Numerical Simulation of the Vascular Solid Tumour Growth Model and Therapy – Parallel Implementation

  • Krzysztof Psiuk-Maksymowicz
  • Damian Borys
  • Sebastian Student
  • Andrzej Świerniak
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 283)


The main interest of the authors was to develop vascular solid tumour growth model, implement efficient numerical methods for simulations towards finding a solution of the model and trying to optimise influence of different types of therapies. A system of partial differential equations was introduced in order to simulate growth of tumour and normal cells as well as the dynamics of the diffusing nutrient and anti-angiogenic or chemotherapeutic factors within the tissue. Numerical simulations of our model were executed toward finding a suboptimal therapy protocol using stable FDTD (finite difference time-domain) method implementation. Combined therapy protocols has been selected by means of meta-heuristic algorithms. In this work we have selected genetics algorithms, ant colony algorithms and simulated annealing method. For those algorithms convergence to suboptimal solution was examined and compared, as well as average tumour size, average iteration count and average execution time.


tumour growth model parallel implementation therapy optimisation heuristic methods 


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  1. 1.
    Araujo, R.P., McElwain, D.L.S.: A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull. Math. Biol. 66, 1039–1091 (2004)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Dormann, S., Deutsch, A.: Modeling of self-organized avascular tumor growth with a hybrid cellular automaton. Silico Biology 2, 393–406 (2002)Google Scholar
  3. 3.
    Billy, F., Clairambault, J., Delaunay, F., Feillet, C., Robert, N.: Age-structured cell population model to study the influence of growth factors on cell cycle dynamics. Math. Biosci. Eng. 10, 1–17 (2013)MATHMathSciNetGoogle Scholar
  4. 4.
    Rejniak, K.A.: A single-cell approach in modeling the dynamics of tumor microregions. Math. Biosci. Eng. 2, 643–655 (2005)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Lowengrub, J.S., Frieboes, H.B., Jin, F., Chuang, Y.-L., Li, X., Macklin, P., Wise, S.M., Cristini, V.: Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23, R1–R9 (2010)Google Scholar
  6. 6.
    Mamontov, E., Koptioug, A., Psiuk-Maksymowicz, K.: The minimal, phase-transition model for the cell-number maintenance by the hyperplasia-extended homeorhesis. Acta Biotheor. 54, 61–101 (2006)CrossRefGoogle Scholar
  7. 7.
    Bankhead III, A., Magnuson, N.S., Heckendorn, R.B.: Cellular automaton simulation examining progenitor hierarchy structure effects on mammary ductal carcinoma in situ. J. Theor. Biol. 246, 491–498 (2007)CrossRefGoogle Scholar
  8. 8.
    Kam, Y., Rejniak, K.A., Anderson, A.R.: Cellular modeling of cancer invasion: integration of in silico and in vitro approaches. J. Cell Physiol. 227, 431–438 (2012)CrossRefGoogle Scholar
  9. 9.
    Chaplain, M.A., Anderson, A.R.: Mathematical modelling, simulation and prediction of tumour-induced angiogenesis. Invasion Metastasis 16, 222–234 (1996)Google Scholar
  10. 10.
    Sherratt, J.A.: Predictive mathematical modeling in metastasis. Methods Mol. Med. 57, 309–315 (2001)Google Scholar
  11. 11.
    Evans, G., Blackledge, J., Yardley, P.: Numerical Methods for Partial Differential Equations. Springer, London (1999)Google Scholar
  12. 12.
    The Message Passing Interface (MPI) standard,
  13. 13.
  14. 14.
    Larsson, S., Thomee, V.: Finite Difference Methods for Hyperbolic Equations. Springer, Heidelberg (2003)Google Scholar
  15. 15.
    Macklin, P., Lowengrub, J.: Nonlinear simulation of the effect of microenvironment on tumor growth. J. Theor. Biol. 245, 677–704 (2007)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Kirkpatrick, S., Gelatt Jr, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Bonabeau, E., Dorigo, M., Theraulaz, G.: Swarm Intelligence: From Natural to Artificial Systems. Oxford Univ. Press, New York (1999)MATHGoogle Scholar
  18. 18.
    Cooper, L., Steinberg, D.: Introduction to Methods of Optimization. W. B. Saunders, Co., Philadelphia (1970)MATHGoogle Scholar
  19. 19.
    Byrne, H., Preziosi, L.: Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20, 341–366 (2003)CrossRefMATHGoogle Scholar
  20. 20.
    Psiuk-Maksymowicz, K.: Multiphase modelling of desmoplastic tumour growth. J. Theor. Biol. 329, 52–63 (2013)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Ambrosi, D., Preziosi, L.: On the closure of mass balance models for tumor growth. Math. Models Meth. Appl. Sci. 12, 737–754 (2002)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Krzysztof Psiuk-Maksymowicz
    • 1
  • Damian Borys
    • 1
  • Sebastian Student
    • 1
  • Andrzej Świerniak
    • 1
  1. 1.Institute of Automatic ControlSilesian University of TechnologyGliwicePoland

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