Biomedical Engineering Reference
In-Depth Information
supposed to keep the mechanical properties they had before death and the same
volume fraction
and an inner liquid core
NL
. As we said in Sect.
2
, the actual
presence of the latter structure appears to have some experimental support. As a
matter of fact, by “liquid” we mean a mixture that may contain solid fragments and
macromolecules. The important feature from the mechanical point of view is that
the stress, in static condition, is isotropic. In the biological literature we may find
evidences of more complex states, like, e.g. coagulative necrosis [
45
],which would
require, however, much more complicated constitutive equations.
Dead cells are supposed to degrade into liquid after a fixed time,
ν
θ
D
, upon death.
Such an assumption makes a new interface appear,
r
=
ρ
D
, dividing
NS
from
NL
.
ρ
D
at the stationary state, it is necessary to calculate the velocity
field
u
of the cells in
P
In order to find
∪
Q
∪
NS
. From the mass balance, we have the system
∇
·
u
=
χ
,
in
P
,
(16)
∇
·
u
=
0
,
in
Q
∪
NS
,
(17)
ν
∇
·
v
=
−
χ
−
ν
,
in
P
,
(18)
1
∇
·
=
,
∪
,
v
0
in
Q
N
(19)
which keeps into account the incompressibility of the mixture, i.e
∇
·
[
ν
u
+(
1
−
ν
)
is unknown, and this fact makes it
impossible
to
determine the evolution of the spheroid and the stationary radius by means of
a purely kinematic approach. This was instead possible in the case of a “solid”
necrotic core because in such a case we could impose
u
v
]=
0. Note that
u
(
ρ
D
,
t
)
(
0
,
t
)=
0.
By imposing the global flux continuity at
r
=
ρ
D
, namely
(
ρ
D
)=
ν
(
ρ
D
)+(
(
ρ
D
)
,
v
u
1
−
ν
)
v
(
ρ
D
)=
since
v
(
0
)=
0, which holds by symmetry, and Eq. (
19
) implies
v
0, we get
(
ρ
D
)+(
(
ρ
D
)=
ν
u
1
−
ν
)
v
0
.
Thus, for any
r
∈
(
ρ
D
,
R
)
we have
ν
u
+(
1
−
ν
)
v
=
0
,
(20)
i.e. a global no flux condition holds. Therefore, at the steady state both
u
and
v
vanish at
r
R
. Note that having taken the same density for the cells and for the
liquid, proliferation and degradation do not imply volume changes.
Since at the steady state
u
is zero on
r
=
=
R
, the radial component
u
(
r
)
of the cell
velocity, for a given
R
, can be easily computed:
)=
−
3
r
2
(
R
3
r
3
u
(
r
−
)
,
for
ρ
P
<
r
<
R
,
(21)
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