December  2013, 18(10): 2689-2704. doi: 10.3934/dcdsb.2013.18.2689

Numerical simulation of chemotaxis models on stationary surfaces

1. 

Institut für Angewandte Mathematik, TU Dortmund, 44227 Dortmund, Germany, Germany, Germany

Received  November 2012 Revised  April 2013 Published  October 2013

In this paper we present an implicit finite element method for a class of chemotaxis models, where a new linearized flux-corrected transport (FCT) algorithm is modified in such a way as to keep the density of on-surface living cells nonnegative. Level set techniques are adopted for an implicit description of the surface and for the numerical treatment of the corresponding system of partial differential equations. The presented scheme is able to deliver a robust and accurate solution for a large class of chemotaxis-driven models. The numerical behavior of the proposed scheme is tested on the blow-up model on a sphere and an ellipsoid and on the pattern-forming dynamics model of Escherichia coli on a sphere.
Citation: Andriy Sokolov, Robert Strehl, Stefan Turek. Numerical simulation of chemotaxis models on stationary surfaces. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2689-2704. doi: 10.3934/dcdsb.2013.18.2689
References:
[1]

M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system,, J. London Math. Soc., 74 (2006), 453. doi: 10.1112/S0024610706023015. Google Scholar

[2]

M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system,, Scientiae Mathematicae Japonicae, 59 (2004), 577. Google Scholar

[3]

D. Ambrosi, F. Bussolino and L. Preziosi, A review of vasculogenesis models,, J. Theor. Med., 6 (2005), 1. doi: 10.1080/1027366042000327098. Google Scholar

[4]

A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis,, Journal of Theoretical Medicine, 2 (2000), 129. doi: 10.1080/10273660008833042. Google Scholar

[5]

M. Bergdorf, I. F. Sbalzarini and P. Koumoutsakos, A Lagrangian particle method for reaction-diffusion systems on deforming surfaces,, J. Math. Biol., 61 (2010), 649. doi: 10.1007/s00285-009-0315-2. Google Scholar

[6]

M. A. J. Chaplain, The mathematical modelling of tumour angiogenesis and invasion,, ACTA Biotheoretica, 43 (1995), 387. doi: 10.1007/BF00713561. Google Scholar

[7]

M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis,, Bulletin of Mathematical Biology, 60 (1998), 857. Google Scholar

[8]

M. A. J. Chaplain, Mathematical modelling of angiogenesis,, Journal of Neuro-Oncology, 50 (2000), 37. Google Scholar

[9]

M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor,, IMA Journal of Mathematics Applied in Medicine and Biology, 10 (1993), 149. doi: 10.1093/imammb/10.3.149. Google Scholar

[10]

A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models,, Numer. Math., 111 (2008), 169. doi: 10.1007/s00211-008-0188-0. Google Scholar

[11]

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

[12]

L. Corriasa and B. Perthame, Asymptotic decay for the solutions of the parabolic Keller-Segel chemotaxis system in critical spaces,, Mathematical and Computer Modelling, 47 (2008), 755. doi: 10.1016/j.mcm.2007.06.005. Google Scholar

[13]

G. Dziuk and C. M. Elliott, Eulerian finite element method for parabolic PDEs on implicit surfaces,, Interfaces and Free Boundaries, 10 (2008), 119. doi: 10.4171/IFB/182. Google Scholar

[14]

Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models,, J. Comput. Appl. Math., 224 (2009), 168. doi: 10.1016/j.cam.2008.04.030. Google Scholar

[15]

Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model,, SIAM J. Numer. Anal., 47 (): 386. doi: 10.1137/07070423X. Google Scholar

[16]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model,, Numer. Math., 104 (2006), 457. doi: 10.1007/s00211-006-0024-3. Google Scholar

[17]

H. Gajewski, W. Jäger and A. Koshelev, About loss of regularity and 'blow up' of solutions for quasilinear parabolic systems,, R-Report, (1993). Google Scholar

[18]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77. doi: 10.1002/mana.19981950106. Google Scholar

[19]

A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamics in blood vessels formation,, Phys. Rev. Lett., 90 (2003). doi: 10.1103/PhysRevLett.90.118101. Google Scholar

[20]

