Biomedical Engineering Reference
In-Depth Information
The differences
R
−
ρ
P
,
ρ
P
−
ρ
N
tend to stabilize, as
R
increases, to values
σ
∗
and obtainable by solving the much simpler system in plane
depending on
geometry.
3.1
The “Solid” Necrotic Core Model
We give here a short description of a simple model based on a two-phase approach,
assuming, as in Greenspan [
34
], that the necrotic core is “solid”. The necrotic
core will be then filled by dead cells whose local volume fraction
ν
N
is constant
while they are dissolving into liquid with a rate constant
μ
N
. Supposing that all
the components of the mixture have equal mass density and that
ν
N
=
ν
,themass
balance yields the following equations for
u
and
v
:
∇
·
u
=
χ
,
in
P
,
(7)
∇
·
u
=
0
,
in
Q
,
(8)
∇
·
u
=
−
μ
N
,
in
N
,
(9)
ν
∇
·
v
=
−
χ
−
ν
,
in
P
,
(10)
1
∇
·
v
=
0
,
in
Q
,
(11)
=
μ
N
ν
1
∇
·
v
−
ν
,
in
N
.
(12)
By multiplying Eqs. (
7
)-(
9
)by
ν
and Eqs. (
10
)-(
12
)by1
−
ν
, and summing up, we
obtain
∇
·
(
ν
u
+(
1
−
ν
)
v
)=
0
,
(13)
both in
P
Q
and in
N
.
In spherical symmetry the velocities are expressed by the scalars
u
∪
(
r
,
t
)
and
v
(
r
.
t
)
,
andsymmetryimposes
u
(
0
,
t
)=
0
,
v
(
0
,
t
)=
0
.
(14)
Therefore, taking into account the continuity of the velocities at
r
=
ρ
(
t
)
and at
N
r
=
ρ
(
t
)
, from Eqs. (
7
)to(
9
)and(
14
), the following expression for
u
can be
P
obtained:
⎧
⎨
−
μ
3
r
,
in N
,
3
N
(
t
)
r
2
ρ
−
μ
3
u
(
r
,
t
)=
,
in Q
,
(15)
⎩
3
r
r
2
−
μ
3
ρ
P
(
N
(
ρ
t
)
t
)
−
,
.
in P
r
2
From Eqs. (
13
)and(
14
) it follows that
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