Some Models for the Prediction of Tumor Growth: General Framework and Applications to Metastases in the Lung

  • Thierry Colin
  • Angelo Iollo
  • Damiano Lombardi
  • Olivier SautEmail author
  • Françoise Bonichon
  • Jean Palussière


This chapter presents an example of an application of a mathematical model: the goal is here to help clinicians evaluate the aggressiveness of some metastases to the lung. For this matter, an adequate spatial model is described and two algorithms (one using a reduced model approach and the other one a sensitivity technique) are shown. They allow us to find reasonable values of the parameters of this model for a given patient with a sequence of medical images. The quality of the prognosis obtained through the calibrated model is then illustrated with several clinical cases.


Tumor growth modeling Clinical data assimilation Inverse problem Partial differential equations Scientific computing Numerical prognosis lung cancer metastases reduce model genetic regulation carcinoma clinical decision 


  1. 1.
    Alarcón T, Byrne H, Maini P (2003) A cellular automaton model for tumour growth in inhomogeneous environment. J Theor Biol 225(2):257–274CrossRefGoogle Scholar
  2. 2.
    Ambrosi D, Preziosi L (2002) On the closure of mass balance models for tumor growth. Math Models Meth Appl Sci 12(05):737–754MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anderson ARA, Weaver AM, Cummings PT, Quaranta V (2006) Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127(5): 905–915CrossRefGoogle Scholar
  4. 4.
    Bergmann M, Bruneau CH, Iollo A (2009) Enablers for robust pod models. J Comput Phys 228:516–538MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Billy F, Ribba B, Saut O, Morre-Trouilhet H, Colin T, Bresch D, Boissel J-P, Grenier E, Flandrois J (2009) A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy. J Theor Biol 260(4):545–562. doi:10.1016/j.jtbi.2009.06.026MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bresch D, Colin T, Grenier E, Ribba B, Saut O (2009) A viscoelastic model for avascular tumor growth. Discrete Contin Dyn Syst Suppl 2009, 101–108MathSciNetzbMATHGoogle Scholar
  7. 7.
    Byrd RH, Nocedal J, Schnabe RB (1994) Representations of quasi-Newton matrices and their use in limited memory methods. Math Program 63:129–156CrossRefzbMATHGoogle Scholar
  8. 8.
    Byrne H, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58(4–5):657–687MathSciNetCrossRefGoogle Scholar
  9. 9.
    Byrne HM (2010) Dissecting cancer through mathematics: from the cell to the animal model. Nat Rev Cancer 10(3):221–230CrossRefGoogle Scholar
  10. 10.
    Clatz O, Sermesant M, Bondiau P-Y, Delingette H, Warfield SK, Malandain G, Ayache N (2005) Realistic simulation of the 3-d growth of brain tumors in mr images coupling diffusion with biomechanical deformation. IEEE Trans Med Imaging 24(10):1334–1346CrossRefGoogle Scholar
  11. 11.
    Friedman A (2004) A hierarchy of cancer models and their mathematical challenges. Discrete Contin Math Syst Ser B 4(1):147–160CrossRefzbMATHGoogle Scholar
  12. 12.
    Gatenby R, Gawlinski ET (1996) A reaction-diffusion model of cancer invasion. Cancer Res 56(24):5745–5753Google Scholar
  13. 13.
    Hahnfeldt P, Panigrahy D, Folkman J, Hlatky L (1999) Tumor development under angiogenic signaling. Cancer Res 59(19):4770Google Scholar
  14. 14.
    Hanahan D, Weinberg R (2000) The hallmarks of cancer. Cell 100(1):57–70CrossRefGoogle Scholar
  15. 15.
    Hanke M (1997) A regularizing Levenberg-Marquardt scheme with applications to inverse groundwater filtration problems. Inverse Probl 13:79–95MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hogea C, Davatzikos C, Biros G (2008) An image-driven parameter estimation problem for a reaction–diffusion glioma growth model with mass effects. J Math Biol 56(6):793–825MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Holmes P, Lumley JL, Berkooz G (1996) Turbulence, coherent structures, dynamical systems and symmetry. Cambridge monographs on mechanics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  18. 18.
