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Branching Process Models of Cancer

  • Richard DurrettEmail author
Chapter
Part of the Mathematical Biosciences Institute Lecture Series book series (MBILS, volume 1.1)

Abstract

In this chapter, we will use multitype branching processes with mutation to model cancer. With cancer progression, resistance to therapy, and metastasis in mind, we will investigate τ k , the time of the first type k mutation, and σ k , the time of the first type k mutation that founds a family line that does not die out, as well as the growth of the number of type k cells. The last three sections apply these results to metastasis, ovarian cancer, and tumor heterogeneity. Even though martingales and stable laws are mentioned, these notes should be accessible to a student who is familiar with Poisson processes and continuous time Markov chains.

Keywords

Branching Process Stable Law Family Line Size-biased Permutation Offspring Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. 1.
    Antal, T., and Krapivsky, P.L. (2011) Exact solution of a two-type branching process: models of tumor progression. J. Stat. Mech.: Theory and Experiment arXiv: 1105.1157Google Scholar
  2. 2.
    Armitage, P. (1952) The statistical theory of bacterial populations subject to mutations. J. Royal Statistical Society, B. 14, 1–40zbMATHMathSciNetGoogle Scholar
  3. 3.
    Athreya, K.B., and P.E. Ney (1972) Branching Processes. Springer-Verlag, new YorkGoogle Scholar
  4. 4.
    Bailey, N.T.J. (1964) The Elements of Stochastic Processes. John Wiley and Sons, New YorkzbMATHGoogle Scholar
  5. 5.
    Bozic I., Antal T., Ohtsuki H., Carter H., Kim D., Chen, S., Karchin, R., Kinzler, K.W., Vogelstein, B., and Nowak, M.A. (2010) Accumulation of driver and passenger mutations during tumor progression. Proc. Natl. Acad. Sci. 107, 18545–18550CrossRefGoogle Scholar
  6. 6.
    Crump. K.S., and Hoel, D.G. (1974) Mathematical models for estimating mutation rates in cell populations. Biometrika. 61, 237–252zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Danesh, K., Durrett, R., Havrliesky, L., and Myers, E. (2013) A branching process model of ovarian cancer. J. Theor. Biol. 314, 10–15CrossRefGoogle Scholar
  8. 8.
    Darling, D.A. (1952) The role of the maximum term in the sum of independent random variables. Trans. American Math Society. 72, 85–107MathSciNetGoogle Scholar
  9. 9.
    Durrett, R. (2008) Probability Models for DNA Sequence Evolution. Second Edition. Springer, New YorkzbMATHCrossRefGoogle Scholar
  10. 10.
    Durrett, R. (2010) Probability: Theory and Examples. Fourth edition. Cambridge U. PressCrossRefGoogle Scholar
  11. 11.
    Durrett, R., Foo, J., Leder, K., Mayberry, J., Michor, F. (2010) Evolutionary dynamics of tumor progression with random fitness values. Theor. Popul. Biol. 78, 54–66CrossRefGoogle Scholar
  12. 12.
    Durrett, R., Foo, J., Leder, K., Mayberry, J., Michor, F. (2011) Intratumor heterogeneity in evolutionary models of tumor progresssion. Genetics. 188, 461–477CrossRefGoogle Scholar
  13. 13.
    Durrett, R., and Moseley, S. (2010) Evolution of resistance and progression to disease during clonal expansion of cancer. Theor. Popul. Biol. 77, 42–48CrossRefGoogle Scholar
  14. 14.
    Durrett, R., and Schweinsberg, J.. (2004) Approximating selective sweeps. Theor. Popul. Biol. 66, 129–138zbMATHCrossRefGoogle Scholar
  15. 15.
    Durrett, R., and Schweinsberg, J. (2005) Power laws for family sizes in a gene duplication model. Ann. Probab. 33, 2094–2126zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Foo, Jasmine and Leder, Kevin (2013) Dynamics of cancer recurrence. Ann. Appl. Probab. 23, 1437–1468.zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Foo, J., Leder, K., and Mummenthaler, S. (2013) Cancer as a moving target: understanding the composition and rebound growth kinetics of recurrent tumors. Evolutionary Applications. 6, 54–69CrossRefGoogle Scholar
  18. 18.
    Fuchs, A., Joffe, A., and Teugels, J. (2001) Expectation of the ratio of the sums of squares to the square of the sum: exact and asymptotic results. Theory Probab. Appl. 46, 243–255zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Griffiths, R.C., and Pakes, A.G. (1988) An infinite-alleles version of the simple branching process Adv. Appl. Prob. 20, 489–524Google Scholar
  20. 20.
    Haeno, H., Conen, M., Davis, M.B., Hrman, J.M., Iacobuzio-Donahue, C.A., and Michor, F. (2012) Computational modeling of pancreatic cancer reveals kinetics of metastasis suggesting optimum treatment strategies. Cell. 148, 362–375CrossRefGoogle Scholar
