Biomedical Engineering Reference
In-Depth Information
1
2
v
v
d
d
t
u
K
·
δ
(
1
−
ν
)
·
=
−
(
1
−
ν
)
v
·
∇
p
E
+
v
u
·
u
ν
=
−
(
1
−
ν
)
v
·
∇
p
E
−
−
ν
,
(39)
K
1
where again the left-hand sides can be neglected. Note the presence of the term
−
2
=
ρ
P
.Sucha
singularity is a consequence of the extreme schematization of the transition
P
3
νη
C
∇
χ
·
ˆ
u
, producing a Dirac distribution centred at the interface
r
Q
linked to a threshold of oxygen concentration. In a model with a gradual transition
the jump of ˆ
→
would be replaced by some steep variation, corresponding to a peak
in the derivative, but with no substantial change in the qualitative behaviour. A very
similar remark can be made for the Laplacian of
u
.
Since Eqs. (
38
)and(
39
) derive from the momentum balance, in view of our
purpose we do not have a new piece of information. In [
28
] the equations above
have been used to recognize the dissipation terms, namely
w
C
(
χ
νη
C
D
C
:
D
C
,
representing the power dissipated per unit volume because of cell-cell friction, and
w
E
(
r
)=
2
, due to liquid-cell friction (the other terms are either
negligible or represent power production or transmission). Dissipation due to the
conversion of liquid into cells in the proliferating zone can be checked to be
absolutely negligible compared to
w
C
and to
w
E
. Thus it is possible to calculate
the power globally dissipated,
W
diss
, by summing the two contributions integrated
over the spheroid.
The explicit expression of the cell-cell friction dissipation term is
r
)=
u
·
u
/
[
K
(
1
−
ν
)]
νη
C
χ
−
r
2
νη
C
r
2
2
2
r
2
2
u
2
6
u
2
2
4
r
χ
+
+
=
2
χ
−
,
in
P
,
w
C
(
r
)=
(40)
νη
C
u
2
12
,
in
Q
∪
NS
.
r
2
Integrating over the spheroid, the global cell-cell friction dissipation power is
obtained as
2
R
3
y
3
y
3
2
ρ
1
y
3
1
y
3
1
3
16
9
πνη
C
χ
1
2
1
1
P
ρ
D
W
C
=
−
+
+
−
−
,
(41)
where
y
=
R
/
ρ
P
>
1. The global contribution of liquid-cell friction is
R
5
y
1
y
2
1
y
5
2
4
πχ
1
1
5
1
W
E
=
−
1
−
−
+
−
9
K
(
1
−
ν
)
1
y
1
y
3
2
ρ
P
1
+
−
D
−
.
(42)
ρ
The dissipated power
W
diss
=
W
C
+
W
E
results therefore a function of
R
.
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