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December  2018, 23(10): 4397-4431. doi: 10.3934/dcdsb.2018169

Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model

1. 

Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, 70569, Stuttgart, Germany

2. 

Institute of Mathematics, Johannes Gutenberg-University, 55128, Mainz, Germany

3. 

Institute of Applied Mathematics, Heidelberg University, 69120, Heidelberg, Germany

Received  April 2017 Revised  January 2018 Published  December 2018 Early access  June 2018

Fund Project: J.G. was supported by the Baden-W¨urttemberg Stiftung via the project "Numerical Methods for Multiphase Flows with Strongly Varying Mach Numbers". N.K. was supported by the MaxPlanck Graduate School and the Impuls Fond "Single Cell" of the University of Mainz, M.L. was partially supported by the German Science Foundation (DFG) under the grant TRR 146 "Multiscale Simulation Methods for Soft Matter Systems". N.S was partially supported by the German Science Foundation (DFG) under the grant SFB 873 "Maintenance and Differentiation of Stem Cells in Development and Disease".

We consider a haptotaxis cancer invasion model that includes two families of cancer cells. Both families migrate on the extracellular matrix and proliferate. Moreover the model describes an epithelial-to-mesenchymal-like transition between the two families, as well as a degradation and a self-reconstruction process of the extracellular matrix.

We prove in two dimensional space positivity and conditional global existence and uniqueness of the classical solutions of the problem for large initial data.

Citation: Jan Giesselmann, Niklas Kolbe, Mária Lukáčová-Medvi${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over d} }}$ová, Nikolaos Sfakianakis. Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4397-4431. doi: 10.3934/dcdsb.2018169
References:
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N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differ. Eq., 33 (1979), 201-225.  doi: 10.1016/0022-0396(79)90088-3.

[2]

V. AndasariA. GerischG. LolasA. P. South and M. A. J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141-171.  doi: 10.1007/s00285-010-0369-1.

[3]

A. R. A. AndersonM. A. J. ChaplainE. L. NewmanR. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis, Comput. Math. Method., 2 (2000), 129-154.  doi: 10.1080/10273660008833042.

[4]

N. BellomoN. K. Li and P. K. Maini, On the foundations of cancer modelling: Selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci., 18 (2008), 593-646.  doi: 10.1142/S0218202508002796.

[5]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Diff. Eqns., 44 (2006), 32pp.

[6]

T. BrabletzA. JungS. SpadernaF. Hlubek and T. Kirchner, Opinion: Migrating cancer stem cells - an integrated concept of malignant tumour progression, Nat. Rev. Cancer, 5 (), 744-749. 

[7]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Mod. Meth. Appl. S., 15 (2005), 1685-1734.  doi: 10.1142/S0218202505000947.

[8]

L. CorriasB. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.  doi: 10.1007/s00032-003-0026-x.

[9]

J. Dolbeault and Chr. Schmeiser, The two-dimensional keller-segel model after blow-up, Discrete Cont. Dyn.-B., 25 (2009), 109-121.  doi: 10.3934/dcds.2009.25.109.

[10]

M. Egeblad and J. Werb, New functions for the matrix metalloproteinases in cancer progression, Nat. Rev. Cancer, 2 (2002), 161-174.  doi: 10.1038/nrc745.

[11]

J. Gross and C. Lapiere, Collagenolytic activity in amphibian tissues: A tissue culture assay, Proc Natl Acad Sci USA, 48 (1962), 1014-1022.  doi: 10.1073/pnas.48.6.1014.

[12]

P. B. GuptaC. L. Chaffer and R. A. Weinberg, Cancer stem cells: Mirage or reality?, Nat. Med., 15 (2009), 1010-1012.  doi: 10.1038/nm0909-1010.

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D. Hanahan and R. A. Weinberg, The hallmarks of cancer, Cell, 100 (2000), 52-70.  doi: 10.1093/med/9780199656103.003.0001.

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N. HellmannN. Kolbe and N. Sfakianakis, A mathematical insight in the epithelial-mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix, Bull. Braz. Math. Soc., New Series, 47 (2016), 397-412.  doi: 10.1007/s00574-016-0147-9.

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T. HillenK. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Mod. Meth. Appl. S., 23 (2013), 165-198.  doi: 10.1142/S0218202512500480.

[17]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[18]

V. JeneiM. L. Nystrom and G. J. Thomas, Measuring invasion in an organotypic model, Methods Mo. Biol., 769 (2011), 423-232. 

