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Mathematical Ecology of Cancer

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Managing Complexity, Reducing Perplexity

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 67))

Abstract

It is an emerging understanding that cancer does not describe one disease, or one type of aggressive cell, but, rather, a complicated interaction of many abnormal features and many different cell types, which is situated in a heterogeneous habitat of normal tissue. Hence, as proposed by Gatenby, and Merlo et al., cancer should be seen as an ecosystem; issues such as invasion, competition, predator-prey interaction, mutation, selection, evolution and extinction play an important role in determining outcomes. It is not surprising that many methods from mathematical ecology can be adapted to the modeling of cancer. This paper is a statement about the important connections between ecology and cancer modelling. We present a brief overview about relevant similarities and then focus on three aspects; treatment and control, mutations and evolution, and invasion and metastasis. The goal is to spark curiosity and to bring together mathematical oncology and mathematical ecology to initiate cross fertilization between these fields. We believe that, in the long run, ecological methods and models will enable us to move ahead in the design of treatment to fight this devastating disease.

“The idea of viewing cancer from an ecological perspective has many implications, but fundamentally it means that we cannot just consider cancer as a collection of mutated cells but as part of a complex balance of many interacting cellular and microenvironmental elements”. (quoted from the website of the Anderson Lab, Moffit Cancer Centre, Tampa Bay, USA.)

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References

  1. T. Alarcon, M.R. Owen, H.M. Byrne, P.K. Maini, Multiscale modelling of tumour growth and therapy: the influence of vessel normalisation on chemotherapy. Comput. Math. Methods Med. 7(2–3), 85–119 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. J.W.N. Bachman, T. Hillen, Mathematical optimization of the combination of radiation and differentiation therapies of cancer. Front Oncol (2013, free online). doi:10.3389/fonc.2013.00052

  3. R.S. Cantrell, C. Cosner, V. Hutson, Permanence in ecological systems with spatial heterogeneity. Proc. R. Soc. Edinb. 123A, 533–559 (1993)

    Article  MathSciNet  Google Scholar 

  4. M.A. Chaplain, S.R. McDougall, A.R.A. Anderson, Mathematical modeling of tumor-induced angiogenesis. Annu. Rev. Biomed. Eng 8, 233–257 (2006)

    Article  Google Scholar 

  5. T. Day, P. Taylor, Evolutionary dynamics and stability in discrete and continuous games. Evol. Ecol. Res. 5, 605–613 (2003)

    Google Scholar 

  6. U. Dieckmann, Can adaptive dynamics invade. Trends Ecol. Evol. 12, 128–131 (1997)

    Article  Google Scholar 

  7. O. Diekmann, M. Gyllenberg, J.A.J. Metz, H.R. Thieme, On the formulation and analysis of general deterministic structured population models. J. Math. Biol. 36, 349–388 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Dingli, F. Michor, Successful therapy must eradicate cancer stem cells. Stem Cells 24(12), 2603–2610 (2006)

    Article  Google Scholar 

  9. J.M. Drake, D.M. Lodge, Allee effects, propagule pressure and the probability of establishment: risk analysis for biological invasions. Biol. Invasions 8, 365–375 (2006)

    Article  Google Scholar 

  10. H. Enderling, M. Chaplain, A. Anderson, J. Vaidya, A mathematical model of breast cancer development, local treatment and recurrence. J. Theor. Biol. 246(2), 245–259 (2007)

    Article  MathSciNet  Google Scholar 

  11. H. Enderling, L. Hlatky, P. Hahnfeldt, Migration rules: tumours are conglomerates of self-metastases. Brit. J. Cancer 100(12), 1917–1925 (2009)

    Article  Google Scholar 

  12. R.S. Epanchin-Niell, A. Hastings, Controlling established invaders: integrating economics and spread dynamics to determine optimal management. Ecol. Lett. 13, 528–541 (2010)

    Article  Google Scholar 

  13. W.F. Fagan, M.A. Lewis, M.G. Neubert, P. van den Driessche, Invasion theory and biological control. Ecol. Lett. 5, 148–157 (2002)

    Article  Google Scholar 

  14. R.A. Fisher, The wave of advance of advantageous genes. Ann. Eugen. Lond. 37, 355–369 (1937)

    Article  Google Scholar 

  15. N.S. Forbes, Engineering the perfect (bacterial) cancer therapy. Nat. Rev. Cancer 10, 785–794 (2010)

    Article  Google Scholar 

  16. P. Friedl, E.B. Bröcker, The biology of cell locomotion within three dimensional extracellular matrix. Cell Motil. Life Sci. 57, 41–64 (2000)

    Article  Google Scholar 

  17. P. Friedl, K. Wolf, Tumour-cell invasion and migration: diversity and escape mechanisms. Nat. Rev. 3, 362–374 (2003)

    Article  Google Scholar 

  18. R.A. Gatenby, J. Brown, T. Vincent, Lessons from applied ecology: cancer control using a evolutionary double bind. Perspect. Cancer Res. 69(19), 0F1–4 (2009)

