Traveling wave solutions for a cancer stem cell invasion model

Caleb Mayer; Eric Stachura

doi:10.3934/dcdsb.2020333

Article Contents

2021, Volume 26, Issue 9: 5067-5093. Doi: 10.3934/dcdsb.2020333

This issue Previous Article Stochastic modelling and analysis of harvesting model: Application to "summer fishing moratorium" by intermittent control Next Article Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity

Traveling wave solutions for a cancer stem cell invasion model

Caleb Mayer^1, and
Eric Stachura^2, ,

1.
University of Michigan, 540 Thompson Street, Ann Arbor, MI 48104, USA
2.
Department of Mathematics, Kennesaw State University, 850 Polytechnic Lane, Marietta, GA 30060, USA

^*Corresponding author: E. Stachura
^*Corresponding author: E. Stachura

Received: December 2019

Revised: September 2020

Early access: November 2020

Published: September 2021

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Abstract

In this paper we analyze the dynamics of a cancer invasion model that incorporates the cancer stem cell hypothesis. In particular, we develop a model that includes a cancer stem cell subpopulation of tumor cells. Traveling wave analysis and Geometric Singular Perturbation Theory are used in order to determine existence and persistence of solutions for the model.

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Mathematics Subject Classification: Primary: 34C37, 34E17; Secondary: 35C07.

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Figure 1. Profile of traveling wave solution at $ t = 50 $. The qualitative behavior is similar to the results presented in [7] and [22] for only one invasive cell population, the latter reference taking $ \epsilon = 0 $ in their analysis

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Figure 2. Solution profile of (49)-(51) with wavespeed $ c = 1.5 $, $ u^* = 1 $, and $ w^* = 0.6 $

