Fundamentals of Haemodynamics - Computational Hemodynamics: Theory, Modelling and Applications - page 81

Fig. 4.12 Poiseuilles' Law that relates flow rate with pipe length, pressure, and radius

parabolic shape. Despite this, Poiseulle's Law provides a conceptual estimate to
relate pressure drop to flow rate.
To obtain the correlation we substituting

i

QU A = into Eq. 4.12 and the defini-

avg c

2

c A π

=

tion of a cross-sectional area of a pipe we get

æö

2 4

æ ö

× dP R p R dP

÷

ç

ç ÷

÷

2

ç

Q =- p R = - ÷

÷ ç

(4.17)

ç

÷ ÷

÷ ç

dx ç 8 m ÷ 8 m è ø dx

èø

Poiseuille's Law relates a steady laminar flow rate, Q through a rigid pipe, with the dif-
ference in pressure at the two ends P 1 and P 2 , the radius r and the length L of the pipe
(represented by dx in the equation), and the viscosity µ of the blood (Fig. 4.12 ). The
flow rate is linearly proportional to the pressure, inversely proportional to the length of
the pipe and viscosity, and increases to the power of 4 to the radius of the pipe.
The flow rate is defined as the average velocity of the fluid passing through
a cross-sectional area. An increase in the pipe length increases the overall resis-
tance so reduces flow through the pipe by half. Doubling the pressure increases the
amount of energy that drives the fluid motion, leading to a doubling of the flow rate.
The biggest significant factor on flow rate is the change in radius. Doubling the pipe
radius produces a 16-times increase in flow rate.
The significance of blood vessel radius reduction can be seen by considering a
small amount of arterial occlusion and its effect on the required pressure to main-
tain the same blood flow rate. Under a fixed pressure, an occlusion of 20 % reduces
the flow rate to 41 % of its original rate, and this is even worse when the occlusion
is 50 % where the flow rate is reduced to 6.3 %. To restore the normal flow rate
the corresponding percentage increase in pressure required are 244 % and 1600 %
(Fig. 4.13 ). Such high pressure changes are unlikely to be achieved, and instead
body has the ability to increase the volume flowrate by vasodilation of the small
vessels called arterioles. These smaller vessels provide most of the resistance to
flow, and play a critical role in delivering oxygen and nutrients to the body via the
capillaries that are next to them. A vasodilation of the arteries by 19 % will double
the blood flow.