Biomedical Engineering Reference
In-Depth Information
concluded that predicted shear stress magnitude resulted in differences on the order
of 10% as compared with Newtonian models [
11
].
A non-Newtonian viscosity model for simulating flow in arteries is presented
in this chapter. Considering blood flow an incompressible non-Newtonian flow
and neglecting body forces, the fluid flow is governed by the incompressible
Navier-Stokes equations. There are two potential sources of numerical instability
in the Galerkin finite element solution of these equations. The first one is due to
the numerical treatment of the saddle-point problem arising from the variational
formulation of the incompressible flow equations. The second difficulty is related
to the discretization of the nonlinear convective terms which requires the use of
stabilized finite element formulations to properly treat high Reynolds number flows.
Some of the approaches for the numerical solutions are presented in this study.
To demonstrate the application of the developed finite element technique for
numerical simulations of blood flow in arteries, a flow simulation in the human
carotid artery bifurcation and a search for an optimized geometry of an artificial
bypass graft is addressed here. In medical practice, bypass grafts are commonly
used as an alternative route around strongly stenosed or occluded arteries. When the
arterial flow is high, artificial grafts perform well [
2
,
12
,
13
] and it has been shown
over the years that they provide durable results. Su [
2
] investigated the complexity
of blood flow in the complete model of arterial bypass. He found that flow in the
bypass graft is greatly dependent on the area reduction in the host artery. As the
area reduction increases, higher stress concentration and larger recirculation zones
are formed bringing out the possibility of restenosis. Probst [
14
] concluded that
computing derivatives of the flow solution (and related quantities like shear rate)
with respect to viscosity could not reveal the sensitivity of the optimal graft shape to
the fluid model. So, optimization should be applied to the entire framework, which
would enable to actually compute the optimal shape. In the present work a multi-
objective optimization framework is presented. A genetic algorithm coupled with
the developed finite element methodology for blood flow simulation is considered in
order to reach optimal graft geometries. Numerical results show the benefits of shape
optimization in achieving design improvements before a bypass surgery, minimizing
recirculation zones and flow stagnation.
2
Governing Equations
A number of important phenomena in fluid mechanics are described by the Navier-
Stokes equations. They are a statement of the dynamical effect of the externally
applied forces and the internal forces due to pressure and viscosity of the fluid. The
time dependent flow of a viscous incompressible fluid is governed by the momentum
and mass conservation equations, the Navier-Stokes equations given as:
@
u
u
@t
C
u
:
r
Dr C
f
r
:
u
D
0
(1)