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References

[1]	H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher–KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844. doi: 10.1088/0951-7715/22/12/002.
[2]	R. Courant and D. Hilbert, Methods of Mathematical Physics II, Interscience, 1962.
[3]	P. Domschke, A. Gerisch, M. A. J. Chaplain and D. Trucu, Structured models of cell migration incorporating molecular binding processes, Journal of Mathematical Biology, 75 (2017), 1517-1561. doi: 10.1007/s00285-017-1120-y.
[4]	A. Fasano, A. Mancini and M. Primicerio, Tumours with cancer stem cells: A PDE model, Mathematical Biosciences, 272 (2016), 76-80. doi: 10.1016/j.mbs.2015.12.003.
[5]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.
[6]	A. Ghazaryan, P. Gordon and C. K. R. T. Jones, Traveling waves in porous media combustion: Uniqueness of waves for small thermal diffusivity, Journal of Dynamics and Differential Equations, 19 (2007), 951-966. doi: 10.1007/s10884-007-9079-9.
[7]	K. Harley, P. van Heijster, R. Marangell, G. J. Pettet and M. Wechselberger, Existence of traveling wave solutions for a model of tumor invasion, SIAM Journal on Applied Dynamical Systems, 13 (2014), 366-396. doi: 10.1137/130923129.
[8]	K. Harley, P. van Heijster and G. J. Pettet, A geometric construction of travelling wave solutions to the Keller–Segel model, ANZIAM Journal, 55 (2013), 399-415.
[9]	G. Hek, Geometric singular perturbation theory in biological practice, Journal of Mathematical Biology, 60 (2010), 347-386. doi: 10.1007/s00285-009-0266-7.
[10]	F. Islam, B. Qiao, R. A. Smith, V. Gopalan and A. K.-Y. Lam, Cancer stem cell: Fundamental experimental pathological concepts and updates, Experimental and Molecular Pathology, 98 (2015), 184-191. doi: 10.1016/j.yexmp.2015.02.002.
[11]	H. Jardón-Kojakhmetov, J. M. A. Scherpen and D. del Puerto-Flores, Stabilization of slow-fast systems at fold points, arXiv: 1704.05654.
[12]	C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical systems, Springer, 1995, 44–118. doi: 10.1007/BFb0095239.
[13]	E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.
[14]	D. A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type, SIAM Journal on Applied Mathematics, 34 (1978), 93-103. doi: 10.1137/0134008.
[15]	J. D. Lathia and H. Liu, Overview of cancer stem cells and stemness for community oncologists, Targeted Oncology, 12 (2017), 387-399. doi: 10.1007/s11523-017-0508-3.
[16]	M.-R. Li, Y.-J. Lin and T.-H. Shieh, The space-jump model of the movement of tumor cells and healthy cells, in Abstract and Applied Analysis, 2014 (2014), 840891, 7pp. doi: 10.1155/2014/840891.
[17]	L. Liotta, Tumor invasion and metastases–role of the extracellular matrix: Rhodes memorial award lecture, Cancer Research, 46 (1986), 1-7.
[18]	H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Communications on Pure and Applied Mathematics, 28 (1975), 323-331. doi: 10.1002/cpa.3160280302.
[19]	R. Natalini, M. Ribot and M. Twarogowska, A numerical comparison between degenerate parabolic and quasilinear hyperbolic models of cell movements under chemotaxis, Journal of Scientific Computing, 63 (2015), 654-677. doi: 10.1007/s10915-014-9909-y.
[20]	J. D. O'Flaherty, M. Barr, D. Fennell, D. Richard, J. Reynolds, J. O'Leary and K. O'Byrne, The cancer stem-cell hypothesis: Its emerging role in lung cancer biology and its relevance for future therapy, Journal of Thoracic Oncology, 7 (2012), 1880-1890. doi: 10.1097/JTO.0b013e31826bfbc6.
[21]	K. J. Painter, Modelling cell migration strategies in the extracellular matrix, Journal of Mathematical Biology, 58 (2009), 511-543. doi: 10.1007/s00285-008-0217-8.
[22]	A. J. Perumpanani, B. P. Marchant and J. Norbury, Traveling shock waves arising in a model of malignant invasion, SIAM Journal on Applied Mathematics, 60 (2000), 463-476. doi: 10.1137/S0036139998328034.
[23]	A. J. Perumpanani, J. A. Sherratt, J. Norbury and H. M. Byrne, A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion, Physica D: Nonlinear Phenomena, 126 (1999), 145-159. doi: 10.1016/S0167-2789(98)00272-3.
[24]	R. J. Petrie, A. D. Doyle and K. M. Yamada, Random versus directionally persistent cell migration, Nature Reviews Molecular Cell Biology, 10 (2009), 538-549. doi: 10.1038/nrm2729.
[25]	T. Reya, S. J. Morrison, M. F. Clarke and I. L. Weissman, Stem cells, cancer, and cancer stem cells, Nature, 414 (2001), 105-111. doi: 10.1038/35102167.
[26]	N. Sfakianakis, N. Kolbe, N. Hellmann and M. Lukávčová-Medvid'ová, A multiscale approach to the migration of cancer stem cells: Mathematical modelling and simulations, Bulletin of Mathematical Biology, 79 (2017), 209-235. doi: 10.1007/s11538-016-0233-6.
[27]	W. G. Stetler-Stevenson, S. Aznavoorian and L. A. Liotta, Tumor cell interactions with the extracellular matrix during invasion and metastasis, Annual Review of Cell Biology, 9 (1993), 541-573. doi: 10.1146/annurev.cb.09.110193.002545.
[28]	T. Stiehl, N. Baran, A. D. Ho and A. Marciniak-Czochra, Clonal selection and therapy resistance in acute leukaemias: Mathematical modelling explains different proliferation patterns at diagnosis and relapse, Journal of The Royal Society Interface, 11 (2014), 20140079. doi: 10.1098/rsif.2014.0079.
[29]	T. Stiehl and A. Marciniak-Czochra, Stem cell self-renewal in regeneration and cancer: Insights from mathematical modeling, Current Opinion in Systems Biology, 5 (2017), 112-120. doi: 10.1016/j.coisb.2017.09.006.
[30]	P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, Journal of Differential Equations, 92 (1991), 252-281. doi: 10.1016/0022-0396(91)90049-F.
[31]	P. Szmolyan and M. Wechselberger, Canards in $\mathbb R^3$, J. Differential Equations, 177 (2001), 419-453. doi: 10.1006/jdeq.2001.4001.
[32]	A. Toma, A. Mang, T. A. Schuetz, S. Becker and T. M. Buzug, A novel method for simulating the extracellular matrix in models of tumour growth, Computational and Mathematical Methods in Medicine, (2012), Art. ID 109019, 11 pp. doi: 10.1155/2012/109019.
[33]	V. G. Vaidya and F. J. Alexandro Jr., Evaluation of some mathematical models for tumor growth, International Journal of Bio-Medical Computing, 13 (1982), 19-35. doi: 10.1016/0020-7101(82)90048-4.
[34]	L. Wang, C. Mu, X. Hu and Y. Tian, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Mathematical Methods in the Applied Sciences, 40 (2017), 3000-3016. doi: 10.1002/mma.4216.
[35]	Z.-A. Wang, Mathematics of traveling waves in chemotaxis–review paper, Discrete and Continuous Dynamical Systems, Series B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601.
[36]	M. Wechselberger, Àpropos de canards (Apropos canards), Transactions of the American Mathematical Society, 364 (2012), 3289-3309. doi: 10.1090/S0002-9947-2012-05575-9.
[37]	M. Wechselberger and G. J. Pettet, Folds, canards and shocks in advection–reaction–diffusion models, Nonlinearity, 23 (2010), 1949-1969. doi: 10.1088/0951-7715/23/8/008.
[38]	A. Zagaris, H. G. Kaper and T. J. Kaper, Fast and slow dynamics for the computational singular perturbation method, Multiscale Modeling & Simulation, 2 (2004), 613-638. doi: 10.1137/040603577.
[39]	A. Zhigun, C. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Zeitschrift Für Angewandte Mathematik und Physik, 67 (2016), Art. 146, 29 pp. doi: 10.1007/s00033-016-0741-0.