Biomedical Engineering Reference
In-Depth Information
)= 3 r 2 (
R 3
3
P
u
(
r
ρ
) ,
for
ρ D <
r
< ρ P .
(22)
The latter formula emphasizes the occurrence of a singularity if
D is allowed to
vanish. Following the motion along the velocity field Eq. ( 22 ), we can deduce the
value of
ρ
ρ
D imposing that
ρ N
d r
u
θ D =
) ,
(
r
ρ D
so that
ρ D is given by
D
N
R 3
P
ρ
= ρ
χθ D (
ρ
) .
(23)
Equation ( 23 ) represents a constraint on the system, meaning that R has to be
sufficiently large to allow Eq. ( 23 ) to have a positive solution. Through Eqs. ( 22 )
and ( 23 ) we recognize indeed that a transition from the “solid” to the “liquid” phase
that occurs with a fixed delay from death is not compatible (at the steady state) with
ρ D =
0, i.e. with a necrotic core fully “solid”.
At this point it is clear that the internal structure of the stationary spheroid can be
found once R is known. To proceed further for determining R we must address the
mechanical description of the spheroid.
4
A Mechanical Scheme Based on the Two-Fluid Model
Two-fluid models adopt the point of view that a spheroid is a two-component
mixture consisting of an inviscid fluid (the extracellular fluid) and another fluid
(representing cells) for which an appropriate rheological model has to be chosen.
In some papers (cf. [ 28 , 29 ]) a simple Newtonian scheme is assumed in which
the effect of cell-cell interactions is somehow translated into a viscosity. In other
models cells are treated as an inviscid fluid too [ 41 ] or according to some nonlinear
constitutive law. In this section we present the implications of identifying cells
with a Newtonian fluid. The limitations which are intrinsic to this approach will
be discussed in the next section.
In the Newtonian framework, the Cauchy stress tensors for the two components
are written in the form
uI
2
3 η C ·
T C = ν
p C I
+
2
η C D C
,
(24)
ν )
p E I ,
T E =(
1
(25)
1
T
where D C =
η C is the cell viscosity.
In Eq. ( 24 ) the so-called Stokes' assumption has been used. The pressures p C , p E
have to stay distinct. The reason for that will become apparent when we consider for
instance the conditions at the boundary r
2 [
u
+(
u
)
]
is the cell strain rate tensor, and
R . Each component is incompressible,
but the two velocity fields u , v are not divergence free in the proliferation region, as
we have seen in the previous section. This leads to the definition of the discontinuous
=
 
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