Biomedical Engineering Reference
In-Depth Information
Fig. 1
Carotid artery bifurcation model
a function of
u
i
and h
i
element velocity and length, respectively, and the grid
Peclet number
Pe
e
. The SUPG-method produces a substantial increase in accuracy
as stabilizing artificial diffusivity is added only in the direction of the streamlines
and crosswind diffusion effects are avoided.
The resulting system of nonlinear equations is characterized by a non-symmetric
matrix, and a special solver is adopted in order to reduce the bandwidth and the
storage of the sparse system matrix; in addition the Skyline method is used to some
improvement of the Gauss elimination.
3.4
Carotid Bifurcation Model
The numerical example presented here is a 3D flow simulation in the human carotid
artery bifurcation. Figure
1
shows the geometrical model described by Perktold
[
11
,
18
]. The Navier-Stokes equations are solved using the Finite Element SUPG-
method with implicit Euler backward differences for time derivatives and Picard
iteration for nonlinear terms.
The non-Newtonian property of blood is important in the hemodynamic effect
and plays a significant role in vascular biology and pathology. In this study the
viscosity is empirically obtained using Casson law for the shear stress relation.
Considering
D
II
the second invariant of the strain rate and
c
the red cell concen-
tration, the shear stress given by the generalized Casson relation is:
k
1
.c/
q
2
p
D
II
p
D
k
0
C
(17)
and the apparent dynamic viscosity
D
(c,
D
II
), a function of the red cell concen-
tration,
k
0
C
k
1
.c/
q
2
p
D
II
2
1
2
p
D
II
D
(18)
where parameters
k
0
D
0.6125 and
k
1
D
0.174 were obtained fitting experimental
data, considering
c
D
45% and plasma viscosity
0
D
0.124 Pa s [
11
].
