Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

The mathematics of cancer: integrating quantitative models

Nature Reviews Cancer volume 15, pages 730–745 (2015)Cite this article

Subjects

Key Points

  • Mathematical models have become an integral part of cancer biology. They are useful tools for deriving a mechanistic understanding of dynamic processes in cancer.

  • The somatic evolutionary process, which maintains tissues and can initiate cancer, has served as a hallmark of mathematical descriptions of tumours. Mathematical models have helped in the understanding of interactions among homeostatic mechanisms, environmental factors and mutation accumulation that drive tumorigenesis.

  • Using cell-based hierarchical models of tissue structure, theoretical insights have influenced the prediction of the cell of origin of human cancers, which may drive an understanding of metastasis and treatment response.

  • The temporal order of events during tumour development can be recapitulated using mathematical modelling and genomics data sets.

  • Mathematical models have also been used to explore the role of the tumour microenvironment in cancer progression. Such models help to elucidate important microenvironmental barriers to effective cancer treatment and how to overcome them.

  • Metastasis evolution and immunotherapy have attracted increasing interest but still offer a wide range of opportunities for mathematical modelling.

  • In combination with pharmacological considerations, quantitative models have a decisive role in the exploration of novel treatment modalities of cancer. This includes drug scheduling and the effect of combination therapy to avoid the evolution of resistance.

  • The key role of mathematical modelling in the future will not only be to describe what is known, but also to point to gaps in our understanding of which complex interactions drive tumour growth, treatment dynamics and resistance evolution.

Abstract

Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and metastases as well as intra-tumour heterogeneity, treatment responses and resistance. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research in cancer biology.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Tumour initiation and progression.
Figure 2: Microenvironmental interactions and the emergence of metastases.
Figure 3: Treatment response and dosing strategies.

References

  1. Weinberg, R. A. The Biology of Cancer (Garland Science, 2013).

    Google Scholar 

  2. Vogelstein, B. & Kinzler, K. W. The Genetic Basis of Human Cancer (McGraw–Hill, 1998).

    Google Scholar 

  3. Hanahan, D. & Weinberg, R. A. The hallmarks of cancer. Cell 100, 57–70 (2000).

    Article  CAS  PubMed  Google Scholar 

  4. Armitage, P. & Doll, R. The age distribution of cancer and a multi-stage theory of carcinogenesis. Br. J. Cancer 8, 1–12 (1954).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  5. Nowell, P. C. The clonal evolution of tumor cell populations. Science 194, 23–28 (1976).

    Article  CAS  PubMed  Google Scholar 

  6. Garraway, L. A. & Lander, E. S. Lessons from the cancer genome. Cell 153, 17–37 (2013).

    Article  CAS  PubMed  Google Scholar 

  7. Dunn, G. P., Bruce, A. T., Ikeda, H., Old, L. J. & Schreiber, R. D. Cancer immunoediting: from immunosurveillance to tumor escape. Nat. Immunol. 3, 991–998 (2002).

    Article  CAS  PubMed  Google Scholar 

  8. Pages, F. et al. Effector memory T cells, early metastasis, and survival in colorectal cancer. N. Engl. J. Med. 353, 2654–2666 (2005).

    Article  CAS  PubMed  Google Scholar 

  9. Allan, J. M. & Travis, L. B. Mechanisms of therapy-related carcinogenesis. Nat. Rev. Cancer 5, 943–955 (2005).

    Article  CAS  PubMed  Google Scholar 

  10. Nguyen, D. X., Bos, P. D. & Massague, J. Metastasis: from dissemination to organ-specific colonization. Nat. Rev. Cancer 9, 274–284 (2009).

    Article  CAS  PubMed  Google Scholar 

  11. Kim, M. Y. et al. Tumor self-seeding by circulating cancer cells. Cell 139, 1315–1326 (2009).

    Article  PubMed  PubMed Central  Google Scholar 

  12. Anderson, A. R., Weaver, A. M., Cummings, P. T. & Quaranta, V. Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127, 905–915 (2006). A multiscale mathematical model that attempts to describe cancer evolution and the dynamics of the microenvironment, and shows that both genetic changes and environmental changes can impact cancer invasiveness.

    Article  CAS  PubMed  Google Scholar 

  13. Vaupel, P., Kallinowski, F. & Okunieff, P. Blood flow, oxygen and nutrient supply, and metabolic microenvironment of human tumors: a review. Cancer Res. 49, 6449–6465 (1989).

    CAS  PubMed  Google Scholar 

  14. Anderson, A. R. & Quaranta, V. Integrative mathematical oncology. Nat. Rev. Cancer 8, 227–234 (2008).

    Article  CAS  PubMed  Google Scholar 

  15. Byrne, H. M. Dissecting cancer through mathematics: from the cell to the animal model. Nat. Rev. Cancer 10, 221–230 (2010).

    Article  CAS  PubMed  Google Scholar 

  16. Knudson, A. G. Jr Mutation and cancer: statistical study of retinoblastoma. Proc. Natl Acad. Sci. USA 68, 820–823 (1971).

    Article  PubMed  PubMed Central  Google Scholar 

  17. Cavenee, W. K. et al. Expression of recessive alleles by chromosomal mechanisms in retinoblastoma. Nature 305, 779–784 (1983).

    Article  CAS  PubMed  Google Scholar 

  18. Bozic, I. et al. Evolutionary dynamics of cancer in response to targeted combination therapy. eLife 2, e00747 (2013). A stochastic evolutionary model that identifies probabilities of evolution of resistance to combination therapy.

    Article  PubMed  PubMed Central  Google Scholar 

  19. Lenaerts, T., Pacheco, J. M., Traulsen, A. & Dingli, D. Tyrosine kinase inhibitor therapy can cure chronic myeloid leukemia without hitting leukemic stem cells. Haematologica 95, 900–907 (2010).

    Article  CAS  PubMed  Google Scholar 

  20. Diaz, L. A. Jr et al. The molecular evolution of acquired resistance to targeted EGFR blockade in colorectal cancers. Nature 486, 537–540 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. Jones, S. et al. Comparative lesion sequencing provides insights into tumor evolution. Proc. Natl Acad. Sci. USA 105, 4283–4288 (2008).

    Article  PubMed  PubMed Central  Google Scholar 

  22. Leder, K. et al. Mathematical modeling of PDGF-driven glioblastoma reveals optimized radiation dosing schedules. Cell 156, 603–616 (2014). A mathematical model and optimization approach to identify better radiation scheduling in glioblastoma that led to survival improvement in a mouse trial.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  23. Sherratt, J. A. & Nowak, M. A. Oncogenes, anti-oncogenes and the immune response to cancer: a mathematical model. Proc. Biol. Sci. 248, 261–271 (1992).

