Computational Models for Cancer Phylogenetics

  • Russell SchwartzEmail author
Part of the Computational Biology book series (COBO, volume 29)


Cancer development has long been recognized as a product of aberrant evolution of cell populations, inspiring the idea that phylogenetic algorithms could be a powerful tool for reconstructing progression processes in cancer. Translating that intuition into practice, however, has required extensive work on adapting phylogenetic models and algorithms to represent more accurately the many ways tumor evolution is distinct from classic species evolution. The result has been a large and growing body of problems and theory on phylogenetics as it applies specifically to cancers. This chapter surveys some of the key ideas and variants on phylogeny problem that have arisen in the development of tumor phylogeny methods. Its purpose is to introduce readers to some of the space of approaches in current practice in phylogenetics and help them appreciate how models have developed, and continue to develop, to better capture the peculiar nature of evolution in cancers.


Cancer progression Tumor Phylogenetics Algorithms 



Russell Schwartz was supported in part by U.S. National Institutes of Health award R21CA216452 and Pennsylvania Dept. of Health award 4100070287. The Pennsylvania Department of Health specifically disclaims responsibility for any analyses, interpretations or conclusions.


  1. 1.
    Alexandrov, L., Nik-Zainal, S., Wedge, D.C., Aparicio, S.A.J.R., Behjati, S., Biankin, A.V., Bignell, G.R., Bolli, N., Å. Borg, Børresen-Dale, A., Boyault, S., Burkhardt, B., Butler, A.P., Caldas, C., Davies, H.R., Desmedt, C., Eils, R., Eyfjörd, J.E., Foekens, J.A., Greaves, M., Hosoda, F., Hutter, B., Ilicic, T., Imbeaud, S., Imielinski, M., Jger, N., Jones, D.T.W., Jones, D., Knappskog, S., Kool, M., Lakhani, S.R., López-Otín, C., Martin, S., Munsh, N.C., Nakamura, H., Northcott, P.A., Pajic, M., Papaemmanuil, E., Paradiso, A., Pearson, J.V., Puente, X.S., Raine, K., Ramakrishna, M., Richardson, A.L., Richter, J., Rosenstiel, P., Schlesner, M., Schumacher, T.N., Spa, P.N., Teague, J.W., Totoki, Y., Tutt, A.N.J., Valdés-Mas, R., van Buuren, M.M., van ’t Veer, L., Vincent-Salomon, A., Waddell, N., Yates, L.R., Australian Pancreatic Cancer Genome Initiative, ICGC Breast Cancer Consortium, ICGC MMML-Seq Consortium, ICGC PedBrain, Zucman-Rossi, J., Futreal, P.A., McDermott, U., Lichter, P., Meyerson, M., Grimmond, S.M., Siebert, R., Campo, E., Tatsuhiro: Signatures of mutation processes in human cancers. Nature 500(7463), 415–421 (2013)Google Scholar
  2. 2.
    Alexandrov, L.B., Stratton, M.R.: Mutational signatures: the patterns of somatic mutations hidden in cancer genomes. Curr. Opin. Genet. Devel. 24, 52–60 (2014)Google Scholar
  3. 3.
    Attolini, C.S., Cheng, Y.K., Beroukhim, R., Getz, G., Abdel-Wahab, O., Levine, R.L., Mellinghoff, I.K., Michor, F.: A mathematical framework to determine the temporal sequence of somatic genetic events in cancer. Proc. Natl. Acad. Sci. USA 107(41), 17604–17609 (2010)zbMATHGoogle Scholar
  4. 4.
    Bader, D.A., Moret, B.M., Yan, M.: A linear-time algorithm for computing inversion distance between signed permutations with an experimental study. J. Comput. Biol. 8(5), 483–491 (2001)zbMATHGoogle Scholar
  5. 5.
    Bandelt, H., Forster, P., Röhl, A.: Median-joining networks for inferring intraspecific phylogenies. Mol. Biol. Evol. 16(1), 37–48 (1999)Google Scholar
  6. 6.
    Bandelt, H.J., Forster, P., Sykes, B.C., Richards, M.B.: Mitochondrial portraits of human populations using median networks. Genetics 141(2), 743–753 (1995)Google Scholar
  7. 7.
    Beerenwinkel, N., Rahnenführer, J., Däumer, M., Hoffmann, D., Kaiser, R., Selbig, J., Lengauer, T.: Learning multiple evolutionary pathways from cross-sectional data. J. Comput. Biol. 12(6), 584–598 (2005)Google Scholar
  8. 8.
