Biomedical Engineering Reference
In-Depth Information
ν
v
(
r
,
t
)=
−
u
(
r
,
t
)
.
1
−
ν
The evolution of the outer radius is determined by the equation
R
(
t
)=
u
(
R
(
t
)
,
t
)
,
and a steady state exists for the
R
value such that
χ
R
3
)
=
μ
N
ˆ
3
P
3
N
−
ρ
ˆ
(
R
ρ
(
R
)
.
Note that at the interface
r
=
ρ
N
the velocity
u
is negative and
v
is positive: so, there
is continual
loss of liquid
from the necrotic core induced by the constraint
ν
N
=
N
v
ν
=
constant
. It can be easily verified that this volumetric loss 4
πρ
(
ρ
N
,
t
)(
1
−
ν
)
4
N
is equal to
0, so that all the
liquid mass necessary for cell proliferation comes from the necrotic core. Assuming
a first-order kinetics for the degradation of dead cells corresponds to supposing that
degradation occurs randomly according to the Poisson distribution, i.e. that the time
interval from cell death to cell dissolution is exponentially distributed with mean
value equal to 1
3
πρ
νμ
N
. At the steady state, it is
u
(
R
)=
v
(
R
)=
/
μ
N
.In[
12
], a Gamma distribution for the degradation time was
considered, and the distributed delay of cell dissolution was modelled by the passage
of dead cells through a chain of
n
equal stages with Poisson exit, the last one marking
the actual transition to the liquid waste. According to that model, Eq. (
9
) is changed
into the following set of equations:
∂ν
N
1
∂
ν
N
1
)=
−
μ
N
ν
N
1
,
t
+
∇
·
(
u
∂ν
N
2
∂
ν
N
2
)=
μ
N
ν
N
1
−
μ
N
ν
N
2
,
t
+
∇
·
(
u
.
∂ν
N
n
∂
)=
μ
N
ν
N
n
−
1
−
μ
N
ν
N
n
,
+
∇
·
(
u
ν
N
n
t
where
ν
N
i
(
r
,
t
)
is the local volume fraction of dead cells in the
i
-th subcompartment,
μ
N
is the exit rate constant from each subcompartment. For the
volume fractions of cells in different stages of death, the constraint
i
=
1
,...,
n
,and
∑
ν
N
i
=
ν
=
constant
is assumed, so that the velocity
u
can still be determined by
u
(
0
,
t
)=
0.
3.2
The NS/NL Partition of the Necrotic Core
In [
28
,
29
], as previously mentioned, we have assumed that the necrotic core at
the steady state is partitioned into two zones: a “solid” domain
NS
where cells are
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