Fast Parameter Estimation for Cancer Cell Progression and Response to Therapy



We investigate a mathematical model describing the development of cancer cells in the mammalian cell division cycle under therapy and present an efficient strategy for fast numerical simulations and effective treatment programs in parallel computing environments. We perform a series of computations implemented on a cluster of computers with multicore processors for which the model equations are algorithmically split and solved over independent processors working in parallel and examine the speedup gained. In our implementation, the time interval is split into a number of subintervals corresponding to the number of available processors and the parallelization is invoked across time so that the number of processors to be used is unrestricted. We also extend our implementation to a generalized model of human tumor growth in vivo and demonstrate the computational efficiency of the algorithm and the associated decrease in computational time as the number of utilized processors increases by means of a series of numerical experiments performed in a parallel computing environment.


  1. [AfBe14]
    Afraites, L., Bellouquid, A.: Global optimization approaches to parameters identification in an immune competition model. Commun. Appl. Ind. Math. 5, e-466, 1–19 (2014)Google Scholar
  2. [AC15]
    American Cancer Society: Cancer Facts & Figures 2015. American Cancer Society, Atlanta (2015)Google Scholar
  3. [BaEtAl03]
    Basse, B., Baguley, B.C., Marshall, E.S., Joseph, W.R., van Brunt, B., Wake, G.C., Wall, D.J.N.: A mathematical model for analysis of the cell cycle in cell lines derived from human tumours. J. Math. Biol. 47, 295–312 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. [Be08]
    Bellomo, N.: Modeling complex living systems. A kinetic theory and stochastic game approach. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Inc., Boston (2008)Google Scholar
  5. [BeEtAl04]
    Bellomo, N., Bellouquid, A., Delitala, M.: Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition. Math. Models Methods Appl. Sci. 14, 1683–1733 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. [BeEtAl08]
    Bellomo, N., Li, N.K, Maini, P.K.: On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 18, 593–646 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. [BeCh14]
    Bellouquid, A., CH-Chaoui, M.: Asymptotic analysis of a nonlinear integro-differential system modeling the immune response. Comput. Math. Appl. 68, 905–914 (2014)Google Scholar
  8. [BeDe06]
    Bellouquid, A., Delitala, M.: Mathematical modeling of complex biological systems. A kinetic theory approach. With a preface by Nicola Bellomo. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Inc., Boston (2006)Google Scholar
  9. [ChEtAl01]
    Chandra, R., Dagum, L., Kohr, D., Maydan, D., McDonald, J., Menon, R.: Parallel Programming in OpenMP. Morgan Kaufmann Publishers, San Francisco (2001)Google Scholar
  10. [DrEtAl10]
    Drucis, K., Kolev, M., Majda, W., Zubik-Kowal, B.: Nonlinear modeling with mammographic evidence of carcinoma. Nonlinear Anal. Real World Appl. 11, 4326–4334 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. [JaEtAl09]
    Jackiewicz, Z., Kuang, Y., Thalhauser, C., Zubik-Kowal, B.: Numerical solution of a model for brain cancer progression after therapy. Math. Model. Anal. 14, 43–56 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. [JcEtAl09]
    Jackiewicz, Z., Zubik-Kowal, B., Basse, B.: Finite-difference and pseudospectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Math. Biosci. Eng. 6, 561–572 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. [JoEtAl12]
    Jorcyk, C.L., Kolev, M., Tawara, K., Zubik-Kowal, B.: Experimental versus numerical data for breast cancer progression. Nonlinear Anal. Real World Appl. 13, 78–84 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. [KoEtAl13]
    Kolev, M., Nawrocki, S., Zubik-Kowal, B.: Numerical simulations for tumor and cellular immune system interactions in lung cancer treatment. Commun. Nonlinear Sci. Numer. Simul. 18, 1473–1480 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. [NaZu15]
    Nawrocki, S., Zubik-Kowal, B.: Clinical study and numerical simulation of brain cancer dynamics under radiotherapy. Commun. Nonlinear Sci. Numer. Simul. 22, 564–573 (2015)MathSciNetCrossRefGoogle Scholar
  16. [NV15]
    NVIDIA Corporation: CUDA Programming Guide (2015). NVIDIA Corporation available at
  17. [Op13]
    OpenACC: The OpenACC Application Programming Interface (2013).
  18. [Pa96]
    Pacheco, P.: Parallel Programming with MPI. Morgan Kaufmann, San Frncisco (1996)MATHGoogle Scholar
  19. [WiEtAl88]
    Wilson, G.D., McNally, N.J., Dische, S., Saunders, M.I., Des Rochers, C., Lewis, A.A., Bennett, M.H.: Measurement of cell kinetics in human tumours in vivo using bromo-deoxyuridine incorporation and flow cytometry. Br. J. Cancer 58, 423–431 (1988)CrossRefGoogle Scholar
  20. [Zu13]
    Zubik-Kowal, B.: Numerical algorithm for the growth of human tumor cells and their responses to therapy. Appl. Math. Comput. 230, 174–179 (2014)MathSciNetGoogle Scholar
  21. [Zu14]
    Zubik-Kowal, B.: A fast parallel algorithm for delay partial differential equations modeling the cell cycle in cell lines derived from human tumors. In: Hartung, F., Pituk, M. (eds.) Recent Advances in Delay Differential and Difference Equations, vol. 94, pp. 251–260. Springer Proceedings in Mathematics & Statistics. Springer, Cham (2014)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Mathematics, Maria Curie–Skłodowska UniversityLublinPoland
  2. 2.Department of MathematicsBoise State UniversityBoiseUSA

Personalised recommendations