D. Horstmann, Generalizing the Keller-Segel Model: Lyapunov Functionals, Steady State Analysis, and Blow-Up Results for Multi-species Chemotaxis Models in the Presence of Attraction and Repulsion Between Competitive Interacting Species,, J. Nonlinear Sci., 21 (2011), 231. doi: 10.1007/s00332-010-9082-x. Google Scholar

[21]

D. Horstmann and M. Lucia, Uniqueness and symmetry of equilibria in a chemotaxis model,, Journal für die Reine und angewandte Mathematik (Crelle's Journal), 654 (2011), 83. doi: 10.1515/CRELLE.2011.030. Google Scholar

[22]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European Journal of Applied Mathematics, 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar

[23]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, Journal of Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar

[24]

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

[25]

E. Keller and L. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[26]

D. Kuzmin, R. Löhner and S. Turek, Flux-Corrected Transport,, Springer, (2012). Google Scholar

[27]

D. Kuzmin and M. Möller, Algebraic flux correction I. Scalar conservation laws,, in Flux-Corrected Transport: Principles, (2005), 155. doi: 10.1007/3-540-27206-2_6. Google Scholar

[28]

D. Kuzmin and S. Turek, Flux correction tools for finite elements,, J. Comput. Phys., 175 (2002), 525. doi: 10.1006/jcph.2001.6955. Google Scholar

[29]

D. Kuzmin, Explicit and implicit FEM-TVD algorithms with flux linearization,, J. Comput. Phys., 228 (2009), 2517. doi: 10.1016/j.jcp.2008.12.011. Google Scholar

[30]

D. Kuzmin, Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes,, Journal of Computational and Applied Mathematics, 236 (2012), 2317. doi: 10.1016/j.cam.2011.11.019. Google Scholar

[31]

R. J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems,, J. Comput. Phys., 131 (1997), 327. Google Scholar

[32]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth,, Physica A, 230 (1996), 499. Google Scholar

[33]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains,, J. Inequal. Appl., 6 (2001), 37. doi: 10.1155/S1025583401000042. Google Scholar

[34]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcialaj Ekvacioj, 44 (2001), 441. Google Scholar

[35]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis,, IMA J. Numer. Anal., 27 (2007), 332. doi: 10.1093/imanum/drl018. Google Scholar

[36]

T. Senba and T. Suzuki, Parabolic System of Chemotaxis: Blowup in a Finite and the Infinite Time,, IMS Workshop on Reaction-Diffusion Systems (Shatin, 8 (2001), 349. Google Scholar

[37]

G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi and F. Bussolino, Modeling the early stages of vascular network assembly,, The EMBO Journal, 22 (2003), 1771. doi: 10.1093/emboj/cdg176. Google Scholar

[38]

R. Strehl, A. Sokolov, D. Kuzmin and S. Turek, A flux-corrected finite element method for chemotaxis problems,, Computational methods in applied mathematics, 10 (2010), 219. doi: 10.2478/cmam-2010-0013. Google Scholar

[39]

R. Strehl, A. Sokolov, D. Kuzmin, D. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D,, Journal of Computational and Applied Mathematics, 239 (2013), 290. doi: 10.1016/j.cam.2012.09.041. Google Scholar

[40]

R. Strehl, A. Sokolov and S. Turek, Efficient, accurate and flexible Finite Element solvers for Chemotaxis problems,, Computers and Mathematics with Applications, 64 (2012), 175. doi: 10.1016/j.camwa.2011.12.040. Google Scholar

[41]

S. Turek, Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach,, Springer, (1999). doi: 10.1007/3-540-48092-7. Google Scholar

[42]

R. Tyson, S. R. Lubkin and J. D. Murray, A minimal mechanism for bacterial pattern formation,, Proc. R. Soc. Lond. B, 266 (1999), 299. doi: 10.1098/rspb.1999.0637. Google Scholar

[43]

R. Tyson, S. R. Lubkin and J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium,, J. Math. Biol., 38 (1999), 359. doi: 10.1007/s002850050153. Google Scholar

[44]

R. Tyson, L. G. Stern and R. J. LeVeque, Fractional step methods applied to a chemotaxis model,, J. Math. Biol., 41 (2000), 455. doi: 10.1007/s002850000038. Google Scholar

[45]

C. M. Elliott, B. Stinner and C. Venkataraman, Modelling cell motility and chemotaxis with evolving surface finite elements,, J. R. Soc. Interface, 9 (2012), 3027. doi: 10.1098/rsif.2012.0276. Google Scholar

