Advertisement

Stochastic and Deterministic Models for the Metastatic Emission Process: Formalisms and Crosslinks

Protocol
Part of the Methods in Molecular Biology book series (MIMB, volume 1711)

Abstract

Although the detection of metastases radically changes prognosis of and treatment decisions for a cancer patient, clinically undetectable micrometastases hamper a consistent classification into localized or metastatic disease. This chapter discusses mathematical modeling efforts that could help to estimate the metastatic risk in such a situation. We focus on two approaches: (1) a stochastic framework describing metastatic emission events at random times, formalized via Poisson processes, and (2) a deterministic framework describing the micrometastatic state through a size-structured density function in a partial differential equation model. Three aspects are addressed in this chapter. First, a motivation for the Poisson process framework is presented and modeling hypotheses and mechanisms are introduced. Second, we extend the Poisson model to account for secondary metastatic emission. Third, we highlight an inherent crosslink between the stochastic and deterministic frameworks and discuss its implications. For increased accessibility the chapter is split into an informal presentation of the results using a minimum of mathematical formalism and a rigorous mathematical treatment for more theoretically interested readers.

Key words

Poisson process Structured population equation Metastasis Mathematical modeling 

Notes

Acknowledgements

The authors thank Florence Hubert, Charlotte Kloft, and Andrea Henrich for suggestions and critical reading of the manuscript. NH gratefully acknowledges financial support by the Agence Nationale de la Recherche under grant ANR-09-BLAN-0217-01.

