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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  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 & Continuous Dynamical Systems - B, 2018, 23 (10) : 4397-4431. doi: 10.3934/dcdsb.2018169
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.   Google Scholar

[34]

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

[35]

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

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.   Google Scholar

[34]

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

[35]

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

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

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

[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.  Google Scholar

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]
[1]

J. Ignacio Tello. On a mathematical model of tumor growth based on cancer stem cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 263-278. doi: 10.3934/mbe.2013.10.263

[2]

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