Biomedical Engineering Reference
In-Depth Information
cell proliferation and death are assumed to depend on the concentration of a single
critical chemical (generally oxygen), diffusing from the external medium into the
spheroid mass. According to this view, the boundary between the viable rim and
the necrotic core is often defined as the level set of the oxygen concentration
corresponding to a given threshold. However, the formation mechanism of the
central necrotic region in multicellular spheroids is a much debated and a not
yet well-understood process. The diffusion of both glucose and oxygen has been
included in the spheroid models proposed in [
21
,
39
,
55
]. More recently, the cell
energy metabolism, i.e. the intracellular ATP production involving glucose, oxygen
and lactate, has been incorporated in models of spheroids [
10
,
11
,
61
], as well as
in various models of tumour growth [
7
,
8
,
33
,
57
]. In [
10
,
11
], the formation of the
spheroid necrotic region was described by assuming that cell death occurs when
the ATP production rate falls below a critical value. The possible role of acidity in
determining the onset of the central necrosis in tumours was investigated in [
13
].
With only a few exceptions [
1
,
2
,
48
], the final attainment of a steady state during
the sheroid growth has been associated, in the modelling literature, with a loss of
volume from the necrotic core that balances the new cellular volume created in the
viable rim by cell proliferation. The experimental evidence of this mechanism, how-
ever, is indirect and relies on the observation of active cell proliferation even when
the growth rate of the spheroid is very small or vanishes [
27
]. From a biological
point of view the way the necrotic core is modelled may look to be a minor question.
Nevertheless, the structure attributed to the necrotic zone has a crucial influence on
the general mechanical behaviour of the entire spheroid and hence on its evolution.
In the present chapter, we first give a brief survey (Sect.
2
) of the modelling
options proposed in the literature for describing the necrotic core and for explaining
the balance between live and dead cells in the steady state of tumour spheroids.
Next we review our recent work on this topic (Sects.
3
-
5
). We adopted the
two-fluid scheme, in which the extracellular fluid and cells are schematized as
incompressible fluids (inviscid and viscous, respectively), and we introduced several
free boundaries, having the role of sharp transition interfaces marking a change
in the state of cells. In particular, the necrotic region was subdivided in a shell
of dead cells surrounding a purely liquid core. The advantage of sharp interfaces
is the possibility of writing the velocity field of each fluid in an explicit way.
The equilibrium size of the spheroid (if it exists) can then be obtained through
the analysis of the stress accompanying the flow of the two fluids, generated by
proliferation. Although the two-fluid approach is certainly naıve, it allowed to reach
meaningful quantitative results, with no need of postulating any mechanism for the
removal of necrotic material. Some concluding remarks are given in Sect.
6
.
2
A Brief History of the Necrotic Core Modelling
In the influential paper by Greenspan [
34
], the necrotic core, composed of “dead
cells and cellular material in various stages of disintegration” is viewed as a “jelly-
like” material “capable of supporting the pressure exerted on it by the outer viable
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