References

  1. 1.
    Çınlar E (2011) Probability and stochastics. Graduate texts in mathematics, vol 261. Springer, New YorkGoogle Scholar
  2. 2.
    Yu M, Bardia A, Wittner BS, Stott SL, Smas ME, Ting DT, Isakoff SJ, Ciciliano JC, Wells MN, Shah AM, Concannon KF, Donaldson MC, Sequist LV, Brachtel E, Sgroi D, Baselga J, Ramaswamy S, Toner M (2013) Circulating breast tumor cells exhibit dynamic changes in epithelial and mesenchymal composition. Science 339(6119):580–584Google Scholar
  3. 3.
    Nguyen DX, Bos PD, Massagué J (2009) Metastasis: from dissemination to organ-specific colonization. Nat Rev Cancer 9(4):274–284Google Scholar
  4. 4.
    Sahai E (2007) Illuminating the metastatic process. Nat Rev Cancer 7(10):737–749Google Scholar
  5. 5.
    WHO (2015) Cancer fact sheet. http://www.who.int/mediacentre/factsheets/fs297/en/ . Accessed 14 Jan 2016Google Scholar
  6. 6.
    Pantel K, Cote RJ, Fodstad O (1999) Detection and clinical importance of micrometastatic disease. J Natl Cancer Inst 91(13):1113–1124Google Scholar
  7. 7.
    Scott JG, Gerlee P, Basanta D, Fletcher AG, Maini PK, Anderson ARA (2013) Mathematical modeling of the metastatic process. In: Malek A (ed) Experimental metastasis: modeling and analysis. Springer, Dordrecht, pp 189–208Google Scholar
  8. 8.
    Gupta GP, Massagué J (2006) Cancer metastasis: building a framework. Cell 127(4):679–695Google Scholar
  9. 9.
    Hanahan D, Weinberg RA (2011) Hallmarks of cancer: the next generation. Cell 144(5):646–674Google Scholar
  10. 10.
    Michor F, Nowak MA, Iwasa Y (2006) Stochastic dynamics of metastasis formation. J Theor Biol 240(4):521–530Google Scholar
  11. 11.
    Haeno H, Michor F (2010) The evolution of tumor metastases during clonal expansion. J Theor Biol 263(1):30–44Google Scholar
  12. 12.
    Anderson AR, Quaranta V (2008) Integrative mathematical oncology. Nat Rev Cancer 8(3):227–234Google Scholar
  13. 13.
    Koscielny S, Tubiana M, Lê MG, Valleron J, Mouriesse H, Contesso G, Sarrazin D (1984) Breast cancer: relationship between the size of the primary tumour and the probability of metastatic dissemination. Br J Cancer 49(6):709–715Google Scholar
  14. 14.
    Michaelson JS, Silverstein M, Wyatt J, Weber G, Moore R, Halpern E, Kopans DB, Hughes K (2002) Predicting the survival of patients with breast carcinoma using tumor size. Cancer 95(4):713–723Google Scholar
  15. 15.
    van de Vijver MJ, He YD, van’t Veer LJ, Dai H, Hart AAM, Voskuil DW, Schreiber GJ, Peterse JL, Roberts C, Marton MJ, Parrish M, Atsma D, Witteveen A, Glas A, Delahaye L, van der Velde T, Bartelink H, Rodenhuis S, Rutgers ET, Friend SH, Bernards R (2002) A gene-expression signature as a predictor of survival in breast cancer. N Engl J Med 347(25):1999–2009Google Scholar
  16. 16.
    Hahnfeldt P, Panigrahy D, Folkman J, Hlatky L (1999) Tumor development under angiogenic signaling: a dynamical theory of tumor growth, response and postvascular dormancy. Cancer Res 59:4770–5PubMedGoogle Scholar
  17. 17.
    Norton L (1988) A Gompertzian model of human breast cancer growth. Cancer Res 48:7067–7071Google Scholar
  18. 18.
    Verga F (2010) Modélisation mathématique de processus métastatiques. Ph.D. thesis, Aix-Marseille UniversitéGoogle Scholar
  19. 19.
    Hart D, Shochat E, Agur Z (1998) The growth law of primary breast cancer as inferred from mammography screening trials data. Br J Cancer 78:382–387Google Scholar
  20. 20.
    Benzekry S, Lamont C, Beheshti A, Tracz A, Ebos JML, Hlatky L, Hahnfeldt P (2014) Classical mathematical models for description and prediction of experimental tumor growth. PLoS Comput Biol 10(8):e1003800Google Scholar
  21. 21.
    Bartoszyński R, Edler L, Hanin L, Kopp-Schneider A, Pavlova L, Tsodikov A, Zorin A, Yakovlev A (2001) Modeling cancer detection: tumor size as a source of information on unobservable stages of carcinogenesis. Math Biosci 171:113–142Google Scholar
  22. 22.
    Hanin L, Rose J, Zaider M (2006) A stochastic model for the sizes of detectable metastases. J Theor Biol 243:407–417Google Scholar
  23. 23.
    Iwata K, Kawasaki K, Shigesada N (2000) A dynamical model for the growth and size distribution of multiple metastatic tumors. J Theor Biol 203:177–186CrossRefPubMedGoogle Scholar
  24. 24.
    Hartung N, Mollard S, Barbolosi D, Benabdallah A, Chapuisat G, Henry G, Giacometti S, Iliadis A, Ciccolini J, Faivre C, Hubert F (2014) Mathematical modeling of tumor growth and metastatic spreading: validation in tumor-bearing mice. Cancer Res 74:6397–6407Google Scholar
  25. 25.