  21. 21.
    Haeno, H., Iwasa, Y., and Michor, F. (2007) The evolution of two mutations during clonal expansion. Genetics. 177, 2209–2221CrossRefGoogle Scholar
  22. 22.
    Haeno, H., and Michor, F. (2010) The evolution of tumor metastases during clonal expansion. J Theor. Biol. 263, 30–44MathSciNetCrossRefGoogle Scholar
  23. 23.
    Harris, T.E. (1948) Branching processes. Ann. Math. Statist. 19, 474–494zbMATHCrossRefGoogle Scholar
  24. 24.
    Iwasa, Y., Nowak, M.A., and Michor, F. (2006) Evolution of resistance during clonal expansion. Genetics. 172, 2557–2566CrossRefGoogle Scholar
  25. 25.
    Kingman, J.F.C. (1975) Random discrete distributions. J. Royal Statistical Society, B. 37, 1–22zbMATHMathSciNetGoogle Scholar
  26. 26.
    Komarova, N.L., Wu, Lin, and Baldi, P. (2007) The fixed-size Luria-Delbruck model with a non-zero death rate. Mathematical Biosciences. 210, 253–290zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Logan, B.F., Mallows, C.L., Rice, S.O., and Shepp, L.A. (1973) Limit distributionsof self-normalized random sums. Annals of Probability. 1, 788–809zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Lea, E.A., and Coulson, C.A. (1949) The distribution of the number of mutants in bacterial populations. Journal of Genetics. 49, 264–285CrossRefGoogle Scholar
  29. 29.
    Leder, K., Foo, J., Skaggs, B., Gorre, M., Sawyers, C.L., and Michor, F. (2011) Fitness conferred by BCR-ABL kinase domain mutations determines the risk of pre-existing resistance in chronic myeloid leukemia. PLoS One. 6, paper e27682Google Scholar
  30. 30.
    Luria, S.E., and Delbruck, M. (1943) Mutations of bacteria from virus sensitivity to virus resistance. Genetics. 28, 491–511Google Scholar
  31. 31.
    Michor, F, et al. (2005) Dynamics of chronic myeloid leukemia. Nature. 435, 1267–1270CrossRefGoogle Scholar
  32. 32.
    O’Connell, N. (1993) Yule approximation for the skeleton of a branching process. J. Appl. Prob. 30, 725–729zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Parzen, E. (1962) Stochastic Processes. Holden-Day, San FranciscozbMATHGoogle Scholar
  34. 34.
    Pitman, J., and Yor, M. (1997) The two parameter Poisson-Dirichlet distribution derived from a stabel subordinator. Annals of Probability. 25, 855–900zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Slatkin, M., and Hudson, R.R. (1991) Pairwise comparisons of mitochondrial DNA sequences in stable and exponentially growing populations. Genetics. 129, 555–562Google Scholar
  36. 36.
    Tomasetti, C., and Levy, D. (2010) Roles of symmetric and asymmetric division of stem cells in developing drug resistance. Proc. natl. Acad. Sci. 107, 16766–16771CrossRefGoogle Scholar
  37. 37.
    Zheng, Q. (1999) Progress of a half-century in the study of the Luria-Delbrück distribution. Mathematical Biosciences. 162, 1–32zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Zheng, Q. (2009) Remarks on the asymptotics of the Luria-Delbruck and related distributions. J. Appl. Prob. 46, 1221–1224 Cancer Biology Google Scholar
  39. 39.
    Armitage, P. (1985) Multistage models of carcinogenesis. Environmental health Perspectives. 63, 195–201CrossRefGoogle Scholar
  40. 40.
    Armitage, P., and Doll, R. (1954) The age distribution of cancer and a multi-stage theory of carcinogenesis. British J. Cancer. 8, 1–12CrossRefGoogle Scholar
  41. 41.
    Brown, P.O., and Palmer, C. (2009) The preclinical natural history of serous ovarian cancer: defining the target for early detection. PLoS Medicine. 6(7):e1000114.CrossRefGoogle Scholar
  42. 42.
    Buys SS, Partridge E, Black A, et al. (2011) Effect of screening on ovarian cancer mortality The prostate, lung, colorectal and ovarian (PLCO) cancer screening randomized controlled trial. JAMA 305(22): 2295–2303. doi:10.1001/jama.2011.766.CrossRefGoogle Scholar
  43. 43.
    Collisson, E.A., Cho, R.J., and Gray, J.W. (2012) What are we learning from the cancer genome? Nature Reviews. Clinical Oncology. 9, 621–630CrossRefGoogle Scholar
  44. 44.
    Decruze, S.B., and Kirwan, J.M. (2006) Ovarian cancer. Current Obstetrics and Gynecology. 16(3): 161–167CrossRefGoogle Scholar
  45. 45.
    Fearon, E.F. (2011) Molevular genetics of colon cancer. Annu. Rev. Pathol. Mech. Dis. 6, 479–507CrossRefGoogle Scholar
  46. 46.
    Fearon, E.R., and Vogelstein, B. (1990) A genetic model fro colorectal tumorigenesis. Cell. 87, 759–767CrossRefGoogle Scholar
