The Steady State of Multicellular Tumour Spheroids: A Modelling Challenge - Mathematical Methods and Models in Biomedicine - page 189

6 Conclusion and Discussion

It is well known that multicellular spheroids in an advanced stage of their evolution
contain a considerable fraction of dead cells and debris, mostly concentrated in the
central region. It is therefore quite natural to speak of a necrotic core. Describing the
structure of that region, as it results from cells degradation, turns out to be a crucial
step in modelling the whole system, since, irrespective of the constitutive equations
selected for the various components, the growth of the spheroid will depend on how
the viable region interacts mechanically with the necrotic core. Various schemes
have been proposed in the literature since the early paper by Greenspan [ 34 ],
ranging between two extremes: from a completely solid to a completely liquid core.
The necrotic zone is frequently described as a region bounded by a sharp interface.
Clearly, the presence of interfaces within a spheroid separating cells in different
states is an extrapolation which is frequently adopted (the spherical symmetry itself
is an idealization). In our opinion the sharp boundary approach is quite sensible,
since it simplifies the conceptual geometrical scheme without deeply altering the
actual cell distribution.
In this chapter we paid special attention to the modelling of the necrotic zone,
confining ourselves to the analysis of the steady state (when it exists). We started
with a short summary of the relevant literature, trying to point out which specific
assumption in each of the considered models (most of the times related with the
necrotic region) guarantees the existence of a steady state. After having illustrated
the implications of assuming a completely solid necrotic core, we review some
theories developed in the papers [ 28 , 29 ] in which the necrotic region consists of
a solid shell encasing a liquid nucleus. We took this opportunity for carrying out a
critical analysis not only of the approach of those papers, in which we have adopted
a two-fluid scheme, but also of the general conceptual difficulties accompanying
such a representation of the spheroids mechanics. However, we emphasize that
there are good motivations for selecting a relatively simple model, since not only
the number of parameters to be determined increases with the complexity of
the mathematical scheme, but also the uncertainty of their identification becomes
more and more serious. On the other hand, there is a price to pay for simplicity, since
one cannot expect that models with few parameters can give a particularly accurate
description of intrinsically complex systems. Thus, as a general rule in mathematical
biology, the focal point in modelling is to find a reasonable compromise. Of course
it is only the comparison with experimental data which can say to what extent the
compromise is acceptable.
In this respect we found that the approach followed in [ 28 ],basedonenergy
balance, and the one used in [ 29 ], imposing the continuity of normal stress
throughout the system, perform in a comparable way, despite the fact that—strictly
speaking—they are not mutually compatible. Indeed, while the principle of normal
stress continuity always applies, the energy balance proposed in [ 28 ] involved the
conjecture that all proliferating cells deliver the same amount w P of mechanical
power. Such a conjecture is not confirmed by the results of [ 29 ]. This is an indication