A numerical approach to modelling avascular tumour evolution with white noise

Keng-Cheng Ang, Liang-Soon Tan


A model for avascular tumour growth with white noise is presented. The model is a set of partial differential equations describing the spatio-temporal evolutions of cell concentrations based on reaction-diffusion dynamics and the law of mass conservation. Perturbations in the form of white noise are introduced to model the effects of random processes on distinct time scales. Numerical simulations in one and two space dimensions are presented. Numerical results indicate that the proposed model is a reasonable approach that may be used to examine the effects of nutrient supply in tumour growth dynamics.

  • Abercrombie, M., Contact inhibition in tissue culture, In vitro., 6, 1970, 128--140. doi:10.1007/BF02616114
  • Adam, J. A., A simplified mathematical model of tumour growth, Mathematical Biosciences, 81, 1986, 224--229. doi:10.1016/0025-5564(86)90119-7
  • Albano, G. and Giorno, V., A stochastic model in tumor growth, Journal of Theoretical Biology, 242, 2006, 329--336. doi:10.1016/j.jtbi.2006.03.001
  • Burton, A. C., Rate of growth of solid tumours as a problem of diffusion, Growth, 30, 1966, 157--176.
  • Dorie, M., Kallman, R. and Coyne, M., Effect of cytochalasin b, nocodazole and irradiation on migration and internalization of cells and microspheres in tumor cell spheroids, Experimental Cell Research, 166, 1986, 370--378. doi:10.1016/0014-4827(86)90483-0
  • Folkman, J., and Hochberg, M., Self-regulation of growth in three dimensions, Journal of Experimental Medicine, 138, 1973, 745-753.
  • Folkman, J., Tumour Angiogenesis, Advances in Cancer Research, 43, 1985, 175--203.
  • Higham, D. J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43, 2001, 525--546. doi:10.1137/S0036144500378302
  • Lo, C. F., Stochastic Gompertz model of tumour cell growth, Journal of Theoretical Biology, 248, 2007, 317--321. doi:10.1016/j.jtbi.2007.04.024
  • Nirmala, C., Rao, J. S., Ruifrok, A. C., Langford, L. A. and Obeyesekere, M., Growth characteristics of glioblastoma spheroids, International Journal of Oncology, 19, 2001, 1109--1115.
  • Sheratt, J. A. and Chaplain, M. A. J., A new mathematical model for avascular tumour growth, Journal of Mathematical Biology, 43, 2001, 291--312. doi:10.1007/s002850100088
  • Tan, L. S. and Ang, K. C., A numerical simulation of avascular tumour growth, ANZIAM Journal, 46(E), 2005, C902--C917. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/997
  • Ward, J. P., and King, J. R., Mathematical modelling of avascular tumour-growth, IMA Journal of Mathematics Applied in Medicine and Biology, 14, 1997, 39--69. doi:10.1093/imammb/14.1.39

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DOI: http://dx.doi.org/10.21914/anziamj.v50i0.1362

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