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September  2021, 26(9): 5067-5093. doi: 10.3934/dcdsb.2020333

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

Received  December 2019 Revised  September 2020 Published  September 2021 Early access  November 2020

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.

Keywords: Traveling wave solutions, geometric singular perturbation theory, canards, cancer stem cells, extracellular matrix.
Mathematics Subject Classification: Primary: 34C37, 34E17; Secondary: 35C07.
Citation: Caleb Mayer, Eric Stachura. Traveling wave solutions for a cancer stem cell invasion model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5067-5093. doi: 10.3934/dcdsb.2020333
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.

show all references

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.

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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