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Traveling wave solutions for a cancer stem cell invasion model
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 |
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.
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. |
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