Biomedical Engineering Reference
In-Depth Information
The idea proposed in [
28
] was to define the
average power production
of
a proliferating cells,
w
cell
, as an independent cell
parameter
, thus computing
independently the global power production,
W
prod
, as a quantity simply proportional
to the volume of the proliferating region (in turn a function of
R
). The new
equation, from which the stationary radius can be determined, was then obtained by
equating
W
diss
(
R
)
to
W
prod
(
R
)
,i.e.to
w
cell
multiplied by the number of proliferating
cells.
In [
28
] we considered, as reference, a spheroid having at the steady state
σ
∗
=
R
=
1 mm when the outer oxygen concentration is
0
.
28 mM. The selected
48 h
−
1
,
parameter values were the following:
ν
=
0
.
6,
χ
=
log 2
/
θ
D
=
48 h,
D
O
2
=
10
−
5
cm
2
/s [
49
],
1
.
82
·
σ
P
=
0
.
05 mM,
σ
N
=
0
.
01 mM,
m
=
2[
14
]. The right-hand
side of Eq. (
1
) was written as
nQ
,where
Q
is the maximum
oxygen consumption rate per cell and
n
is the cell concentration, and we assumed
Q
σ
(
r
)
/
(
H
+
σ
(
r
))
10
−
3
mM
[
20
]. In models in which the volume loss of dead cells occurred after a chain of
successive stages, the mean time for this process was estimated greater than
10
−
17
mol/(cell s) [
31
],
n
10
8
cell/cm
3
=
8
.
3
·
=
5
·
[
31
]and
H
=
4
.
64
·
80 h
by fitting growth curves of treated sheroids [
12
] or xenografts [
56
]. We chose for
∼
θ
D
a shorter value, taking into account that the above estimates reflect the full process
of volume loss and account also for the dynamics of the effect of treatment [
56
].
Concerning the choice of the parameters
η
C
and
K
, we managed to obtain
values for the two quantities
W
C
,
W
E
of the same order. This was achieved by
assuming
10
4
Poise according to [
38
] (compared to 10
−
2
Poise for water
at room temperature) and
K
η
C
=
10
−
7
cm
3
s/g (i.e. a permeability of 10
−
9
cm
2
,
typical of a moderately permeable material). Tumours in vivo have a much lower
permeability (two orders of magnitude less), as healthy tissues do [
53
],butthatis
due to a considerable compactness provided by a substantial extracellular matrix.
In spheroids extracellular matrix is a much lighter structure [
35
], and we have even
neglected its volume fraction. This justifies the assumption of a relatively large value
for
K
. The value taken for viscosity may also look quite large (in the viscosity
range of a paste). In the Newtonian scheme, however, viscosity mimics not just
pure friction in the relative motion of cells, but also the influence of the forming and
breaking of intercellular links (which suggest that a Bingham scheme would be more
appropriate). Here we are in a domain of large uncertainty and the choices made in
[
28
], especially for the
K
value, are certainly questionable. Their main motivation
was to keep both kinds of dissipation in the game, waiting for the acquisition of
more precise information.
=
Remark 5.1.
The dimensionless ratio of the factors multiplying the brackets in the
expressions of
W
C
,
W
E
is 4
R
2
. A typical value for
ν
(
1
−
ν
)
η
C
K
/
ν
in spheroids is
0
η
C
K
is
reduced (which can easily be the case), then dissipation is dominated by
W
E
.
The value of
w
cell
might be estimated from the knowledge of the steady-state
radius at a given oxygen concentration (provided the values of the other parameters
are known). In [
28
], for instance, for a spheroid having radius
R
.
6, thus when
R
=
1 mm we obtain a value close to 0
.
1. If the product
=
1mm at
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