Biomedical Engineering Reference
In-Depth Information
R
3
r
+
2
R
2
r
2
2
−
χ
3
p
E
(
r
)=
p
ext
+
,
for
ρ
P
≤
r
≤
R
,
(44)
3
K
(
1
−
ν
)
1
r
−
R
3
3
p
E
(
ρ
P
)+
χ
(
−
ρ
P
)
1
ρ
P
p
E
(
r
)=
,
for
ρ
D
≤
r
<
ρ
P
,
(45)
3
K
(
1
−
ν
)
R
3
r
+
2
R
2
r
2
2
−
2
R
−
χ
3
K
3
p
C
(
r
)=
p
+
,
for
ρ
P
<
r
<
R
,
(46)
ν
1
r
−
R
3
3
P
p
C
(
ρ
P
)
−
χ
(
−
ρ
)
1
ρ
P
p
C
(
)=
,
<
<
ρ
.
r
for
ρ
r
(47)
D
P
3
K
ν
The jump relation
4
3
η
C
χ
p
C
(
ρ
P
)
−
p
C
(
ρ
P
)=
(48)
is generated by the Dirac distribution in Eq. (
34
), that we have already commented.
In Eq. (
44
)
p
ext
is the value taken by
p
E
at
r
R
, supposed to coincide with the
pressure of the water component in the outer medium (which can be just water or
a gel, containing in any case a preponderant water fraction). In Eq. (
46
)
=
γ
denotes
R
, that needs
some explanation. Its value has to be found by imposing the balance of normal stress
when passing from the spheroid to the external medium. This operation may not be
trivial.
If the spheroid is grown in water, then the external normal stress reduces to
the pressure
p
ext
. If the outer medium is a gel, then on the cells there will be an
extra action due to the deformation of the polymer network making the skeleton
of the gel. Spheroids which are subjected to such an extra compression have
been reported to exhibit a reduced growth [
36
]. This question would deserve a
deeper investigation, since it can be related to inhibition of proliferation (which
would make the proliferation
2
the surface tension, and a new quantity appears, namely
p
=
p
C
(
R
)
−
depend on pressure [
22
]), but can also have an
independent mechanical origin, as we shall see. It has to be emphasized that the
extra compression, coming from the solid component of gel, acts only on the solid
component of the spheroid (namely the cells, despite their schematization as a fluid).
Similarly, surface tension acts exclusively on the cells.
Thus, while we just have pressure continuity for the liquid component, the boundary
condition for
p
C
can be stated by imposing the following jump condition to the total
stress
T
, relative to radial direction (see, e.g. [
40
]):
Te
χ
e
r
=
R
=
−
ν
2
R
,
·
(49)
where
e
is the radial unit vector. Equation (
49
) is equivalent to
p
C
2
3
η
2
R
−
C
u
(
−
ν
(
R
)+
χ
+
2
νη
R
)
−
(
1
−
ν
)
p
ext
=
−
ν
p
gel
−
ν
p
ext
,
(50)
C
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