Blood Flow in Arteries and Veins - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 151

where E is the elastic modulus of the vessel wall and h is the wall thickness. If more of the
assumptions are relaxed, we can develop more realistic approximations for the wave prop-
agation speed within an arterial wall, but that is beyond the scope of this discussion.

Example
Calculate the wave speed using Equations 5.17 and 5.18 to determine if the assumptions that
were made to obtain these formulas are acceptable. Assume that the pressure within a blood ves-
sel with a 12 mm radius and a thickness of 100 μ m is 100 mmHg. The compliance of this blood

vessel is 6.7 μ m/Pa, and the Young's Modulus is 15 kPa.

Solution

s

r i

s

Eh
4 pr i

c 5

c 5

2 α p

s

12 mm

s

15 kPa ð 100 μ m Þ

5 5 25 : 9cm = s 5 5 4 : 8cm = s

2 ð 6 : 7 μ m = Pa Þð 100 mmHg Þ 4 ð 100 mmHg Þð 12 mm Þ

Considering that we are only concerned with the flow at one instant in time and we are
neglecting the effects of slow velocity, the second approximation is more reasonable than
the first solution. In formulating Equation 5.17 , too many assumptions were made, and
this calculation for wave speed within the blood vessel wall is not close to the physiologi-
cal situation. In vivo , wave speeds are on the order of 1-10 mm/s so the second approxima-
tion is much more reasonable.
We will include a brief discussion on the dissipation of the pulsatility of the pressure
and velocity waveforms in arteries due to the reflection and transmission of wave energy
at bifurcations (and around turns). In large arteries, the discussion up to this point is rela-
tively valid because of the thickness of the vessel wall and the relatively small wave
amplitude as compared to wavelength. Consider a standard bifurcating blood vessel
( Figure 5.13 ). The parent branch not only contains the incident wave, but also contains a

FIGURE 5.13 Pressure wave reflectance and transmit-
tance at a bifurcation. The pressure pulse is reflected back
along the mother branch, but this is normally out of phase
with the next incident wave and therefore the entire pressure
would be reduced. The energy of the transmitted waves do
not summate to the energy of the incident wave.

Daughter wave 1
p d1

B

Incident wave
p p

O

A

Reflected wave
p r

C

Daughter wave 2
p d2