The Steady State of Multicellular Tumour Spheroids: A Modelling Challenge - Mathematical Methods and Models in Biomedicine - page 172

layers”. This “solid” debris, while keeping a constant density, continually dissolves
into “simpler permeable compounds” capable of moving easily through the outer
region of the spheroid. Because of this motion, a volume loss occurs from the region
of necrosis. Although it does not play a direct role in the model, a surface tension
is postulated to maintain the compactness of the aggregate. The degradation of the
necrotic material occurs according to a first-order kinetics, with uniform degradation
rate constant through the whole core. In Greenspan's model, however, the volume
loss is not the only mechanism allowing the attainment of a steady state: another
important role is played by a mitosis inhibitor which is supposed to be produced at a
constant rate in the necrotic core or as a waste from living cells (for comments about
this conjecture, see [ 3 ]). Cell death occurs when the oxygen concentration decreases
to some critical threshold (a feature incorporated in many subsequent models) and
mitosis stops when the inhibitor concentration raises above the inhibition threshold.
According to this picture, the model contains two free boundaries. A substantial gap
of the model is the absence of any mechanical explanation of how the postulated
material loss from the necrotic core can take place. In other words, a study of the
flow of the various components based on the general principles of mechanics is
missing. This kind of analysis came much later in cancer modelling.
The Greespan's viewpoint was largely adopted in the following years (see
Deakin [ 26 ], MacElwain and Ponzo [ 47 ], Maggelakis and Adam [ 44 ], Adam and
Maggelakis [ 3 ], Byrne and Chaplain [ 17 ], Cui and Friedman [ 23 ], Bertuzzi et al.
[ 12 ]), and in the next section we will illustrate a simple model based on it. Also the
model studied by Cui and Friedman in [ 24 ] describes the central zone of a spherical
tumour (although without a sharp interface) as essentially full of dead cells subjected
to degradation according to a uniform rate constant.
A different mechanism for the attainment of a stationary state during the spheroid
growth was proposed by McElwain and Morris [ 48 ]. These authors, following
Burton [ 16 ], assumed the necrotic material immune from degradation (at least in
the time horizon of interest) and supposed that the relevant volume loss happens in
the inner viable rim via cell apoptosis and phagocytosis of the resulting apoptotic
bodies by the viable neighbouring cells. This mechanism then accounts for some
experimental observations of stationary spheroids without central necrosis [ 59 ].
Volume loss was totally absent in the models by Adam [ 1 ] and Adam and Magge-
lakis [ 2 ], where instead a diffusing endogenous mitotic inhibitor, possibly produced
inside the necrotic region [ 2 ], eventually blocks the proliferation of all the cells.
Some different views of the necrotic region derived from the explicit modelling of
the multi-phase nature of the cell aggregates. Ward and King [ 62 ] distinguished
in the spheroid the viable cells (that can occupy a varying local volume fraction)
and a diffusible “cellular material” originated by the immediate degradation of cells
upon death (a death rate is introduced depending on the concentration of a critical
chemical). This material may be reused to sustain the cell proliferation, so that
growth saturation can be achieved. The necrotic core is then a zone deprived of cells
but occupied by the cellular material, and the volume loss from the necrotic core is
given by the diffusing flow of such a material towards the outer region. This view
was reconsidered in [ 60 ], where, however, the lack of reutilization of the material
coming from cell disintegration prevents the saturation of growth.