    Article  CAS  PubMed  Google Scholar 

  24. Komarova, N. L. & Wodarz, D. Drug resistance in cancer: principles of emergence and prevention. Proc. Natl Acad. Sci. USA 102, 9714–9719 (2005). A mathematical modelling contribution towards the understanding of resistance that existed prior to chemotherapy and targeted combination therapy.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  25. Sanga, S. et al. Mathematical modeling of cancer progression and response to chemotherapy. Expert Rev. Anticancer Ther. 6, 1361–1376 (2006).

    Article  CAS  PubMed  Google Scholar 

  26. Swanson, K. R., Alvord, E. C. Jr & Murray, J. D. Virtual brain tumours (gliomas) enhance the reality of medical imaging and highlight inadequacies of current therapy. Br. J. Cancer 86, 14–18 (2002). A hallmark paper that shows how a 'virtual tumour' allows data-driven therapy improvements using mathematical modelling.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  27. Frank, S. A., Iwasa, Y. & Nowak, M. A. Patterns of cell division and the risk of cancer. Genetics 163, 1527–1532 (2003).

    PubMed  PubMed Central  Google Scholar 

  28. Armitage, P. & Doll, R. A two-stage theory of carcinogenesis in relation to the age distribution of human cancer. Br. J. Cancer 11, 161–169 (1957).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  29. Nordling, C. O. A new theory on cancer-inducing mechanism. Br. J. Cancer 7, 68–72 (1953).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  30. Fisher, J. & Hollomon, J. A hypothesis for the origin of cancer foci. Cancer 4, 916–918 (1951).

    Article  CAS  PubMed  Google Scholar 

  31. Stratton, M. R., Campbell, P. J. & Futreal, P. A. The cancer genome. Nature 458, 719–724 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  32. Foy, M., Spitz, M. R., Kimmel, M. & Gorlova, O. Y. A smoking-based carcinogenesis model for lung cancer risk prediction. Int. J. Cancer 129, 1907–1913 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  33. Kimmel, M. & Axelrod, D. Branching Processes in Biology. Interdisciplinary Applied Mathematics (Springer, 2015).

    Google Scholar 

  34. Haccou, P. Branching Processes: Variation, Growth, and Extinction of Populations (Cambridge Univ. Press, 2005).

    Book  Google Scholar 

  35. Durrett, R. Branching Process Models of Cancer (Springer, 2015).

    Book  Google Scholar 

  36. Antal, T. & Krapivsky, P. L. Exact solution of a two-type branching process: models of tumor progression. J. Statist. Mechan.-Theory Exper. (2011).

  37. Parzen, E. Stochastic processes (SIAM, 1999).

    Book  Google Scholar 

  38. Bozic, I. et al. Accumulation of driver and passenger mutations during tumor progression. Proc. Natl Acad. Sci. USA 107, 18545–18550 (2010). A mathematical modelling contribution that studies the accumulation of driver and passenger mutations in cancer.

    Article  PubMed  PubMed Central  Google Scholar 

  39. Bauer, B., Siebert, R. & Traulsen, A. Cancer initiation with epistatic interactions between driver and passenger mutations. J. Theor. Biol. 358, 52–60 (2014).

    Article  PubMed  Google Scholar 

  40. Tomasetti, C., Vogelstein, B. & Parmigiani, G. Half or more of the somatic mutations in cancers of self-renewing tissues originate prior to tumor initiation. Proc. Natl Acad. Sci. USA 110, 1999–2004 (2013). A mathematical model of development and tumorigenesis in which driver and passenger mutations accumulate.

    Article  PubMed  PubMed Central  Google Scholar 

  41. Tomasetti, C. & Vogelstein, B. Variation in cancer risk among tissues can be explained by the number of stem cell divisions. Science 347, 78–81 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  42. Wodarz, D. & Zauber, A. G. Cancer: risk factors and random chances. Nature 517, 563–564 (2015).

    Article  CAS  PubMed  Google Scholar 

  43. Ashford, N. A. et al. Cancer risk: role of environment. Science 347, 727 (2015).

    Article  CAS  PubMed  Google Scholar 

  44. O'Callaghan, M. Cancer risk: accuracy of literature. Science 347, 729 (2015).

    Article  CAS  PubMed  Google Scholar 

  45. Potter, J. D. & Prentice, R. L. Cancer risk: tumors excluded. Science 347, 727 (2015).

    Article  CAS  PubMed  Google Scholar 

  46. Tomasetti, C. & Vogelstein, B. Cancer risk: role of environment-response. Science 347, 729–731 (2015).

    Article  CAS  PubMed  Google Scholar 

  47. Noble, R., Kaltz, O. & Hochberg, M. E. Peto's paradox and human cancers. Phil. Trans. R. Soc. B 370, 11 (2015).

    Article  Google Scholar 

  48. McFarland, C. D., Korolev, K. S., Kryukov, G. V., Sunyaev, S. R. & Mirny, L. A. Impact of deleterious passenger mutations on cancer progression. Proc. Natl Acad. Sci. USA 110, 2910–2915 (2013). A mathematical model that includes slightly deleterious passenger mutations that accumulate between sweeps caused by oncogenic driver mutations.

    Article  PubMed  PubMed Central  Google Scholar 

  49. McFarland, C. D., Mirny, L. A. & Korolev, K. S. Tug-of-war between driver and passenger mutations in cancer and other adaptive processes. Proc. Natl Acad. Sci. USA 111, 15138–15143 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  50. Foo, J., Leder, K. & Michor, F. Stochastic dynamics of cancer initiation. Phys. Biol. 8, 015002 (2011).

    Article  PubMed  PubMed Central  Google Scholar 

  51. Moran, P. A. P. Random processes in genetics. Math. Proc. Cambridge Phil. Soc. 54, 60–71 (1958).

    Article  Google Scholar 

  52. Foo, J. et al. An evolutionary approach for identifying driver mutations in colorectal cancer. PLoS Comput. Biol. 11, e1004350 (2015).

    Article  PubMed  PubMed Central  Google Scholar 

  53. Mumenthaler, S. M. et al. The impact of microenvironmental heterogeneity on the evolution of drug resistance in cancer cells. Cancer Inform. 14, 19–31 (2015).

    CAS  PubMed  PubMed Central  Google Scholar 

  54. Beerenwinkel, N. et al. Genetic progression and the waiting time to cancer. PLoS Comput. Biol. 3, e225 (2007). A mathematical model that allows calculation of the expected waiting time to cancer using a Wright–Fisher process and tumour mutation data.

  55. Werner, B., Dingli, D. & Traulsen, A. A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues. J. R. Soc. Interface 10, 20130349 (2013).

