 3cm/min . This Suppose in wide tube each ball moves at 3cm/min . This means 6 balls leave the wide tube per minuteand enter means 6 balls leave the wide tube per minuteand enter narrower tubesnarrower tubes

Obviously then these 6 six balls must leave the narrower Obviously then these 6 six balls must leave the narrower tubes per minute. This means each ball is moving at a tubes per minute. This means each ball is moving at a

slower speed of 1cm/minslower speed of 1cm/min

Vessel cross sectional area vs Vessel cross sectional area vs velocity of blood flowvelocity of blood flow

Special features of Blood FlowSpecial features of Blood Flow

Fahreus-Lindqvist Effect:Fahreus-Lindqvist Effect:

Relative viscosity of water, serum or plasma is not Relative viscosity of water, serum or plasma is not altered when they are made to flow through tubes of altered when they are made to flow through tubes of different sizesdifferent sizes

But the relative viscosity of blood is altered when it But the relative viscosity of blood is altered when it passes through tubes of different sizes i.e. blood flow in passes through tubes of different sizes i.e. blood flow in very minute vessels exhibit far less viscous effect than it very minute vessels exhibit far less viscous effect than it does in large vessels. This is called Fahreus-Lindqvist does in large vessels. This is called Fahreus-Lindqvist EffectEffect

This effect is caused by alignment of red blood cells as This effect is caused by alignment of red blood cells as they pass through vesselsthey pass through vessels