    Beerenwinkel, N., Rahnenführer, J., Kaiser, R., Hoffmann, D., Selbig, J., Lengauer, T.: Mtreemix: a software package for learning and using mixture models of mutagenetic trees. Bioinformatics 21(9), 2106–2107 (2005)Google Scholar
  9. 9.
    Beerenwinkel, N., Schwarz, R.F., Gerstung, M., Markowetz, F.: Cancer evolution: mathematical models and computational inference. Syst. Biol. 64(1), e1–e25 (2015)Google Scholar
  10. 10.
    Campbell, P.J., Pleasance, E.D., Stephens, P.J., Dicks, E., Rance, R., Goodhead, I., Follows, G.A., Green, A.R., Futreal, P.A., Stratton, M.R.: Subclonal phylogenetic structures in cancer revealed by ultra-deep sequencing. Proc. Natl. Acad. Sci. USA 105(35), 13081–13086 (2008)Google Scholar
  11. 11.
    Catanzaro, D., Ravi, R., Schwartz, R.: A mixed integer linear programming model to reconstruct phylogenies from single nucleotide polymorphism haplotypes under the maximum parsimony criterion. Algorithms Mol. Biol. 8(1), 3 (2013)Google Scholar
  12. 12.
    Cavalli-Sforza, L.L., Edwards, A.W.: Phylogenetic analysis: models and estimation procedures. Evolution 21(3), 550–570 (1967)Google Scholar
  13. 13.
    Chowdhury, S.A., Gertz, E.M., Wangsa, D., Heselmeyer-Haddad, K., Ried, T., Schäffer, A.A., Schwartz, R.: Inferring models of multiscale copy number evolution for single-tumor phylogenetics. Bioinformatics 31(12), i258–i267 (2015)Google Scholar
  14. 14.
    Chowdhury, S.A., Shackney, S.E., Heselmeyer-Haddad, K., Ried, T., Schäffer, A.A., Schwartz, R.: Phylogenetic analysis of multiprobe fluorescence in situ hybridization data from tumor cell populations. Bioinformatics 29(13), i189–i198 (2013)Google Scholar
  15. 15.
    Chowdhury, S.A., Shackney, S.E., Heselmeyer-Haddad, K., Ried, T., Schäffer, A.A., Schwartz, R.: Algorithms to model single gene, single chromosome, and whole genome copy number changes jointly in tumor phylogenetics. PLoS Comp. Biol. 10(7), e1003740 (2014)Google Scholar
  16. 16.
    Christinat, Y., Moret, B.M.: Inferring transcript phylogenies. In: BMC Bioinform. (BioMed Central) 13(9), S1 (2012)Google Scholar
  17. 17.
    Deisboeck, T.S., Wang, Z., Macklin, P., Cristini, V.: Multiscale cancer modeling. Annu. Rev. Biomed. Eng. 13, 127–155 (2011)Google Scholar
  18. 18.
    Deshwar, A.G., Vembu, S., Yung, C.K., Jang, G.H., Stein, L., Morris, Q.: PhyloWGS: reconstructing subclonal composition and evolution from whole-genome sequencing of tumors. Genome Biol. 16, 35 (2015)Google Scholar
  19. 19.
    Desper, R., Jiang, F., Kallioniemi, O.P., Moch, H., Papadimitriou, C.H., Schäffer, A.A.: Inferring tree models of oncogenesis from comparative genomic hybridization data. J. Comput. Biol. 6(1), 37–51 (1999)Google Scholar
  20. 20.
    Desper, R., Jiang, F., Kallioniemi, O.P., Moch, H., Papadimitriou, C.H., Schäffer, A.A.: Distance-based reconstruction of tree models for oncogenesis. J. Comput. Biol. 7(6), 789–803 (2000)Google Scholar
  21. 21.
    Desper, R., Khan, J., Schäffer, A.A.: Tumor classification using phylogenetic methods on expression data. J. Theor. Biol. 228(4), 477–496 (2004)MathSciNetGoogle Scholar
  22. 22.
    Eaton, J., Wang, J., Schwartz, R.: Deconvolution and phylogeny inference of structural variations in tumor genomic samples. Bioinformatics (2018). In pressGoogle Scholar
  23. 23.
    El-Kebir, M., Oesper, L., Acheson-Field, H., Raphael, B.J.: Reconstruction of clonal trees and tumor composition from multi-sample sequencing data. Bioinformatics 31(12), i62–i70 (2015)Google Scholar
  24. 24.
