Biomedical Engineering Reference
In-Depth Information
Table 1 List of normalised model parameters
Parameter
Dimensionless
value
Parameter
Dimensionless
value
Parameter
Dimensionless
value
4
:
32
10
3
D
1
k
3
;
1
0.864
k
1
;
1
0.864
D
2
0.0864
k
1
;
2
2.592
k
2
;
2
0.864
D
3
4
:
32
10
3
k
3
;
2
4.32
u
3
eq
1.0
D
n
2
10
2
k
n
;
n
0.7488
r
n
1.664
u
h
1
0.25
K
10.0
K
1
2
4.1.1 Mathematical Model
We propose an angiogenesis model that incorporates biochemical factors, con-
sisting of a number of coupled reaction-diffusion equations. The primary variables
of the model are the concentrations of oxygen
ð
u
1
Þ
;
macrophage-derived growth
factors -such as VEGF or others, henceforth MDGF-
ð
u
2
Þ
;
capillary density
ð
u
3
Þ
and fibroblasts
ð
n
Þ
:
We have modified the model proposed by Javierre et al. [
61
],
based on the model of Maggelakis [
72
], in order to introduce the effect of fibro-
blasts in the process. The set of equations that describe the behavior of angio-
genesis is:
o
u
1
ot
¼
D
1
r
2
u
1
þ
k
3
;
1
u
3
k
1
;
1
u
1
ð
1
Þ
ou
2
ot
¼
D
2
r
2
u
2
þ
k
1
;
2
Q
ð
u
1
Þ
k
2
;
2
u
2
ð
2
Þ
o
u
3
1
u
3
u
e
e
ot
¼
D
3
r
2
u
3
þ
k
3
;
2
u
2
u
3
ð
3
Þ
ð
4
Þ
u
1
o
n
ot
¼
D
n
r
2
u
n
k
n
;
n
n
þ
r
n
n 1
n
1
u
1
K
1
K
where Q represents the function describing the production of MDGF when the
levels of oxygen are low,
Q
ð
u
1
Þ¼
1
u
1
(
if u
1
\u
h
2
1
;
u
h
2
1
ð
5
Þ
0
otherwise
:
It is known that fibroblasts are responsible for ECM synthesis, and serve as an
indicator of ECM maturity. Hence, as a novelty from the previous model [
61
], we
have incorporated the fibroblasts equation (Eq.
4
) in which fibroblast proliferation
is modelled with a logistic term with an oxygen dependent rate [
108
]. The values
of the (dimensionless) model parameters are included in Table
1
.
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