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Modeling the connection between primary and metastatic tumors

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Abstract

We put forward a model for cancer metastasis as a migration phenomenon between tumor cell populations coexisting and evolving in two different habitats. One of them is a primary tumor and the other one is a secondary or metastatic tumor. The evolution of the different cell phenotype populations in each habitat is described by means of a simple quasispecies model allowing for a cascade of mutations between the different phenotypes in each habitat. The cell migration event is supposed to be unidirectional and take place continuously in time. The possible clinical outcomes of the model depending on the parameter space are analyzed and the effect of the resection of the primary tumor is studied.

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References

  • Andasari V, Gerish A, Lolas G, South AP, Chaplain MAJ (2010) Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation. J Math Biol 63: 141–171

    Article  Google Scholar 

  • Baake E, Wagner H (2001) Mutation-selection models solved exactly with methods of statistical mechanics. Genet Res 78: 93–117

    Article  Google Scholar 

  • Basanta D, Hatzikirou H, Deutsch A (2008) Studying the emergence of invasiveness in tumors using game theory. Eur Phys J B 63: 393–397

    Article  MathSciNet  MATH  Google Scholar 

  • Bearer EL, Lowengrub JS, Frieboes HB, Chuang Y-L, Jin F, Wise SM, Ferrari M, Agus DB, Cristini V (2009) Multiparameter computational modeling of tumor invasion. Cancer Res 69: 4493–4501

    Article  Google Scholar 

  • Bernards R, Weinberg RA (2002) Metastasis genes: a progression puzzle. Nature 418: 823–823

    Article  Google Scholar 

  • Boushaba K, Levine HA, Nilsen-Hamilton M (2006) A mathematical model for the regulation of tumor dormancy based on enzyme kinetics. Bull Math Biol 68: 1495–1526

    Article  MathSciNet  Google Scholar 

  • Cairns RA, Harris IS, Mak TW (2011) Regulation of cancer cell metabolism. Nat Rev Cancer 11: 85–95

    Article  Google Scholar 

  • Chaffer CL, Weinberg RA (2011) A perspective on cancer cell metastasis. Science 331: 1559–1564 sites Nat Rev Cancer 2:563–572

    Article  Google Scholar 

  • Chambers A, Naumov G, Vantyghem S, Tuck A (2000) Molecular biology of breast cancer biology: clinical implications of experimental studies on metastatic inefficiency. Breast Can Res 2: 400–407

    Article  Google Scholar 

  • Chambers A, Naumov GN, Varghese HJ, Nadkarni KV, MacDonald IC, Groom AC (2001) Critical steps in hematogenous metastasis: an overview. Surg Oncol Clin North Am 10: 243–255

    Google Scholar 

  • Chambers AF, Groom AC, MacDonald IC (2002) Dissemination and growth of cancer cells in metastatic sites. Nat Rev Cancer 2: 563–572

    Article  Google Scholar 

  • Chen LL, Christakis NA, Barabási A-L, Deisboeck TS (2009) Cancer metastasis networks and the prediction of progression patterns. Br J Cancer 101: 749–758

    Article  Google Scholar 

  • Coghlin C, Murray GI (2010) Current and emerging concepts in tumour metastasis. J Pathol 222: 1–15

    Article  Google Scholar 

  • Cook LM, Hurst DR, Welch DR (2011) Metastasis suppressors and the tumor microenvironment. Semin Cancer Biol 21: 113–122

    Article  Google Scholar 

  • Dattoli G, Guiot C, Delsanto PP, Ottaviani PL, Pagnutti S, Deisboeck TS (2009) Cancer metabolism and the dynamics of metastasis. J Theor Biol 256: 305–310

    Article  MathSciNet  Google Scholar 

  • Demicheli R, Retsky MW, Hrushesky WJM, Baum M, Gukas ID (2008) The effects of surgery on tumor growth: a century of investigations. Ann Oncol 19: 1821–1828

    Article  Google Scholar 

  • Dingli D, Michor F, Antal T, Pacheco JM (2007) The emergence of tumor metastases. Cancer Biol Ther 6(3): 383–390

    Article  Google Scholar 

  • DiSibio G, French SW (2008) Metastatic patterns of cancer: results from a large autopsy study. Arch Pathol Lab Med 132: 931–939

    Google Scholar 

  • Ebos JML, Lee CR, Cruz-Munoz W, Bjarnason GA, Christensen JG (2009) Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis. Cancer Cell 15: 232–239

    Article  Google Scholar 

  • Eikenberry S, Thalhauser C, Kuang Y (2009) Tumor-immune interaction, surgical treatment and cancer recurrence in a mathematical model of melanoma. Plos Comput Biol 5: e1000362

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Folkman J (2002) Role of angiogenesis in tumor growth and metastasis. Semin Oncol 29: 15–18

