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
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
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: 1105.1157
Armitage, P. (1952) The statistical theory of bacterial populations subject to mutations. J. Royal Statistical Society, B. 14, 1–40
Athreya, K.B., and P.E. Ney (1972) Branching Processes. Springer-Verlag, new York
Bailey, N.T.J. (1964) The Elements of Stochastic Processes. John Wiley and Sons, New York
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, 18545–18550
Crump. K.S., and Hoel, D.G. (1974) Mathematical models for estimating mutation rates in cell populations. Biometrika. 61, 237–252
Danesh, K., Durrett, R., Havrliesky, L., and Myers, E. (2013) A branching process model of ovarian cancer. J. Theor. Biol. 314, 10–15
Darling, D.A. (1952) The role of the maximum term in the sum of independent random variables. Trans. American Math Society. 72, 85–107
Durrett, R. (2008) Probability Models for DNA Sequence Evolution. Second Edition. Springer, New York
Durrett, R. (2010) Probability: Theory and Examples. Fourth edition. Cambridge U. Press
Durrett, R., Foo, J., Leder, K., Mayberry, J., Michor, F. (2010) Evolutionary dynamics of tumor progression with random fitness values. Theor. Popul. Biol. 78, 54–66
Durrett, R., Foo, J., Leder, K., Mayberry, J., Michor, F. (2011) Intratumor heterogeneity in evolutionary models of tumor progresssion. Genetics. 188, 461–477
Durrett, R., and Moseley, S. (2010) Evolution of resistance and progression to disease during clonal expansion of cancer. Theor. Popul. Biol. 77, 42–48
Durrett, R., and Schweinsberg, J.. (2004) Approximating selective sweeps. Theor. Popul. Biol. 66, 129–138
Durrett, R., and Schweinsberg, J. (2005) Power laws for family sizes in a gene duplication model. Ann. Probab. 33, 2094–2126
Foo, Jasmine and Leder, Kevin (2013) Dynamics of cancer recurrence. Ann. Appl. Probab. 23, 1437–1468.
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, 54–69
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, 243–255
Griffiths, R.C., and Pakes, A.G. (1988) An infinite-alleles version of the simple branching process Adv. Appl. Prob. 20, 489–524
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, 362–375
Haeno, H., Iwasa, Y., and Michor, F. (2007) The evolution of two mutations during clonal expansion. Genetics. 177, 2209–2221
Haeno, H., and Michor, F. (2010) The evolution of tumor metastases during clonal expansion. J Theor. Biol. 263, 30–44
Harris, T.E. (1948) Branching processes. Ann. Math. Statist. 19, 474–494
Iwasa, Y., Nowak, M.A., and Michor, F. (2006) Evolution of resistance during clonal expansion. Genetics. 172, 2557–2566
Kingman, J.F.C. (1975) Random discrete distributions. J. Royal Statistical Society, B. 37, 1–22
Komarova, N.L., Wu, Lin, and Baldi, P. (2007) The fixed-size Luria-Delbruck model with a non-zero death rate. Mathematical Biosciences. 210, 253–290
Logan, B.F., Mallows, C.L., Rice, S.O., and Shepp, L.A. (1973) Limit distributionsof self-normalized random sums. Annals of Probability. 1, 788–809
Lea, E.A., and Coulson, C.A. (1949) The distribution of the number of mutants in bacterial populations. Journal of Genetics. 49, 264–285
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 e27682
Luria, S.E., and Delbruck, M. (1943) Mutations of bacteria from virus sensitivity to virus resistance. Genetics. 28, 491–511
Michor, F, et al. (2005) Dynamics of chronic myeloid leukemia. Nature. 435, 1267–1270
O’Connell, N. (1993) Yule approximation for the skeleton of a branching process. J. Appl. Prob. 30, 725–729
Parzen, E. (1962) Stochastic Processes. Holden-Day, San Francisco
Pitman, J., and Yor, M. (1997) The two parameter Poisson-Dirichlet distribution derived from a stabel subordinator. Annals of Probability. 25, 855–900
Slatkin, M., and Hudson, R.R. (1991) Pairwise comparisons of mitochondrial DNA sequences in stable and exponentially growing populations. Genetics. 129, 555–562
Tomasetti, C., and Levy, D. (2010) Roles of symmetric and asymmetric division of stem cells in developing drug resistance. Proc. natl. Acad. Sci. 107, 16766–16771
Zheng, Q. (1999) Progress of a half-century in the study of the Luria-Delbrück distribution. Mathematical Biosciences. 162, 1–32
Zheng, Q. (2009) Remarks on the asymptotics of the Luria-Delbruck and related distributions. J. Appl. Prob. 46, 1221–1224 Cancer Biology
Armitage, P. (1985) Multistage models of carcinogenesis. Environmental health Perspectives. 63, 195–201
Armitage, P., and Doll, R. (1954) The age distribution of cancer and a multi-stage theory of carcinogenesis. British J. Cancer. 8, 1–12
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.