    Article  PubMed  PubMed Central  Google Scholar 

  56. Werner, B., Dingli, D., Lenaerts, T., Pacheco, J. M. & Traulsen, A. Dynamics of mutant cells in hierarchical organized tissues. PLoS Comput. Biol. 7, e1002290 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  57. Weekes, S. L. et al. A multicompartment mathematical model of cancer stem cell-driven tumor growth dynamics. Bull. Math. Biol. 76, 1762–1782 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  58. Michor, F., Nowak, M. A., Frank, S. A. & Iwasa, Y. Stochastic elimination of cancer cells. Proc. Biol. Sci. 270, 2017–2024 (2003).

    Article  PubMed  PubMed Central  Google Scholar 

  59. Roeder, I. et al. Dynamic modeling of imatinib-treated chronic myeloid leukemia: functional insights and clinical implications. Nat. Med. 12, 1181–1184 (2006).

    Article  CAS  PubMed  Google Scholar 

  60. Haeno, H., Levine, R. L., Gilliland, D. G. & Michor, F. A progenitor cell origin of myeloid malignancies. Proc. Natl Acad. Sci. USA 106, 16616–16621 (2009). Mathematical modelling of the different evolutionary pathways leading to cancer, which calculates the likelihood of individual cell types serving as the cell of origin.

    Article  PubMed  PubMed Central  Google Scholar 

  61. Ganguly, R. & Puri, I. K. Mathematical model for the cancer stem cell hypothesis. Cell Prolif. 39, 3–14 (2006).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  62. Hambardzumyan, D., Cheng, Y. K., Haeno, H., Holland, E. C. & Michor, F. The probable cell of origin of NF1- and PDGF-driven glioblastomas. PLoS ONE 6, e24454 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  63. Fearon, E. R. & Vogelstein, B. A genetic model for colorectal tumorigenesis. Cell 61, 759–767 (1990).

    Article  CAS  PubMed  Google Scholar 

  64. Vogelstein, B. & Kinzler, K. W. The Genetic Basis of Human Cancer (McGraw, 2002).

    Google Scholar 

  65. Vogelstein, B. et al. Genetic alterations during colorectal-tumor development. N. Engl. J. Med. 319, 525–532 (1988).

    Article  CAS  PubMed  Google Scholar 

  66. Cancer Genome Atlas Network. Comprehensive molecular characterization of human colon and rectal cancer. Nature 487, 330–337 (2012).

  67. Cancer Genome Atlas Research Network. Integrated genomic analyses of ovarian carcinoma. Nature 474, 609–615 (2011).

  68. Cancer Genome Atlas Research Network. Comprehensive genomic characterization of squamous cell lung cancers. Nature 489, 519–525 (2012).

  69. Cancer Genome Atlas Research Network. Genomic and epigenomic landscapes of adult de novo acute myeloid leukemia. N. Engl. J. Med. 368, 2059–2074 (2013).

  70. Forbes, S. A. et al. COSMIC: mining complete cancer genomes in the Catalogue of Somatic Mutations in Cancer. Nucleic Acids Res. 39, D945–D950 (2011).

    Article  CAS  PubMed  Google Scholar 

  71. International Cancer Genome Consortium et al. International network of cancer genome projects. Nature 464, 993–998 (2010).

  72. Sjoblom, T. et al. The consensus coding sequences of human breast and colorectal cancers. Science 314, 268–274 (2006).

    Article  CAS  PubMed  Google Scholar 

  73. Stransky, N. et al. The mutational landscape of head and neck squamous cell carcinoma. Science 333, 1157–1160 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  74. Desper, R. et al. Inferring tree models for oncogenesis from comparative genome hybridization data. J. Comput. Biol. 6, 37–51 (1999).

    Article  CAS  PubMed  Google Scholar 

  75. Desper, R. et al. Distance-based reconstruction of tree models for oncogenesis. J. Comput. Biol. 7, 789–803 (2000).

    Article  CAS  PubMed  Google Scholar 

  76. Hoglund, M., Frigyesi, A., Sall, T., Gisselsson, D. & Mitelman, F. Statistical behavior of complex cancer karyotypes. Genes Chromosomes Cancer 42, 327–341 (2005).

    Article  CAS  PubMed  Google Scholar 

  77. Beerenwinkel, N. et al. Learning multiple evolutionary pathways from cross-sectional data. J. Comput. Biol. 12, 584–598 (2005). The introduction of a mixture model of multiple evolutionary trees that allows detailed characterization of mutations leading to cancer.

    Article  CAS  PubMed  Google Scholar 

  78. Huang, Z. et al. Construction of tree models for pathogenesis of nasopharyngeal carcinoma. Genes Chromosomes Cancer 40, 307–315 (2004).

    Article  CAS  PubMed  Google Scholar 

  79. Pathare, S., Schaffer, A. A., Beerenwinkel, N. & Mahimkar, M. Construction of oncogenetic tree models reveals multiple pathways of oral cancer progression. Int. J. Cancer 124, 2864–2871 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  80. Hjelm, M., Hoglund, M. & Lagergren, J. New probabilistic network models and algorithms for oncogenesis. J. Comput. Biol. 13, 853–865 (2006).

    Article  CAS  PubMed  Google Scholar 

  81. Gerstung, M., Baudis, M., Moch, H. & Beerenwinkel, N. Quantifying cancer progression with conjunctive Bayesian networks. Bioinformatics 25, 2809–2815 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  82. Simon, R. et al. Chromosome abnormalities in ovarian adenocarcinoma: III. Using breakpoint data to infer and test mathematical models for oncogenesis. Genes Chromosomes Cancer 28, 106–120 (2000).

    Article  CAS  PubMed  Google Scholar 

  83. Gerstung, M., Eriksson, N., Lin, J., Vogelstein, B. & Beerenwinkel, N. The temporal order of genetic and pathway alterations in tumorigenesis. PLoS ONE 6, e27136 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  84. Attolini, C. S. et al. A mathematical framework to determine the temporal sequence of somatic genetic events in cancer. Proc. Natl Acad. Sci. USA 107, 17604–17609 (2010). The introduction of an evolutionary modelling approach allowing the identification of the order of mutations fuelling tumour development.

    Article  PubMed  PubMed Central  Google Scholar 

  85. Cheng, Y. K. et al. A mathematical methodology for determining the temporal order of pathway alterations arising during gliomagenesis. PLoS Comput. Biol. 8, e1002337 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  86. Antal, T. & Scheuring, I. Fixation of strategies for an evolutionary game in finite populations. Bull. Math. Biol. 68, 1923–1944 (2006).

