Blood Flow in Arteries and Veins - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 150

where r i ,0 is the radius of the blood vessel at a zero external pressure and is a constant

that describes the material properties of the blood vessel. Equations 5.10, 5.11, and 5.12
govern the wave propagation speed throughout a blood vessel (with the assumptions that
we have made, note that Equation 5.12 is overly simplifying the biological mechanical
properties/responses to forces.). Substituting the area into Equation 5.11 and remembering
that the fluid velocity is relatively slow, Equations 5.10 and 5.11 simplify to

α

1
ρ

@ p

x 1 @

u
@

0 ð 5 : 13 Þ

5

@ t

r i Þ r i

@

5 @ðπ 1 @

@

r i @

r i
@

π @ r i @

u

r i @

r i
@

r i @

u
@

2

r i @

r i
@ t 1 @

u

r i Þ 5 0

x ð

u π 2 π u

x 1 π

2 π

t 1 π

ð 5 : 14 Þ

t 1 x 5 x 5 @ t @ @ x

For clarity, the two appropriate assumptions are that the blood velocity is close to zero
( u

0) and that the wave amplitude is much smaller than the wavelength. Combining
Equation 5.12 with Equation 5.14 , we get

5

r i @

2 r i

@ t 1 @

u
@

2

r i @

Þ 1 @

u 2 r i @ p

@ t 1 @

u

0

t ð

r i; 0

1 α

p ð 5 : 15 Þ

5

x 5 x 5 @ @ @ x

To solve this in the form of the wave equation, we must take the temporal derivative of
Equation 5.15 and subtract this from the spatial derivative of Equation 5.13 . With some
algebraic modifications of the proportionality constants, we can get the wave speed (c):

2 p 2 u

@
@

1
ρ

@ p

x 1 @

u
@

1
ρ

@

x 2 1 @

0 0

5 - 5

x @ t @ @ x @ t

@
@

r i @ p

@ t 1 @

u r i @

2 p

t 2 1 @

2 u

2 2

0 0

5 - 5

t @ x @ @ t @ x

2 p 2 u 2 p

@

2 u 2 p 2 p

@

2 p 2 p

@

1
ρ

@

x 2 1 @

2 r i @

t 2 2 @

1
ρ

@ 2 r i @

t 2 5 @

2 αρ

r i

@

0 ð 5 : 16 Þ

5

x 2 x 5 x 2 2 x 2 2 t 2

@ @ t @ @ t @ @ @

Putting Equation 5.16 into the form of the wave equation,

2 p 2 p

@

5 @

c 2 @

1

0

x 2 2 t 2 @

r

r i

c ð 5 : 17 Þ

5

2 α p

Using the same analysis we can derive the equation for wave speed in various types of
blood vessels. For instance, if the blood vessel wall is purely elastic and relatively thin
compared to the diameter, then the wave speed is

s

Eh
4 pr i

c ð 5 : 18 Þ

5