[19]

M. D. JohnstonP. K. MainiS Jonathan-ChapmanC. M. Edwards and W. F. Bodmer, On the proportion of cancer stem cells in a tumour, J. Theor. Biol., 266 (2010), 708-711.  doi: 10.1016/j.jtbi.2010.07.031.

[20]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[21]

C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44 (2003), 139-181.  doi: 10.1016/S0168-9274(02)00138-1.

[22]

N. KolbeJ. Kat'uchováN. SfakianakisN. Hellmann and M. Lukáčová-Medvid'ová, A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion: The urokinase model, Appl. Math. Comput., 273 (2016), 353-376.  doi: 10.1016/j.amc.2015.08.023.

[23]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.

[24]

A. N. Krylov, On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined, Otdel. mat. i estest. nauk., 4 (1931), 491-539. 

[25]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1988.

[26]

S. A. ManiW. Guo and M. J. Liao, The epithelial-mesenchymal transition generates cells with properties of stem cells, Cell, 133 (2008), 704-715.  doi: 10.1016/j.cell.2008.03.027.

[27]

A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model, Math. Mod. Meth. Appl. S., 20 (2010), 449-476.  doi: 10.1142/S0218202510004301.

[28]

F. Michor, Mathematical models of cancer stem cells, J. Clin. Oncol., 39 (2008), 3-14.  doi: 10.1200/JCO.2007.15.2421.

[29]

N. L. NystromG. J. ThomasM. StoneI. R. MarshallJ. F. Mackenzie and I. C. Hart, Development of a quantitative method to analyse tumour cell invasion in organotypic culture, J. Pathol., 205 (2005), 468-475.  doi: 10.1002/path.1716.

[30]

L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129-155.  doi: 10.1007/s10915-004-4636-4.

[31]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.

[32]

B. PerthameF. Quiros and J. L. Vazquez, The hele-shaw asymptotics for mechanical models of tumor growth, Arch. Rational Mech. Anal., 212 (2014), 93-127.  doi: 10.1007/s00205-013-0704-y.

[33]

A. J. PerumpananiJ. A. SherrattJ. Norbury and H. M. Byrne, Biological inferences from a mathematical model for malignant invasion, Invas. Metast., 16 (1996), 209-221. 

[34]

L. Preziosi, Cancer Modelling and Simulation, CRC Press, 2003. doi: 10.1201/9780203494899.

[35]

J. S. Rao, Molecular mechanisms of glioma invasiveness: The role of proteases, Nat. Rev. Cancer, 3 (2003), 489-501.  doi: 10.1038/nrc1121.

[36]

T. ReyaS. J. MorrisonM. F. Clarke and I. L. Weissman, Stem cells, cancer, and cancer stem cells, Nature, 414 (2001), 105-111. 

[37]

T. RooseS. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007), 179-208.  doi: 10.1137/S0036144504446291.

[38]

E. T. RoussosJ. S. Condeelis and A. Patsialou, Chemotaxis in cancer, Nat. Rev. Cancer, 11 (2011), 573-587.  doi: 10.1038/nrc3078.

[39]

N. SfakianakisN. KolbeN. Hellmann and M. Lukáčová-Medvid'ová, A multiscale approach to the migration of cancer stem cells: Mathematical modelling and simulations, Bull. Math. Biol., 79 (2017), 209-235.  doi: 10.1007/s11538-016-0233-6.

[40]

A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212.  doi: 10.1137/S0036139998342065.

[41]

C. StinnerC. Surulescu and A. Uatay, Global existence for a go-or-grow multiscale model for tumor invasion with therapy, Math. Mod. Meth. Appl. S., 26 (2016), 2163-2201.  doi: 10.1142/S021820251640011X.

[42]

C. StinnerC. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X.

[43]

Z. SzymanskaC. M. RodrigoM. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math Mod.Methods Appl. Sci., 19 (2009), 257-281.  doi: 10.1142/S0218202509003425.

[44]

Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69.  doi: 10.1016/j.jmaa.2008.12.039.

[45]

Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal.-Real, 12 (2011), 418-435.  doi: 10.1016/j.nonrwa.2010.06.027.

[46]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Diff. Eq., 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014.

[47]

V. VainsteinO. U. Kirnasovsky and Y. K. Zvia Agur, Strategies for cancer stem cell elimination: Insights from mathematical modelling, J. Theor. Biol., 298 (2012), 32-41.  doi: 10.1016/j.jtbi.2011.12.016.

[48]

H. A. van der Vorst,. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Comput., 13 (1992), 631–644. doi: 10.1137/0913035.

