Biomedical Engineering Reference
In-Depth Information
gravitational and time-varying terms that contain r. Similarly, for Newtonian fluids, viscosity
m is constant. In summary, although we can write perfectly general equations of motion, the
difficulty of solving these equations requires making reasonable simplifying assumptions.
Two reasonable assumptions for blood flow in major vessels are those of Newtonian
and incompressible behavior. These assumptions reduce Eq. (4.59) to the Navier-Stokes
equations:
2
2
2
g
x
@
p
m
@
u
@
x
2
þ
@
u
@
y
2
þ
@
u
@
z
r
du
dt
r
@
x
þ
¼
ð
4
:
63
Þ
2
2
2
2
g
y
@
p
@
v
@
x
2
þ
@
v
@
y
2
þ
@
v
r
dv
dt
r
@
y
þ
m
¼
ð
4
:
64
Þ
@
z
2
2
2
2
g
z
@
p
@
w
@
x
þ
@
w
@
y
þ
@
w
@
z
r
dw
dt
r
@
z
þ
m
¼
ð
4
:
65
Þ
2
2
2
Blood vessels are more easily described using a cylindrical coordinate system rather than
a rectangular one. Hence, the coordinates
x, y
, and
z
may be transformed to radius
r
, angle y,
and longitudinal distance
x
. If we assume irrotational flow, y
¼
0 and two Navier-Stokes
equations suffice:
2
2
dP
r
dw
dt
þ
u
dw
dr
þ
w
dw
m
d
w
dr
1
r
dw
dr
þ
d
w
dx
dx
¼
2
þ
ð
4
:
66
Þ
2
dx
2
2
dP
r
du
dt
þ
u
du
dr
þ
w
du
m
d
u
dr
1
r
du
dr
þ
d
u
dx
2
u
dr
¼
2
þ
ð
4
:
67
Þ
dx
r
2
where
. Most arterial models
also use the continuity equation, arising from the conservation of mass:
du
w
is longitudinal velocity
dx/dt
, and
u
is radial velocity
dr/dt
dr
þ
u
r
þ
dw
dx
¼
0
ð
4
:
68
Þ
In essence, the net rate of mass storage in a system is equal to the net rate of mass influx
minus the net rate of mass efflux.
Noordergraaf and his colleagues [20] rewrote the Navier-Stokes Eq. (4.66) as
dP
dx
¼
RQ
þ
L
dQ
ð
4
:
69
Þ
dt
where
P
is pressure,
Q
is the volume rate of flow,
R
is an equivalent hydraulic resistance,
and
is fluid inertance. The Navier-Stokes equations describe fluid mechanics within the
blood vessels. Since arterial walls are elastic, equations of motion for the arterial wall are
also required. The latter have evolved from linear elastic and linear viscoelastic, to complex
viscoelastic (see [21]). The most general mechanical description of linear anisotropic arterial
wall material requires 21 parameters (see [10]), most of which have never been measured.
Noordergraaf and colleagues divided the arterial system into short segments and combined
the fluid mechanical equation (Eq. (4.69)) with the continuity equation for each vessel seg-
ment. The arterial wall elasticity leads to a time-varying amount of blood stored in the
L
