Biomedical Engineering Reference
In-Depth Information
portion of the wave that is reflected backward into the parent branch. Consider this the
portion of the velocity that travels perfectly straight and hits the perfectly sharp point
where the two daughter branches meet for this simplified geometry. Each of the two
daughter branches will contain a portion of the wave that is transmitted. There will be
some energy loss because of the energy used when the wave hits the vessel wall, rebounds
off the wall, and causes a distortion to the wall. Mathematically, the pressure must have a
single value at point
O
in
Figure 5.13
, and the flow velocity must be continuous. Therefore,
p
p
2
p
r
5
p
d
1
1
p
d
2
ð
5
:
19
Þ
and
Q
p
2
Q
r
5
Q
d
1
1
Q
d
2
ð
5
:
20
Þ
The volumetric flow rate
Q
can be quantified as
A
ρ
Q
Au
c
p
ð
5
:
21
Þ
5
5
under these conditions, where
is the blood density,
c
is the wave speed, and
p
is the
pressure. The characteristic impedance of the blood vessel is defined as
ρ
5
ρ
c
A
Z
ð
5
:
22
Þ
Interestingly, if we use this new relationship for pressure driven flow, we can calculate
the amplitudes of the pressure waveforms immediately after a simple two-dimensional
bifurcation.
ZQ
p
5
p
p
p
r
Z
p
5
p
d
1
Z
d
1
1
p
d
2
Z
d
2
2
ð
5
:
23
Þ
Therefore,
Z
2
1
p
Z
2
1
d
1
Z
2
1
d
2
2
ð
Þ
1
p
r
p
p
5
ð
5
:
24
Þ
Z
2
1
p
1
ð
Z
2
1
d
1
Z
2
1
d
2
Þ
1
2
Z
2
1
p
p
d
1
p
p
5
p
d
2
p
p
5
ð
5
:
25
Þ
Z
2
1
d
1
Z
2
1
d
2
Z
2
1
p
1
ð
Þ
1
Equations 5.23 through 5.25
show that there will be a reduction in the pressure pulses
through the arterial system, because of energy that is reflected back into the parent branch
and is reflected on the daughter branches.
5.8 FL
OW SEPARATION AT BIFURCATIONS AND AT
WALLS
Before we move into the discussion of flow separation, it is important to first discuss
some of the important flow considerations at bifurcations. We have already learned that a
very efficient way to control blood flow through the vascular system is through small
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