2011, 8(2): 575-589. doi: 10.3934/mbe.2011.8.575

On some models for cancer cell migration through tissue networks

1. 

IANS, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

2. 

INAM, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany

Received  February 2010 Revised  September 2010 Published  April 2011

We propose some models allowing to account for relevant processes at the various scales of cancer cell migration through tissue, ranging from the receptor dynamics on the cell surface over degradation of tissue fibers by protease and soluble ligand production towards the behavior of the entire cell population.
   For a genuinely mesoscopic version of these models we also provide a result on the local existence and uniqueness of a solution for all biologically relevant space dimensions.
Citation: Jan Kelkel, Christina Surulescu. On some models for cancer cell migration through tissue networks. Mathematical Biosciences & Engineering, 2011, 8 (2) : 575-589. doi: 10.3934/mbe.2011.8.575
References:
[1]

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

[2]

V. H. Barocas and R. T. Tranquillo, An anisotropic biphasic theory of tissue-equivalent mechanics: The interplay among cell traction, fibrillar network deformation, fibril alignment and cell contact guidance,, ASME Journal of Biomechanical Engineering, 119 (1997), 137. doi: 10.1115/1.2796072. Google Scholar

[3]

N. Bellomo, A. Bellouquid and M. A. Herrero, From microscopic to macroscopic description of multicellular systems and biological growing tissues,, Computers and Mathematuics with Applications, 53 (2007), 647. doi: 10.1016/j.camwa.2006.02.028. Google Scholar

[4]

H. Berry, Oscillatory behavior of a simple kinetic model for proteolysis during cell invasion,, Biophysical Journal, 77 (1999), 655. doi: 10.1016/S0006-3495(99)76921-3. Google Scholar

[5]

F. Chalub, P. A. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits,, Monatsh. Math., 142 (2004), 123. doi: 10.1007/s00605-004-0234-7. Google Scholar

[6]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogenity,, Networks and Heterogeneous Media, 1 (2006), 399. Google Scholar

[7]

A. Chauviere, T. Hillen and L. Preziosi, Modeling cell movement in anisotropic and heterogeneous network tissues,, Networks and Heterogeneous Media, 2 (2007), 333. doi: 10.3934/nhm.2007.2.333. Google Scholar

[8]

R. Dickinson, A generalized transport model for biased cell migration in an anisotropic environment,, J. Math. Biol., 40 (2000), 97. doi: 10.1007/s002850050006. Google Scholar

[9]

R. Erban and H. G. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multi-scale modeling in biology,, Multiscale Modeling and Simulation, 3 (2005), 362. doi: 10.1137/040603565. Google Scholar

[10]

F. Filbet, P. Laurencot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biology, 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar

[11]

B. Firmani, L. Guerri, and L. Preziosi, Tumor immune system competition with medically induced activation disactivation,, Math. Models Meth. Appl. Sci., 9 (1999), 491. doi: 10.1142/S0218202599000269. Google Scholar

[12]

W. Greenberg, C. van der Mee and V. Protopopescu, "Boundary Value Problems in Abstract Kinetic Theory,", Birkh\, (1987). Google Scholar

[13]

T. Hillen and H. G. Othmer, The Diffusion Limit of Transport Equations Derived From Velocity-Jump Processes,, SIAM J. Applied Math., 61 (2000), 751. doi: 10.1137/S0036139999358167. Google Scholar

[14]

T. Hillen, (M5) Mesoscopic and macroscopic models for mesenchymal motion,, J. Math. Biol., 53 (2006), 585. doi: 10.1007/s00285-006-0017-y. Google Scholar

[15]

H.J. Hwang, K. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosensitive movement,, SIAM Journal Math. Analysis, 36 (2005), 1177. doi: oi:10.1137/S0036141003431888. Google Scholar

[16]

J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks,, Preprint IANS 3/2010., (). Google Scholar

[17]

J. Kelkel and C. Surulescu, On a stochastic reaction-diffusion system modeling pattern formation on seashells,, J. Math. Biol., 60 (2010), 765. doi: 10.1007/s00285-009-0284-5. Google Scholar

[18]

M. Lachowicz, From microscopic to macroscopic descriptions of complex systems,, Comp. Rend. Mecanique, 331 (2003), 733. doi: 10.1016/j.crme.2003.09.003. Google Scholar

[19]

P. K. Maini, Spatial and spatio-temporal patterns in a cell-haptotaxis model,, J. of Math. Biol., 27 (1989), 507. doi: 10.1007/BF00288431. Google Scholar

[20]

D. G. Mallet and G. J. Pettet, A mathematical model of integrin-mediated haptotactic cell migration,, Bull. of Math. Biol., 68 (2006), 231. doi: 10.1007/s11538-005-9032-1. Google Scholar

