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During avascular tumour growth, the balance between cell proliferation and cell loss determines the rate at which the tumour expands. Recent experimental results suggest that growth factors produced during cell proliferation manipulate the rate of natural cell death. In this paper, we extend the standard model of avascular tumour growth to study the effect that the production of such growth factors can have on a tumour's development. We assume that the growth factor is produced in inactive form, and only becomes activated when it binds to a tumour cell. Two dependent variables are introduced to describe the levels of active and inactive growth factor. The model is studied using a combination of analytical and numerical techniques. These results show that the inclusion into the model of growth factors endows the tumour with history dependence, in that its evolution depends not only on its structure at a given instant, but also on its structure at earlier times. Numerical simulations suggest that the manner in which the growth factor acts is crucial to the tumour's evolution. For example, a growth factor which enhances apoptosis does not alter the qualitative behaviour of the tumour: it simply decreases the time taken to reach the equilibrium configuration in which the tumour may be present or absent. By contrast, a growth factor which inhibits apoptosis can dramatically alter the tumour's behaviour, giving rise to asymmetric tumour pulsing. Here a single cycle comprises a long period of slow tumour growth followed by a short period of tumour regression. Using asymptotic analysis, we identify regions of parameter space in which such periodic behaviour arises and show how they separate regions of unstable tumour growth from regions of bounded growth. The implications of our results are also discussed briefly.

Original publication




Journal article


Mathematical and Computer Modelling

Publication Date





35 - 55