Biomedical Engineering Reference
In-Depth Information
flow
FIGURE 6.1
Schematic of the physical model and coordinate system.
and consequently to show the influence of the coils deployment on the veloc-
ity and pressure fields. The endovascular coil filling the sac of the aneurysm
was modeled using a porous medium approximation and the transport equa-
tions, commonly known as the “generalized model,” were solved to determine
flow characteristics in an idealized geometry as shown in Figure 6.1. More-
over, the porous medium is viewed as a continuum with the solid and fluid
phases in thermal equilibrium, isotropic, homogeneous, and saturated with
an incompressible fluid. The generalized model, which was obtained through
local volume averaging and matched asymptotic expansions, is also known as
the Brinkman-Forchheimer-Darcy model and is described in rigorous detail
by Amiri and Vafai [52, 53], Alazmi and Vafai [54], and Khanafer et al. [55].
These equations can be summarized as follows:
Continuity equation:
∇·
v
= 0
(6.1)
Momentum equation:
∂
=
ρ
f
ε
v
f
+
µ
f
µ
f
K
2
+
(
v
·∇
)
v
−∇
P
ε
∇
v
−
v
∂t
ρ
f
Fε
√
K
−
[
v
.
v
]
J
(6.2)
where
ε
is the porosity,
F
is the geometric function,
K
is the permeability,
µ
f
is the fluid dynamic viscosity,
J
=
v
p
/
|
v
p
|
is the unit vector along the pore
f
is the average
pressure. The medium permeability
K
can be properly modeled as shown by
Ergun [56] and Vafai [7,8]. The porosity of the coil may be used as an index
to determine the required minimal packing density of a coil and avoid the
danger of rupture from unneeded excessive coil packing. More experimental
studies are necessary to correlate the porosity of the coil geometry (thickness
and length), shape of the coil (i.e., complex-shaped coils or helical coils), and
the volume of the aneurysm sac.
The fluid motion within the parent artery is governed by the Navier-
Stokes equations with constant density and fluid properties, together with the
continuity equation. In a Cartesian coordinate with a fixed reference frame,
velocity vector
v
p
,
v
is the average velocity vector, and
P
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