Biomedical Engineering Reference
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Remark 5.4. By recalculating the values of w P corresponding to the solutions
predicted by Eq. ( 54 ) for different values of the oxygen concentration
σ , w P
is actually found to vary. So the conjecture of the energy-based approach that
the power w P can be defined as a characteristic cell parameter is disconfirmed.
Nevertheless, in the simulation of [ 29 ] the variation of w P was only of the order
of 15 %.
As a final comment, we want to point out some internal contradictions of the
two-fluid model that we summarize in the following remarks.
Remark 5.5. The sheer fact that we use Darcy's law to describe the motion of the
extracellular fluid relative to cells implies that a fluid-cell friction does exist, though
we have supposed that the fluid is inviscid (which would make it flow among cells
with no resistance). But certainly its viscosity is many orders of magnitude smaller
than
η C , thus the above compromise is reasonable.
Remark 5.6. As we have seen, in many models (including the model in [ 29 ]) the
action of “surface tension” is necessary to reach equilibrium. However, we must
not identify the concept of surface tension in a spheroid with the one arising in a
liquid drop. Indeed cells mutually interact through macromolecular bridges which
can provide some limited tensile stress and evolve according to the dynamical state
of the spheroid. At the outer surface of a growing spheroid such stresses can produce
an effect similar to surface tension, but only if the number of cells in the spheroid
is large enough. Thus it would be wrong to use the classical Laplace formula in
the early stage of the spheroid growth, introducing in the model abnormally high
pressures which simply are not there. The correct way of using surface tension in
spheroids should be instead to let it come into play in a gradual way as the size of
the spheroid grows.
Remark 5.7. The action of intercellular links cannot be fully taken into account in
the framework of the two-fluid model if the “cell fluid” is Newtonian. Thus, in view
of the previous remark, including surface tension is in fact an internal contradiction.
For this reason it may be of interest to study model extensions in which the cell
component is represented by a Bingham fluid (such an extension was preliminarily
considered in [ 29 ]).
However, any model in which mechanics is excessively simplified cannot provide
an accurate description, since the adopted governing laws are in fact trying to
provide a simple representation of phenomena whose nature can be more complex.
At the same time, going deeper in investigating the mechanical structure clashes
inevitably with the practical impossibility of getting experimental information on
the many parameters involved. Therefore, if on one side we must be conscious of the
great limitations of a mechanically naıve approach, on the other we must recognize
that simplicity is the basic component of a practical strategy.
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