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.
Collisson, E.A., Cho, R.J., and Gray, J.W. (2012) What are we learning from the cancer genome? Nature Reviews. Clinical Oncology. 9, 621–630
Decruze, S.B., and Kirwan, J.M. (2006) Ovarian cancer. Current Obstetrics and Gynecology. 16(3): 161–167
Fearon, E.F. (2011) Molevular genetics of colon cancer. Annu. Rev. Pathol. Mech. Dis. 6, 479–507
Fearon, E.R., and Vogelstein, B. (1990) A genetic model fro colorectal tumorigenesis. Cell. 87, 759–767
Feller, L., Kramer, B., and Lemmer, J. (2012) Pathobiology of cancer metastasis: a short account. Caner Cell International. 12, paper 24
Fidler, I.J. (1978) Tumor heterogeneity and the biology of cancer invasion and metastases. Cancer Research. 38, 2651–2660
Fisher, J.C., and Holloman, J.H. (1951) A hypothesis for the origin of cancer foci. British J. Cancer. 7, 407–417
Fisher, R., Pusztai, L., and Swanton, C. (2013) Cancer heterogeneity: implications for targeted therapeutics. Cancer Research.
Gerlinger, M. et al. (2012) Intratumor heterogeneity and branched evolution revealed by multiregion sequencing. New England Journal of Medicine. 366, 883–892
Knudson, A.G., Jr. (1971) Mutation and cancer: Statistical study of retinoblastoma. Proc. Natl. Acad. Sci. 68, 820–823
Knudson, A.G. (2001) Two genetic hits (more or less) to cancer. Nature Reviews Cancer. 1, 157–162
Jones, S., et al. (2008) Core signaling pathways in human pancreatic cancers revealed by global genomic analyses. Science. 321, 1801–1812
Lengyel, E. (2010) Ovarian cancer development and metastasis. The American Journal of Pathology. 177(3): 1053–1064
Luebeck, E.G., and Mollgavkar, S.H. (2002) Multistage carcinogenesis and teh incidence of colorectal cancer. proc. natl. Acad. Sci. 99, 15095–15100
Maley, C.C., et al. (2006) Genetic clonal diversity predicts progresssion to esophageal adenocarcinoma. Nature Genetics. 38, 468–473
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–
Naora, H., and Montell, D.J. (2005) Ovarian cancer metastasis: integrating insights from disparate model organisms. Nature Reviews Cancer. 5(5): 355–366
Navin, N., et al (2011) Tumor evolution inferred from single cell sequencing. Nature. 472, 90–94
Nordling, C.O. (1953) A new theory on cancer inducing mechanism. British J. Cancer. 7, 68–72
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.
Parsons, D.W., et al. (2008) An integrated genotmic analysis of human glioblastome multiforme. Science. 321, 1807–1812
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, 3810–3818
Siegel, R., Naishadham, D., and Jemal, A. (2012) Cancer statistics, 2012. CA: A Cancer Journal for Clinicians. 62: 1029. doi: 10.3322/caac.20138
Sjöblom, T., et al. (2006) The consensus coding sequences of human breast and colorectal cancers. Science. 314, 268–274
Sottoriva, A., et al. (2013) Intratumor heterogeneity in human glioblastoma reflects cancer evolutionary dynamics. Proc. Natl. Acad. Sci. 110, 4009–4014
Surveillance, Epidemiology, and End Results (SEER) Program. http://seer.cancer.gov/.
The Cancer Genome Atlas Research Network (2008) Comprehensive genomic characterization defines human glioblastoma genes and core pathways. Nature. 455, 1061–1068
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, 1999–2004
Valastyan, S., and Weinberg, R.A. (2011) Tumor metastasis: Moecluar insights and evolving pardigms. Cell. 147, 275–292
Wood, L.D., et al. (2007) The genomic landscapes of human breast and colorectal cancers. Science. 318, 1108–1113
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Durrett, R. (2015). Branching Process Models of Cancer. In: Branching Process Models of Cancer. Mathematical Biosciences Institute Lecture Series(), vol 1.1. Springer, Cham. https://doi.org/10.1007/978-3-319-16065-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-16065-8_1
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16064-1
Online ISBN: 978-3-319-16065-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)