[49]

B. Van Leer, Towards the ultimate conservative difference scheme. Ⅳ. A new approach to numerical convection, J. Comput. Phys., 23 (1977), 276-299. 

[50]

C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.  doi: 10.1137/060655122.

show all references

References:
[1]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differ. Eq., 33 (1979), 201-225.  doi: 10.1016/0022-0396(79)90088-3.

[2]

V. AndasariA. GerischG. LolasA. P. South and M. A. J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141-171.  doi: 10.1007/s00285-010-0369-1.

[3]

A. R. A. AndersonM. A. J. ChaplainE. L. NewmanR. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis, Comput. Math. Method., 2 (2000), 129-154.  doi: 10.1080/10273660008833042.

[4]

N. BellomoN. K. Li and P. K. Maini, On the foundations of cancer modelling: Selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci., 18 (2008), 593-646.  doi: 10.1142/S0218202508002796.

[5]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Diff. Eqns., 44 (2006), 32pp.

[6]

T. BrabletzA. JungS. SpadernaF. Hlubek and T. Kirchner, Opinion: Migrating cancer stem cells - an integrated concept of malignant tumour progression, Nat. Rev. Cancer, 5 (), 744-749. 

[7]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Mod. Meth. Appl. S., 15 (2005), 1685-1734.  doi: 10.1142/S0218202505000947.

[8]

L. CorriasB. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.  doi: 10.1007/s00032-003-0026-x.

[9]

J. Dolbeault and Chr. Schmeiser, The two-dimensional keller-segel model after blow-up, Discrete Cont. Dyn.-B., 25 (2009), 109-121.  doi: 10.3934/dcds.2009.25.109.

[10]

M. Egeblad and J. Werb, New functions for the matrix metalloproteinases in cancer progression, Nat. Rev. Cancer, 2 (2002), 161-174.  doi: 10.1038/nrc745.

[11]

J. Gross and C. Lapiere, Collagenolytic activity in amphibian tissues: A tissue culture assay, Proc Natl Acad Sci USA, 48 (1962), 1014-1022.  doi: 10.1073/pnas.48.6.1014.

[12]

P. B. GuptaC. L. Chaffer and R. A. Weinberg, Cancer stem cells: Mirage or reality?, Nat. Med., 15 (2009), 1010-1012.  doi: 10.1038/nm0909-1010.

[13]

D. Hanahan and R. A. Weinberg, The hallmarks of cancer, Cell, 100 (2000), 52-70.  doi: 10.1093/med/9780199656103.003.0001.

[14]

N. HellmannN. Kolbe and N. Sfakianakis, A mathematical insight in the epithelial-mesenchymal-like transition in cancer cells and its effect in the invasion of the extracellular matrix, Bull. Braz. Math. Soc., New Series, 47 (2016), 397-412.  doi: 10.1007/s00574-016-0147-9.

[15]

D. Henry, Geometric Theory of Semilinear Parabolic Systems, volume 840. Springer Lecture notes in mathematics, 1981.

[16]

T. HillenK. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Mod. Meth. Appl. S., 23 (2013), 165-198.  doi: 10.1142/S0218202512500480.

[17]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[18]

V. JeneiM. L. Nystrom and G. J. Thomas, Measuring invasion in an organotypic model, Methods Mo. Biol., 769 (2011), 423-232. 

[19]

M. D. JohnstonP. K. MainiS Jonathan-ChapmanC. M. Edwards and W. F. Bodmer, On the proportion of cancer stem cells in a tumour, J. Theor. Biol., 266 (2010), 708-711.  doi: 10.1016/j.jtbi.2010.07.031.

[20]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[21]

C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44 (2003), 139-181.  doi: 10.1016/S0168-9274(02)00138-1.

[22]

N. KolbeJ. Kat'uchováN. SfakianakisN. Hellmann and M. Lukáčová-Medvid'ová, A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion: The urokinase model, Appl. Math. Comput., 273 (2016), 353-376.  doi: 10.1016/j.amc.2015.08.023.

[23]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.

[24]

A. N. Krylov, On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined, Otdel. mat. i estest. nauk., 4 (1931), 491-539. 

[25]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1988.

[26]

S. A. ManiW. Guo and M. J. Liao, The epithelial-mesenchymal transition generates cells with properties of stem cells, Cell, 133 (2008), 704-715.  doi: 10.1016/j.cell.2008.03.027.

[27]

A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model, Math. Mod. Meth. Appl. S., 20 (2010), 449-476.  doi: 10.1142/S0218202510004301.

