Biomedical Engineering Reference
In-Depth Information
an
(
−
1) /
a
µ µ
=+− +
(
µ µ
)[1
(
λγ
)
]
,
(7.6)
∞
0
∞
where
η
∞
is the viscosity for an infinite shear rate, and
η
0
is the plasma viscosity at
zero shear rate.
λ
,
n
, and
a
are fitting parameters, which are borrowed from the ex-
perimental data based on a well-tested blood-mimicking fluid (Gijsen et al. 1999a,
b). As an example, these parameters can have the following values:
η
∞
= 2.2 × 10
-3
Pa
s,
η
0
= 22 × 10
-3
Pa s,
λ
= 0.11 s, n = 0.392,
a
= 0.644, and
ρ
= 1410 kg m
−3
.
Estimates of patient-specific haemodynamics parameters have been attained
(Long et al. 2000; Steinman et al. 2002a). Some researchers employed non-New-
tonian fluid models to aptly characterize the rheological effect in blood flows
(Chen and Lu 2006; Gonzalez and Moraga 2005; Lee and Steinman 2007). How-
ever, many numerical investigations considered Newtonian fluid flows in rigid
stenosed arteries (Farmakis et al. 2004; Giannoglou et al. 2005; Siauw et al. 2000;
Zhao et al. 2002). In these studies, the fluid flow weres considered steady,
isothermal, incompressible and Newtonian, consistent with experimental studies
in the literature. Wall boundaries of the computational model were assumed rigid
and impermeable. The blood was Newtonian with a density
ρ
of 1050 kg m
−3
and
a dynamic viscosity μ of 3.5 × 10
−3
Pa•s (Ufuk Olgac et al. 2008).
Boundary conditions for the computational surfaces need to be defined. While
the surface walls are easily understood in terms of its definition as a rigid or elastic
wall boundary, the common carotid artery (CCA) inlet, as well as the internal ca-
rotid artery (ICA) and external carotid artery (ECA) outlets provide the user with
more options for definition. In this case study, a uniform flow perpendicular to the
CCA inlet was specified.
This is enforced as we are making direct comparisons with experimental results
that used a fixed flow rate. It is important to maintain similar settings and to keep
the flow rates the same between the computer and experimental model. A pressure
boundary condition could have also been used for modelling the blood pumping
cycle. Other boundary conditions commonly used are a velocity or mass flow rate
boundary, if they are known. The cardiac cycle is caused by the blood pressure dif-
ference induced by the heart chamber dilating or contracting to increase/decrease
the volume of the chamber. Therefore, it is natural to assign the CCA inlet, and the
ICA and ECA outlets as pressure conditions. The inlet pressure condition is used
to define the total pressure at the flow inlet. It is an ideal condition to use when the
pressure at the inlet is known but the flow rate and/or velocity is unknown. The
pressure outlets are used to define the static pressures at the flow outlets (and also
other scalar variables, in case of backflow). However, the problem with the pressure
outlets is that it is unknown in relation to the CCA inlet to induce the required flow
rate. A modelling strategy to overcome this is to first simulate the flow field with
the known mass flow rate in mind using a mass flow rate condition (Figure
7.15
).
Upon completion of the simulation, the pressure difference between the CCA inlet
and ICA outlet as well as between the CCA inlet and ECA outlet will be known, and
thus the pressure at the inlets and outlet can be defined.
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