    Article  PubMed  Google Scholar 

  87. Li, X. et al. Temporal and spatial evolution of somatic chromosomal alterations: a case-cohort study of Barrett's esophagus. Cancer Prev. Res. (Phila) 7, 114–127 (2014).

    Article  Google Scholar 

  88. Sprouffske, K., Pepper, J. W. & Maley, C. C. Accurate reconstruction of the temporal order of mutations in neoplastic progression. Cancer Prev. Res. (Phila) 4, 1135–1144 (2011).

    Article  Google Scholar 

  89. Wang, Y. et al. Clonal evolution in breast cancer revealed by single nucleus genome sequencing. Nature 512, 155–160 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  90. Martins, F. C. et al. Evolutionary pathways in BRCA1-associated breast tumors. Cancer Discov. 2, 503–511 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  91. Durinck, S. et al. Temporal dissection of tumorigenesis in primary cancers. Cancer Discov. 1, 137–143 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  92. Greenman, C. D. et al. Estimation of rearrangement phylogeny for cancer genomes. Genome Res. 22, 346–361 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  93. Purdom, E. et al. Methods and challenges in timing chromosomal abnormalities within cancer samples. Bioinformatics 29, 3113–3120 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  94. Sottoriva, A. et al. A Big Bang model of human colorectal tumor growth. Nat. Genet. 47, 209–216 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  95. Fidler, I. J. The pathogenesis of cancer metastasis: the 'seed and soil' hypothesis revisited. Nat. Rev. Cancer 3, 453–458 (2003).

    Article  CAS  PubMed  Google Scholar 

  96. Liotta, L. A. & Kohn, E. C. The microenvironment of the tumour–host interface. Nature 411, 375–379 (2001).

    Article  CAS  PubMed  Google Scholar 

  97. Mueller, M. M. & Fusenig, N. E. Friends or foes – bipolar effects of the tumour stroma in cancer. Nat. Rev. Cancer 4, 839–849 (2004).

    Article  CAS  PubMed  Google Scholar 

  98. Paget, S. The distribution of secondary growths in cancer of the breast. 1889. Cancer Metastasis Rev. 8, 98–101 (1989).

    CAS  PubMed  Google Scholar 

  99. Byrne, H. M. & Chaplain, M. Growth of necrotic tumors in the presence and absence of inhibitors. Math. Biosci. 135, 187–216 (1996).

    Article  CAS  PubMed  Google Scholar 

  100. Sherratt, J. A. & Chaplain, M. A. A new mathematical model for avascular tumour growth. J. Math. Biol. 43, 291–312 (2001).

    Article  CAS  PubMed  Google Scholar 

  101. Ferreira, S. C. Jr., Martins, M. L. & Vilela, M. J. Reaction-diffusion model for the growth of avascular tumor. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65, 021907 (2002).

    Article  CAS  PubMed  Google Scholar 

  102. Orme, M. E. & Chaplain, M. A. Two-dimensional models of tumour angiogenesis and anti-angiogenesis strategies. IMA J. Math. Appl. Med. Biol. 14, 189–205 (1997).

    Article  CAS  PubMed  Google Scholar 

  103. Stokes, C. L. & Lauffenburger, D. A. Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. Theor. Biol. 152, 377–403 (1991).

    Article  CAS  PubMed  Google Scholar 

  104. Macklin, P. et al. Multiscale modelling and nonlinear simulation of vascular tumour growth. J. Math. Biol. 58, 765–798 (2009).

    Article  PubMed  Google Scholar 

  105. Rejniak, K. A. & Anderson, A. R. Hybrid models of tumor growth. Wiley Interdiscip. Rev. Syst. Biol. Med. 3, 115–125 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  106. Hatzikirou, H., Basanta, D., Simon, M., Schaller, K. & Deutsch, A. 'Go or grow': the key to the emergence of invasion in tumour progression? Math. Med. Biol. 29, 49–65 (2012).

    Article  CAS  PubMed  Google Scholar 

  107. Anderson, A. R. & Chaplain, M. A. Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60, 857–899 (1998).

    Article  CAS  PubMed  Google Scholar 

  108. McDougall, S. R., Anderson, A. R. & Chaplain, M. A. Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies. J. Theor. Biol. 241, 564–589 (2006).

    Article  PubMed  Google Scholar 

  109. Kim, Y., Stolarska, M. A. & Othmer, H. G. A hybrid model for tumor spheroid growth in vitro I: theoretical development and early results. Math. Models Methods Appl. Sci. 17, 1773–1798 (2007).

    Article  CAS  Google Scholar 

  110. Anderson, A. R. A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math. Med. Biol. 22, 163–186 (2005). One of the first papers to use multiscale hybrid models in cancer.

    Article  PubMed  Google Scholar 

  111. Araujo, A., Cook, L. M., Lynch, C. C. & Basanta, D. An integrated computational model of the bone microenvironment in bone-metastatic prostate cancer. Cancer Res. 74, 2391–2401 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  112. Li, X. et al. Nonlinear three-dimensional simulation of solid tumor growth. Discrete Continuous Dyn. Sys. Ser. B 7, 581–604 (2007).

    Article  Google Scholar 

  113. Frieboes, H. B. et al. Three-dimensional multispecies nonlinear tumor growth-II: tumor invasion and angiogenesis. J. Theor. Biol. 264, 1254–1278 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  114. Wise, S. M., Lowengrub, J. S., Frieboes, H. B. & Cristini, V. Three-dimensional multispecies nonlinear tumor growth—I Model and numerical method. J. Theor. Biol. 253, 524–543 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  115. Frieboes, H. B. et al. Prediction of drug response in breast cancer using integrative experimental/computational modeling. Cancer Res. 69, 4484–4492 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  116. Waclaw, B. et al. A spatial model predicts that dispersal and cell turnover limit intratumour heterogeneity. Nature (2015). A model that combines spatially explicit evolutionary processes with dynamics of mutation accumulation.

  117. Warburg, O., Wind, F. & Negelein, E. The metabolism of tumors in the body. J. Gen. Physiol. 8, 519–530 (1927).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  118. Gatenby, R. A. & Gawlinski, E. T. A reaction-diffusion model of cancer invasion. Cancer Res. 56, 5745–5753 (1996).

    CAS  PubMed  Google Scholar 

  119. Patel, A. A., Gawlinski, E. T., Lemieux, S. K. & Gatenby, R. A. A cellular automaton model of early tumor growth and invasion. J. Theor. Biol. 213, 315–331 (2001).

    Article  CAS  PubMed  Google Scholar 

  120. Gerlee, P. & Anderson, A. R. A hybrid cellular automaton model of clonal evolution in cancer: the emergence of the glycolytic phenotype. J. Theor. Biol. 250, 705–722 (2008).