[46]

C. Landsberg, F. Stenger, A. Deutsch, M. Gelinsky, A. Rösen-Wolff and A. Voigt, Chemotaxis of mesenchymal stem cells within 3D biomimetic scaffolds-a modeling approach,, J. Biomech, 44 (2011), 359. Google Scholar

[47]

S. T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids,, J. Comput. Phys., 31 (1979), 335. doi: 10.1016/0021-9991(79)90051-2. Google Scholar

show all references

References:
[1]

M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system,, J. London Math. Soc., 74 (2006), 453. doi: 10.1112/S0024610706023015. Google Scholar

[2]

M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system,, Scientiae Mathematicae Japonicae, 59 (2004), 577. Google Scholar

[3]

D. Ambrosi, F. Bussolino and L. Preziosi, A review of vasculogenesis models,, J. Theor. Med., 6 (2005), 1. doi: 10.1080/1027366042000327098. Google Scholar

[4]

A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis,, Journal of Theoretical Medicine, 2 (2000), 129. doi: 10.1080/10273660008833042. Google Scholar

[5]

M. Bergdorf, I. F. Sbalzarini and P. Koumoutsakos, A Lagrangian particle method for reaction-diffusion systems on deforming surfaces,, J. Math. Biol., 61 (2010), 649. doi: 10.1007/s00285-009-0315-2. Google Scholar

[6]

M. A. J. Chaplain, The mathematical modelling of tumour angiogenesis and invasion,, ACTA Biotheoretica, 43 (1995), 387. doi: 10.1007/BF00713561. Google Scholar

[7]

M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis,, Bulletin of Mathematical Biology, 60 (1998), 857. Google Scholar

[8]

M. A. J. Chaplain, Mathematical modelling of angiogenesis,, Journal of Neuro-Oncology, 50 (2000), 37. Google Scholar

[9]

M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor,, IMA Journal of Mathematics Applied in Medicine and Biology, 10 (1993), 149. doi: 10.1093/imammb/10.3.149. Google Scholar

[10]

A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models,, Numer. Math., 111 (2008), 169. doi: 10.1007/s00211-008-0188-0. Google Scholar

[11]

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

[12]

L. Corriasa and B. Perthame, Asymptotic decay for the solutions of the parabolic Keller-Segel chemotaxis system in critical spaces,, Mathematical and Computer Modelling, 47 (2008), 755. doi: 10.1016/j.mcm.2007.06.005. Google Scholar

[13]

G. Dziuk and C. M. Elliott, Eulerian finite element method for parabolic PDEs on implicit surfaces,, Interfaces and Free Boundaries, 10 (2008), 119. doi: 10.4171/IFB/182. Google Scholar

[14]

Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models,, J. Comput. Appl. Math., 224 (2009), 168. doi: 10.1016/j.cam.2008.04.030. Google Scholar

[15]

Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model,, SIAM J. Numer. Anal., 47 (): 386. doi: 10.1137/07070423X. Google Scholar

[16]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model,, Numer. Math., 104 (2006), 457. doi: 10.1007/s00211-006-0024-3. Google Scholar

[17]

H. Gajewski, W. Jäger and A. Koshelev, About loss of regularity and 'blow up' of solutions for quasilinear parabolic systems,, R-Report, (1993). Google Scholar

[18]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77. doi: 10.1002/mana.19981950106. Google Scholar

[19]

A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamics in blood vessels formation,, Phys. Rev. Lett., 90 (2003). doi: 10.1103/PhysRevLett.90.118101. Google Scholar

[20]

D. Horstmann, Generalizing the Keller-Segel Model: Lyapunov Functionals, Steady State Analysis, and Blow-Up Results for Multi-species Chemotaxis Models in the Presence of Attraction and Repulsion Between Competitive Interacting Species,, J. Nonlinear Sci., 21 (2011), 231. doi: 10.1007/s00332-010-9082-x. Google Scholar

[21]

D. Horstmann and M. Lucia, Uniqueness and symmetry of equilibria in a chemotaxis model,, Journal für die Reine und angewandte Mathematik (Crelle's Journal), 654 (2011), 83. doi: 10.1515/CRELLE.2011.030. Google Scholar

[22]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European Journal of Applied Mathematics, 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar

[23]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, Journal of Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar

[24]

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

[25]

E. Keller and L. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[26]

D. Kuzmin, R. Löhner and S. Turek, Flux-Corrected Transport,, Springer, (2012). Google Scholar

[27]

D. Kuzmin and M. Möller, Algebraic flux correction I. Scalar conservation laws,, in Flux-Corrected Transport: Principles, (2005), 155. doi: 10.1007/3-540-27206-2_6. Google Scholar

[28]

D. Kuzmin and S. Turek, Flux correction tools for finite elements,, J. Comput. Phys., 175 (2002), 525. doi: 10.1006/jcph.2001.6955. Google Scholar

[29]

D. Kuzmin, Explicit and implicit FEM-TVD algorithms with flux linearization,, J. Comput. Phys., 228 (2009), 2517. doi: 10.1016/j.jcp.2008.12.011. Google Scholar

[30]

D. Kuzmin, Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes,, Journal of Computational and Applied Mathematics, 236 (2012), 2317. doi: 10.1016/j.cam.2011.11.019. Google Scholar

[31]

R. J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems,, J. Comput. Phys., 131 (1997), 327. Google Scholar

[32]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth,, Physica A, 230 (1996), 499. Google Scholar

[33]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains,, J. Inequal. Appl., 6 (2001), 37. doi: 10.1155/S1025583401000042. Google Scholar

[34]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcialaj Ekvacioj, 44 (2001), 441. Google Scholar

[35]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis,, IMA J. Numer. Anal., 27 (2007), 332. doi: 10.1093/imanum/drl018. Google Scholar

[36]

T. Senba and T. Suzuki, Parabolic System of Chemotaxis: Blowup in a Finite and the Infinite Time,, IMS Workshop on Reaction-Diffusion Systems (Shatin, 8 (2001), 349. Google Scholar

[37]

G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi and F. Bussolino, Modeling the early stages of vascular network assembly,, The EMBO Journal, 22 (2003), 1771. doi: 10.1093/emboj/cdg176. Google Scholar

[38]

R. Strehl, A. Sokolov, D. Kuzmin and S. Turek, A flux-corrected finite element method for chemotaxis problems,, Computational methods in applied mathematics, 10 (2010), 219. doi: 10.2478/cmam-2010-0013. Google Scholar

[39]

R. Strehl, A. Sokolov, D. Kuzmin, D. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D,, Journal of Computational and Applied Mathematics, 239 (2013), 290. doi: 10.1016/j.cam.2012.09.041. Google Scholar

[40]

R. Strehl, A. Sokolov and S. Turek, Efficient, accurate and flexible Finite Element solvers for Chemotaxis problems,, Computers and Mathematics with Applications, 64 (2012), 175. doi: 10.1016/j.camwa.2011.12.040. Google Scholar

[41]

S. Turek, Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach,, Springer, (1999). doi: 10.1007/3-540-48092-7. Google Scholar

[42]

R. Tyson, S. R. Lubkin and J. D. Murray, A minimal mechanism for bacterial pattern formation,, Proc. R. Soc. Lond. B, 266 (1999), 299. doi: 10.1098/rspb.1999.0637. Google Scholar

[43]

R. Tyson, S. R. Lubkin and J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium,, J. Math. Biol., 38 (1999), 359. doi: 10.1007/s002850050153. Google Scholar

[44]

R. Tyson, L. G. Stern and R. J. LeVeque, Fractional step methods applied to a chemotaxis model,, J. Math. Biol., 41 (2000), 455. doi: 10.1007/s002850000038. Google Scholar

[45]

C. M. Elliott, B. Stinner and C. Venkataraman, Modelling cell motility and chemotaxis with evolving surface finite elements,, J. R. Soc. Interface, 9 (2012), 3027. doi: 10.1098/rsif.2012.0276. Google Scholar

[46]

C. Landsberg, F. Stenger, A. Deutsch, M. Gelinsky, A. Rösen-Wolff and A. Voigt, Chemotaxis of mesenchymal stem cells within 3D biomimetic scaffolds-a modeling approach,, J. Biomech, 44 (2011), 359. Google Scholar

[47]

S. T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids,, J. Comput. Phys., 31 (1979), 335. doi: 10.1016/0021-9991(79)90051-2. Google Scholar

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