A Continuum Model and Numerical Simulation for Avascular Tumor Growth

  • Sounak SadhukhanEmail author
  • S. K. Basu
  • Naveen Kumar
Conference paper
Part of the Learning and Analytics in Intelligent Systems book series (LAIS, volume 3)


A spatio-temporal continuum model is developed for avascular tumor growth in two dimensions using fractional advection-diffusion equation as the transportation in biological systems is heterogeneous and anomalous in nature (non-Fickian). The model handles skewness with a suitable parameter. We study the behavior of this model with a set of parameters, and suitable initial and boundary conditions. It is found that the fractional advection-diffusion equation based model is more realistic as it provides more insightful information for tumor growth at the macroscopic level.


Avascular tumor growth Anomalous diffusion Fractional advection-diffusion equation 



We are thankful to University Grant Commission, Government of India for supporting the first author with a Junior Research Fellowship.


  1. 1.
    Sutherland RM (1988) Cell and environment interactions in tumor microregions: the multicell spheroid model. Science 240(4849):177–184CrossRefGoogle Scholar
  2. 2.
    van Kempen LCL, Leenders WPJ (2006) Tumours can adapt to anti-angiogenic therapy depending on the stromal context: lessons from endothelial cell biology. Eur J Cell Biol 85(2):61–68CrossRefGoogle Scholar
  3. 3.
    Orme ME, Chaplain MAJ (1996) A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching. Math Med Biol: J IMA 13(2):73–98zbMATHCrossRefGoogle Scholar
  4. 4.
    Hystad ME, Rofstad EK (1994) Oxygen consumption rate and mitochondrial density in human melanoma monolayer cultures and multicellular spheroids. Int J Cancer 57(4):532–537CrossRefGoogle Scholar
  5. 5.
    Greenspan HP (1972) Models for the growth of a solid tumor by diffusion. Stud Appl Math 51(4):317–340zbMATHCrossRefGoogle Scholar
  6. 6.
    Ward JP, King JR (1997) Mathematical modelling of avascular-tumour growth. Math Med Biol: J IMA 14(1):39–69zbMATHCrossRefGoogle Scholar
  7. 7.
    Ward JP, King JR (1999) Mathematical modelling of avascular-tumour growth II: modelling growth saturation. Math Med Biol: J IMA 16(2):171–211zbMATHCrossRefGoogle Scholar
  8. 8.
    Sherratt JA, Chaplain MAJ (2001) A new mathematical model for avascular tumour growth. J Math Biol 43(4):291–312MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gal N, Weihs D (2010) Experimental evidence of strong anomalous diffusion in living cells. Phys Rev E81(2):020903Google Scholar
  10. 10.
    Caputo M, Cametti C (2008) Diffusion with memory in two cases of biological interest. J Theor Biol 254(3):697–703MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Morales-Casique E, Neuman SP, Guadagnini A (2006) Non-local and localized analyses of non-reactive solute transport in bounded randomly heterogeneous porous media: theoretical framework. Adv Water Resour 29(8):1238–1255CrossRefGoogle Scholar
  12. 12.
    Cushman JH, Ginn TR (2000) Fractional advection-dispersion equation: a classical mass balance with convolution-Fickian flux. Water Resour Res 36(12):3763–3766CrossRefGoogle Scholar
  13. 13.
    Roop JP (2006) Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2. J Comput Appl Math 193(1):243–268MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Chen W, Sun H, Zhang X, Korošak D (2010) Anomalous diffusion modeling by fractal and fractional derivatives. Comput Math Appl 59(5):1754–1758MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Meerschaert MM, Tadjeran C (2006) Finite difference approximations for two-sided space-fractional partial differential equations. Appl Numer Math 56(1):80–90MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Sherratt JA, Murray JD (1991) Mathematical analysis of a basic model for epidermal wound healing. J Math Biol 29(5):389–404zbMATHCrossRefGoogle Scholar
  17. 17.
    Casciari JJ, Sotirchos SV, Sutherland RM (1988) Glucose diffusivity in multicellular tumor spheroids. Can Res 48(14):3905–3909Google Scholar
  18. 18.
    Burton AC (1966) Rate of growth of solid tumours as a problem of diffusion. Growth 30(2):157–176Google Scholar
  19. 19.
    Busini V, Arosio P, Masi M (2007) Mechanistic modelling of avascular tumor growth and pharmacokinetics influence—Part I. Chem Eng Sci 62(7):1877–1886CrossRefGoogle Scholar
  20. 20.

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Computer ScienceBanaras Hindu UniversityVaranasiIndia
  2. 2.Department of MathematicsBanaras Hindu UniversityVaranasiIndia

Personalised recommendations