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A Fast Parallel Algorithm for Delay Partial Differential Equations Modeling the Cell Cycle in Cell Lines Derived from Human Tumors

  • Barbara Zubik-KowalEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 94)

Abstract

We present a fast numerical algorithm for solving delay partial differential equations that model the growth of human tumor cells. The undetermined model parameters need to be estimated according to experimental data and it is desired to shorten the computational time needed in estimating them. To speed up the computations, we present an algorithm invoking parallelization designed for arbitrary numbers of available processors. The presented numerical results demonstrate the efficiency of the algorithm.

Keywords

Delay partial differential equations Cell division cycle Parallel algorithm Parallel computations Computational efficiency 

Notes

Acknowledgements

The work was partially supported by BSU COAS grant.

References

  1. 1.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    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
  3. 3.
    Bellomo, N., Bellouquid, A.: From a class of kinetic models to macroscopic equations for multicellular systems in biology. Discrete Contin. Dyn. Syst. B 4, 59–80 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bellomo, N., De Angelis, E.: Strategies of applied mathematics towards an immuno-mathematical theory on tumors and immune system interactions. Math. Model Methods Appl. Sci. 8(8), 1403–1429 (1998)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bellouquid, A., De Angelis, E.: From kinetic models of multicellular growing systems to macroscopic biological tissue models. Nonlinear Anal. Real World Appl. 12, 1111–1122 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bellouquid, A., Delitala, M.: Mathematical methods and tools of kinetic theory towards modelling complex biological systems. Math. Model Methods Appl. Sci. 15, 1639–1666 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bellouquid, A., Delitala, M.: Modelling Complex Biological Systems: A Kinetic Theory Approach. Birkhäuser, Boston (2006)Google Scholar
  8. 8.
    Bellomo, N., Forni, G.: Dynamics of tumor interaction with the host immune system, original research article. Math. Comput. Model. 20, 107–122 (1994)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bellomo, N., Bellouquid, A., De Angelis, E.: The modelling of the immune competition by generalized kinetic (Boltzmann) models; review and research perspectives. Math. Comput. Model. 37(1–2), 65–86 (2003)CrossRefzbMATHGoogle Scholar
  10. 10.
    Bellomo, N., Bellouquid, A., Soler, J.: From the mathematical kinetic theory for active particles on the derivation of hyperbolic macroscopic tissue models. Math. Comput. Model. 49(11–12), 2083–2093 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Bianca, C., Bellomo, N.: Towards a Mathematical Theory of Complex Biological Systems. Series in Mathematical Biology and Medicine, vol. 11. World Scientific Publishing Co. Pte. Ltd., Hackensack (2011)Google Scholar
  12. 12.
    Drucis, K., Kolev, M., Majda, W., Zubik-Kowal, B.: Nonlinear modeling with mammographic evidence of carcinoma. Nonlinear Anal. Real World Appl. 11(5), 4326–4334 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Jackiewicz, Z., Jorcyk, C.L., Kolev, M., Zubik-Kowal, B.: Correlation between animal and mathematical models for prostate cancer progression. Comput. Math. Methods Med. 10(4), 241–252 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    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(3), 561–572 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Jorcyk, C.L., Kolev, M., Tawara, K., Zubik-Kowal, B.: Experimental versus numerical data for breast cancer progression. Nonlinear Anal. Real World Appl. 13(1), 78–84 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kolev, M.: Mathematical modelling of the competition between tumors and immune system considering the role of the antibodies. Math. Comput. Model. 37(11), 1143–1152 (2003)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kolev, M., Kozłowska, E., Lachowicz, M.: A mathematical model for single cell cancer-immune system dynamics. Math. Comput. Model. 41, 1083–1095 (2005)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kolev, M., Nawrocki, S., Zubik-Kowal, B.: Numerical simulations for tumor and cellular immune system interactions in lung cancer treatment. Commun. Nonlinear Sci. Numer. Simulat. 18, 1473–1480 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Terasima, T., Tolmach, L.J.: Variations in survival responses of HeLa cells to X-irradiation during the division cycle. Biophys. J. 3, 11–33 (1963)CrossRefGoogle Scholar
  20. 20.
    Zubik-Kowal, B.: Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations. Appl. Numer. Math. 34(2–3), 309–328 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Zubik-Kowal, B.: Solutions for the cell cycle in cell lines derived from human tumors. Comput. Math. Methods Med. 7(4), 215–228 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Zubik-Kowal, B.: Numerical algorithm for the growth of human tumor cells and their responses to therapy. Appl. Math. Comput. 230, 174–179 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsBoise State UniversityBoiseUSA

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