Biomedical Engineering Reference
In-Depth Information
The objective functions need to be carefully chosen to capture the physics of
the underlying problem. Regarding the choice of suitable objective functions for the
graft optimization problem, several different approaches have been pursued in the
literature [
2
,
12
,
14
]. The most frequently considered quantities in the context of
blood flow are based on either shear stress and its gradient or the flow rate. Many
authors choose to minimize the integral of the squared shear rate over the entire
simulated domain .
b
/. This integral is also called dissipation integral because it
measures the dissipation of energy due to viscous effects, expressed in terms of the
rate of strain tensor,
Z
1
2
:
2
dx
(25)
.
b
/
The minimization of this function is related to flow efficiency. Flow efficiency
can equivalently be measured by computing the maximum pressure variation in the
domain. In this project we chose to optimize the flow efficiency by minimizing the
pressure variation quantified as:
®
1
.
b
/
D
p
D j
p
max
p
min
j
(26)
The second chosen objective function is related to the minimization of reversed
flow and residence times along the arterial bypass system. For each idealized bypass
graft geometry simulation, a domain
.
b
/ has been identified indicating where
reversed flow and residence times are enhanced. Then, to minimize residence times
is equivalent to maximize the longitudinal velocity
u
x
in that critical domain.
The following objective function is considered for the minimization problem
investigated in this work,
X
®
2
.
b
/
D
u
x
(27)
.b/
The artery is simulated using a fixed diameter tube of 10 mm. Design parameters
are considered for the coupled graft presenting a sinusoidal geometry. The graft
mesh does not maintain the same width along its whole length. At the centre line
of the graft, nodes move in the radial direction preserving their distance to the
deforming centreline. The graft is properly connected to the artery always in the
same region but the graft diameter will vary. Due to the sinusoidal shape, the graft
artery junction is always larger than the width at the graft center line.
The developed computer program modelled blood flow in artery and graft
using 2,261 nodes and 2,024 four-node linear elements for a two-dimensional
finite element approximation. Figure
5
presents the geometry and finite element
mesh considered for the idealized arterial bypass system simulation with the flow
proceeding from left to right. The boundary conditions for the flow field are
parabolic inlet velocity corresponding to a Reynolds number equal to 300, no-slip
boundary conditions including the graft and a parallel flow condition at the outlet.