  47. 47.
    Feller, L., Kramer, B., and Lemmer, J. (2012) Pathobiology of cancer metastasis: a short account. Caner Cell International. 12, paper 24Google Scholar
  48. 48.
    Fidler, I.J. (1978) Tumor heterogeneity and the biology of cancer invasion and metastases. Cancer Research. 38, 2651–2660Google Scholar
  49. 49.
    Fisher, J.C., and Holloman, J.H. (1951) A hypothesis for the origin of cancer foci. British J. Cancer. 7, 407–417Google Scholar
  50. 50.
    Fisher, R., Pusztai, L., and Swanton, C. (2013) Cancer heterogeneity: implications for targeted therapeutics. Cancer Research. Google Scholar
  51. 51.
    Gerlinger, M. et al. (2012) Intratumor heterogeneity and branched evolution revealed by multiregion sequencing. New England Journal of Medicine. 366, 883–892CrossRefGoogle Scholar
  52. 52.
    Knudson, A.G., Jr. (1971) Mutation and cancer: Statistical study of retinoblastoma. Proc. Natl. Acad. Sci. 68, 820–823CrossRefGoogle Scholar
  53. 53.
    Knudson, A.G. (2001) Two genetic hits (more or less) to cancer. Nature Reviews Cancer. 1, 157–162CrossRefGoogle Scholar
  54. 54.
    Jones, S., et al. (2008) Core signaling pathways in human pancreatic cancers revealed by global genomic analyses. Science. 321, 1801–1812CrossRefGoogle Scholar
  55. 55.
    Lengyel, E. (2010) Ovarian cancer development and metastasis. The American Journal of Pathology. 177(3): 1053–1064CrossRefGoogle Scholar
  56. 56.
    Luebeck, E.G., and Mollgavkar, S.H. (2002) Multistage carcinogenesis and teh incidence of colorectal cancer. proc. natl. Acad. Sci. 99, 15095–15100Google Scholar
  57. 57.
    Maley, C.C., et al. (2006) Genetic clonal diversity predicts progresssion to esophageal adenocarcinoma. Nature Genetics. 38, 468–473CrossRefGoogle Scholar
  58. 58.
    Merlo, L.M.F., et al (2010) A comprehensive survey of clonal diversity measures in Barrett’s esophagus as biomarkers of progression to esophageal adenocarcinoma. Cancer Prevention Research. 3, 1388–Google Scholar
  59. 59.
    Naora, H., and Montell, D.J. (2005) Ovarian cancer metastasis: integrating insights from disparate model organisms. Nature Reviews Cancer. 5(5): 355–366CrossRefGoogle Scholar
  60. 60.
    Navin, N., et al (2011) Tumor evolution inferred from single cell sequencing. Nature. 472, 90–94CrossRefGoogle Scholar
  61. 61.
    Nordling, C.O. (1953) A new theory on cancer inducing mechanism. British J. Cancer. 7, 68–72CrossRefGoogle Scholar
  62. 62.
    Park, S.Y., Gönen, M, Kim, H.J., Michor, F., and Polyak, K. (2010) Cellular and genetic diversity in the progression of in situ human breast cancer to an invasive phenotype.Google Scholar
  63. 63.
    Parsons, D.W., et al. (2008) An integrated genotmic analysis of human glioblastome multiforme. Science. 321, 1807–1812CrossRefGoogle Scholar
  64. 64.
    Russnes, H.G., Navin, N., Hicks, J., and Borrensen-Dale, A.L. (2011) Insight into the heterogeniety of breast cancer inferred through next generation sequencing. J. Clin. Invest. 121, 3810–3818CrossRefGoogle Scholar
  65. 65.
    Siegel, R., Naishadham, D., and Jemal, A. (2012) Cancer statistics, 2012. CA: A Cancer Journal for Clinicians. 62: 1029. doi: 10.3322/caac.20138Google Scholar
  66. 66.
    Sjöblom, T., et al. (2006) The consensus coding sequences of human breast and colorectal cancers. Science. 314, 268–274CrossRefGoogle Scholar
  67. 67.
    Sottoriva, A., et al. (2013) Intratumor heterogeneity in human glioblastoma reflects cancer evolutionary dynamics. Proc. Natl. Acad. Sci. 110, 4009–4014CrossRefGoogle Scholar
  68. 68.
    Surveillance, Epidemiology, and End Results (SEER) Program. http://seer.cancer.gov/.
  69. 69.
    The Cancer Genome Atlas Research Network (2008) Comprehensive genomic characterization defines human glioblastoma genes and core pathways. Nature. 455, 1061–1068CrossRefGoogle Scholar
  70. 70.
    Tomasettim C., Vogelstein, B., and Parmigiani, G. (2013) Half or more somatic mutations in cancers of self-renewing tissues originate prior to tumor initiation. Proc. Natl. Acad. Sci. 110, 1999–2004CrossRefGoogle Scholar
  71. 71.
    Valastyan, S., and Weinberg, R.A. (2011) Tumor metastasis: Moecluar insights and evolving pardigms. Cell. 147, 275–292CrossRefGoogle Scholar
  72. 72.
    Wood, L.D., et al. (2007) The genomic landscapes of human breast and colorectal cancers. Science. 318, 1108–1113CrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA

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