[21]

H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems., J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[22]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Applied Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar

[23]

A. J. Perumpanani, D. L. Simmonsm, A. J. H. Gearing, K. M. Miller, G. Ward, J. Norbury, M. Schneemann and J. A. Sherratt, Extracellular matrix-mediated chemotaxis can impede cell migration,, Proc. R. Soc. Lond. B, 265 (1998), 2347. doi: 10.1098/rspb.1998.0582. Google Scholar

[24]

A. Tosin and L. Preziosi, Multiphase modeling of tumor growth with matrix remodeling and fibrosis,, Math. Comput. Modelling, 52 (2009), 969. doi: 10.1016/j.mcm.2010.01.015. Google Scholar

show all references

References:
[1]

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

[2]

V. H. Barocas and R. T. Tranquillo, An anisotropic biphasic theory of tissue-equivalent mechanics: The interplay among cell traction, fibrillar network deformation, fibril alignment and cell contact guidance,, ASME Journal of Biomechanical Engineering, 119 (1997), 137. doi: 10.1115/1.2796072. Google Scholar

[3]

N. Bellomo, A. Bellouquid and M. A. Herrero, From microscopic to macroscopic description of multicellular systems and biological growing tissues,, Computers and Mathematuics with Applications, 53 (2007), 647. doi: 10.1016/j.camwa.2006.02.028. Google Scholar

[4]

H. Berry, Oscillatory behavior of a simple kinetic model for proteolysis during cell invasion,, Biophysical Journal, 77 (1999), 655. doi: 10.1016/S0006-3495(99)76921-3. Google Scholar

[5]

F. Chalub, P. A. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits,, Monatsh. Math., 142 (2004), 123. doi: 10.1007/s00605-004-0234-7. Google Scholar

[6]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogenity,, Networks and Heterogeneous Media, 1 (2006), 399. Google Scholar

[7]

A. Chauviere, T. Hillen and L. Preziosi, Modeling cell movement in anisotropic and heterogeneous network tissues,, Networks and Heterogeneous Media, 2 (2007), 333. doi: 10.3934/nhm.2007.2.333. Google Scholar

[8]

R. Dickinson, A generalized transport model for biased cell migration in an anisotropic environment,, J. Math. Biol., 40 (2000), 97. doi: 10.1007/s002850050006. Google Scholar

[9]

R. Erban and H. G. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multi-scale modeling in biology,, Multiscale Modeling and Simulation, 3 (2005), 362. doi: 10.1137/040603565. Google Scholar

[10]

F. Filbet, P. Laurencot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biology, 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar

[11]

B. Firmani, L. Guerri, and L. Preziosi, Tumor immune system competition with medically induced activation disactivation,, Math. Models Meth. Appl. Sci., 9 (1999), 491. doi: 10.1142/S0218202599000269. Google Scholar

[12]

W. Greenberg, C. van der Mee and V. Protopopescu, "Boundary Value Problems in Abstract Kinetic Theory,", Birkh\, (1987). Google Scholar

[13]

T. Hillen and H. G. Othmer, The Diffusion Limit of Transport Equations Derived From Velocity-Jump Processes,, SIAM J. Applied Math., 61 (2000), 751. doi: 10.1137/S0036139999358167. Google Scholar

[14]

T. Hillen, (M5) Mesoscopic and macroscopic models for mesenchymal motion,, J. Math. Biol., 53 (2006), 585. doi: 10.1007/s00285-006-0017-y. Google Scholar

[15]

H.J. Hwang, K. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosensitive movement,, SIAM Journal Math. Analysis, 36 (2005), 1177. doi: oi:10.1137/S0036141003431888. Google Scholar

[16]

J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks,, Preprint IANS 3/2010., (). Google Scholar

[17]

J. Kelkel and C. Surulescu, On a stochastic reaction-diffusion system modeling pattern formation on seashells,, J. Math. Biol., 60 (2010), 765. doi: 10.1007/s00285-009-0284-5. Google Scholar

[18]

M. Lachowicz, From microscopic to macroscopic descriptions of complex systems,, Comp. Rend. Mecanique, 331 (2003), 733. doi: 10.1016/j.crme.2003.09.003. Google Scholar

[19]

P. K. Maini, Spatial and spatio-temporal patterns in a cell-haptotaxis model,, J. of Math. Biol., 27 (1989), 507. doi: 10.1007/BF00288431. Google Scholar

[20]

D. G. Mallet and G. J. Pettet, A mathematical model of integrin-mediated haptotactic cell migration,, Bull. of Math. Biol., 68 (2006), 231. doi: 10.1007/s11538-005-9032-1. Google Scholar

[21]

