Skip to main content
SpringerLink
Log in

Traveling Wave Solutions in a Model for Tumor Invasion with the Acid-Mediation Hypothesis

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this manuscript, we prove the existence of slow and fast traveling wave solutions in the original Gatenby–Gawlinski model. We prove the existence of a slow traveling wave solution with an interstitial gap. This interstitial gap has previously been observed experimentally, and here we derive its origin from a mathematical perspective. We give a geometric interpretation of the formal asymptotic analysis of the interstitial gap and show that it is determined by the distance between a layer transition of the tumor and a dynamical transcritical bifurcation of two components of the critical manifold. This distance depends, in a nonlinear fashion, on the destructive influence of the acid and the rate at which the acid is being pumped.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. See the discussion in Sect. 5 regarding using the techniques of this manuscript to analyze TW solutions found in (3) and (4).

  2. The cases \(\alpha =1\) and \(\alpha =2\) are the border values for which the characteristics of the slow TW solution change, see Fig. 3. Therefore, they are excluded from Theorem 1.2 as, for instance, for \(\alpha =2\) the layer transition now occurs at the same time as the transcritical bifurcation. This loss of normal hyperbolicity of the critical manifold at the layer transition complicates the proof of the theorem and is hence omitted, see Sect. 4 for more details. That being said, we fully anticipate that the result also holds for \(\alpha =1\) and \(\alpha =2\). That is, for \(\alpha =1\) we expect that \(U=0\) only in the limit \(x \rightarrow -\infty \), while for \(\alpha =2\) the normal cell density is expected to start to grow at the tumor front.

  3. Note that the slow and fast in slow-fast system is not related to the slow and fast in slow TW solution and fast TW solution. This terminology is standard in the GSPT literature and we decided not to change it.

  4. Recall that the slow in slow formulation is not related to the slow in slow TW solution, that is, (12) is the slow formulation of the ODEs associated to both the slow TW solutions with \(p=1/2\) and the fast TW solutions with \(p=0\).

  5. We rearranged the order of the equations in (15) to emphasize the slow-fast structure of the problem.

  6. \(\varepsilon ^{1/4} \ll \varepsilon ^{-(3/8-1/2)}\).

  7. This does not come as a surprise since the V-component of the original PDE (1), in the fast variable y and for \(U=1-\frac{1}{2} \alpha \), is the Fisher–KPP equation \(V_\tau = \beta V (1 - V) + \frac{\alpha }{2} V_{yy}.\)

  8. The expression for \(c_{min}\) also arose from the formal analysis of [4].

  9. The first “2” originates from the number of eigenvalues (22) on \(S_{\mathrm{S}}^2\) with negative real part (i.e the number of fast stable eigenvalues), while the second “2” comes from the number of slow variables.

  10. Here, slow-fast refers to the difference in asymptotic scaling of the (nonlinear) diffusion coefficient of (1).

References

  1. Davis, P.N., van Heijster, P., Marangell, R.: Absolute instabilities of travelling wave solutions in a Keller–Segel model. Nonlinearity 30, 4029–4061 (2017)

    Article  MathSciNet  Google Scholar 

  2. Davis, P.N., van Heijster, P., Marangell, R.: Spectral stability of travelling wave solutions in a Keller-Segel model. Appl. Numer. Math. 141, 54–61 (2019)

    Article  MathSciNet  Google Scholar 

  3. Doelman, A., Gardner, R.A., Kaper, T.J.: Large stable pulse solutions in reaction–diffusion equations. Indiana Univ. Math. J. 50, 443–507 (2001)

    Article  MathSciNet  Google Scholar 

  4. Fasano, A., Herrero, M.A., Rodrigo, M.R.: Slow and fast invasion waves in a model of acid-mediated tumour growth. Math. Biosci. 220, 45–56 (2009)

    Article  MathSciNet  Google Scholar 

  5. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98 (1979)

    Article  MathSciNet  Google Scholar 

  6. Gatenby, R.A., Gawlinski, E.T.: A reaction–diffusion model for cancer invasion. Cancer Res. 56, 5745–5753 (1996)

    Google Scholar 

  7. Gatenby, R.A., Gillies, R.J.: Why do cancers have high aerobic glycolysis? Nat. Rev. Cancer 4, 891–899 (2004)

    Article  Google Scholar 

  8. Hanahan, D., Weinberg, R.A.: Hallmarks of cancer: the next generation. Cell 144, 646–674 (2011)

    Article  Google Scholar 

  9. Harley, K., van Heijster, P., Marangell, R., Pettet, G.J., Wechselberger, M.: Existence of traveling wave solutions for a model of tumor invasion. SIAM J. Appl. Dyn. Syst. 13, 366–396 (2014)

    Article  MathSciNet  Google Scholar 

  10. Harley, K., van Heijster, P., Marangell, R., Pettet, G.J., Wechselberger, M.: Novel solutions for a model of wound healing angiogenesis. Nonlinearity 27, 2975–3003 (2014)

    Article  MathSciNet  Google Scholar 

  11. Harley, K., van Heijster, P., Marangell, R., Pettet, G.J., Wechselberger, M.: Numerical computation of an Evans function for travelling waves. Math. Biosci. 266, 36–51 (2015)