    El-Kebir, M., Raphael, B.J., Shamir, R., Sharan, R., Zaccaria, S., Zehavi, M., Zeira, R.: Complexity and algorithms for copy-number evolution problems. Algorithms Mol. Biol. 12(1), 13 (2017)zbMATHGoogle Scholar
  25. 25.
    El-Kebir, M., Satas, G., Oesper, L., Raphael, B.J.: Inferring the mutational history of a tumor using multi-state perfect phylogeny mixtures. Cell Syst. 3(1), 43–53 (2016)Google Scholar
  26. 26.
    Fearon, E., Vogelstein, B.: A genetic model for colorectal tumorigenesis. Cell 61(5), 759–767 (1990)Google Scholar
  27. 27.
    Felsenstein, J.: PHYLIP-phylogeny inference package (version 3.2). Cladistics 5(163), 6 (1989)Google Scholar
  28. 28.
    Felsenstein, J.: Inferring Phylogenies. Sinauer Associates Inc, Sunderland, MA (2004)Google Scholar
  29. 29.
    Frumkin, D., Wasserstrom, A., Itzkovitz, S., Stern, T., Harmelin, A., Eilam, R., Rechavi, G., Shapiro, E.: Cell lineage analysis of a mouse tumor. Cancer Res. 68(14), 5924–5931 (2008)Google Scholar
  30. 30.
    Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Gerlinger, M., Rowan, A.J., Horswell, S., Larkin, J., Endesfelder, D., Gronroos, E., Martinez, P., Matthews, N., Stewart, A., Tarpey, P., Varela, I., Phillimore, B., Begum, S., McDonald, N.Q., Butler, A., Jones, D., Raine, K., Latimer, C., Santos, C.R., Nohadani, M., Eklund, A.C., Spencer-Dene, B., Clark, G., Pickering, L., Stamp, G., Gore, M., Szallasi, Z., Downward, J., Futreal, P.A., Swanton, C.: Intratumor heterogeneity and branched evolution revealed by multiregion sequencing. N. Engl. J. Med. 366(10), 883–892 (2012)Google Scholar
  32. 32.
    Gertz, E.M., Chowdhury, S.A., Lee, W., Wangsa, D., Heselmeyer-Haddad, K., Ried, T., Schwartz, R., Schäffer, A.A.: FISHtrees 3.0: Tumor phylogenetics using a ploidy probe. PLoS ONE 11(6), e0158569 (2016)Google Scholar
  33. 33.
    Greaves, M., Maley, C.C.: Clonal evolution in cancer. Nature 481(7381), 306–313 (2012)Google Scholar
  34. 34.
    Gusfield, D.: Haplotyping as perfect phylogeny: conceptual framework and efficient solutions. In: Proceedings of the Sixth Annual International Conference on Computational Biology, pp. 166–175. ACM (2002)Google Scholar
  35. 35.
    Hajirasouliha, I., Mahmoody, A., Raphael, B.J.: A combinatorial approach for analyzing intra-tumor heterogeneity from high-throughput sequencing data. Bioinformatics 30(12), i78–i86 (2014)Google Scholar
  36. 36.
    Halperin, E., Eskin, E.: Haplotype reconstruction from genotype data using imperfect phylogeny. Bioinformatics 20(12), 1842–1849 (2004)Google Scholar
  37. 37.
    Hastings, P., Lupski, J.R., Rosenberg, S.M., Ira, G.: Mechanisms of change in gene copy number. Nat. Rev. Genet. 10(8), 551–564 (2009)Google Scholar
  38. 38.
    Jiang, Y., Qiu, Y., Minn, A.J., Zhang, N.R.: Assessing intratumor heterogeneity and tracking longitudinal and spatial clonal evolutionary history by next-generation sequencing. Proc. Natl. Acad. Sci. USA 113(37), E5528–E5537 (2016). Scholar
  39. 39.
    Jiao, W., Vembu, S., Deshwar, A.G., Stein, L., Morris, Q.: Inferring clonal evolution of tumors from single nucleotide somatic mutations. BMC Bioinform. 15, 35 (2014)Google Scholar
  40. 40.
    Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Springer (1972)Google Scholar
  41. 41.