    Macklin P, Lowengrub JS (2005) Evolving interfaces via gradients of geometry-dependent interior poisson problems: application to tumor growth. J Comput Phys 203(1):191–220MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mansury Y, Kimura M, Lobo J, Deisboeck T (2002) Emerging patterns in tumor systems: simulating the dynamics of multicellular clusters with an agent-based spatial agglomeration model. J Theor Biol 219(3):343–370MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mantzaris N, Webb S, Othmer H (2004) Mathematical modeling of tumor-induced angiogenesis. J Math Biol 49(2):111–187MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Marzouk YM, Najm HN (2009) Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J Comput Phys 228(6):1862–1902MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Othmer HG, Stevens A (1997) Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J Appl Math 57(4):1044–1081MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Reinboldt W (1993) On the sensitivity of solutions of parametrized equations. SIAM J Numer Anal 30:305–320MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ribba B, Saut O, Colin T, Bresch D, Grenier E, Boissel J-P (2006) A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents. J Theor Biol 243(4):532–541MathSciNetCrossRefGoogle Scholar
  25. 25.
    Robbins RJ, Wan Q, Grewal RK, Reibke R, Gonen M, Strauss HW, Tuttle RM, Drucker W, Larson SM (2006) Real-time prognosis for metastatic thyroid carcinoma based on 2-[18F]fluoro-2-deoxy-D-glucose-positron emission tomography scanning. J Clin Endocrinol Metab 91(2):498–505CrossRefGoogle Scholar
  26. 26.
    Roose T, Chapman SJ, Maini PK (2007) Mathematical models of avascular tumor growth. SIAM Rev 49(2):179–208MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sachs LR, Hahnfeldt P (2001) Simple ODE models of tumor growth and anti-angiogenic or radiation treatment. Math Comput Model 33(12–13):1297–1305MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Schlumberger M, Sherman SI (2009) Clinical trials for progressive differentiated thyroid cancer: patient selection, study design, and recent advances. Thyroid 19(12):1393–1400CrossRefGoogle Scholar
  29. 29.
    Sherratt J, Chaplain M (2001) A new mathematical model for avascular tumour growth. J Math Biol 43(4):291–312MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shiraishi F, Tomita T, Iwata M, Berrada AA, Hirayama H (2009) A reliable taylor series-based computational method for the calculation of dynamic sensitivities in large-scale metabolic reaction systems: algorithm and software evaluation. Math Biosci 222:73–85MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Simeoni M, Magni P, Cammia C, De Nicolao G, Croci V, Pesenti E, Germani M, Poggesi I, Rocchetti M (2004) Predictive pharmacokinetic-pharmacodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents. Cancer Res 64(3): 1094–1101CrossRefGoogle Scholar
  32. 32.
    Sirovich L (1989) Low dimensional description of complicated phenomena. Contemp Math 99:277–305MathSciNetCrossRefGoogle Scholar
  33. 33.
    Swanson KR, Alvord EC, Murray JD (2002) Quantifying efficacy of chemotherapy of brain tumors with homogeneous and heterogeneous drug delivery. Acta Biotheor 50(4):223–237CrossRefGoogle Scholar
  34. 34.
    Tenoroio L (2001) Statistical regularization of inverse problems. SIAM Rev 43:347–366MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer New York 2014

Authors and Affiliations

  • Thierry Colin
    • 1
  • Angelo Iollo
    • 1
  • Damiano Lombardi
    • 1
  • Olivier Saut
    • 1
    Email author
  • Françoise Bonichon
    • 2
  • Jean Palussière
    • 2
  1. 1.Institut de Mathématiques de Bordeaux UMR 5251Université de Bordeaux and INRIA Bordeaux Sud-OuestTalenceFrance
  2. 2.Institut BergoniéBordeauxFrance

Personalised recommendations