Biomedical Engineering Reference
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where we have denoted by
p
gel
the pressure selectively exerted on cells by the gel
polymeric network. Such a quantity should be represented by a monotone function
of
R
, stabilizing to an asymptotic finite value, since the network deformation can
only have a somehow localized influence. Here
p
ext
is atmospheric pressure. We
recall that
u
(
R
)=
χ
. Hence Eq. (
50
) implies
4
3
η
C
χ
.
p
=
p
gel
+
p
ext
+
(51)
=
ρ
Next we turn our attention to the normal stress continuity at
r
D
,whichtakes
the form
(
ρ
D
)
−
C
u
(
ρ
D
)+(
−
ν
)
(
ρ
)=
(
ρ
)
,
ν
p
C
2
νη
1
p
E
p
E
(52)
D
D
i.e.
(
ρ
D
)=
C
u
(
ρ
D
)
,
(
ρ
)+
p
C
p
E
2
η
(53)
D
finally leading, for
p
gel
=
0, to
R
ρ
D
1
3
1
2
ρ
P
R
ρ
P
R
R
3
3
K
1
χ
3
2
2
γ
=
−
−
−
ν
(
1
−
ν
)
R
R
ρ
D
3
1
3
ρ
P
R
4
3
η
C
χ
+
−
.
(54)
The study of Eq. (
54
) has been performed in [
29
] (actually a slightly different
expression was used there, corresponding to a simplified definition of
p
). The right-
hand side of Eq. (
54
) is a function of
R
tending to infinity both for
R
→
∞
and in
correspondence of the critical value of
R
for which
ρ
D
→
0 and below which the
interface
r
=
ρ
D
is not defined. There is only one minimum, which we may call
γ
∗
, which defines a critical value of the surface tension, discriminating between
existence and non-existence of a steady state. Thus the problem of finding
R
is
solvable if and only if
2
γ
>
γ
∗
.In[
29
] we found that a value slightly greater than
0
.
01 dyne/cm for
γ
is compatible with our reference situation with
R
=
1 mm.
05 dyne/cm, if
K
is reduced to 10
−
8
cm
3
s/g.
The value of
γ
increases to about 0
.
γ
>
γ
∗
Eq. (
54
) has two solutions. Since the spheroid grows to a
steady state from a small initial size, we can say that the physical solution is the
smaller.
Clearly when
Remark 5.2.
In the liquid-dominated case (see Remark
5.1
) the term in Eq. (
54
)
containing viscosity can be neglected. In such a case the solution is going to depend
just on the product
K
γ
.
Remark 5.3.
Equation (
54
) shows that, keeping
γ
fixed, equilibrium becomes
impossible when
η
C
is raised above some threshold. This fact has a physical
interpretation. Indeed a steady-state configuration requires that all cells possess
a radial inward directed velocity. If viscosity is too large, the inward motion is
hindered and the spheroid tends to grow indefinitely to the exterior.
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