    Korbel, J.O., Kim, P.M., Chen, X., Urban, A.E., Weissman, S., Snyder, M., Gerstein, M.B.: The current excitement about copy-number variation: how it relates to gene duplications and protein families. Curr. Opin. Struct. Biol. 18(3), 366–374 (2008)Google Scholar
  42. 42.
    Malikic, S., Jahn, K., Kuipers, J., Sahinalp, C., Beerenwinkel, N.: Integrative inference of subclonal tumour evolution from single-cell and bulk sequencing data. bioRxiv (2017).
  43. 43.
    Malikic, S., McPherson, A.A., Donmez, N., Sahinalp, C.S.: Clonality inference in multiple tumor samples using phylogeny. Bioinformatics 31(9), 1349–1356 (2015)Google Scholar
  44. 44.
    Mardis, E.R., Wilson, R.K.: Cancer genome sequencing: a review. Hum. Mol. Gen. 18(R2), R163–R168 (2009)Google Scholar
  45. 45.
    Merlo, L.M.F., Pepper, J.W., Ried, B.J., Maley, C.C.: Cancer as an evolutionary and ecological process. Nat. Rev. Cancer 6(12), 924–935 (2006)Google Scholar
  46. 46.
    Miller, C.A., White, B.S., Dees, N.D., Griffith, M., Welch, J.S., Griffith, O.L., R., V., Tomasson, M.H., Graubert, T.A., Walter, M.J., Ellis, M.J., Schierding, W., DiPersio, J.F., Ley, T.J., Mardis, E.R., Wilson, R.K., Ding, L.: SciClone: inferring clonal architecture and tracking the spatial and temporal patterns of tumor evolution. PLoS Comput. Biol. 10(8), e1003665 (2014)Google Scholar
  47. 47.
    Misra, N., Szczurek, E., Vingron, M.: Inferring the paths of somatic evolution in cancer. Bioinformatics 30(17), 2456–2463 (2014)Google Scholar
  48. 48.
    Moret, B.M.: Towards a discipline of experimental algorithmics. In: Data Structures, Near Neighbor Searches, and Methodology: Fifth and Sixth DIMACS Implementation Challenges, Vol. 59, pp. 197–213 (2002)Google Scholar
  49. 49.
    Moret, B.M., Shapiro, H.D.: An empirical analysis of algorithms for constructing a minimum spanning tree. In: Workshop on Algorithms and Data Structures, pp. 400–411. Springer (1991)Google Scholar
  50. 50.
    Moret, B.M., Shapiro, H.D.: Algorithms and experiments: the new (and old) methodology. J. Univ. Comput. Sci. 7(5), 434–446 (2001)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Nair, N.U., Lin, Y., Manasovska, A., Antic, J., Grnarova, P., Sahu, A.D., Bucher, P., Moret, B.M.: Study of cell differentiation by phylogenetic analysis using histone modification data. BMC Bioinform. 15(1), 269 (2014)Google Scholar
  52. 52.
    Navin, N., Kendall, J., Troge, J., Andrews, P., Rodgers, L., McIndoo, J., Cook, K., Stepansky, A., Levy, D., Esposito, D., et al.: Tumour evolution inferred by single-cell sequencing. Nature 472(7341), 90–94 (2011)Google Scholar
  53. 53.
    Navin, N., Krasnitz, A., Rodgers, L., Cook, K., Meth, J., Kendall, J., Riggs, M., Eberling, Y., Troge, J., Grubor, V., Levy, D., Lundin, P., Må nér, S., Zetterberg, A., Hicks, J., Wigler, M.: Inferring tumor progression from genomic heterogeneity. Genome Res. 20(1), 68–80 (2010)Google Scholar
  54. 54.
    Nowell, P.C.: The clonal evolution of tumor cell populations. Science 194(4260), 23–28 (1976)Google Scholar
  55. 55.
    Park, Y., Shackney, S., Schwartz, R.: Network-based inference of cancer progression from microarray data. IEEE/ACM Trans. Comput. Biol. Bioinform. 6(2), 200–212 (2009)Google Scholar
  56. 56.
    Pattengale, N.D., Alipour, M., Bininda-Emonds, O.R., Moret, B.M., Stamatakis, A.: How many bootstrap replicates are necessary? J. Comput. Biol. 17(3), 337–354 (2010)MathSciNetGoogle Scholar
  57. 57.
    Pennington, G., Smith, C.A., Shackney, S., Schwartz, R.: Expectation-maximization method for reconstructing tumor phylogenies from single-cell data. In: Computational Systems Bioinformatics Conference, pp. 371–380 (2006)Google Scholar
  58. 58.
