Multicellular spheroids (MCS) are clusters of cancer cells, used in
the laboratory to study the early stages of avascular tumour
growth. Mature MCS possess a well-defined
structure, comprising a central core of necrotic, or dead, cells,
surrounded by a layer of non-proliferating, quiescent cells, with
proliferating cells restricted to the outer, nutrient-rich layer of
the tumour. As such, they are often used
to assess the efficacy of new anti-cancer drugs and treatment
therapies.
Progress in modelling tumour growth has been largely guided by
experimental results. Thus the majority of mathematical models focus
on the growth of MCS or avascular tumour growth. However,
several models have been proposed to describe angiogenesis.
This process marks the transition from the relatively harmless,
and localised, avascular state described above to the more dangerous
vascular state, wherein the tumour possesses the ability to invade
surrounding tissue and metastasise to distant parts of the body.
Angiogenesis is the process by which tumours
induce blood vessels from the host tissue to sprout capillary
tips which migrate towards and ultimately penetrate the tumour,
providing it with a circulating blood supply and, therefore,
an almost limitless source of nutrients.
The vascular growth phase which follows angiogenesis is marked by a
rapid increase in cell proliferation and is usually accompanied by an
increase in the pressure at the centre of the tumour. This may be
sufficient to occlude blood vessels and, thereby, to restrict drug
delivery to the tumour.
Recent work has focused on the evolution of a MCS growing in
response to a single, externally-supplied nutrient,
such as oxygen or glucose, and
two growth inhibitors. We assume that
the tumour adopts a multi-layered strucutre, with proliferating
cells in the outer shell, quiescent cells occupying the adjacent, middle
shell, and necrotic cells confined to the central core. The proportion
of each cell type changes as the tumour grows.
Consider, for example, the growth of a tumour that
initially comprises only proliferating cells.
As the tumour grows, cells
towards its centre, being deprived of vital nutrients,
cease proliferating and become quiescent.
(Quiescent cells are not dead: they simply do not divide. If the
environmental conditions improve, i.e. nutrient levels increase,
then such cells may recommence proliferating. Thus quiescence is a
reversible state
of a tumour cell.) Further growth of the tumour
is accompanied by an increase in the quiescent cell population
and a further reduction in the minimum nutrient
concentration until eventually tumour cells towards the centre of
the tumour, being starved of vital nutrients, become necrotic.
Mathematical models of MCS typically consist of an ordinary differential equation (ODE) coupled to one or more reaction-diffusion equations (RDEs). The ODE is derived from mass conservation and describes the evolution of the outer tumour boundary, whereas the RDEs describe the distribution within the tumour of vital nutrients such as oxygen and glucose and growth inhibitors. These models accurately reproduce the growth of spheroids and also the macroscopic heterogeneity which is the hallmark of spheroids. The model developed in this paper exhibits similar features. However it also enables us to assess a tumour's potential for invasion. The key feature that determines this potential is the balance between an internal expansive force (caused by cell proliferation) and a restraining force (caused by forces of adhesion which exist between cells on the tumour boundary).
The tumuor is essentially viewed as an incompressible fluid so that local changes in the cell population caused by cell death or birth will generate pressure gradients which drive cell motion and induce expansion of the tumour. No a priori assumptions about the width of the proliferating rim or the tumour's internal structure.
Attention is also restricted to the moving domain defined by the tumour volume and it is assumed that the nutrient concentration satisfies the Gibbs-Thomson relation on the boundary. This relation states that the nutrient concentration at a point on the tumour boundary is less than the external concentration by a factor which depends on the local curvature there. This energy is needed by cells on the periphery in order to preserve the forces of adhesion which exist to maintain the tumour's compactness. Clear experimental evidence for these effects can be found in \cite{Miyasaka}, \cite{Nagle} and references therein.
{\em In vitro} observations suggest that in the early stages solid tumours remain approximately spherical as they grow. Therefore, our analysis focuses on the existence and stability of radially-symmetric solutions of the model equations. We show how the pressure can be eliminated from the model when radial symmetry is assumed and how a reduced system of equations, which is similar in form to existing models of MCS, can be recovered. We believe that our derivation of this simplified model indicates how models of avascular tumour growth arise as special cases of more physically based models.
Solution of the radially symmetric model shows how the existence and stability with respect to time-dependent perturbations depends on the system parameters. For example, if the strength of the cell-cell adhesion bonds is increased then the tumour is likely to disappear. In contrast when cell-cell adhesion bonds are weak the tumour persists and may even expand.
Our analysis of the reduced model also investigates how small deviations from radial symmetry develop in time. If all asymmetric perturbations ultimately disappear then we conclude that the tumour will simply grow as a radially-symmetric mass. In contrast, the growth of a perturbation implies that the tumour has a propensity for asymmetric local invasion, the growth rate of the perturbation indicating the degree of aggression.