    Article  CAS  PubMed  Google Scholar 

  121. Swanson, K. R., Bridge, C., Murray, J. & Alvord Jr, E. C. Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J. Neurol. Sci. 216, 1–10 (2003).

    Article  PubMed  Google Scholar 

  122. Cocosco, C. A., Kollokian, V., Kwan, R. K.-S., Pike, G. B. & Evans, A. C. BrainWeb: online interface to a 3D MRI simulated brain database. NeuroImage 5, S425 (1997).

    Google Scholar 

  123. Harpold, H. L., Alvord, E. C. Jr & Swanson, K. R. The evolution of mathematical modeling of glioma proliferation and invasion. J. Neuropathol. Exp. Neurol. 66, 1–9 (2007).

    Article  PubMed  Google Scholar 

  124. Wang, C. H. et al. Prognostic significance of growth kinetics in newly diagnosed glioblastomas revealed by combining serial imaging with a novel biomathematical model. Cancer Res. 69, 9133–9140 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  125. Swanson, K. R. et al. Quantifying the role of angiogenesis in malignant progression of gliomas: in silico modeling integrates imaging and histology. Cancer Res. 71, 7366–7375 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  126. Atuegwu, N. C. et al. Incorporation of diffusion-weighted magnetic resonance imaging data into a simple mathematical model of tumor growth. Phys. Med. Biol. 57, 225–240 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  127. Yankeelov, T. E. et al. Clinically relevant modeling of tumor growth and treatment response. Sci. Transl Med. 5, 187ps9 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  128. Yankeelov, T. E., Quaranta, V., Evans, K. J. & Rericha, E. C. Toward a science of tumor forecasting for clinical oncology. Cancer Res. 75, 918–923 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  129. Deming, D. A. et al. PIK3CA and APC mutations are synergistic in the development of intestinal cancers. Oncogene 33, 2245–2254 (2014).

    Article  CAS  PubMed  Google Scholar 

  130. Mahabeleshwar, G. H., Feng, W., Reddy, K., Plow, E. F. & Byzova, T. V. Mechanisms of integrin-vascular endothelial growth factor receptor cross-activation in angiogenesis. Circ. Res. 101, 570–580 (2007).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  131. Archetti, M., Ferraro, D. A. & Christofori, G. Heterogeneity for IGF-II production maintained by public goods dynamics in neuroendocrine pancreatic cancer. Proc. Natl Acad. Sci. USA 112, 1833–1838 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  132. Gerlee, P. & Altrock, P. M. Complexity and stability in growing cancer cell populations. Proc. Natl Acad. Sci. USA 112, E2742–E2743 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  133. Marusyk, A. et al. Non-cell-autonomous driving of tumour growth supports sub-clonal heterogeneity. Nature 514, 54–58 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  135. Carmeliet, P. & Jain, R. K. Angiogenesis in cancer and other diseases. Nature 407, 249–257 (2000).

    Article  CAS  PubMed  Google Scholar 

  136. Chapman, S. J., Shipley, R. J. & Jawad, R. Multiscale modeling of fluid transport in tumors. Bull. Math. Biol. 70, 2334–2357 (2008).

    Article  PubMed  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  138. Alarcon, T., Byrne, H. M. & Maini, P. K. A multiple scale model for tumor growth. Multiscale Model. Simul. 3, 440–475 (2005).

    Article  CAS  Google Scholar 

  139. Rockne, R. et al. Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach. Phys. Med. Biol. 55, 3271–3285 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  140. Jackson, T., Komarova, N. & Swanson, K. R. Mathematical oncology: using mathematics to enable cancer discoveries. Am. Math. Mon. 121, 840–856 (2014).

    Article  Google Scholar 

  141. Dale, R. G. The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy. Br. J. Radiol 58, 515–528 (1985).

    Article  CAS  PubMed  Google Scholar 

  142. Rosenberg, S. A., Yang, J. C. & Restifo, N. P. Cancer immunotherapy: moving beyond current vaccines. Nat. Med. 10, 909–915 (2004).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  143. Kalos, M. et al. T cells with chimeric antigen receptors have potent antitumor effects and can establish memory in patients with advanced leukemia. Sci. Transl Med. 3, 95ra73 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  144. Blattman, J. N. & Greenberg, P. D. Cancer immunotherapy: a treatment for the masses. Science 305, 200–205 (2004).

    Article  CAS  PubMed  Google Scholar 

  145. Pardoll, D. M. The blockade of immune checkpoints in cancer immunotherapy. Nat. Rev. Cancer 12, 252–264 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  146. Old, L. J. Cancer immunology: the search for specificity—G. H. A. Clowes Memorial lecture. Cancer Res. 41, 361–375 (1981).

    CAS  PubMed  Google Scholar 

  147. Abbas, A. K., Lichtman, A. H. & Pillai, S. Cellular and Molecular Immunology (Elsevier Health Sciences, 1994).

    Google Scholar 

  148. Kuznetsov, V. in A Survey of Models for Tumor-Immune System Dynamics (eds Adam, J. A. & Bellomo, N.) 237–294 (Springer, 1997).

    Book  Google Scholar 

  149. Bellomo, N. & Preziosi, L. Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Computer Modell. 32, 413–452 (2000).

    Article  Google Scholar 

  150. Kolev, M. Mathematical modelling of the competition between tumors and immune system considering the role of the antibodies. Math. Computer Modell. 37, 1143–1152 (2003).

    Article  Google Scholar 

  151. de Pillis, L. G. & Radunskaya, A. A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. Computat. Math. Methods Med. 3, 79–100 (2001).

    Google Scholar 

  152. de Pillis, L. G., Radunskaya, A. E. & Wiseman, C. L. A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res. 65, 7950–7958 (2005).

    Article  CAS  PubMed  Google Scholar 

  153. d'Onofrio, A. Metamodeling tumor–immune system interaction, tumor evasion and immunotherapy. Math. Computer Modell. 47, 614–637 (2008).

    Article  Google Scholar 

  154. d'Onofrio, A. A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences. Phys. D Nonlin. Phenomena 208, 220–235 (2005).

    Article  Google Scholar 

  155. Sivakumar, P. V., Foster, D. C. & Clegg, C. H. Interleukin-21 is a T-helper cytokine that regulates humoral immunity and cell-mediated anti-tumour responses. Immunology 112, 177–182 (2004).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  156. Cappuccio, A., Elishmereni, M. & Agur, Z. Cancer immunotherapy by interleukin-21: potential treatment strategies evaluated in a mathematical model. Cancer Res. 66, 7293–7300 (2006).

