Biomedical Engineering Reference
In-Depth Information
Vessel Extension.
The movement (extension) of tips follows a Langevin model; at
any
t
T
i
>
and for any
k
∈{
1
,...,
N
(
t
)
}
we have
d
X
i
v
i
X
i
(
t
)=
(
t
)(
1
−
γ
I
X
(
t
)
(
(
t
))
d
t
,
)=
−
F
C
))
d
t
d
v
i
kv
i
X
i
X
i
d
W
i
(
t
(
t
)+
(
t
,
(
t
))
,
f
(
t
,
(
t
+
σ
(
t
)
.
(8)
kv
i
The advection term includes the typical inertial component
−
(
t
)
, while, accord-
ing to a typical chemotaxis, velocity
v
i
is driven by a function
F
of the underlying
fields, the TAF
C
and the fibronectin
f
.Asin[
34
,
36
], we take the bias depending
on the TAF and the ficronectin fields
F
C
(
t
)
))
=
X
i
X
i
X
i
X
i
(
t
,
(
t
))
,
f
(
t
,
(
t
d
C
(
C
(
t
,
(
t
)))
∇
C
(
t
,
(
t
))
X
i
X
i
+
d
f
(
f
(
t
,
(
t
)))
∇
f
(
t
,
(
t
))
.
d
C
,
d
f
,
are turning coefficients, modelled as follows:
∇
X
k
C
(
t
,
(
t
))
X
k
d
C
(
C
(
t
,
(
t
))) =
d
1
q
2
,
X
k
(
1
+
q
1
C
(
t
,
(
t
))
d
2
∇
,
X
k
X
k
d
f
(
f
(
t
,
(
t
))) =
f
(
t
,
(
t
))
with
q
1
,
0. So the reorientation of cells increases as a function of the magnitude
of the chemotactic, haptotactic gradient; furthermore cells become desensitized
to chemotactic gradients at high attractant concentrations, as stressed in [
1
,
23
].
In Eq. (
8
) the parameter
q
2
≥
γ
may assume only the values 0 and 1;
γ
=
0 means
that no impingement is considered; otherwise, for
1 the phenomenon of
anastomosis is taken into account (for further information see [
7
] and references
therein).
γ
=
Vessel Branching.
In literature two kinds of branching have been identified, either
from a tip or from a mature vessel (see, e.g., [
1
,
25
,
34
]). Here we describe
explicitly the branching processes from a mathematical point of view only for the tip
branching. The reader may refer to [
7
] for a possible mathematical modelling of the
mature vessel branching. We consider, as it is always supposed, that the branching
rates depend on the field of concentration of TAF. The tip vessel branching is
described by a process
Φ
=
∑
i
ε
(
T
i
,
X
i
)
on the
σ
-algebra
B
(R
+
,
X
i
,i.e.,amarked
)
counting process such that, for any measurable set
A
⊆B
R
+
×
R
d
,
=
∑
i
ε
(
T
i
,
X
i
)
(
A
)=
card
{
n
:
(
T
n
Y
n
Φ
(
A
)
:
,
)
∈
A
}
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