    Article  MathSciNet  Google Scholar 

  12. Harley, K., van Heijster, P., Pettet, G.J.: A geometric construction of traveling wave solutions to the Keller–Segel model. ANZIAM J. 55(EMAC2013), C399–C415 (2014)

    Article  Google Scholar 

  13. Hek, G.: Geometric singular perturbation theory in biological practice. J. Math. Biol. 60, 347–386 (2010)

    Article  MathSciNet  Google Scholar 

  14. Holder, A.B., Rodrigo, M.R., Herrero, M.A.: A model for acid-mediated tumour growth with a nonlinear acid production term. Appl. Math. Comput. 227, 176–198 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Jones, C.K.R.T.: Geometric singular perturbation theory. In: Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Math. 1609, 44–118 (1995)

  16. Kaper, T.J.: An introduction to geometric methods and dynamical systems theory for singular perturbation problems. Analyzing Multiscale Phenomena Using Singular Perturbation Methods. Proc. Sympos. Appl. Math., 56, 85–131 (1999)

  17. Kolmogorov, A., Petrovsky, I., Piscounov, N.: Étude de l’equation de la diffusion avec croissance de lat quantité de matière et son application à un problèm biologique. Moscow Univ. Math. Bull. 1, 1–25 (1937)

    Google Scholar 

  18. Krupa, M., Szmolyan, P.: Extending slow manifolds near transcritical and pitchfork singularities. Nonlinearity 14, 1473–1491 (2001)

    Article  MathSciNet  Google Scholar 

  19. Kuehn, C.: Multiple Time Scale Dynamics, vol. 191. Springer, Berlin (2015)

    Book  Google Scholar 

  20. Larson, D.A.: Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type. SIAM J. Appl. Math. 34, 93–103 (1978)

    Article  MathSciNet  Google Scholar 

  21. Liu, W., Xiao, D., Yi, Y.: Relaxation oscillations in a class of predator–prey systems. J. Differ. Equ. 188, 306–331 (2003)

    Article  MathSciNet  Google Scholar 

  22. McGillen, J.B., Gaffney, E.A., Martin, N.K., Maini, P.K.: A general reaction–diffusion model of acidity in cancer invasion. J. Math. Biol. 68, 1199–1224 (2014)

    Article  MathSciNet  Google Scholar 

  23. McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Commun. Pure Appl. Math. 28, 323–331 (1975)

    Article  MathSciNet  Google Scholar 

  24. Murray, J.D.: Mathematical Biology. I, Interdisciplinary Applied Mathematics, vol. 17. Springer, New York (2002)

  25. Robinson, C.: Sustained resonance for a nonlinear system with slowly varying coefficients. SIAM J. Math. Anal. 5, 847–860 (1983)

    Article  MathSciNet  Google Scholar 

  26. Sewalt, L., Harley, K., van Heijster, P., Balasuriya, S.: Influences of Allee effects in the spreading of malignant tumours. J. Theor. Biol. 394, 77–92 (2016)

    Article  MathSciNet  Google Scholar 

  27. Szmolyan, P.: Transversal heteroclinic and homoclinic orbits in singular perturbation problems. J. Differ. Equ. 92, 252–281 (1991)

    Article  MathSciNet  Google Scholar 

  28. van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386, 29–222 (2003)

    Article  Google Scholar 

  29. Warburg, O.: The Metabolism of Tumors. Arnold Constable, London (1930)

    Google Scholar 

  30. Wechselberger, M., Pettet, G.J.: Folds, canards and shocks in advection–reaction–diffusion models. Nonlinearity 23, 1949–1969 (2010)

    Article  MathSciNet  Google Scholar 

  31. Zhang, W., Kirk, V., Sneyd, J., Wechselberger, M.: Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales. J. Math. Neurosci. 1, 1 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

PD and PvH acknowledge support under the Australian Research Council’s Discovery Early Career Researcher Award Funding Scheme DE140100741. PD acknowledges the support of the Grant No. 20-11027X financed by Czech Science Foundation (GAČR). PvH and RM acknowledge support under the Australian Research Council’s Discovery Project DP200102130. The authors would like to thank Martin Wechselberger, Hinke Osinga, and Bernd Krauskopf for their insightful and productive discussions. PvH and RM would also like to thank the University of Wollongong for their hospitality. This research was initiated during the first Joint Australia-Japan Workshop on Dynamical Systems with Applications in Life Sciences at Queensland University of Technology.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, 4000, Australia

    Paige N. Davis & Peter van Heijster

  2. Faculty of Mathematics and Physics, Mathematical Institute, Charles University, Sokolovska 83, 186 75, Prague 8, Czech Republic

    Paige N. Davis

  3. Mathematical and Statistical Methods - Biometris, Wageningen University & Research, 6708 PB, Wageningen, The Netherlands

    Peter van Heijster

  4. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

    Robert Marangell

  5. School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia

    Marianito R. Rodrigo

Authors
  1. Paige N. Davis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter van Heijster
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robert Marangell
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Marianito R. Rodrigo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paige N. Davis.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davis, P.N., van Heijster, P., Marangell, R. et al. Traveling Wave Solutions in a Model for Tumor Invasion with the Acid-Mediation Hypothesis. J Dyn Diff Equat 34, 1325–1347 (2022). https://doi.org/10.1007/s10884-021-10003-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-021-10003-7

Keywords

Search

Navigation