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A general reaction–diffusion model of acidity in cancer invasion

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Abstract

We model the metabolism and behaviour of a developing cancer tumour in the context of its microenvironment, with the aim of elucidating the consequences of altered energy metabolism. Of particular interest is the Warburg Effect, a widespread preference in tumours for cytosolic glycolysis rather than oxidative phosphorylation for glucose breakdown, as yet incompletely understood. We examine a candidate explanation for the prevalence of the Warburg Effect in tumours, the acid-mediated invasion hypothesis, by generalising a canonical non-linear reaction–diffusion model of acid-mediated tumour invasion to consider additional biological features of potential importance. We apply both numerical methods and a non-standard asymptotic analysis in a travelling wave framework to obtain an explicit understanding of the range of tumour behaviours produced by the model and how fundamental parameters govern the speed and shape of invading tumour waves. Comparison with conclusions drawn under the original system—a special case of our generalised system—allows us to comment on the structural stability and predictive power of the modelling framework.

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Acknowledgments

JBM, NKM, and PKM were partially supported by the National Cancer Institute, National Institutes of Health grant U54CA143970. JBM was partially supported by an Overseas Graduate Scholarship from St. Catherine’s College at Oxford University and would like to thank the Systems Biology Doctoral Training Centre for its hospitality. NKM and PKM were partially supported by the National Cancer Institute, National Institutes of Health grant U56CA113004. PKM was partially supported by a Royal Society Wolfson Research Merit award. This work is produced by NKM under the terms of the postdoctoral research training fellowship issued by the NIHR. The views expressed in this publication are those of the authors and not necessarily those of the NHS, The National Institute for Health Research or the Department of Health.

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Authors and Affiliations

  1. Centre for Mathematical Biology, Mathematical Institute, Oxford University, 24-29 St Giles’, Oxford, OX1 3LB, UK

    Jessica B. McGillen, Eamonn A. Gaffney & Philip K. Maini

  2. Department of Social Medicine, University of Bristol, Canynge Hall, 39 Whatley Road, Bristol, BS8 2PS, UK

    Natasha K. Martin

  3. Department of Global Health and Development, London School of Hygiene and Tropical Medicine, Keppel Street, London, WCIE 7HT, UK

    Natasha K. Martin

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  1. Jessica B. McGillen
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  2. Eamonn A. Gaffney
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  3. Natasha K. Martin
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  4. Philip K. Maini
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Correspondence to Philip K. Maini.

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McGillen, J.B., Gaffney, E.A., Martin, N.K. et al. A general reaction–diffusion model of acidity in cancer invasion. J. Math. Biol. 68, 1199–1224 (2014). https://doi.org/10.1007/s00285-013-0665-7

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  • DOI: https://doi.org/10.1007/s00285-013-0665-7

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