Biomedical Engineering Reference
In-Depth Information
6.4 The pressure gradient across capillaries is relatively low, not more than 5 mmHg. However,
there is a large pressure gradient in the arterioles, which effectively forces the blood into
the microvascular beds. However, if one looks at the pressure gradient in small segments of
blood vessels, approximately 100-
μ
m lengths, the pressure gradient across the capillaries is
much larger than other blood vessels. Under disease conditions, the mean pressure within
capillaries remains the same. The changes in pressure under these conditions are largely
accounted for by the arterioles. The cardiac pressure pulse is marginally seen within the
microvasculature, but this usually does not confound results or analysis techniques.
6.5 The mean velocity in capillaries is the slowest of all blood vessels and can be approximated
by the blood cell velocity. Flow through the capillaries is dominated by the viscous fluid
effects and not the inertial effects (i.e., there is a low Reynolds number). Under these condi-
tions, the Navier-Stokes solution can be simplified to the Stokes equations, which are
0
@
1
A 5 μr
@
@
2 u
x 2 1 @
2 u
y 2 1 @
2 u
p
2 u
x 5 μ
@
@
@
@
z 2
0
@
1
A 5 μr
2 v
2 v
2 v
@
p
@
x 2 1 @
y 2 1 @
2 v
y 5 μ
@
@
@
@
z 2
0
@
1
A 5 μr
@
@
2 w
x 2 1 @
2 w
y 2 1 @
2 w
@
p
2 w
z 5 μ
@
@
@
z 2
or
2
4
0
@
1
A 1
3
5
@
@
@
r @
r 2 @
2 v r
r 2 @
1 @
2 v r
@
p
1
1
2
v θ
r 5 μ
r ð
rv r Þ
2
2
@
r
@
z 2
2
4
0
@
1
A 1
3
5
@
r @
r @
r 2 @
2 v z
1 @
2 v z
@
p
1
v z
@
1
z 5 μ
2
@
@
r
r
z 2
2
4
0
@
1
A 1
3
5
2 v θ
2 v θ
@
@
p
5 μ
@
@
1
r @
r 2 @
1
r 2 @
2
1 @
v r
r ð
rv θ Þ
2
r
@
2
z 2
The Stokes equations are relevant for many gas flows. Due to the cellular matter taking a
larger portion of the fluid cross-sectional area, the cellular elements may experience signifi-
cant drag forces which can be quantified using
6
πμ
rv
C D 5
1
=
2
ρ
v 2
π
r 2
assuming that the cellular elements are spherical.
6.6 The hydrostatic pressure within the capillaries and the interstitial space, as well as the col-
loidal osmotic pressure of the plasma and the interstitial space, dictate the net direction of
water movement into or out of the capillary. This can be formulated as
m
5
K p ð
P B 2
P I 2 Π B 1 Π I Þ
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