    Pennington, G., Smith, C.A., Shackney, S., Schwartz, R.: Reconstructing tumor phylogenies from heterogeneous single-cell data. J. Bioinform. Comput. Biol. 5(2a), 407–427 (2007)Google Scholar
  59. 59.
    Popic, V., Salari, R., Hajirasouliha, I., Kashef-Haghighi, D., West, R.B., Batzoglou, S.: Fast and scalable inference of multi-sample cancer lineages. Genome Biol. 16, 91 (2015)Google Scholar
  60. 60.
    Potter, N.E., Ermini, L., Papaemmanuil, E., Cazzaniga, G., Vijayaraghavan, G., Titley, I., Ford, A., Campbell, P., L., K., Greaves, M.: Single cell mutational profiling and clonal phylogeny in cancer. Genome Res. 23(12), 2115–2125 (2013)Google Scholar
  61. 61.
    Riester, M., Attolini, C., Downey, R.J., Singer, S., Michor, F.: A differentiation-based phylogeny of cancer subtypes. PLoS Comp. Biol. 6(5), e1000,777 (2010)Google Scholar
  62. 62.
    Roman, T., Nayyeri, A., Fasy, B.T., Schwartz, R.: A simplicial complex-based approach to unmixing tumor progression data. BMC Bioinform. 16(1), 254 (2015)Google Scholar
  63. 63.
    Roman, T., Xie, L., Schwartz, R.: Automated deconvolution of structured mixtures from heterogeneous tumor genomic data. PLoS Comput. Biol. 13(10), e1005815 (2017)Google Scholar
  64. 64.
    Schwartz, R., Schäffer, A.A.: The evolution of tumour phylogenetics: principles and practice. Nat. Rev. Genet. 18(4), 213 (2017)Google Scholar
  65. 65.
    Schwartz, R., Shackney, S.E.: Applying unmixing to gene expression data for tumor phylogeny inference. BMC Bioinform. 11, 42 (2010)Google Scholar
  66. 66.
    Schwarz, R.F., Trinh, A., Sipos, B., Brenton, J.D., Goldman, N., Markowetz, F.: Phylogenetic quantification of intra-tumour heterogeneity. PLoS Comput. Biol. 10(4), e1003535 (2014)Google Scholar
  67. 67.
    Shlush, L.I., Chapal-Ilani, N., Adar, R., Pery, N., Maruvka, Y., Shouval, Spiro A., R., Rowe, J., Tzukerman, M., Bercovich, D., Izraeli, S., Marcucci, G., Bloomfield, C., Zuckerman T. Skorecki, K., Shapiro, E.: Cell lineage analysis of acute leukemia relapse uncovers the role of replication-rate heterogeneity and microsatellite instability. Blood 120(3), 603–612 (2012)Google Scholar
  68. 68.
    Sottoriva, A., Spiteri, I., Shibata, D., Curtis, C., Tavaré, S.: Single-molecule genomic data delineate patient-specific tumor profiles and cancer stem cell organization. Cancer Res. 73(1), 41–49 (2013)Google Scholar
  69. 69.
    Sridhar, S., Blelloch, G.E., Ravi, R., Schwartz, R.: Optimal imperfect phylogeny reconstruction and haplotyping (IPPH). In: Computational Systems Bioinformatics, pp. 199–210. World Scientific (2006)Google Scholar
  70. 70.
    Sridhar, S., Dhamdhere, K., Blelloch, G., Halperin, E., Ravi, R., Schwartz, R.: Algorithms for efficient near-perfect phylogenetic tree reconstruction in theory and practice. IEEE/ACM Trans. Comput. Biol. Bioinform. 4(4), 561–571 (2007)Google Scholar
  71. 71.
    Sridhar, S., Lam, F., Blelloch, G.E., Ravi, R., Schwartz, R.: Mixed integer linear programming for maximum-parsimony phylogeny inference. IEEE/ACM Trans. Comput. Biol. Bioinform. 5(3), 323–331 (2008)Google Scholar
  72. 72.
    Storchova, Z., Pellman, D.: From polyploidy to aneuploidy, genome instability and cancer. Nat. Rev. Mol. Cell Biol. 5(1), 45 (2004)Google Scholar
  73. 73.
    Swenson, K.M., Rajan, V., Lin, Y., Moret, B.M.: Sorting signed permutations by inversions in \(o(n \log n) \) time. In: Annual International Conference on Research in Computational Molecular Biology (RECOMB), pp. 386–399. Springer (2009)Google Scholar
  74. 74.