    Article  CAS  PubMed  Google Scholar 

  157. Pappalardo, F. et al. SimB16: modeling induced immune system response against B16-melanoma. PLoS ONE 6, e26523 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  158. Pienta, K. J., Robertson, B. A., Coffey, D. S. & Taichman, R. S. The cancer diaspora: metastasis beyond the seed and soil hypothesis. Clin. Cancer Res. 19, 5849–5855 (2013).

    Article  PubMed  Google Scholar 

  159. Comen, E. & Norton, L. Self-seeding in cancer. Recent Results Cancer Res. 195, 13–23 (2012).

    Article  PubMed  Google Scholar 

  160. Saidel, G. M., Liotta, L. A. & Kleinerman, J. System dynamics of metastatic process from an implanted tumor. J. Theor. Biol. 56, 417–434 (1976).

    Article  CAS  PubMed  Google Scholar 

  161. Panetta, J. C. A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment. Bull. Math. Biol. 58, 425–447 (1996).

    Article  CAS  PubMed  Google Scholar 

  162. Enderling, H. et al. Paradoxical dependencies of tumor dormancy and progression on basic cell kinetics. Cancer Res. 69, 8814–8821 (2009). A modelling contribution that combines concepts of a differentiation hierarchy with spatial cell migration and therapy effects.

    Article  CAS  PubMed  Google Scholar 

  163. Almendro, V. et al. Genetic and phenotypic diversity in breast tumor metastases. Cancer Res. 74, 1338–1348 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  164. Scott, J., Kuhn, P. & Anderson, A. R. Unifying metastasis — integrating intravasation, circulation and end-organ colonization. Nat. Rev. Cancer 12, 445–446 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  165. Weiss, L. Comments on hematogenous metastatic patterns in humans as revealed by autopsy. Clin. Exp. Metastasis 10, 191–199 (1992).

    Article  CAS  PubMed  Google Scholar 

  166. Scott, J. G., Basanta, D., Anderson, A. R. & Gerlee, P. A mathematical model of tumour self-seeding reveals secondary metastatic deposits as drivers of primary tumour growth. J. R. Soc. Interface 10, 20130011 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  167. Scott, J. G., Fletcher, A. G., Maini, P. K., Anderson, A. R. & Gerlee, P. A filter-flow perspective of haematogenous metastasis offers a non-genetic paradigm for personalised cancer therapy. Eur. J. Cancer 50, 3068–3075 (2014).

    Article  PubMed  PubMed Central  Google Scholar 

  168. Fu, F., Nowak, M. A. & Bonhoeffer, S. Spatial heterogeneity in drug concentrations can facilitate the emergence of resistance to cancer therapy. PLoS Comput. Biol. 11, e1004142 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  169. Haeno, H. et al. Computational modeling of pancreatic cancer reveals kinetics of metastasis suggesting optimum treatment strategies. Cell 148, 362–375 (2012). A stochastic process model that considers tumour growth, death, mutation and dissemination events parameterized using pancreatic cancer patient data to identify improved treatment strategies.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  170. Haeno, H. & Michor, F. The evolution of tumor metastases during clonal expansion. J. Theor. Biol. 263, 30–44 (2010).

    Article  CAS  PubMed  Google Scholar 

  171. Newton, P. K. et al. A stochastic Markov chain model to describe lung cancer growth and metastasis. PLoS ONE 7, e34637 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  172. Newton, P. K. et al. Spreaders and sponges define metastasis in lung cancer: a Markov chain Monte Carlo mathematical model. Cancer Res. 73, 2760–2769 (2013). A mathematical model parameterized using tumour autopsy data to study the metastatic process, which allows different tumour sites to be classified as spreaders or sponges.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  173. Bretcha-Boix, P., Rami-Porta, R., Mateu-Navarro, M., Hoyuela-Alonso, C. & Marco-Molina, C. Surgical treatment of lung cancer with adrenal metastasis. Lung Cancer 27, 101–105 (2000).

    Article  CAS  PubMed  Google Scholar 

  174. Bazhenova, L. et al. Adrenal metastases in lung cancer: clinical implications of a mathematical model. J. Thorac. Oncol. 9, 442–446 (2014).

    Article  PubMed  PubMed Central  Google Scholar 

  175. Pathak, A. & Kumar, S. Independent regulation of tumor cell migration by matrix stiffness and confinement. Proc. Natl Acad. Sci. USA 109, 10334–10339 (2012).

    Article  PubMed  PubMed Central  Google Scholar 

  176. Michor, F., Liphardt, J., Ferrari, M. & Widom, J. What does physics have to do with cancer? Nat. Rev. Cancer 11, 657–670 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  177. Jain, R. K., Tong, R. T. & Munn, L. L. Effect of vascular normalization by antiangiogenic therapy on interstitial hypertension, peritumor edema, and lymphatic metastasis: insights from a mathematical model. Cancer Res. 67, 2729–2735 (2007).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  178. Jain, R. K. Normalizing tumor microenvironment to treat cancer: bench to bedside to biomarkers. J. Clin. Oncol. 31, 2205–2218 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  179. Coldman, A. J. & Goldie, J. H. A stochastic model for the origin and treatment of tumors containing drug-resistant cells. Bull. Math. Biol. 48, 279–292 (1986).

    Article  CAS  PubMed  Google Scholar 

  180. Goldie, J. H. & Coldman, A. J. Quantitative model for multiple levels of drug resistance in clinical tumors. Cancer Treat. Rep. 67, 923–931 (1983).

    CAS  PubMed  Google Scholar 

  181. Goldie, J. H. & Coldman, A. J. The genetic origin of drug resistance in neoplasms: implications for systemic therapy. Cancer Res. 44, 3643–3653 (1984).

    CAS  PubMed  Google Scholar 

  182. Norton, L. & Simon, R. Growth curve of an experimental solid tumor following radiotherapy. J. Natl Cancer Inst. 58, 1735–1741 (1977).

    Article  CAS  PubMed  Google Scholar 

  183. Citron, M. L. et al. Randomized trial of dose-dense versus conventionally scheduled and sequential versus concurrent combination chemotherapy as postoperative adjuvant treatment of node-positive primary breast cancer: first report of Intergroup Trial C9741/Cancer and Leukemia Group B Trial 9741. J. Clin. Oncol. 21, 1431–1439 (2003).

    Article  CAS  PubMed  Google Scholar 

  184. Goldie, J. H. & Coldman, A. J. A mathematic model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Cancer Treat. Rep. 63, 1727–1733 (1979).

    CAS  PubMed  Google Scholar 

  185. Bonadonna, G., Zambetti, M., Moliterni, A., Gianni, L. & Valagussa, P. Clinical relevance of different sequencing of doxorubicin and cyclophosphamide, methotrexate, and fluorouracil in operable breast cancer. J. Clin. Oncol. 22, 1614–1620 (2004).

