Biomedical Engineering Reference
In-Depth Information
2.3.2 Flow Across Bifurcations
To evaluate the flow in an arterial tree, a bifurcation model needs to be further
incorporated. If we assume that the parent vessel is a, the two daughter vessels are
b and c, then according to the mass conservation law:
F
a
¼ F
b
þ F
c
(5)
where
F
stands for flow rate. If the length of the vessel segment a is
l
a
, the area of
the segment is
S
a
, then from Newton's second law of motion:
DP ¼ l
a
r
@
V
a
@t
(6)
Similar equations can be written for vessel segments b and c. These equations
are expanded using a central difference scheme about each time step. The resulting
equations, together with (
5
), are solved by using a Newton-Raphson iterative
scheme.
The numerical methods for (
1
)-(
5
) are implemented in our inhouse research
code HemoSim [
11
].
3 Results
3.1 Normal Arterial Tree
r
n
When solving the governing equations, the density
of the blood
were set as 1.05 g/cm
3
and 3.2 cm
2
/s, respectively. The initial velocity at all vessel
segments was 0 mm/s. The initial pressure was 10.6 kPa (80 mmHg). The data of
vessel wall elasticity
E
was taken from [
8
]. The spatial size
Dx
of the finite
difference grid was configured as 1 mm. The temporal step
and viscosity
Drt
was 0.1 millisec-
ond. The number of total time steps in a cardiac cycle (assumed to be 1 second)
therefore was 10,000. A physiologically realistic pressure waveform of the aorta
(adopted from [
8
]) ranging from 80 to 120 mmHg was used as the inflow boundary
condition. A fixed pressure of 80 mmHg was imposed at all outlets. The pressure
gradient between the inlet and outlets therefore drove the flow in the vascular
system.
It took about sixteen minutes to solve blood flow in the arterial tree over four
cardiac cycles. The computational results at the last cycle, which included vessel
radius, blood pressure and flow velocity (i.e.,
R
,
P
,
V
in (
1
), (
2
) and (
4
)) for all grid
points at every time step, were used for further numerical analysis. The pressure
distribution across the tree at 4 instants (T1, T2, T3, T4
0.2 s, 0.5 s, 0.7 s and
0.95 s) spans both systole and diastole, and is post-processed in Fig.
5
. It can be
¼