    Google Scholar 

  19. R.A. Gatenby, R.J. Gillies, A microenvironmental model of carcinogenesis. Nat. Rev. Cancer 8(1), 56–61 (2008)

    Article  Google Scholar 

  20. R.A. Gatenby, R.J. Gillies, Of cancer and cavefish. Nat. Rev. Cancer 11, 237–238 (2011)

    Article  Google Scholar 

  21. J. Gong, Tumor control probability models. Ph.D. thesis, University of Alberta, Canada (2011)

    Google Scholar 

  22. J. Gong, M. dos Santos, C. Finlay, T. Hillen, Are more complicated tumor control probability models better? Math. Med. Biol. 19 (2011). doi:10.1093/imammb/dqr023. Accessed 17, Oct 2011

  23. D. Hanahan, R. Weinberg, The hallmarks of cancer. Cell 100(1), 57–70 (2000)

    Article  Google Scholar 

  24. D. Hanahan, R.A. Weinberg, Hallmarks of cancer: the next generation. Cell 144, 646–674 (2011)

    Article  Google Scholar 

  25. L.G. Hanin, A stochastic model of tumor response to fractionated radiation: limit theorems and rate of convergence. Math. Biosci. 91(1), 1–17 (2004)

    Article  MathSciNet  Google Scholar 

  26. L.G. Hanin, Iterated birth and death process as a model of radiation cell survival. Math. Biosci. 169(1), 89–107 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. M.P. Hassell, The dynamics of arthropod predator-prey systems (Princeton University Press, Princeton, 1978)

    MATH  Google Scholar 

  28. A. Hastings, Models of spatial spread: is the theory compete? Ecology 77(6), 1675–1679 (1996)

    Google Scholar 

  29. H. Hatzikirou, L. Brusch, C. Schaller, M. Simon, A. Deutsch, Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion. Comput. Math. Appl. 59, 2326–2339 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. T. Hillen, \({M}^5\) mesoscopic and macroscopic models for mesenchymal motion. J. Math. Biol. 53(4), 585–616 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. T. Hillen, G. de Vries, J. Gong, C. Finlay, From cell population models to tumour control probability: including cell cycle effects. Acta Oncol. 49, 1315–1323 (2010)

    Google Scholar 

  32. T. Hillen, H. Enderling, P. Hahnfeldt, The tumor growth paradox and immune system-mediated selection for cancer stem cells. Bull. Math Biol. 75(1), 161–184 (2013)

    Google Scholar 

  33. T. Hillen, K. Painter, in Dispersal, Individual Movement and Spatial Ecology: A Mathematical Perspective, ed. by M. Lewis, P. Maini, S. Petrovskii. Transport and Anisotropic Diffusion Models for Movement in Oriented Habitats (Springer, Heidelberg, 2012), p. 46

    Google Scholar 

  34. J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamical Systems. London Mathematical Society Student Texts (Cambridge University Press, Cambridge, 1988)

    Google Scholar 

  35. Y. Iwasa, M.A. Nowak, F. Michor, Evolution of resistance during clonal expansion. Genetics 172, 2557–2566 (2006)

    Article  Google Scholar 

  36. A. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K.R. Swanson, M. Pelegrini-Issac, R. Guillevin, H. Benali, Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging. Magn. Reson. Med. 54, 616–624 (2005)

    Google Scholar 

  37. W.S. Kendal, A closed-form description of tumour control with fractionated radiotherapy and repopulation. Int. J. Radiat. Biol. 73(2), 207–210 (1998)

    Article  Google Scholar 

  38. N.L. Komarova, D. Wodarz, Drug resistance in cancer: principles of emergence and prevention. Proc. Natl. Acad. Sci. USA 102, 9714–9719 (2005)

    Google Scholar 

  39. E. Konukoglu, O. Clatz, P.Y. Bondiau, H. Delignette, N. Ayache, Extrapolation glioma invasion margin in brain magnetic resonance images: suggesting new irradiation margins. Med. Image Anal. 14, 111–125 (2010)

    Article  Google Scholar 

  40. M. Kot, M.A. Lewis, P. van den Driessche, Dispersal data and the spread of invading organisms. Ecology 77(7), 2027–2042 (1996)

    Article  Google Scholar 

  41. S. Lenhart, J.T. Workman, Optimal Control Applied to Biological Models (Chapman Hall/CRC Press, London, 2007)

    MATH  Google Scholar 

  42. S.A. Levin, The problem of pattern and scale in ecology. Ecology 73(6), 1943–1967 (1992)

    Article  Google Scholar 

  43. A. Maler, F. Lutscher, Cell cycle times and the tumor control probability. Math. Med. Biol. 27(4), 313–342 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  44. J. Maynard-Smith, The theory of games and animal conflict. J. Theor. Biol. 47, 209–209 (1974)

    Article  MathSciNet  Google Scholar 

  45. H.W. McKenzie, E.H. Merrill, R.J. Spiteri, M.A. Lewis, Linear features affect predator search time; implications for the functional response. Roy. Soc. Interface Focus 2, 205–216 (2012)

