The Steady State of Multicellular Tumour Spheroids: A Modelling Challenge - Mathematical Methods and Models in Biomedicine - page 173

Following some ideas of the model in [ 54 ], Landman and Please [ 41 ] described
the spheroid as a liquid-cells mixture whose mechanics is borrowed from a model
for suspensions [ 42 ]. The force balance equation is explicitly included together with
the mass balance, and not only the liquid, but also the cell component has isotropic
stress tensor. Thus, stresses are expressed by two pressures: the liquid pressure and
the intercellular pressure. The net proliferation and death rate are expressed as a
function of oxygen concentration, switching sign across a critical threshold. Cells
immediately degrade into liquid after death (then all cells are living cells), whereas
maintaining a constant local volume fraction (and a compact arrangement) until
the cellular pressure is greater than the liquid pressure. Complete mass exchange
between liquid and cellular phase occurs during cell proliferation and at cell death.
An interesting feature introduced in [ 41 ] is claiming that when the cellular pressure
tends to drop below the liquid pressure, cells detach from the compact arrangement
and “float” in the liquid. Thus the necrotic core is essentially described as a liquid
with a small fraction of viable cells committed to death. This fraction vanishes at
the steady state, when the necrotic core is purely liquid. However, the existence of
the steady state is related to the presence of a suitable surface tension: if the surface
tension is insufficient, the spheroid eventually will grow linearly.
A two-phase model based on a more complex mechanics was proposed by
Byrne and Preziosi [ 18 ](seealso[ 4 , 15 , 19 ]). In this “two-fluid” model, cells are
represented by a viscous fluid whose pressure contains an extra-term depending
on the cell volume fraction and describing the cell-to-cell interaction, whereas
the extracellular liquid is represented by an inviscid fluid. Again, in this model
cells degrade instantaneously into liquid after death, and complete mass exchange
between liquid and cellular phase occurs at cell proliferation and cell death, which
are under the control of a critical nutrient. At the steady state, the local volume
fraction of (living) cells continuously decreases towards the centre of the spheroid,
at which it does not vanish, so that the necrotic core is mimicked by a region in
which the density of living cells is reduced. A similar view is also present in the
model by Ambrosi and Preziosi [ 5 ], in which the cell component is represented by
a visco-elasto-plastic fluid, and in the model proposed by Cristini et al. [ 25 ], which
was focussed on the derivation of the interaction potential.
In two recent papers [ 28 , 29 ], we have proposed that at the steady state the
necrotic core ( N ) may be partitioned into two zones: a “solid” domain NS where
dead cells are supposed to keep the mechanical properties they had before death and
the same volume fraction and an inner core NL simply liquid. This partition follows
from the assumption that cell membrane degradation occurs after a fixed time from
cell death, and that this degradation marks the transition from “solid” to “liquid”.
Some support to the NS

NL partition comes from nuclear magnetic resonance
(NMR) measurements of the self-diffusion of water in EMT-6 spheroids [ 52 ].
These measurements have shown that whereas in the viable rim water appears
confined into two compartments with different diffusion coefficients (intracellular
and extracellular water), the central part of the necrotic core looks like a single

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