Abstract

We investigate a recently proposed cross-diffusion system modeling the growth of gliobastoma taking into account size exclusion in both the migration and the proliferation process. In addition to degenerate nonlinear cross-diffusion, the model includes reaction terms as in the Fisher--Kolmogorov equation and linear ones modeling transition between states of proliferation and migration. We discuss the mathematical structure of the system and provide a complete existence analysis in spatial dimension one. The proof is based on exploiting partial entropy dissipation techniques and fully implicit time discretizations. In order to prove the existence of the latter, appropriate variational and fixed-point techniques are used, together with suitable a priori estimates. Moreover, we review the existence of traveling waves and their relation to potential growth or decay of glioblastoma. Finally, we provide extensive numerical studies in one and two spatial dimensions, including the effect of anisotropic diffusions as found in neural tissues.

Keywords

  1. partial differential equation
  2. glioblastoma
  3. mathematical biology
  4. analysis
  5. numerical simulations

MSC codes

  1. 35K45
  2. 35K55
  3. 35K65
  4. 92Bxx

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
J. Belmonte-Beitia, G. Calvo, and V. Pérez-García, Effective particle methods for Fisher--Kolmogorov equations: Theory and applications to brain tumor dynamics, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), pp. 3267--3283.
2.
J. Berendsen, M. Burger, and J.-F. Pietschmann, On a cross-diffusion model for multiple species with nonlocal interaction and size exclusion, Nonlinear Anal., 159 (2017), pp. 10--39, https://doi.org/10.1016/j.na.2017.03.010,
3.
A. Braides, Gamma-Convergence for Beginners, Vol. 22, Clarendon Press, Oxford, 2002.
4.
M. Bramson, Convergence of Solutions of the Kolmogorov Equation to Travelling Waves, Vol. 285, American Mathematical Society, Providence, RI, 1983.
5.
M. Bruna, M. Burger, H. Ranetbauer, and M.-T. Wolfram, Cross-diffusion systems with excluded-volume effects and asymptotic gradient flow structures, J. Nonlinear Sci., 27 (2017), pp. 687--719.
6.
M. Burger, M. D. Francesco, J.-F. Pietschmann, and B. Schlake, Nonlinear cross-diffusion with size exclusion, SIAM J. Math. Anal., 42 (2010), pp. 2842--2871, https://doi.org/10.1137/100783674.
7.
M. Burger, S. Hittmeir, H. Ranetbauer, and M. Wolfram, Lane formation by side-stepping, SIAM J. Math. Anal., 48 (2016), pp. 981--1005.
8.
A. Champneys, S. Harris, J. Toland, J. Warren, and D. Williams, Algebra, analysis and probability for a coupled system of reaction-diffusion equations, Philos. Trans. Roy. Soc. A, 350 (1995), pp. 69--112.
9.
X. Chen, A. Jüngel, and J. Liu, A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., (2013).
10.
B. Contri, Fisher-KPP equations and applications to a model in medical sciences, Netw. Heterog. Media, 13 (2018), p. 119, https://doi.org/10.3934/nhm.2018006.
11.
E. Dahlman and Y. Watanabe, Su-f-t-109: A shortcoming of the Fisher-Kolmogorov reaction-diffusion equation for modeling tumor growth, Med. Phys., 43 (2016), pp. 3486--3487, https://doi.org/10.1118/1.4956245.
12.
E. Emmrich, Discrete versions of Gronwall's lemma and their application to the numerical analysis of parabolic problems, Preprint No. 637, Fachbereich Mathematik, TU, Berlin, 1999.
13.
C. Engwer, T. Hillen, M. Knappitsch, and C. Surulescu, Glioma follow white matter tracts: A multiscale DTI-based model, J. Math. Biol., 71 (2015), pp. 551--582.
14.
L. C. Evans, Partial Differential Equations, 2nd ed., Graduate Stud. Math. 19, American Mathematical Society, Providence, RI, 2010, https://doi.org/10.1090/gsm/019.
15.