    Article  Google Scholar 

  46. L. Merlo, J. Pepper, B. Reid, C. Maley, Cancer as an evolutionary and ecological process. Nat. Rev. Cancer 6, 924–935 (2006)

    Article  Google Scholar 

  47. F. Mollica, L. Preziosi, and K.R. Rajagopal, (eds.), Modelling of Biological Material (Birkhauser, New York, 2007)

    Google Scholar 

  48. W.F. Morris, D.F. Doak, Quantitative Conservation Biology: Theory and Practice of Population Viability Analysis (Sinauer Associates Inc., Sunderland, 2002)

    Google Scholar 

  49. J.D. Nagy, The ecology and evolutionary biology of cancer: a review of mathematical models for necrosis and tumor cell diversity. Math. Biosci. Eng. 2(2), 381–418 (2005)

    Article  MathSciNet  Google Scholar 

  50. M.G. Neubert, H. Caswell, Demography and dispersal: calculation and sensitivity analysis of invasion speed for stage-structured populations. Ecology 81, 1613–1628 (2000)

    Article  Google Scholar 

  51. H.G. Othmer, S.R. Dunbar, W. Alt, Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  52. K.J. Painter, Modelling migration strategies in the extracellular matrix. J. Math. Biol. 58, 511–543 (2009)

    Article  MathSciNet  Google Scholar 

  53. K.J. Painter, T. Hillen, Mathematical modelling of glioma growth: the use of diffusion tensor imaging DTI data to predict the anisotropic pathways of cancer invasion (2012) (submitted)

    Google Scholar 

  54. A.B. Potapov, M.A. Lewis, D.C. Finnoff, Optimal control of biological invasions in lake networkds. Nat. Resour. Model. 20, 351–380 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  55. L. Preziosi (ed.), Cancer Modelling and Simulation (Chapman Hall/CRC Press, Boca Raton, 2003)

    Google Scholar 

  56. K.A. Rejniak, A.R.A. Anderson, Hybrid models of tumor growth. WIREs Syst. Biol. Med. 3, 115–125 (2011)

    Article  Google Scholar 

  57. E. Renshaw, Modelling Biological Populations in Space and Time (Cambridge University Press, Cambridge, 1991)

    Book  MATH  Google Scholar 

  58. N. Shigesada, K. Kawasaki, Biological Invasions: Theory and Practice (Oxford University Press, Oxford, 1997)

    Google Scholar 

  59. J.G. Skellam, Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  60. H. Smith, The Theory of the Chemostat (Cambridge University Press, Cambridge, 1995)

    Google Scholar 

  61. N.A. Stavreva, P.V. Stavrev, B. Warkentin, B.G. Fallone, Investigating the effect of cell repopulation on the tumor response to fractionated external radiotherapy. Med. Phys. 30(5), 735–742 (2003)

    Article  Google Scholar 

  62. K.R. Swanson, C. Bridge, J.D. Murray, E.C. Jr Alvord, Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J. Neurol. Sci. 216, 110 (2003)

    Google Scholar 

  63. T.E. Weldon, Mathematical Models in Cancer Research (Adam Hilger, Philadelphia, 1988)

    Google Scholar 

  64. S.M. Wise, J.A. Lowengrub, H.B. Frieboes, V. Cristini, Three-dimensional multispecies nonlinear tumor growth—I. J. Theor. Biol. 253, 524–543 (2008)

    Article  MathSciNet  Google Scholar 

  65. H. Youssefpour, X. Li, A.D. Lander, J.S. Lowengrub, Multispecies model of cell lineages and feedback control in solid tumors. J. Theor. Biol. 304, 39–59 (2012)

    Google Scholar 

  66. M. Zaider, G.N. Minerbo, Tumor control probability: a formulation applicable to any temporal protocol of dose delivery. Phys. Med. Biol. 45, 279–293 (2000)

    Article  Google Scholar 

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Acknowledgments

We are grateful for discussions with R. Gatenby and P. Hinow, which have motivated us to look deeper into the connection of cancer and ecology. TH acknowledges an NSERC Discovery Grant. MAL acknowledges NSERC Discovery and Accelerator Grants and a Canada Research Chair.

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Authors and Affiliations

  1. Centre for Mathematical Biology, University of Alberta, Edmonton, Canada

    Thomas Hillen & Mark A. Lewis

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  1. Thomas Hillen
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  2. Mark A. Lewis
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Correspondence to Thomas Hillen .

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Editors and Affiliations

  1. Department of Mathematical Sciences, Politecnico di Torino, Torino, Italy

    Marcello Delitala

  2. OECD, Paris, France

    Giulia Ajmone Marsan

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Hillen, T., Lewis, M.A. (2014). Mathematical Ecology of Cancer. In: Delitala, M., Ajmone Marsan, G. (eds) Managing Complexity, Reducing Perplexity. Springer Proceedings in Mathematics & Statistics, vol 67. Springer, Cham. https://doi.org/10.1007/978-3-319-03759-2_1

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