    Benzekry S, Tracz A, Mastri M, Corbelli R, Barbolosi D, Ebos JML (2016) Modeling spontaneous metastasis following surgery: an in vivo-in silico approach. Cancer Res 76(3):535–547Google Scholar
  26. 26.
    Chaffer CL, Weinberg RA (2011) A perspective on cancer cell metastasis. Science 331(6024):1559–1564Google Scholar
  27. 27.
    Newton PK, Mason J, Bethel K, Bazhenova LA, Nieva J, Kuhn P (2012) A stochastic Markov chain model to describe lung cancer growth and metastasis. PLoS One 7(4):e34637Google Scholar
  28. 28.
    Newton PK, Mason J, Bethel K, Bazhenova L, Nieva J, Norton L, Kuhn P (2013) Spreaders and sponges define metastasis in lung cancer: a Markov chain Monte Carlo mathematical model. Cancer Res 73(9):2760–2769Google Scholar
  29. 29.
    Comen E, Norton L, Massague J (2011) Clinical implications of cancer self-seeding. Nat Rev Clin Oncol 8(6):369–377Google Scholar
  30. 30.
    Scott JG, Basanta D, Anderson AR, Gerlee P (2013) A mathematical model of tumour self-seeding reveals secondary metastatic deposits as drivers of primary tumour growth. J R Soc Interface 10(82):20130011Google Scholar
  31. 31.
    Hanin L, Zaider M (2011) Effects of surgery and chemotherapy on metastatic progression of prostate cancer: evidence from the natural history of the disease reconstructed through mathematical modeling. Cancers 3(3):3632–3660Google Scholar
  32. 32.
    Wheldon TE (1988) Mathematical models in cancer research. Medical science series. Adam Hilger, Bristol/PhiladelphiaGoogle Scholar
  33. 33.
    Benzekry S, Gandolfi A, Hahnfeldt P (2014) Global dormancy of metastases due to systemic inhibition of angiogenesis. PLoS One 9(1):e84249Google Scholar
  34. 34.
    Bethge A, Schumacher U, Wedemann G (2015) Simulation of metastatic progression using a computer model including chemotherapy and radiation therapy. J Biomed Inform 57:74–87Google Scholar
  35. 35.
    Lewis PAW, Shedler GS (1979) Simulation of nonhomogeneous poisson processes by thinning. Nav Res Log Q 26(3):403Google Scholar
  36. 36.
    Sadahiro S, Suzuki T, Ishikawa K, Nakamura T, Tanaka Y, Masuda T, Mukoyama S, Yasuda S, Tajima T, Makuuchi H, Murayama C (2003) Recurrence patterns after curative resection of colorectal cancer in patients followed for a minimum of ten years. Hepatogastroenterology 50(53):1362–1366Google Scholar
  37. 37.
    Siegel R, DeSantis C, Virgo K, Stein K, Mariotto A, Smith T, Cooper D, Gansler T, Lerro C, Fedewa S, Lin C, Leach C, Cannady RS, Cho H, Scoppa S, Hachey M, Kirch R, Jemal A, Ward E (2012) Cancer treatment and survivorship statistics, 2012. CA Cancer J Clin 62(4):220–241Google Scholar
  38. 38.
    Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, CambridgeGoogle Scholar
  39. 39.
    Barbolosi D, Benabdallah B, Hubert F, Verga F (2009) Mathematical and numerical analysis for a model of growing metastatic tumors. Math Biosci 218:1–14Google Scholar
  40. 40.
    Hartung N (2015) Efficient resolution of metastatic tumour growth models by reformulation into integral equations. Discrete Contin Dyn Syst B 20:445–467Google Scholar
  41. 41.
    Lavielle M (2014) Mixed effects models for the population approach. models, tasks, methods and tools. Chapman & Hall/CRC biostatistics series. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  42. 42.
    Tornøe CW, Overgaard RV, Agersø H, Nielsen HA, Madsen H, Jonsson EN (2005) Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations. Pharm Res 22(8):1247–1258Google Scholar
  43. 43.
    Bulfoni M, Gerratana L, Del Ben F, Marzinotto S, Sorrentino M, Turetta M, Scoles G, Toffoletto B, Isola M, Beltrami CA, Di Loreto C, Beltrami AP, Puglisi F, Cesselli D (2016) In patients with metastatic breast cancer the identification of circulating tumor cells in epithelial-to-mesenchymal transition is associated with a poor prognosis. Breast Cancer Res 18(1):30Google Scholar
  44. 44.
    Paoletti C, Hayes DF (2016) Circulating tumor cells. Adv Exp Med Biol 882:235–258Google Scholar
  45. 45.
    Chen LL, Blumm N, Christakis NA, Barabasi AL, Deisboeck TA (2009) Cancer metastasis networks and the prediction of progression patterns. Br J Cancer 101(5):749–758Google Scholar

Copyright information

© Springer Science+Business Media LLC 2018

Authors and Affiliations

  1. 1.Aix Marseille UniversitéCNRS, Centrale Marseille, I2M UMR 7373MarseilleFrance
  2. 2.Department of Clinical Pharmacy and BiochemistryFreie Universität BerlinBerlinGermany
  3. 3.Present address: Institute of MathematicsUniversität PotsdamPotsdamGermany

Personalised recommendations