Dynamics of Blood Flow

Transport System A closed double-pump system: Systemic Circulation Lung Circulation Left side of heart Right side of heart

Transport System Branching of blood vessels –Ateries branch into arterioles, veins into venules Arteries Arterioles Capillaries Venules Veins Heart

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

Hagen–Poiseuille law or Poiseuille law In fluid dynamics, the Hagen–Poiseuille equation is a physical law that states that for steady laminar flow of a Newtonian fluid through a cylindrical tube, the flow rate is directly proportional to the pressure drop, fourth power of radius of tube and inversely to the length and viscosity of fluid or or Where ∆P is the pressure drop L is the length of pipe, ƞ is the dynamic viscosity, Q is the volumetric flow rate and r is the radius of the pipe L r P1P1 P2P2  P= P 1 - P 2

Limitations of Poiseuille law The assumptions of the equation are A long rigid cylinder with length much greater than the radius A 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 turbulent Steady Laminar flow that is not pulsatile and turbulent The fluid velocity at the edges of tube is zero The fluid velocity at the edges of tube is zero Poiseuille law has certain limitations when applied to circulating blood in vivo Blood vessels are not rigid tubes and are quite distensible so that their size depends on the blood pressure within them as well as upon the contraction of smooth muscles in the vessel walls Blood vessels are not rigid tubes and are quite distensible so that their size depends on the blood pressure within them as well as upon the contraction of smooth muscles in the vessel walls Blood is a Non-Newtonian fluid and fluid viscosity is not constant Blood is a Non-Newtonian fluid and fluid viscosity is not constant The flow is not steady but pulsatile in most parts of the vascular bed The flow is not steady but pulsatile in most parts of the vascular bed

Blood volume flow rate Q Liquid flows along the lumen of a rigid tube from a higher to lower hydrostatic pressure In the vascular system, the rate of blood flow (volume/unit time) is proportional to the hydrostatic pressure gradient (∆P) across the vessel and inversely to the resistance (R) offered to its flow Analogous to Ohms law for electrical circuits (I=V/R) we can write Blood flow rate Q = ∆P / R = (P a -P v ) / R

Vascular Resistance Vascular resistance is a term used to define the resistance to flow that must be overcome to push blood through the circulatory system. The resistance offered by the peripheral circulation is known as the systemic vascular resistance (SVR) The systemic vascular resistance may also be referred to as the total peripheral resistance The systemic vascular resistance may also be referred to as the total peripheral resistance

Resistance R Resistance is dependent on the vessel’s dimensions and the viscosity of blood From Poiseuille law, The resistance decreases rapidly as r increases R = ΔP/Q = 8 L η / π r 4 A narrowing of an artery leads to a large increase in the resistance to blood flow because of 1/ r 4 term Vasoconstriction (i.e., decrease in blood vessel diameter) increases SVR, whereas vasodilation (increase in diameter) decreases SVR Vasoconstriction (i.e., decrease in blood vessel diameter) increases SVR, whereas vasodilation (increase in diameter) decreases SVR Peripheral resistance can be equated to DC resistance in electrical circuits

Arrangement of vessels also determines resistance. When the vessels are arranged in series, the total resistance to flow through all the vessels is the sum of individual resistances, whereas when they are arranged in parallel the reciprocal of the total resistance is the sum of all the reciprocals of the individual resistance Less resistance is offered to blood flow when vessels are arranged in parallel rather than in series

Volume Flow Rate Often convenient to define a resistance, R to flow, such that  P=QR P1P1 P2P2 P3P3 R1R1 R2R2 R3R3  P=  P 1 +  P 2 +  P 3 =QR 1 +QR 2 +QR 3 =QR  R=R 1 +R 2 +R 3 SeriesParallel R 1,Q 1 R 2,Q 2 Q=Q 1 +Q 2 =  P/R 1 +  P/R 2 =  P/R  R=1/R 1 +1/R 2

Resistances in series add directly while resistances in parallel add in reciprocals Arteries, arterioles, capillaries, venules and veins are in general arranged in series with respect to each other. However, the vascular supply to the various organs and the vessels e.g. capillaries within an organ are 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 flow Blood leaves heart at ~ 30 cm/s In capillaries, flow slows to ~ 1mm/s –Surprising - continuity should imply higher flow

Equation of continuity, Bernoulli effect a 1 and a 2 are areas of cross section and v 1 and v 2 are velocities If cross sectional area is large, velocity is low and pressure is high If cross sectional area of pipe is small, velocity is high and pressure is low

Cross sectional area of various blood vessels

Linear velocity of blood (cm/s) With cross sectional area of 2.5 cm 2,linear velocity of blood in aorta is 22.5cm/s On the other hand, in capillaries with cross sectional area of 2500 cm 2, linear velocity of blood is simply 0.05cm/s

Hence aorta has smallest cross sectional area but the mean flow velocity is highest Each capillary is tiny, but since the overall capillary bed contains many billions of vessel, it has total cross sectional area several hundred times that of the aortaand hence the mean blood flow velocity falls several folds Linear velocity of blood (cm/s)

Vessel cross sectional area versus velocity of blood flow

Vessel cross sectional area vs velocity of blood flow To understand the effect of cross sectional area on flow velocity, a mechanical model has been suggested Here a series of 1cm diameter balls are depicted as being pushed down a single tube. The tube branches into narrower tubes. Each tributary tube has a area of cross section much smaller than that of the wider tube Suppose in wide tube each ball moves at 3cm/min. This means 6 balls leave the wide tube per minuteand enter narrower tubes Obviously then these 6 six balls must leave the narrower tubes per minute. This means each ball is moving at a slower speed of 1cm/min

Vessel cross sectional area vs velocity of blood flow

Special features of Blood Flow Fahreus-Lindqvist Effect: Relative viscosity of water, serum or plasma is not altered when they are made to flow through tubes of different sizes But the relative viscosity of blood is altered when it passes through tubes of different sizes i.e. blood flow in very minute vessels exhibit far less viscous effect than it does in large vessels. This is called Fahreus-Lindqvist Effect This effect is caused by alignment of red blood cells as they pass through vessels