Biomedical Engineering Reference
In-Depth Information
5
Looking for Steady States
As we said at the end of Sect. 3 , if we are given the spheroid radius R , we can find all
other unknowns. Thus we need just one more equation to find R . The most natural
way of proceeding is to impose that the normal component of the total stress is
continuous across the critical interface, i.e. r
= ρ D . This is the technique we used
in [ 29 ].
However, before we come to that, we want to discuss the possibility that the
missing equation could be derived from considerations based on power dissipation.
Looking at the problem from the point of view of energy is advantageous because
it highlights the relative contribution of cell-cell friction and of liquid-cell friction,
ultimately related with the coefficients
η C and K , respectively.
5.1
The Energy Based Approach
Coming back to our initial goal of obtaining one more equation from energy
considerations, we may think of different options. First of all, we must look at
the spheroid in its equilibrium configuration as an “engine”, in which mechanical
power is produced by proliferating cells and then dissipated by the internal motion
so generated. At the stationary state, indeed, cells move inwards until they reach the
interface r
= ρ D whereas extracellular liquid moves in the opposite way, the liquid
in NL staying at rest (see Sect. 3.2 ). A very tempting criterion is to say that the
radius at equilibrium corresponds to the minimal energy produced and dissipated.
In [ 28 ], instead, we started from the general principle that the equilibrium size must
guarantee the balance of the power dissipated and the one produced by proliferating
cells. Note that power balance for each species can be derived from the momentum
balance equations ( 26 )and( 27 ). Indeed we can write the equations
1
2 u
u
d
d t
δν
·
= · (
T C ·
u
)
T C : D C +
m C ·
u
,
(36)
1
2 v
v
d
d t
δ (
1
ν )
·
= · (
T E
·
v
)
T E : D E
+
m E
·
v
,
(37)
which take the explicit form
1
2 u
u
d
d t
2
3 νη C ∇χ P ·
δν
·
= ν
u
·
p C
u
+
2
νη C · (
D C ·
u
)
2
νη C D C : D C
1
K +
u
ν
χδ
ˆ
·
u
,
(38)
1
ν
 
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