    Article  CAS  PubMed  Google Scholar 

  186. Foo, J. & Michor, F. Evolution of acquired resistance to anti-cancer therapy. J. Theor. Biol. 355, 10–20 (2014).

    Article  PubMed  Google Scholar 

  187. Foo, J. & Michor, F. Evolution of resistance to targeted anti-cancer therapies during continuous and pulsed administration strategies. PLoS Comput. Biol. 5, e1000557 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  188. Chmielecki, J. et al. Optimization of dosing for EGFR-mutant non-small cell lung cancer with evolutionary cancer modeling. Sci. Transl Med. 3, 90ra59 (2011). An evolutionary mathematical model that identifies optimum dosing strategies of targeted drugs to delay the emergence of resistance.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  189. Foo, J., Chmielecki, J., Pao, W. & Michor, F. Effects of pharmacokinetic processes and varied dosing schedules on the dynamics of acquired resistance to erlotinib in EGFR-mutant lung cancer. J. Thorac. Oncol. 7, 1583–1593 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  190. Michor, F. et al. Dynamics of chronic myeloid leukaemia. Nature 435, 1267–1270 (2005).

    Article  CAS  PubMed  Google Scholar 

  191. Neal, M. L. et al. Response classification based on a minimal model of glioblastoma growth is prognostic for clinical outcomes and distinguishes progression from pseudoprogression. Cancer Res. 73, 2976–2986 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  192. Corwin, D. et al. Toward patient-specific, biologically optimized radiation therapy plans for the treatment of glioblastoma. PLoS ONE 8, e79115 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  193. Jain, H. V., Clinton, S. K., Bhinder, A. & Friedman, A. Mathematical modeling of prostate cancer progression in response to androgen ablation therapy. Proc. Natl Acad. Sci. USA 108, 19701–19706 (2011).

    Article  PubMed  PubMed Central  Google Scholar 

  194. Gatenby, R. A., Silva, A. S., Gillies, R. J. & Frieden, B. R. Adaptive therapy. Cancer Res. 69, 4894–4903 (2009). A mathematical model that suggests that outcomes can be improved by maintaining a stable tumour burden in which resistant tumour cells are suppressed by sensitive cells.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  195. Gatenby, R. A. & Frieden, B. R. Inducing catastrophe in malignant growth. Math. Med. Biol. 25, 267–283 (2008).

    Article  PubMed  Google Scholar 

  196. Luria, S. E. & Delbruck, M. Mutations of bacteria from virus sensitivity to virus resistance. Genetics 28, 491–511 (1943).

    CAS  PubMed  PubMed Central  Google Scholar 

  197. Frank, S. A. Somatic mosaicism and cancer: inference based on a conditional Luria-Delbruck distribution. J. Theor. Biol. 223, 405–412 (2003).

    Article  PubMed  Google Scholar 

  198. Haeno, H., Iwasa, Y. & Michor, F. The evolution of two mutations during clonal expansion. Genetics 177, 2209–2221 (2007).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  199. Durrett, R. & Moseley, S. Evolution of resistance and progression to disease during clonal expansion of cancer. Theor. Popul. Biol. 77, 42–48 (2010).

    Article  PubMed  Google Scholar 

  200. Iwasa, Y., Michor, F. & Nowak, M. A. Evolutionary dynamics of escape from biomedical intervention. Proc. Biol. Sci. 270, 2573–2578 (2003).

    Article  PubMed  PubMed Central  Google Scholar 

  201. Komarova, N. L. & Wodarz, D. Stochastic modeling of cellular colonies with quiescence: an application to drug resistance in cancer. Theor. Popul. Biol. 72, 523–538 (2007).

    Article  PubMed  Google Scholar 

  202. Bozic, I. & Nowak, M. A. Timing and heterogeneity of mutations associated with drug resistance in metastatic cancers. Proc. Natl Acad. Sci. USA 111, 15964–15968 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  203. Mumenthaler, S. M. et al. Evolutionary modeling of combination treatment strategies to overcome resistance to tyrosine kinase inhibitors in non-small cell lung cancer. Mol. Pharm. 8, 2069–2079 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  204. Komarova, N. Stochastic modeling of drug resistance in cancer. J. Theor. Biol. 239, 351–366 (2006).

    Article  CAS  PubMed  Google Scholar 

  205. Komarova, N. L., Burger, J. A. & Wodarz, D. Evolution of ibrutinib resistance in chronic lymphocytic leukemia (CLL). Proc. Natl Acad. Sci. USA 111, 13906–13911 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  206. Altrock, P. M. & Traulsen, A. Deterministic evolutionary game dynamics in finite populations. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80, 011909 (2009).

    Article  CAS  PubMed  Google Scholar 

  207. Werner, B., Lutz, D., Brummendorf, T. H., Traulsen, A. & Balabanov, S. Dynamics of resistance development to imatinib under increasing selection pressure: a combination of mathematical models and in vitro data. PLoS ONE 6, e28955 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  208. Gallaher, J. & Anderson, A. R. Evolution of intratumoral phenotypic heterogeneity: the role of trait inheritance. Interface Focus 3, 20130016 (2013).

    Article  PubMed  PubMed Central  Google Scholar 

  209. Huang, W., Haubold, B., Hauert, C. & Traulsen, A. Emergence of stable polymorphisms driven by evolutionary games between mutants. Nat. Commun. 3, 919 (2012).

    Article  CAS  PubMed  Google Scholar 

  210. Huang, W., Hauert, C. & Traulsen, A. Stochastic game dynamics under demographic fluctuations. Proc. Natl Acad. Sci. USA 112, 9064–9069 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  211. Archetti, M., Ferraro, D. A. & Christofori, G. Reply to Gerlee and Altrock: diffusion and population size in game theory models of cancer. Proc. Natl Acad. Sci. USA 112, E2744 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  212. Tabassum, D. P. & Polyak, K. Tumorigenesis: it takes a village. Nat. Rev. Cancer 15, 473–483 (2015).

    Article  CAS  PubMed  Google Scholar 

  213. Pacheco, J. M., Santos, F. C. & Dingli, D. The ecology of cancer from an evolutionary game theory perspective. Interface Focus 4, 20140019 (2014).

    Article  PubMed  PubMed Central  Google Scholar 

  214. Axelrod, R., Axelrod, D. E. & Pienta, K. J. Evolution of cooperation among tumor cells. Proc. Natl Acad. Sci. USA 103, 13474–13479 (2006).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  215. Archetti, M. Evolutionary game theory of growth factor production: implications for tumour heterogeneity and resistance to therapies. Br. J. Cancer 109, 1056–1062 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  216. Basanta, D. et al. Investigating prostate cancer tumour-stroma interactions: clinical and biological insights from an evolutionary game. Br. J. Cancer 106, 174–181 (2012).

