A Viscoelastic Model for Glioma Growth

J. R. Branco, J. A. Ferreira, P. de Oliveira

2013

Abstract

In this paper we propose a mathematical model to describe the evolution of glioma cells in the brain taking into account the viscoelastic properties of brain tissue. The mathematical model is established considering that the glioma cells are of two phenotypes: migratory and proliferative. The evolution of the migratory cells is described by a diffusion-reaction equation of non Fickian type deduced considering a mass conservation law with a non Fickian migratory mass flux. The evolution of the proliferation cells is described by a reaction equation. Numerical simulations that illustrate the behaviour of the mathematical model are included.

References

  1. (2002). Mathematical Biology- An Introduction. Springer Verlag.
  2. Edward, D. and Cohen, D. (1995). A mathematical model for a dissolving polymer. AIChE Journal, 41:2345- 2355.
  3. Edwards, D. (1996). Non-fickian diffusion in thin polymer films. Journal of Polymer Science, Part B: Polymer Physics Edition, 34:981-997.
  4. Edwards, D. A. (2001). A spatially nonlocal model for polymer-penetrant diffusion. Journal Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 52:254-288.
  5. Edwards, D. A. and Cohen, D. S. (1995). An unusual moving boundary condition arising in anomalous diffusion problems. SIAM Journal on Applied Mathematics, 55:662-676.
  6. Fedotov, S. (1998). Traveling waves in a reaction-diffusion system: diffusion with finite velocity and kolmogorov 10 15 5 10 15 - petrovskii - piskunov kinetics. Pysical Review E, 58:5143-5145.
  7. Fedotov, S. (1999). Nonuniform reaction rate distribution for the generalized fisher equation: Ignition ahead of the reaction front. Pysical Review E, 60:4958-4961.
  8. Fedotov, S. and Iomin, A. (2007). Migration and proliferation dichotomy in tumor-cell invasion. Physical Review Letters, 77:1031911(1)-(10).
  9. Fedotov, S. and Iomin, A. (2008). Probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion. Physical Review E, 77:1031911(1)-(10).
  10. G.Franceschini (2006). The mechanics of humain brain tissue. PhD thesis, University of Trento.
  11. Giese, A., Kluwe, L., Laube, B., Meissner, H., Berens, M., and Westphal, M. (1996). Migration of human glioma cells on myelin. Neurosurgery, 38:755-764.
  12. Habib, S., Molina-París, C., and Deisboeck, T. (2003). Complex dynamics of tumors: modeling an energing brain tumor system with coupled reaction-diffusion equations. Physica A, 327:501-524.
  13. Harpold, H., Jr, E. A., and Swanson, K. (2007). The evolution of mathematical modeling of glioma proliferation and invasion. Journal of Neuropathology and Experimental Neurology, 66:1-9.
  14. Hassanizadeh, S. (1996). On the transient non fickian dispersion theory. Transport in Porous Media, 23:107- 124.
  15. Humphrey, J. (2003). Continuum biomechanics of soft biological tissues. Proceedings of RoyalSociety London, 459:3-46.
  16. Joseph, D. and Preziosi, L. (1989). Heat waves. Review of Modern Physics, 61:47-71.
  17. Mehrabian, A. and Abousleiman, Y. (2011). A general solution to poroviscoelastic model of hydrocephalic humain brain tissue. Journal of Theoretical Biology, 29:105-118.
  18. Neuman, S. P. and Tartakovsky, D. M. (2009). Perspective on theories of anomalous transport in heterogeneous media. Advances in Water Resources, 32:670-680.
  19. Rockne, R., Jr, E. A., Rockhill, J., and Swanson, K. (2009). A mathematical model for brain tumor response to radiation therapy. Journal of Mathematical Biology, 58:561-578.
  20. Shaw, S. and Whiteman, J. R. (1998). Some partial differential volterra equation problems arising in viscoelasticity. In Proceeding of the Conference on Differential Equations and their Applications, Brno, August 2529, 1997, ed. R. P. Agarwal, F. Neuman, J.Vosmansky.
  21. Swanson, K., Alvord, E., and Murray, J. (2000). A quantitative model for differential motility of gliomas in grey and white matter. Cell Proliferation, 33:317-330.
  22. Swanson, K., Bridge, C., Murray, J., and Alvord, E. (2003). Virtual and real brain tumors: using mathematical modelig to quantify glioma growth and invasion. Journal of the Neurological Sciences, 216:17-31.
  23. Tracqui, P., Cruywagen, G., Woodward, D., Bartoo, G., Murray, J., and Jr, E. A. (1995). A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth. Cell Proliferation, 28:17- 31.
Download


Paper Citation


in Harvard Style

R. Branco J., Ferreira J. and de Oliveira P. (2013). A Viscoelastic Model for Glioma Growth . In Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: BIOMED, (SIMULTECH 2013) ISBN 978-989-8565-69-3, pages 689-695. DOI: 10.5220/0004632406890695


in Bibtex Style

@conference{biomed13,
author={J. R. Branco and J. A. Ferreira and P. de Oliveira},
title={A Viscoelastic Model for Glioma Growth},
booktitle={Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: BIOMED, (SIMULTECH 2013)},
year={2013},
pages={689-695},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0004632406890695},
isbn={978-989-8565-69-3},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: BIOMED, (SIMULTECH 2013)
TI - A Viscoelastic Model for Glioma Growth
SN - 978-989-8565-69-3
AU - R. Branco J.
AU - Ferreira J.
AU - de Oliveira P.
PY - 2013
SP - 689
EP - 695
DO - 10.5220/0004632406890695