    Google Scholar 

  • Frieboes HB, Zheng X, Sun C-H, Tromberg B, Gatenby RA, Cristiny V (2006) An integrated computational/experimental model of tumor invasion. Cancer Res 66: 1597–1604

    Article  Google Scholar 

  • Gatenby RA, Gawlinski ET (1996) A reaction–diffusion model of cancer invasion. Cancer Res 56: 5745–5753

    Google Scholar 

  • Gatenby RJ, Gillies RJ (2004) Why do cancers have high aerobic glycolysis. Nat Rev Cancer 4: 891–899

    Article  Google Scholar 

  • Gerisch A, Chaplain MAJ (2008) Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. J Theor Biol 250: 684–704

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Hanahan D, Weinberg RA (2000) The hallmarks of cancer. Cell 100: 57–70

    Article  Google Scholar 

  • Hanahan D, Weinberg RA (2011) The hallmarks of cancer: the next generation. Cell 144: 646–674

    Article  Google Scholar 

  • Hanin L, Korosteleva O (2010) Does extirpation of the primary breast tumor give boost to growth of metastases? Evidence revealed by mathematical modeling. Math Biosci 223: 133–141

    Article  MathSciNet  MATH  Google Scholar 

  • Hayashida Y, Hirai T, Morishita S, Kitajima M, Murakami R, Korogi Y, Makino K, Nakamura H, Ikushima I, Yamura M, Kochi M, Kuratsu JI, Yamashita Y (2006) Diffusion-weighted imaging of metastatic brain tumors: comparison with histologic type and tumor cellularity. Am J Neuroradiol 27: 1419–1425

    Google Scholar 

  • Hess KR, Varadhachary GR, Taylor SH, Wei W, Raber MN, Lenzi R, Abbruzzese JL (2006) Metastatic patterns in adenocarcinoma. Cancer 106: 1624–1633

    Article  Google Scholar 

  • Hüsemann Y, Geigl JB, Schubert F, Musiani P, Meyer M, Burghart E, Forni G, Eils R, Fehm T, Riethmüller G, CityKlein CA (2008) Systemic spread is an early step in breast cancer. Cancer Cell 13: 58–68

    Article  Google Scholar 

  • Hurst DR, Welch DR (2011) Metastasis suppressor genes: at the interface between the environment and tumor cell growth. Int Rev Cell Mol Biol 286: 107–180

    Article  Google Scholar 

  • Iwasa Y, Michor F, Nowak MA (2004) Evolutionary dynamics of invasion and escape. J Theor Biol 226: 205–214

    Article  MathSciNet  Google Scholar 

  • Jain RK, Tong RT, Munn LL (2007) 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

    Article  Google Scholar 

  • Kendal WS (2001) The size distribution of human hematogenous metastases. J Theor Biol 211: 29–38

    Article  Google Scholar 

  • Kim MY, Oskarsson T, Acharyya S, Nguyen DX, Zhang XHF, Norton L, Massagué J (2009) Tumor self-seeding by circulating cancer cells. Cell 139: 1315–1326

    Article  Google Scholar 

  • Koppenol WH, Bounds PL, Dang CV (2011) Otto Warburg’s contribution to current concepts of cancer metabolism. Nat Rev Cancer 11: 325–337

    Article  Google Scholar 

  • Koscielny S, Tubiana M, Valleron A (1985) A simulation model of the natural history of human breast cancer. Br J cancer 52: 515–524

    Article  Google Scholar 

  • Malanchi I, Santamaria-Martínez A, Susanto E, Peng H, Lehr HA, Delaloye JF, Huelsken J (2012) Interactions between cancer stem cells and their nicher govern metastatic colonization. Nature 481: 85–89

    Article  Google Scholar 

  • Mendoza-Juez B, Martínez-González A, Calvo GF, Pérez-García VM (2011) A mathematical model for the glucose–lactate metabolism of in vitro cancer cells. Bull Math Biol. doi:10.1007/s11538-011-9711-z

  • Meng S et al (2004) Circulating tumor cells in patients with breast cancer dormancy. Clin Cancer Res 10: 8152–8162

    Article  Google Scholar 

  • Michaelson JS, Cheongsiatmoy JA, Dewey F, Silverstein MJ, Sgroi D, Smith B, Tanabe KK (2005) Spread of human cancer cells occurs with probabilities indicative of a nongenetic mechanism. Br J Cancer 93: 1244–1249

    Article  Google Scholar 

  • Michor F, Nowak MA, Iwasa Y (2006) Stochastic dynamics of metastasis formation. J Theor Biol 240: 521–530

    Article  MathSciNet  Google Scholar 

  • Pantel K, Brakenhoff RH (2004) Dissecting the metastatic cascade. Nat Rev Cancer 4: 448–456

    Article  Google Scholar 

  • Peeters CF, Westphal JR, de Waal RM, Ruiter DJ, Wobbes T, Ruers TJ (2004) Vascular density in colorectal liver metastases increases after removal of the primary tumor in human cancer patients. Int J Cancer 112: 554–559

