Growth of non-necrotic tumours in the presence and absence of inhibitors.pdf

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Growth of non-necrotic tumours in the presence and absence of inhibitors.pdf

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Growth of non-necrotic tumours in the presence and absence of inhibitors

Growth of Non-necrotic Tumours in the Presence andAbsence of InhibitorsH.M. Byrne M.A.J. ChaplainSchool of Mathematical SciencesUniversity of BathClaverton DownBath BA2 7AYphone: +(44) 01225 826242FAX: +(44) 01225 826492e-mail: hmb,majc@maths.bath.ac.ukrunning title: non-necrotic tumour growthAbstractIn this paper a model for the evolution of a spherically symmetric, non-necrotictumour is presented. The eects of nutrients and inhibitors on the existence andstability of time-independent solutions are studied. With a single nutrient and no in-hibitors present the trivial solution, which corresponds to a state in which no tumouris present, persists for all parameter values whereas the nontrivial solution, whichcorresponds to a tumour of nite size, exists for only a prescribed range of parame-ters, corresponding to a balance between cell proliferation and cell death. Stability1 2 H.M. Byrne M.A.J. Chaplainanalysis, based on a two-timing method, suggests that, where it exists, the nontrivialsolution is stable and the trivial solution unstable. Otherwise, the trivial solution isstable. Modications to these characteristic states brought about by the presence ofdierent types of inhibitors are also investigated and shown to have signicant eect.Implications of the model for the treatment of cancer are also discussed.Revised Version Submitted to Mathematical Biosciences December 13, 1994 Non-necrotic Tumour Growth 31 IntroductionOver the past twenty years an increasing number of mathematical models describing solidtumour growth have appeared in the literature [1], [2], [3], [4], [5], [6], [7]. Analysis ofsuch models enables the relative importance of the dierent mechanisms involved in thegrowth process to be examined and may also assist in assessing the relative merits ofdierent courses of drug treatment or chemotherapy [8], [9], [10], [11], [12]. Many of theexisting deterministic tumour models consist of an ordinary dierential equation, coupledto one, or more, reac