    Tolliver, D., Tsourakakis, C., Subramanian, A., Shackney, S., Schwartz, R.: Robust unmixing of tumor states in array comparative genomic hybridization data. Bioinformatics 26(12), i106–i114 (2010)Google Scholar
  75. 75.
    Tsao, J., Zhang, J., Salovaara, R., Li, Z.H., Järvinen, H.J., Mecklin, J., Aaltonen, L., Shibata, D.: Tracing cell fates in human colorectal tumors from somatic microsatellite mutations: evidence of adenomas with stem cell architecture. Am. J. Pathol. 153(4), 1189–1200 (1998)Google Scholar
  76. 76.
    Weinberg, R.: The Biology of Cancer. Garland science (2013)Google Scholar
  77. 77.
    Wilgenbusch, J.C., Swofford, D.: Inferring evolutionary trees with paup. Curr. Protoc. Bioinform. 6–4 (2003)Google Scholar
  78. 78.
    Williams, T.L., Moret, B.M.: An investigation of phylogenetic likelihood methods. In: Proceedings of the Third IEEE Symposium on Bioinformatics and Bioengineering, pp. 79–86. IEEE (2003)Google Scholar
  79. 79.
    Xu, X., Hou, Y., Yin, X., Bao, L., Tang, A., Song, L., Li, F., Tsang, S., Wu, K., Wu, H., He, W., Zeng, L., Xing, M., Wu, R., Jiang, H., Liu, X., Cao, D., Guo, G., Hu, X., Gui, Y., Li, Z., Xie, W., Sun, X., Shi, M., Cai, Z., Wang, B., Zhong, M., Li, J., Lu, Z., Gu, N., Zhang, X., Goodman, L., Bolund, L., Wang, J., Yang, H., Kristiansen, K., Dean, M., Li, Y., Wang, J.: Single-cell exome sequencing reveals single-nucleotide mutation characteristics of a kidney tumor. Cell 148(5), 886–895 (2012)Google Scholar
  80. 80.
    Ye, M., Racz, G.C., Jiang, Q., Zhang, X., Moret, B.M.: NEMo: an evolutionary model with modularity for PPI networks. In: International Symposium on Bioinformatics Research and Applications, pp. 224–236. Springer (2016)Google Scholar
  81. 81.
    Yuan, K., Sakoparnig, T., Markowetz, F., Beerenwinkel, N.: BitPhylogeny: a probabilistic framework for reconstructing intra-tumor phylogenies. Genome Biol. 16, 36 (2015)Google Scholar
  82. 82.
    Zaccaria, S., El-Kebir, M., Klau, G.W., Raphael, B.J.: The copy-number tree mixture deconvolution problem and applications to multi-sample bulk sequencing tumor data. In: International Conference on Research in Computational Molecular Biology (RECOMB), pp. 318–335. Springer (2017)Google Scholar
  83. 83.
    Zaccaria, S., El-Kebir, M., Klau, G.W., Raphael, B.J.: Phylogenetic copy-number factorization of multiple tumor samples. J. Comput. Biol. (2018). Scholar
  84. 84.
    Zack, T.I., Schumacher, S.E., Carter, S.L., Cherniack, A.D., Saksena, G., Tabak, B., Lawrence, M.S., Zhang, C.Z., Wala, J., Mermel, C.H., Sougnez, C., Gabriel, S.B., Hernandez, B., Shen, H., Laird, P.W., Getz, G., Meyerson, M., Beroukhim, R.: Pan-cancer patterns of somatic copy number alteration. Nat. Genet. 45(10), 1134 (2013)Google Scholar
  85. 85.
    Zafar, H., Navin, N., Nakhleh, L., Chen, K.: Computational approaches for inferring tumor evolution from single-cell genomic data. Curr. Opin. Syst. Biol. (2017)Google Scholar
  86. 86.
    Zakov, S., Kinsella, M., Bafna, V.: An algorithmic approach for breakage-fusion-bridge detection in tumor genomes. Proc. Natl. Acad. Sci. USA 110(14), 5546–5551 (2013)MathSciNetGoogle Scholar
  87. 87.
    Zeira, R., Zehavi, M., Shamir, R.: A linear-time algorithm for the copy number transformation problem. J. Comput. Biol. 24(12), 1179–1194 (2017)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Carnegie Mellon UniversityPittsburghUSA

Personalised recommendations