Document Type : Original Article

Authors

Department of Physics, Faculty of Physics, Tehran University, Tehran, Iran

Abstract

We develop a model to study how invasion front depends on the relevant properties of a cellular environment. To do so, we use a nonlinear reaction-diffusion equation, the Fisher equation, to model the population dynamics. Our study is intended to understand how heterogeneity in the cellular environment's stiffness, as well as spatial correlations in its morphology,  given that the existence of both has been demonstrated by experiments, affects the properties of  the invasion front. It is demonstrated that three important factors affect the properties of the front; these are  the spatial distribution of the local diffusion coefficients, the correlations between them, and R/D, the ratio of  the cells' duplication rate R to  the cells' average diffusion coefficient D. Analyzing the scaling properties of  the Fisher equation invasion front, we show that , contrary to several previous claims, invasion fronts, including those of tumors and cancerous cells colonies, cannot be described by the well-known model of kinetic growth, such as the Kardar-Parisi-Zhang equation.

Keywords

  1. K S Korolev, J B Xavier, and J Gore, Nature Reviews Cancer 14, 5 (2014) 371.
  2. J D Murray, “Mathematical biology: I. An introduction” Springer Science & Business Media. (2007).
  3. D L DeAngelis and V Grimm, “Individual-based models in ecology after four decades.” F1000prime reports 6 (2014)
  4. A Morozov and J C Poggiale, Ecological Complexity 10 (2012) 1.
  5. R A Fisher, 1937. Annals of eugenics 7, 4 (1937) 355.
  6. G Birzu, O Hallatschek, and K S Korolev, Proceedings of the National Academy of Sciences 115, 16 (2018) E3645.
  7. Y Azimzade, M Sasar, and V M P García, “Environmental Disorder Regulation of Invasion and Genetic Loss”. arXiv preprint arXiv:1908.02532, (2019).
  8. K M A Yong, Z Li, S D Merajver, and J Fu, Scientific reports 7, 1 (2017) 1.
  9. A Brú, S Albertos, J L Subiza, J L García-Asenjo, and I Brú, Biophysical journal 85, 5 (2003) 2948.
  10. M A C Huergo, M A Pasquale, A E Bolzán, A J Arvia, and P H González, Phys. Rev. E 82 (2010) 031903 .
  11. J Pérez-Beteta, D Molina-García, A Martínez-González, A Henares-Molina, M Amo, B Luque, E Arregui, M Calvo, J M Borrás, J Martino, et al., European Radiology (2018) 1.
  12. D Wirtz, K Konstantopoulos, and P C Searson, Nature Reviews Cancer 11, 7 (2011) 512.
  13. M Plodinec, M Loparic, C A Monnier, E C Obermann, R Zanetti-Dallenbach, P Oertle, J T Hyotyla, U Aebi, M Bentires-Alj, R Y Lim, et al., Nature Nanotechnology 7, 11 (2012) 757.
  14. S Kondo and T Miura, science 329, 5999 (2010) 1616.
  15. A A Anderson, Math.Med. Biol. 22 (2005) 163.
  16. P Haridas, C J Penington, J A McGovern, D S McElwain, and M J Simpson, J. Theor. Biol. 423 (2017) 13.
  17. T H Keitt, Landscape Ecology 15, 5 (2000) 479.

Y. Azimzade, A A. Saberi, and M Sahimi,. 2018. Scientific reports 8, 1 (2018) 1.