[28]

F. Michor, Mathematical models of cancer stem cells, J. Clin. Oncol., 39 (2008), 3-14.  doi: 10.1200/JCO.2007.15.2421.

[29]

N. L. NystromG. J. ThomasM. StoneI. R. MarshallJ. F. Mackenzie and I. C. Hart, Development of a quantitative method to analyse tumour cell invasion in organotypic culture, J. Pathol., 205 (2005), 468-475.  doi: 10.1002/path.1716.

[30]

L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129-155.  doi: 10.1007/s10915-004-4636-4.

[31]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.

[32]

B. PerthameF. Quiros and J. L. Vazquez, The hele-shaw asymptotics for mechanical models of tumor growth, Arch. Rational Mech. Anal., 212 (2014), 93-127.  doi: 10.1007/s00205-013-0704-y.

[33]

A. J. PerumpananiJ. A. SherrattJ. Norbury and H. M. Byrne, Biological inferences from a mathematical model for malignant invasion, Invas. Metast., 16 (1996), 209-221. 

[34]

L. Preziosi, Cancer Modelling and Simulation, CRC Press, 2003. doi: 10.1201/9780203494899.

[35]

J. S. Rao, Molecular mechanisms of glioma invasiveness: The role of proteases, Nat. Rev. Cancer, 3 (2003), 489-501.  doi: 10.1038/nrc1121.

[36]

T. ReyaS. J. MorrisonM. F. Clarke and I. L. Weissman, Stem cells, cancer, and cancer stem cells, Nature, 414 (2001), 105-111. 

[37]

T. RooseS. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007), 179-208.  doi: 10.1137/S0036144504446291.

[38]

E. T. RoussosJ. S. Condeelis and A. Patsialou, Chemotaxis in cancer, Nat. Rev. Cancer, 11 (2011), 573-587.  doi: 10.1038/nrc3078.

[39]

N. SfakianakisN. KolbeN. Hellmann and M. Lukáčová-Medvid'ová, A multiscale approach to the migration of cancer stem cells: Mathematical modelling and simulations, Bull. Math. Biol., 79 (2017), 209-235.  doi: 10.1007/s11538-016-0233-6.

[40]

A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212.  doi: 10.1137/S0036139998342065.

[41]

C. StinnerC. Surulescu and A. Uatay, Global existence for a go-or-grow multiscale model for tumor invasion with therapy, Math. Mod. Meth. Appl. S., 26 (2016), 2163-2201.  doi: 10.1142/S021820251640011X.

[42]

C. StinnerC. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X.

[43]

Z. SzymanskaC. M. RodrigoM. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math Mod.Methods Appl. Sci., 19 (2009), 257-281.  doi: 10.1142/S0218202509003425.

[44]

Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69.  doi: 10.1016/j.jmaa.2008.12.039.

[45]

Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal.-Real, 12 (2011), 418-435.  doi: 10.1016/j.nonrwa.2010.06.027.

[46]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Diff. Eq., 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014.

[47]

V. VainsteinO. U. Kirnasovsky and Y. K. Zvia Agur, Strategies for cancer stem cell elimination: Insights from mathematical modelling, J. Theor. Biol., 298 (2012), 32-41.  doi: 10.1016/j.jtbi.2011.12.016.

[48]

H. A. van der Vorst,. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Comput., 13 (1992), 631–644. doi: 10.1137/0913035.

[49]

B. Van Leer, Towards the ultimate conservative difference scheme. Ⅳ. A new approach to numerical convection, J. Comput. Phys., 23 (1977), 276-299. 

[50]

C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.  doi: 10.1137/060655122.

Figure 1.  Graphical description on the model (1.1). The more aggressive CSCs escape the main body of the tumour and invade the ECM faster than the DCCs. At the same time, cancer secreted MMPs degrade the ECM
Figure 2.  Simulation results of the Experiment 1. Showing here the spatial distribution of the DCC and CSC densities at several time instances. The CSCs invade the ECM by forming smooth "islands" that merge and smear-out further with time. On the other hand, the evolution of the DCCs is mostly diffusion dominated
Figure 3.  Numerical simulation of the Experiment 2. Showing here the spatial distribution of the densities of the DCCs, CSCs components at several time instances. As opposed to Experiment 1, the particular choice of parameters leads to a highly dynamic invasion of the ECM by the CSCs. Thin waves interact and lead to complex invasion landscape
Table 1.  Butcher tableaux for the explicit (upper) and the implicit (lower) parts of the third order IMEX scheme (6.4), see also [21]
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