Biomedical Engineering Reference
In-Depth Information
can only emerge if sufficient space is available. Around the base of each sprout, a
radius of exclusion is defined, in which new sprouts cannot occur. For the vessel tip
cells, we define
pð
i
!
j
Þ
as the probability of moving from
i
to
j
,tobe
c
VEGF
;
j
c
VEGF
;
i
d
ij
Dx
ð
N
m
N
j
Þ
þ g
1
Þ¼
D
tD
d
ij
D
k2O
i
ð
pð
i
!
j
N
m
M
c
;
(3.12)
N
m
N
k
Þþ
N
m
N
i
þ
x
2
for
i
2 O
i
. Herein,
D
represents the cell motility;
Dx
the CA-lattice size;
N
m
is
the maximal carrying capacity of the cell type attempting to move;
N
i
is the number
of cells;
M
c
is a constant and
c
VEGF,
i
the VEGF level at site
i
6¼
j
is the chemotactic
sensitivity and
O
i
is the set of sites in the neighbourhood of
i
, not including
i
itself.
The probabilities are weighted with the distance between lattice site
i
and
j
, written
as
d
ij
. In the three-dimensional case,
O
i
has at most 26 neighbour elements for each
lattice point
i
. We set the probability to zero if an endothelial cell crosses a vessel.
The probability of not moving is
g
X
pð
i
!
i
Þ¼
1
pð
j
!
k
Þ:
(3.13)
j
;
k
2 O
i
j
6¼
k
Whenever a tip cell moves to another location, an endothelial cell remains at
the former lattice site. This is equivalent to the snail-trail concept also applied in
[
5
,
34
,
23
]. A sprout dies if it does not connect to another sprout or the existing
vasculature within a certain time period.
3.2.3 Diffusible Layer
The diffusible layer facilitates the coupling between the vascular and subcellular
layers. We consider two diffusible components in our model, namely, oxygen and
VEGF, and denote their concentrations by
c
VEGF
and
c
02
, respectively. The vascular
system acts as an oxygen source, while the normal and tumour cells act as sinks.
This behaviour is described by the following, quasi-stationary, reaction-diffusion
equation:
2
c
02
þ
2
p R
c
blood
02
D
02
r
ð
t
;
x
Þ
P
02
ð
c
02
Þ
k
02
ð
t
;
x
Þ
c
02
¼
0
;
(3.14)
with the diffusion coefficient of oxygen,
D
02
.In(
3.14
), the vessel radius indicator
function
R
returns the vessel radius if a vessel is present at position
x
and zero
otherwise. Equation (
3.14
) also accounts for the vessel permeability to oxygen
(
P
02
), the blood oxygen concentration
c
02
blood
and the cell-type-dependent oxygen
consumption rate
k
02
. If cells become hypoxic or quiescent, they start to secrete
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