A. Farin, S. O. Suzuki, M. Weiker, J. E. Goldman, J. N. Bruce, and P. Canoll, Transplanted glioma cells migrate and proliferate on host brain vasculature: A dynamic analysis, Glia, 53 (2006), pp. 799--808, https://doi.org/10.1002/glia.20334.
16.
R. Fisher, The wave of advance of advantageous genes, Ann. Hum. Genet., 7 (1937), pp. 355--369.
17.
M. Freidlin, Wave front propagation for KPP-type equations, Surv. Appl. Math., 2 (1995), pp. 1--62.
18.
P. Gerlee and S. Nelander, The impact of phenotypic switching on glioblastoma growth and invasion, PLoS Comput. Biol., 8 (2012), pp. 1--12.
19.
P. Gerlee and S. Nelander, Travelling wave analysis of a mathematical model of glioblastoma growth, Math. Biosci., 276 (2016), pp. 75--81.
20.
A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), p. 1963, http://stacks.iop.org/0951-7715/28/i=6/a=1963.
21.
A. N. Kolmogorov, I. Petrovsky, and N. Piskunov, Etude de l’équation de la diffusion avec croissance de la quantité de matiere et son applicationa un probleme biologique, Moscow Univ. Math. Bull., 1 (1937), p. 129.
22.
E. Konukoglu, O. Clatz, P.-Y. Bondiau, H. Delingette, and N. Ayache, Extrapolating glioma invasion margin in brain magnetic resonance images: Suggesting new irradiation margins, Med. Image Anal., 14 (2010), pp. 111--125.
23.
E. Konukoglu, O. Clatz, P.-Y. Bondiau, M. Sermesant, M. Delingette, and N. Ayache, Towards an identification of tumor growth parameters from time series of images, in Medical Image Computing and Computer-Assisted Intervention--MICCAI 2007, (2007), pp. 549--556.
24.
A. E. Kyprianou, Travelling wave solutions to the KPP equation: Alternatives to Simon Harris' probabilistic analysis, in Annales de l'Institut Henri Poincare (B) Probability and Statistics, Vol. 40, Elsevier, Amsterdam, 2004, pp. 53--72.
25.
M. A. Lewis and G. Schmitz, Biological invasion of an organism with separate mobile and stationary states: Modeling and analysis, Forma, 11 (1996), pp. 1--25.
26.
J. L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires/[par] J. L. Lions, Dunod Paris, 1969.
27.
N. L. Martirosyan, E. M. Rutter, W. L. Rameys, E. J. Kostelich, Y. Kuang, and M. C. Preul, Mathematically modeling the biological properties of gliomas: A review, Math. Biosci. Eng., 12 (2015), p. 879, https://doi.org/10.3934/mbe.2015.12.879.
28.
K. Painter and T. Hillen, Mathematical modelling of glioma growth: The use of diffusion tensor imaging (DTI) data to predict the anisotropic pathways of cancer invasion, J. Theoret. Biol., 323 (2013), pp. 25--39, https://doi.org/10.1016/j.jtbi.2013.01.014.
29.
B. Schweizer, Partielle Differentialgleichungen, Springer Spektrum, Berlin, 2013.
30.
J. A. Sherratt, On the transition from initial data to travelling waves in the Fisher-KPP equation, Dyn. Stab. Syst., 13 (1998), pp. 167--174.
31.
F. Winkler, Y. Kienast, M. Fuhrmann, L. Von Baumgarten, S. Burgold, G. Mitteregger, H. Kretzschmar, and J. Herms, Imaging glioma cell invasion in vivo reveals mechanisms of dissemination and peritumoral angiogenesis, Glia, 57 (2009), pp. 1306--1315, https://doi.org/10.1002/glia.20850.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 160 - 182
ISSN (online): 1095-712X

History

Submitted: 15 June 2018
Accepted: 24 September 2019
Published online: 14 January 2020

Keywords

  1. partial differential equation
  2. glioblastoma
  3. mathematical biology
  4. analysis
  5. numerical simulations

MSC codes

  1. 35K45
  2. 35K55
  3. 35K65
  4. 92Bxx

Authors

Affiliations

Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : 1073/1-2
H2020 European Research Council https://doi.org/10.13039/100010663 : 615216
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/K032208/1

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media