Biomedical Engineering Reference
In-Depth Information
3
The Internal Structure of a Multicellular Spheroid
We assume the spheroid be a mixture of two components, cells, and extracellular
liquid, whose velocities are denoted respectively by
u
and
v
. The local volume
fractions of living cells, dead cells and extracellular liquid are denoted by
ν
C
,
ν
N
and
ν
E
respectively. Assuming no voids, we have
1. We will consider
oxygen as the limiting nutrient, so oversimplifying the description of metabolism in
order to concentrate on the mechanical aspects. In spherical symmetry, we denote by
σ
(
ν
C
+
ν
N
+
ν
E
=
the oxygen concentration,
r
being the radial distance from the spheroid centre
and
t
the time. Let
R
r
,
t
)
be the outer radius of the spheroid. To gain some conceptual
simplification, it is convenient to divide the spheroid into spherically symmetric
domains, separated by sharp interfaces. The partition of the spheroid is obtained by
introducing thresholds for the oxygen concentration. More precisely, we introduce
a proliferation threshold
(
t
)
σ
P
and a necrosis threshold
σ
N
<
σ
P
assuming that all
cells die when the oxygen concentration reaches
σ
N
. So, all cells in the region
P
=
{
r
:
σ
(
r
,
t
)
>
σ
P
}
are proliferating, while the cells in the region
Q
=
{
r
:
σ
N
<
σ
(
r
,
t
)
<
σ
P
}
are quiescent. The necrotic region is given by
N
=
{
r
:
σ
(
r
,
t
)=
σ
N
}
.
We will assume that in
P
and in
Q
it is
constant
. For simplicity, we
take that all cells in
P
consume oxygen at the same rate and proliferate with a
common constant proliferation rate
ν
C
=
ν
=
χ
. The above scheme of a spheroid includes
two interfaces:
r
=
ρ
P
,the
P
−
Q
interface
r
=
ρ
N
,the
Q
−
N
interface
The determination of
ρ
N
requires the solution of the following
oxygen diffusion-
consumption problem
: given the radius
R
of the spheroid, find a piecewise twice
continuously differentiable function
ρ
P
,
σ
(
r
)
,and
ρ
P
,
ρ
N
, such that
D
O
2
Δσ
(
r
)=
f
(
σ
(
r
))
ν
,
in
P
,
(1)
1
m
f
D
O
2
Δσ
(
)=
(
σ
(
))
ν
,
r
r
in
Q
,
(2)
)=
σ
∗
,
σ
(
R
(3)
σ
(
ρ
P
)=
σ
P
,
(4)
σ
(
ρ
N
)=
σ
N
,
(5)
σ
r
(
ρ
N
)=
0
.
(6)
In the above equations,
D
O
2
is the oxygen diffusivity in the spheroid,
Δ
=
d
r
r
2
d
r
is the Laplacian operator,
f
1
r
2
d
(
σ
(
r
))
is the consumption rate per unit
σ
∗
is the given oxygen
cell volume in
P
, reduced by the factor 1
/
m
<
1in
Q
,and
σ
∗
>
σ
P
). This problem is not trivial, but it
can be proved (with the techniques of [
9
]) that:
concentration at the external boundary (
For any given
R
sufficiently large there exists one and only one solution
ρ
P
=
ˆ
ρ
P
(
ˆ
ρ
N
(
R
)
,
ρ
N
=
R
)
(otherwise at least one of the interface does not exist).
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