    Article  Google Scholar 

  • Peeters CFJM, Westphal JR, De Waal RMW, Ruiter DJ, Wobbes T, Oyen WJG, Ruers TJ (2005) Decrease in circulating anti-angiogenic factors (angiostatin and endostatin) after surgical removal of primary colorectal carcinoma coincides with increased metabolic activity of liver metastases. Surgery 137: 246–249

    Article  Google Scholar 

  • Peeters CF, de Waal RM, Wobbes T, Westphal JR, Ruers TJ (2006) Outgrowth of human liver metastases after resection of the primary colorectal tumor: a shift in the balance between apoptosis and proliferation. Int J Cancer 119: 1249–1253

    Article  Google Scholar 

  • Peeters CF, de Waal RM, Ruiter DJ, Wobbes T, Ruers TJ (2008) Metastatic dormancy imposed by the primary tumor: does it exist in humans. Ann Surg Oncol 15: 3308–3315

    Article  Google Scholar 

  • Podsypanina K, Du Y-C N, Jechlinger M, Beverly LJ, Hambardzumyan D, Varmus H (2008) Seeding and propagation of untransformed mouse mammary cells in the lung. Science 321: 1841–1844

    Article  Google Scholar 

  • Ramis-Conde I, Chaplain MAJ, Anderson ARA (2008) Mathematical modelling of cancer cell invasion of tissue. Math Comput Model 47: 533–545

    Article  MathSciNet  MATH  Google Scholar 

  • Retsky MW, Demicheli R, Swartzendruber DE, Bame PD, Wardwell RH, Bonadonna G, Speer JF, Valagussa P (1997) Computer simulation of a breast cancer metastasis model. Breast Cancer Res Treat 45: 193–202

    Article  Google Scholar 

  • Smallbone K, Gatenby RA, Maini PK (2008) Mathematical modelling of tumour acidity. J Theor Biol 255: 106–112

    Article  MathSciNet  Google Scholar 

  • Sonveaux P, Végran F, Schroeder T, Wergin MC, Verrax J, Rabbani ZN, de Saedeleer CJ, Kennedy KM, Diepart C, Jordan BF, Kelley MJ, Gallez B, Wahl ML, Feron O, Dewhirst MW (2008) Targeting lactate-fueled respiration selectively kills hypoxic tumor cells in mice. J Clin Invest 118: 3930–3942

    Google Scholar 

  • Traulsen A, Nowak MA, Pacheco JM (2006) Stochastic dynamics of invasion and fixation. Phys Rev E 74: 011909

    Article  Google Scholar 

  • Vander-Heiden MG, Locasale JW, Swanson KD, Sharfi H, Heffron GJ, Amador-Noguez D, Christofk HR, Wagner G, Rabinowitz JD, Asara JM, Cantley LC (2010) Evidence for an alternative glycolytic pathway in rapidly proliferating cells. Science 329: 1492–1499

    Article  Google Scholar 

  • Waclaw R, Allen RJ, Evans R (2010) Dynamical phase transition in a model for evolution with migration. Phys Rev Lett 105: 268101

    Article  Google Scholar 

  • Warburg O (1956) On the origin of cancer cells. Science 123: 309–314

    Article  Google Scholar 

  • Warburg O, Posener K, Negelein E (1924) On the metabolism of carcinoma cells. Biochem Z 152: 309–344

    Google Scholar 

  • Weinberg RA (2008) Leaving home early: re-examination of the canonical models of tumour progression. Cancer Cell 14: 283–284

    Article  Google Scholar 

  • Weiss L (2000) Metastasis of cancer: a conceptual history from antiquity to the 1990s. Cancer Metastasis Rev 19: 193–383

    Article  Google Scholar 

  • Wilson RW, Hay PH (2011) Targeting hypoxia in cancer therapy. Nat Rev Cancer 11: 393–410

    Article  Google Scholar 

  • Yorke ED, Fuks Z, Norton L, Whitmore W, Ling CC (1993) Modeling the development of metastases from primary and locally recurrent tumors: comparison with a clinical data base for prostatic cancer. Cancer Res 53: 2987–2993

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemáticas, E.T.S.I. Industriales and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería, Universidad de Castilla-La Mancha, 13071, Ciudad Real, Spain

    David Diego & Víctor M. Pérez-García

  2. Departamento de Matemáticas, E.T.S.I. Caminos, Canales y Puertos and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería, Universidad de Castilla-La Mancha, 13071, Ciudad Real, Spain

    Gabriel F. Calvo

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  1. David Diego
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  2. Gabriel F. Calvo
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Correspondence to David Diego.

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Diego, D., Calvo, G.F. & Pérez-García, V.M. Modeling the connection between primary and metastatic tumors. J. Math. Biol. 67, 657–692 (2013). https://doi.org/10.1007/s00285-012-0565-2

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  • DOI: https://doi.org/10.1007/s00285-012-0565-2

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