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Modeling Growth of Tumors and Their Spreading Behavior Using Mathematical Functions

  • Bertin Hoffmann
  • Thorsten Frenzel
  • Rüdiger Schmitz
  • Udo Schumacher
  • Gero WedemannEmail author
Protocol
Part of the Methods in Molecular Biology book series (MIMB, volume 1878)

Abstract

Computer simulations of the spread of malignant tumor cells in an entire organism provide important insights into the mechanisms of metastatic progression. Key elements for the usefulness of these models are the adequate selection of appropriate mathematical models describing the tumor growth and its parametrization as well as a proper choice of the fractal dimension of the blood vessels in the primary tumor. In addition, survival in the bloodstream and evasion into the connective spaces of the target organ of the future metastasis have to be modeled. Determination of these from experimental models is complicated by systematic and unsystematic experimental errors which are difficult to assess. In this chapter, we demonstrate how to select the best-suited mathematical function to describe tumor growth for experimental xenograft mouse tumor models and how to parametrize them. Common pitfalls and problems are described as well as methods to avoid them.

Key words

Tumor growth Metastatic progression Spreading behavior Mathematical model Parametrization 

References

  1. 1.
    Valastyan S, Weinberg RA (2011) Tumor metastasis: molecular insights and evolving paradigms. Cell 147:275–292.  https://doi.org/10.1016/j.cell.2011.09.024CrossRefPubMedPubMedCentralGoogle Scholar
  2. 2.
    Bethge A, Schumacher U, Wree A, Wedemann G (2012) Are metastases from metastases clinical relevant? Computer modelling of cancer spread in a case of hepatocellular carcinoma. PLoS One 7:e35689.  https://doi.org/10.1371/journal.pone.0035689CrossRefPubMedPubMedCentralGoogle Scholar
  3. 3.
    Brodbeck T, Nehmann N, Bethge A, Wedemann G, Schumacher U (2014) Perforin-dependent direct cytotoxicity in natural killer cells induces considerable knockdown of spontaneous lung metastases and computer modelling-proven tumor cell dormancy in a HT29 human colon cancer xenograft mouse model. Mol Cancer 13:244.  https://doi.org/10.1186/1476-4598-13-244CrossRefPubMedPubMedCentralGoogle Scholar
  4. 4.
    Bethge A, Schumacher U, Wedemann G (2015) Simulation of metastatic progression using a computer model including chemotherapy and radiation therapy. J Biomed Inform 57:74–87.  https://doi.org/10.1016/j.jbi.2015.07.011CrossRefPubMedGoogle Scholar
  5. 5.
    Benzekry S, Tracz A, Mastri M, Corbelli R, Barbolosi D, Ebos JML (2015) Modeling spontaneous metastasis following surgery: an in vivo-in silico approach. Cancer Res 76:535–547.  https://doi.org/10.1158/0008-5472.CAN-15-1389CrossRefPubMedPubMedCentralGoogle Scholar
  6. 6.
    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 mathematical model. Cancer Res 73:2760–2769.  https://doi.org/10.1158/0008-5472.CAN-12-4488CrossRefPubMedPubMedCentralGoogle Scholar
  7. 7.
    Ferrante L, Bompadre S, Possati L, Leone L (2000) Parameter estimation in a Gompertzian stochastic model for tumor growth. Biometrics 56:1076–1081.  https://doi.org/10.1111/j.0006-341X.2000.01076.xCrossRefPubMedGoogle Scholar
  8. 8.
    Witten M, Satzer W (1992) Gompertz survival model parameters: estimation and sensitivity. Appl Math Lett 5:7–12.  https://doi.org/10.1016/0893-9659(92)90125-SCrossRefGoogle Scholar
  9. 9.
    Tan W-Y, Ke W, Webb G (2009) A stochastic and state space model for tumour growth and applications. Comput Math Methods Med 10:117–138.  https://doi.org/10.1080/17486700802200784CrossRefGoogle Scholar
  10. 10.
    Hahnfeldt P, Panigrahy D, Folkman J, Hlatky L (1999) Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res 59:4770–4775Google Scholar
  11. 11.
    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:e1003800.  https://doi.org/10.1371/journal.pcbi.1003800CrossRefPubMedPubMedCentralGoogle Scholar
  12. 12.
    Barbolosi D, Benabdallah A, Hubert F, Verga F (2009) Mathematical and numerical analysis for a model of growing metastatic tumors. Math Biosci 218:1–14.  https://doi.org/10.1016/j.mbs.2008.11.008CrossRefPubMedGoogle Scholar
  13. 13.
    Iwata K, Kawasaki K, Shigesada N (2000) A dynamical model for the growth and size distribution of multiple metastatic tumors. J Theor Biol 203:177–186.  https://doi.org/10.1006/jtbi.2000.1075CrossRefPubMedGoogle Scholar
  14. 14.
    Kozusko F, Bajzer Ž (2003) Combining Gompertzian growth and cell population dynamics. Math Biosci 185:153–167.  https://doi.org/10.1016/S0025-5564(03)00094-4CrossRefPubMedGoogle Scholar
  15. 15.
    Bethge A, Wedemann G (2014) CaTSiT - Computer simulation of metastatic progression and treatments. In: CaTSiT - Computer simulation of metastatic progression and treatments. http://bioinformatics.hochschule-stralsund.de/catsit/. Accessed 12 Nov 2015
  16. 16.
    Gazit Y, Baish JW, Safabakhsh N, Leuning M, Baxter LT, Jain RK (1997) Fractal characteristics of tumor vascular architecture during tumor growth and regression, microcirculation, Informa healthcare. Microcirculation 4:395–402.  https://doi.org/10.3109/10739689709146803CrossRefPubMedGoogle Scholar
  17. 17.
    Jurczyszyn K, Osiecka BJ, Ziółkowski P (2012) The use of fractal dimension analysis in estimation of blood vessels shape in transplantable mammary adenocarcinoma in Wistar rats after photodynamic therapy combined with cysteine protease inhibitors. Comput Math Methods Med 2012:793291.  https://doi.org/10.1155/2012/793291CrossRefPubMedPubMedCentralGoogle Scholar
  18. 18.
    Frenzel T, Hoffmann B, Schmitz R, Bethge A, Schumacher U, Wedemann G (2017) Radiotherapy and chemotherapy change vessel tree geometry and metastatic spread in a small cell lung cancer xenograft mouse tumor model. PLOS ONE 12(11):e0187144.  https://doi.org/10.1371/journal.pone.0187144CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Bertin Hoffmann
    • 1
  • Thorsten Frenzel
    • 2
  • Rüdiger Schmitz
    • 2
  • Udo Schumacher
    • 2
  • Gero Wedemann
    • 1
    Email author
  1. 1.Competence Center Bioinformatics, Institute for Applied Computer ScienceUniversity of Applied Sciences StralsundStralsundGermany
  2. 2.Institute for Anatomy and Experimental MorphologyUniversity Cancer Center Hamburg-EppendorfHamburgGermany

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