Biomedical Engineering Reference
In-Depth Information
3
50
45
2.5
40
2
35
30
1.5
25
1
20
15
0.5
10
0
5
-0.5
0
-100
-50
0
50
100
-100
-50
0
50
100
Radius
Radius
80
1600
70
1400
60
1200
50
1000
40
800
30
600
20
400
10
200
0
0
-10
-200
-100
-50
0
50
100
-100
-50
0
50
100
Radius
Radius
Fig. 6
Simulations of the time evolution of the profiles of type 2 cell (
up-left
), type 1 cells (
up-
right
), VEGF
g
(
down-left
) and nutrient
u
(
down-right
) densities of the fully deterministic model
4. Nutrient
u
(down-right) and growth factor
g
(down-left) concentrations reflect
the densities of cells by which they are produced: diffusion makes their profiles
much smoother.
The time evolution shows how type 2 cells move progressively from the center to the
frontier; we may also notice the expansion of the vascularization by building new
vessels at the boundary region without a significant modification of the profile in
the central area. In the internal part the nutrient concentration
u
increases (supplied
to the tissue cells) and the growth factor starts to decrease because type 2 cells
have moved towards the frontier. Hence, if we were not interested in the detailed
geometric structure of the capillary network, the deterministic model would be able
to catch some of the relevant features of the qualitative behavior of the phenomenon
under study.
The Hybrid System.
Let us consider now the hybrid model in which the stochastic
evolution Eqs. (
59
)and(
60
) of the system of cells
X
k
C
k
)
}
k
is driven with the
deterministic dynamics Eqs. (
54
)and(
55
) of the underlying fields
g
{
(
,
,
u
,
obtained as
a result of the mesoscale averaging.
Clearly, in these simulations, we may use the same numerical results for the
underlying fields obtained above. Hence, as before, the homogenized underlying
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