Biomedical Engineering Reference
In-Depth Information
In order to solve this equation, a substitution can be made: so that Eq.
7
becomes:
d
2
u
dr
2
þ
1
du
dr
þ
i
3
n
u
¼
A
l
:
ð
9
Þ
r
t
This equation can now be solved with a Bessel function of order zero:
!
J
1
ai
3
=
2
A
ixq
u
¼
1
ð
10
Þ
ai
3
=
2
J
o
ai
3
=
2
ð
Þ
where J
0
and J
1
are Bessel functions of the order zero and one, and a
¼
R
p
or
the Womersley number. The function in the brackets was termed 1
F
1
½
and the
values were tabulated by Womersley. Then, allowing the real portion of the
pressure gradient to be M cos xt
/
x
=
t
ð
Þ;
we get a flow rate of:
Q
¼
pR
2
xq
M 1
F
10
½
sin xt
/
ð
Þ:
ð
11
Þ
Equation
8
can be put in a more physical form by putting it in terms of the
modulus
and the phase e
1
ð ;
these values were tabulated by Womersley
based on the Womersley Number.
M
0
10
Q
¼
pR
4
M
l
M
10
a
2
sin xt
/
e
10
ð
Þ:
ð
12
Þ
For a parallel plate flow chamber this equation becomes:
Q
¼
h
3
wM
l
M
10
a
2
sin xt
/
e
10
ð
Þ:
ð
13
Þ
6 Endothelial Cell Physiology
Endothelial cells (ECs) line the inner surface of blood vessels throughout the entire
cardiac circulation; they form a layer between the circulating blood lumen and the
rest of the arterial wall. ECs are continuously exposed to shear stress, pressure and
strain. Shear stress is the most influential factor on ECs; the application of shear
orients the cells in the direction of flow and affects their functions. The endothelium
is essential for vasodilatation and vasoconstriction in response to blood flow related
shear stress [
45
]. The endothelium normally releases vasoconstrictors and growth
promoting factors or vasodilators and growth-inhibiting factors to regulate vascular
homeostasis. The release of these factors enables the ECs to regulate underlying
vascular SMC behavior and contraction. These mediators are thought to control
pulmonary vascular tone, homeostasis and vascular injury repair and growth.
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