Abstract
During the initial avascular phase of solid tumour growth, it is the balance between cell proliferation and cell loss that determines whether the tumour colony expands or regresses. Experimentalists have identified two distinct mechanisms that contribute to cell loss. These are apoptosis and nicrosis. Cell loss due to apoptosis may be riferred to as programmed‐cell‐death, occurring,for example, when a cell exceeds its natural lifespan. In contrast, cell loss due to necrosis is induced by changes in the cells microenvironment,occurring, for example, in nutrient‐depleted regions of the tumour.In this paper we present a mathematical model that describes the growth of an avascular tumour which compuises a centual core of necrotic cells, surrounded by an outer annulus of puoliferating cells. The model distinguishes between apoptisis and necrosis. Using a combination of numerical and analytical techniques we present results which suggest how the relative importance of apoptisis and necrosis changes as the tumour develops. The implications of these results are discussed buiefly.
Highlights
In vivo cancer growth is a complex phenomenon, involving many inter-related processes and the mathematical modelling of such processes is very difficult
The mathematical model presented below describes the evolution of a multicellular spheroid growing in response to an externally-supplied nutrient such as oxygen or glucose
In order to characterise the size of the necrotic core directly after the onset of necrosis we introduce the small parameter c (0 < t
Summary
In vivo cancer growth is a complex phenomenon, involving many inter-related processes and the mathematical modelling of such processes is very difficult. The ODE derives from mass conservation applied to the tumour and describes the evolution of the tumour boundary whereas the RDEs describe the distribution within the tumour of nutrients such as oxygen and glucose and growth inhibitory factors such as chalones Any interior boundaries, such as the interface between the necrotic core and the quiescent region, are defined implicitly, occurring, for example. The model of McElwain and Morris (1978) admits steady solutions which do not possess a necrotic core: the steady state tumour comprises a central core, in which the rates of nutrient consumption and cell proliferation fall, surrounded by an outer, proliferating rim. In this paper we use a combination of numerical and analytical techniques to study the manner in which the relative importance of apoptosis and necrosis as distinct cell loss mechanisms changes as the tumour develops. The paper may be viewed as a summary of existing work, with the emphasis placed on interpretation of the main results in a way which is accessible to mathematicians and biologists alike
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