Biomedical Engineering Reference
In-Depth Information
!
1
1
e
0
:
15
R
C
¼
C
ð
R
Þ¼
:
þ
þ
þ
12
:
0
8
1
12
10
11
10
11
1
þ
ð
2
R
Þ
1
þ
ð
2
R
Þ
(3.20)
Using (
3.16
)-(
3.20
), we can calculate the flux through each vessel segment in terms
of the pressure at each junction of the vascular tree. At any node of the vascular
network, the total flow into that node must balance the total flow out of that node.
With the pressures at each inlet and outlet (
P
in
and
P
out
, respectively) prescribed,
we obtain a linear system of equations for the pressures at each vessel node. This
system is solved with the direct SuperLU solver.
2
When updating the vascular network, there are two different timescales of
interest, the timescale for flow and the timescale for vascular adaptation. While
changes in flow may be rapid, we assume that vascular adaptation occurs on the
same timescale as endothelial cell movement and cell division. Consequently, we
model the temporal evolution of a vessel segment's radius by applying the follow-
ing discretised ODEs
R
ð
t
þ D
t
Þ¼
R
ð
t
Þþa
R
D
tR
ð
t
Þð
S
h
þ
S
m
k
s
Þ;
(3.21)
where
Dt
is the timestep size and the updated radius must satisfy the constraint
R
min
a
R
that appears in (
3.21
) relates the stimuli
to our timestep size
Dt
. In the absence of any details on the rate of vascular
adaptation (since all previous studies of which we are aware consider quasi-steady
state vessel radii), we set
a
R
¼
R
(
t
þ Dt
)
R
max
. The factor
10
6
min
1
so that the rate of change of the
vessel radius is typically less than 10 % per hour.
k
s
is the shrinking tendency of a
vessel which takes into account that vessels tend to regress in the absence of stimuli.
S
h
and
S
m
are haemodynamic and metabolic stimuli for vascular adaptation:
•
Haemodynamic stimulus:
3
:
3
S
h
¼
log
ðt
w
þ t
ref
Þ
k
p
log
ððtð
P
ÞÞ;
(3.22)
with the WSS
R
D
P
L
;
t
w
¼
(3.23)
the constant reference WSS
t
ref
and the corresponding set point pressure of the
WSS
t
(
P
), described by the empirical function
5
:
4
tð
P
Þ¼
100
86 exp
5000 log
½
ð
log
P
Þ
:
(3.24)
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