The Steady State of Multicellular Tumour Spheroids: A Modelling Challenge - Mathematical Methods and Models in Biomedicine - page 190

that defining a quantity like w P may not be correct. As usual, one has to be
cautious in stating what is or is not correct, since it has to be considered that
all these results have been obtained in a framework of a model that we already
know to be largely approximated. In the same spirit we cannot even refuse the
third (quite appealing) option that equilibrium is characterized as the configuration
minimizing the dissipation of mechanical power, which does not appear to be met
by any of the previous models.
That said, once it is made clear that we cannot ask too much to a mechanically
simple model, the studies performed both in [ 28 ]andin[ 29 ] lead to results that are
quantitatively acceptable and qualitatively interesting, since they permit to ascertain
the influence of the basic parameters on the possible attainment of equilibrium,
and on the size eventually reached by the spheroid under specified environmental
conditions. A quantity which may play a critical role is the so called tumour surface
tension, which in some of the models reviewed is necessary for equilibrium to be
attained. This is indeed the case of the model in [ 29 ].
A step further, already envisaged in [ 29 ](seealso[ 5 ]), consists in modelling
the cell component as a Bingham fluid, thus possessing a yield stress acting as
a threshold to allow deformations (and hence the radial flow typical of cells in a
spheroid). The reason to shift to a Bingham flow is to better represent the effect of
intercellular bonds that can bear some limit tension and have to be broken to allow
deformations, a responsibility that is totally assigned to viscosity in the Newtonian
framework. We find the Bingham approach particularly stimulating and we plan to
study the evolution of “Bingham spheroids” in a future paper. We may anticipate
that the selection of a Bingham-like constitutive law is a delicate issue, owing to the
peculiar feature of the “fluid” considered, which is actually incompressible, but in
which volume is not preserved due to proliferation.
As a general conclusion we may say that the scheme in which the necrotic
core has an interface separating a liquid nucleus from a solid shell allows to
describe the complex radial flow within a spheroid at equilibrium, either in a fully
Newtonian framework, or adopting a Bingham scheme for the fluid representing
cells. Despite all the limitations accompanying the two-fluid models, which we have
carefully pointed out, the results obtained are meaningful and they do not require
the arbitrary definition of any special mass removing mechanism from the necrotic
core. The predictions about the influence of the model mechanical parameters on
the equilibrium size could be in principle verified on the basis of ad hoc designed
experiments. Based on the positive outcome of the approach of [ 29 ], we believe
that the whole analysis carried out for the steady state can be extended to describe
the spheroid evolution from an early fully proliferative state to its asymptotic
configuration. We plan to do it in a forthcoming paper.
Possible extensions of the model may include other aspects that are biologically
important, among them the cell inhibition by contact. In the review section of this
chapter we have mentioned models which include inhibitors of proliferation gen-
erated by dead cell disaggregation. A possibly more important cause of inhibition
is related with crowding. Since the cell volume fraction does not vary much within
the spheroid, crowding can be sensed via the stress. For instance in the Newtonian
scheme cell proliferation can be assumed to stop when p C exceeds some threshold.