Mathematical Biosciences and Engineering (MBE)

Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells
Pages: 79 - 93, Issue 1, February 2017

doi:10.3934/mbe.2017006      Abstract        References        Full text (919.5K)           Related Articles

Marcello Delitala - Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (email)
Tommaso Lorenzi - School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, United Kingdom (email)

1 W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.       
2 A. R. A. Anderson, M. Chaplain, E. Newman, R. Steele and E. Thompson, Mathematical modelling of tumour invasion and metastasis, J. Theor. Med., 2 (2000), 129-154.
3 N. Bellomo, A. Bellouquid and M. Delitala, Methods and tools of the mathematical kinetic theory toward modeling complex biological systems, in Transport Phenomena and Kinetic Theory, Eds. C. Cercignani and E. Gabetta, Birkhäuser (Boston), (2007), 175-193.       
4 N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game theory for active particles to modelling mutations, onset, progression and immune competition of cancer cells, Phys. Life Rev., 5 (2008), 183-206.
5 N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Modelling chemotaxis from $L^2$-closure moments in kinetic theory of active particles, Discrete Contin. Dyn. Systems B, 18 (2013), 847-863.       
6 R. Callard, A. J. George and J. Stark, Cytokines, chaos, and complexity, Immunity, 11 (1999), 507-513.
7 F. Cerreti, B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Waves for an hyperbolic Keller-Segel model and branching instabilities, Math. Models and Meth. in Appl. Sci., 21 (2011), 825-842.       
8 F. A. C. C. Chalub, P. A. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.       
9 A. Chauviere, T. Hillen and L. Preziosi, Modeling cell movement in anisotropic and heterogeneous network tissues, Netw. Heterog. Media, 2 (2007), 333-357.       
10 R. H. Chisholm, B. D. Hughes, K. A. Landman and M. Zaman, Analytic study of three-dimensional single cell migration with and without proteolytic enzymes, Cell. Mol. Bioeng, 6 (2013), 239-249.
11 R. H. Chisholm, B. D. Hughes and K. A. Landman, Building a morphogen gradient without diffusion in a growing tissue, PLoS ONE, 5 (2010), e12857.
12 R. H. Chisholm, T. Lorenzi, A. Lorz, A. K. Larsen, L. Neves de Almeida, A. Escargueil and J. Clairambault, Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, non-genetic instability and stress-induced adaptation, Cancer Res., 75 (2015), 930-939.
13 S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl., 255 (2001), 636-677.       
14 J. C. Dallon, J. A. Sherratt and P. K. Maini, Mathematical modelling of extracellular matrix dynamics using discrete cells: fiber orientation and tissue regeneration, J. Theoret. Biol., 199 (1999), 449-471.
15 M. Delitala and T. Lorenzi, A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions, J. Theoret. Biol., 297 (2012), 88-102.       
16 M. Delitala and T. Lorenzi, A mathematical model for progression and heterogeneity in colorectal cancer dynamics, Theor. Popul. Biol., 79 (2011), 130-138.
17 R. Dickinson, A generalized transport model for biased cell migration in an anisotropic environment, J. Math. Biol., 40 (2000), 97-135.       
18 R. Erban and H. G. Othmer, Taxis equations for amoeboid cells, J. Math. Biol., 54 (2007), 847-885.       
19 P. Friedl and K. Wolf, Tumour-cell invasion and migration: Diversity and escape mechanims, Nat. Rev. Cancer, 3 (2003), 362-374.
20 N. Gavert and A. Ben-Ze'ev, Epithelial-mesenchymal transition and the invasive potential of tumors, Trends Mol. Med., 14 (2008), 199-209.
21 T. Hillen and K. Painter, A user's guide to PDE Models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.       
22 T. Hillen, M5 mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585-616.       
23 D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Part II, Jahresber. Dtsch. Math.-Ver., 106 (2004), 51-69.       
24 D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Part I, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.       
25 M. Ishizaki, K. Ashida, T. Higashi, H. Nakatsukasa, T. Kaneyoshi, K. Fujiwara, K. Nouso, Y. Kobayashi, M. Uemura, S. Nakamura and T. Tsuji, The formation of capsule and septum in human hepatocellular carcinoma, Virchows Arch., 438 (2001), 574-580.
26 A. J. Kabla, Collective cell migration: Leadership, invasion and segregation, Journal of the Royal Society Interface, 77 (2012), 3268-3278.
27 R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Philadelphia, SIAM, 2007.       
28 A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bulletin of Mathematical Biology, 77 (2015), 1-22.       
29 A. Lorz, T. Lorenzi, M. E. Hochberg, J. Clairambault and B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, Math. Model. Numer. Anal., 47 (2013), 377-399.       
30 E. Mahes, E. Mones, V. Nameth and T. Vicsek, Collective motion of cells mediates segregation and pattern formation in co-cultures, PLoS ONE, 7 (2012), e31711.
31 J. Murray, Mathematical Biology II: Spatial Models and Biochemical Applications, 3rd edn. Springer, New York, 2003.       
32 G. Naldi, L. Pareschi and G. Toscani (Eds.), Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences, Birkhäuser, Basel, 2010.       
33 K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147.       
34 K. J. Painter, Modelling cell migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543.       
35 K. J. Painter and T. Hillen, Volume-filling and quorum sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2003), 501-543.       
36 B. Perthame, Transport Equations in Biology, Birkhäuser, Basel, 2007.       
37 Z. Szymanska, C. M. Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models and Meth. in Appl. Sci., 19 (2009), 257-281.       
38 C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations, Bull. Math. Biol., 73 (2011), 1695-1733.       
39 M. Yilmaz and G. Christofori, EMT, the cytoskeleton, and cancer cell invasion, Cancer Metastasis Rev., 28 (2009), 15-33.
40 F. van Zijl, S. Mall, G. Machat, C. Pirker, R. Zeillinger, A. Weinhäusel, M. Bilban, W. Berger and W. Mikulits, A human model of epithelial to mesenchymal transition to monitor drug efficacy in hepatocellular carcinoma progression, Mol. Cancer Ther., 10 (2011), 850-860.

Go to top