Biomedical Engineering Reference
In-Depth Information
•
Metabolic stimulus:
!
Q
ref
S
m
¼
k
m
ð
c
VEGF
Þ
a Q
ref
þ
;
log
1
(3.25)
QH
þ
with
c
VEGF
V
0
þ
k
m
k
VEGF
m
k
m
ð
c
VEGF
Þ¼
1
þ
;
(3.26)
c
VEGF
where
Q
ref
,
a
,
k
m
0
,
k
m
VEGF
and
V
0
are parameters.
Q
ref
is a reference flow, and the
term
a Q
ref
(where
a
is a small parameter) in the denominator of (
3.25
)is
introduced to avoid extreme vessel dilation in poorly perfused vessels (and
hence,
S
m
differs slightly from the original model [
20
,
25
]).
In addition to being created, new vessels can also be removed by pruning. If a
vessel is exposed to a WSS that is below a threshold (
t
w
crit
) for a certain time
(
T
prune
), we remove that vessel from the system. In general, the vascular adaptation
algorithm includes the following steps. First, the vessel radii evolve according to
(
3.21
). Then we calculate the flows in the network. In contrast to the previous model
[
20
], the evolution of the radius is considered separately and is not iterated until a
steady state is reached.
3.2.5 Computational Algorithm
The basis of our model is a regular grid that subdivides the simulation domain into
cellular automaton lattice sites. Each lattice site can be occupied by several
biological cells and vessels. Figure
3.1
shows the high degree of coupling between
the different spatial scales. We enumerate the main steps below:
1.
Initialisation (vascular and cellular layers)
We specify an initial vascular
network as a system of straight pipes with fixed inflow and outflow nodes and
prescribed pressures. We also prescribe the amount of haematocrit that enters
through each inlet, and the initial location of the different cell types in the
cellular automaton domain.
2.
Update cells, oxygen and VEGF (diffusible, cellular and subcellular layers)
•
Calculation of oxygen concentration (diffusible layer)
The reaction-
diffusion equation (
3.14
) is used to calculate the quasi-stationary oxygen
concentration
c
02
(
t
,
x
) in the simulation domain. Oxygen consumption by
normal and cancer cells is included in (
3.14
) via sink terms, assuming first-
order kinetics. On the other hand, perfused blood vessels deliver oxygen to
the tissue and thus account for oxygen sources.
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