    Article  CAS  PubMed  Google Scholar 

  217. Kianercy, A., Veltri, R. & Pienta, K. J. Critical transitions in a game theoretic model of tumour metabolism. Interface Focus 4, 20140014 (2014).

    Article  PubMed  PubMed Central  Google Scholar 

  218. Basanta, D., Simon, M., Hatzikirou, H. & Deutsch, A. Evolutionary game theory elucidates the role of glycolysis in glioma progression and invasion. Cell Prolif. 41, 980–987 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

Download references

Author information

Author notes

  1. Philipp M. Altrock and Lin L. Liu: These authors contributed equally to this work.

Authors and Affiliations

  1. Department of Biostatistics and Computational Biology, Dana-Farber Cancer Institute and Department of Biostatistics, Harvard T.H. Chan School of Public Health, 450 Brookline Avenue, Boston, 02115, Massachusetts, USA

    Philipp M. Altrock, Lin L. Liu & Franziska Michor

  2. Program for Evolutionary Dynamics, Harvard University, 1 Brattle Square, Suite 6, Cambridge, 02138, Massachusetts, USA

    Philipp M. Altrock

Authors
  1. Philipp M. Altrock
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lin L. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Franziska Michor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Franziska Michor.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

PowerPoint slides

Glossary

Mathematical models

Models can describe a system by means of abstraction and mathematical formalism. They enable extrapolation beyond situations originally analysed, quantitative predictions, inferrence of mechanisms, falsification of underlying biological hypotheses and quantitative description of relationships between different components of a system.

Hybrid models

A modelling approach that combines several modelling techniques in one. For example, a hybrid model that describes the tumour microenvironment in which stromal cells follow a continuous nonlinear description, whereas tumour cells obey a discrete stochastic process.

Branching process

A stochastic process model of cell division, mutation events and cell death that leads on average to an exponential increase or decrease in the total population size. The branching process is based on the assumption that each individual event occurs at the same rate, independently of, for example, the population size or composition, or the point in time. The branching process is a Markov process; that is, the probability of the next event happening depends only on the current state of the population, and not on its earlier history.

Passenger mutations

Genetic changes that have no obvious or direct effect on cell fitness or cancer development, and may occur and potentially vanish again during any stage of tissue development and homeostasis. According to some definitions, passengers might also be (slightly) deleterious.

Driver mutations

Genetic changes that are causally involved in cancer development, typically conferring a functional change and a somatic evolutionary advantage.

Epistatic interactions

Interactions that occur when the functional effect of one genetic alteration depends on the genetic background of the cell; that is, the state of one or more other genes.

Homeostasis

A property of a system in which variables are regulated so that internal conditions remain stable and relatively constant. An example is the constant tissue size of most organs in the absence of neoplasms.

Deterministic model

Given a specific initial condition, a deterministic process always yields the same output, and no randomness is involved. Deterministic processes can be chaotic in that a small deviation in the initial condition may yield a large deviation after some time. However, this effect is different from the effect of a stochastic process in which the same initial condition can lead to different results.

Hierarchical tissue structures

Structures according to which most tissues are organized, ranging from slowly proliferating stem and progenitor cells to more quickly proliferating precursors and terminally differentiated cells.

Phylogenetic tree

A branching, tree-structured graph that represents the evolutionary relationships among different (mutational) stages of a tumour cell population, quantified by some measure of distance between individual cells or patient samples.

Graphical models

Mathematical structures that describes pairwise relations (called edges) between objects (called nodes), possibly on several layers. An acyclic graph does not have any cycles. Undirected graphs imply that there is no direction in the relationships along any edge. A tree-like graph has the property that every node can be traced back to a central node, called the root node, while final nodes of a tree are called leaves.

Longitudinal data

Repeated observations of the same system or set of systems over time.

Agent-based simulation

A computational approach that models complex systems consisting of interacting discretized items or 'agents'. In cancer modelling, these agents often represent cells, which can mutate into other types, divide into two cells, die or move in space. These simulations can be implemented according to either probabilistic or deterministic laws.

Stochastic process

This describes how a random variable (or set of random variables) changes over time and/or space. A stochastic process ascribes a probability to each event and allows for the prediction of the probability of a certain outcome. In contrast to a deterministic process, the initial condition yields an entire probability distribution over possible events at any later point in time.

Markov process

A memoryless stochastic process in which the conditional probability distribution over all future events depends only on the present state. A Markov chain explicitly addresses stochastic dynamics between discrete states in discrete time, thus allowing for a full characterization using a transition matrix in which the entries describe the probability of transitioning from one state to another.

Biased random walk

The movements of an object or changes in a variable that on average follow a specific pattern or trend.

Linear-quadratic model

A prominent heuristic to describe cell survival under radiation. The number of surviving cells after a certain dose of radiation has been administered takes the form of an exponential function with a linear and a quadratic term in its argument.

Predator–prey models

These models, also known as Lotka–Volterra dynamics, are used to describe the dynamics of ecological species, or types, as a nonlinear deterministic process. They were originally used to describe population dynamics of predators and prey, taking into account abundance, interactions, and population growth and diminution. They can also be used to describe mutualistic and competitive evolutionary dynamics; for example, between cellular types.

Sigmoidal growth curve

An S-shaped growth pattern in which the population size starts from a low density with positive acceleration, then transitions to negative acceleration at high density. An equilibrium population size can be characterized, for instance, by a proliferation–self renewal–death balance, or by a carrying capacity. Examples are Gompertzian growth and logistic growth.

Power-law growth model

A functional relationship between two quantities (for example, time and tumour size), where one quantity varies with the power (that is, exponent) of the other. The exponent can typically be inferred from linear regression analysis of a doubly logarithmic transformation of the data.

Luria–Delbrück model

The Luria-Delbrück experiment investigated whether mutations occur independently from, or owing to, selection. Data from growth experiments in which Escherichia coli were challenged with a virus were compared to a stochastic process model used to calculate the probability of having a certain number of resistant mutants. The findings suggested that mutations occurred randomly over time and were not a response to selection.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Altrock, P., Liu, L. & Michor, F. The mathematics of cancer: integrating quantitative models. Nat Rev Cancer 15, 730–745 (2015). https://doi.org/10.1038/nrc4029

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nrc4029

This article is cited by

Search

Advanced search

Quick links

Nature Briefing: Cancer

Sign up for the Nature Briefing: Cancer newsletter — what matters in cancer research, free to your inbox weekly.

Get what matters in cancer research, free to your inbox weekly. Sign up for Nature Briefing: Cancer