H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems., J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[22]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Applied Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar

[23]

A. J. Perumpanani, D. L. Simmonsm, A. J. H. Gearing, K. M. Miller, G. Ward, J. Norbury, M. Schneemann and J. A. Sherratt, Extracellular matrix-mediated chemotaxis can impede cell migration,, Proc. R. Soc. Lond. B, 265 (1998), 2347. doi: 10.1098/rspb.1998.0582. Google Scholar

[24]

A. Tosin and L. Preziosi, Multiphase modeling of tumor growth with matrix remodeling and fibrosis,, Math. Comput. Modelling, 52 (2009), 969. doi: 10.1016/j.mcm.2010.01.015. Google Scholar

[1]

Jiawei Dou, Lan-sun Chen, Kaitai Li. A monotone-iterative method for finding periodic solutions of an impulsive competition system on tumor-normal cell interaction. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 555-562. doi: 10.3934/dcdsb.2004.4.555

[2]

Gülnihal Meral, Christian Stinner, Christina Surulescu. On a multiscale model involving cell contractivity and its effects on tumor invasion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 189-213. doi: 10.3934/dcdsb.2015.20.189

[3]

Yangjin Kim, Hans G. Othmer. Hybrid models of cell and tissue dynamics in tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1141-1156. doi: 10.3934/mbe.2015.12.1141

[4]

Julien Dambrine, Nicolas Meunier, Bertrand Maury, Aude Roudneff-Chupin. A congestion model for cell migration. Communications on Pure & Applied Analysis, 2012, 11 (1) : 243-260. doi: 10.3934/cpaa.2012.11.243

[5]

Jian-Guo Liu, Min Tang, Li Wang, Zhennan Zhou. Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3011-3035. doi: 10.3934/dcdsb.2018297

[6]

Avner Friedman, Yangjin Kim. Tumor cells proliferation and migration under the influence of their microenvironment. Mathematical Biosciences & Engineering, 2011, 8 (2) : 371-383. doi: 10.3934/mbe.2011.8.371

[7]

John D. Nagy. The Ecology and Evolutionary Biology of Cancer: A Review of Mathematical Models of Necrosis and Tumor Cell Diversity. Mathematical Biosciences & Engineering, 2005, 2 (2) : 381-418. doi: 10.3934/mbe.2005.2.381

[8]

Yangjin Kim, Soyeon Roh. A hybrid model for cell proliferation and migration in glioblastoma. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 969-1015. doi: 10.3934/dcdsb.2013.18.969

[9]

Zhan Chen, Yuting Zou. A multiscale model for heterogeneous tumor spheroid in vitro. Mathematical Biosciences & Engineering, 2018, 15 (2) : 361-392. doi: 10.3934/mbe.2018016

[10]

Janet Dyson, Rosanna Villella-Bressan, G. F. Webb. The evolution of a tumor cord cell population. Communications on Pure & Applied Analysis, 2004, 3 (3) : 331-352. doi: 10.3934/cpaa.2004.3.331

[11]

Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201

[12]

Reinhard Bürger. A survey of migration-selection models in population genetics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 883-959. doi: 10.3934/dcdsb.2014.19.883

[13]

Natalia L. Komarova. Spatial stochastic models of cancer: Fitness, migration, invasion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 761-775. doi: 10.3934/mbe.2013.10.761

[14]

Yuan Lou, Thomas Nagylaki, Wei-Ming Ni. An introduction to migration-selection PDE models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4349-4373. doi: 10.3934/dcds.2013.33.4349

[15]

Marco Scianna, Luigi Preziosi, Katarina Wolf. A Cellular Potts model simulating cell migration on and in matrix environments. Mathematical Biosciences & Engineering, 2013, 10 (1) : 235-261. doi: 10.3934/mbe.2013.10.235

[16]

Marco Scianna, Luca Munaron. Multiscale model of tumor-derived capillary-like network formation. Networks & Heterogeneous Media, 2011, 6 (4) : 597-624. doi: 10.3934/nhm.2011.6.597

[17]

Christian Engwer, Markus Knappitsch, Christina Surulescu. A multiscale model for glioma spread including cell-tissue interactions and proliferation. Mathematical Biosciences & Engineering, 2016, 13 (2) : 443-460. doi: 10.3934/mbe.2015011

[18]

Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631

[19]

Frederike Kissling, Christian Rohde. The computation of nonclassical shock waves with a heterogeneous multiscale method. Networks & Heterogeneous Media, 2010, 5 (3) : 661-674. doi: 10.3934/nhm.2010.5.661

[20]

Morten Brøns. An iterative method for the canard explosion in general planar systems. Conference Publications, 2013, 2013 (special) : 77-83. doi: 10.3934/proc.2013.2013.77

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]