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Global solvability and explicit bounds for non-local adhesion models

Published online by Cambridge University Press:  23 November 2017

T. HILLEN ,
K. J. PAINTER  and
M. WINKLER
T. HILLEN
Affiliation:
Centre for Mathematical Biology, University of Alberta, Edmonton, Canada email: thillen@ualberta.ca
K. J. PAINTER
Affiliation:
Heriot-Watt University, Edinburgh, UK & Department of Mathematics, Politecnico di Torino, Torino, Italy email: K.Painter@hw.ac.uk
M. WINKLER
Affiliation:
Institut für Mathematik, Universität Paderborn, Paderborn, Germany email: michael.winkler@math.uni-paderborn.de
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Abstract

Adhesion between cells and other cells (cell–cell adhesion) or other tissue components (cell–matrix adhesion) is an intrinsically non-local phenomenon. Consequently, a number of recently developed mathematical models for cell adhesion have taken the form of non-local partial differential equations, where the non-local term arises inside a spatial derivative. The mathematical properties of such a non-local gradient term are not yet well understood. Here we use sophisticated estimation techniques to show local and global existence of classical solutions for such examples of adhesion-type models, and we provide a uniform upper bound for the solutions. Further, we discuss the significance of these results to applications in cell sorting and in cancer invasion and support the theoretical results through numerical simulations.

Keywords

adhesionnon-local PDEexistence and uniquenessuniform boundscancer invasion35Q9235K5792B05
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 4 , August 2018 , pp. 645 - 684
Copyright
Copyright © Cambridge University Press 2017 

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References

[1] Alikakos, N. D. (1979) Lp bounds of solutions of reaction-diffusion equations. Commun. Part. Diff. Eq. 4 (8), 827–868.CrossRefGoogle Scholar
[2] Andasari, V., Gerisch, A., Lolas, G., South, A. & Chaplain, M. (2011) Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation. J. Math. Biol. 63 (1), 141–172.Google Scholar
[3] Anguige, K. & Schmeiser, C. (2009) A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion. J. Math. Biol. 58 (3), 395–427.Google Scholar
[4] Armstrong, N., Painter, K. J. & Sherratt, J. (2006) A continuum approach to modelling cell-cell adhesion. J. Theor. Biol. 248 (1), 98–113.CrossRefGoogle Scholar
[5] Bellomo, N., Bellouquid, A., Tao, Y. & Winkler, M. (2015) Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Mod. Meth. Appl. Sci. 25 (9), 1663–1763.CrossRefGoogle Scholar
[6] Berx, G. & Van Roy, F. (2009) Involvement of members of the cadherin superfamily in cancer. Cold Spring Harbor Persp. Biol. 1 (6), a003129.Google Scholar
[7] Buttenschoen, A. (2017) Integro-Partial Differential Equation Models for Cell-Cell Adhesion and its Application, PhD thesis, University of Alberta.Google Scholar
[8] Buttenschoen, A., Hillen, T., Gerisch, A. & Painter, K. J. (2017) A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis. J. Math. Biol., DOI: 10.1007/s00285-017-1144-3.Google Scholar
[9] Chaplain, M. A. J., Lachowicz, M., Szymańska, Z. & Wrzosek, D. (2011) Mathematical modelling of cancer invasion: The importance of cell–cell adhesion and cell–matrix adhesion. Math. Mod. Meth. Appl. Sci. 21 (4), 719–743.Google Scholar
[10] Domschke, P., Trucu, D., Gerisch, A. & Chaplain, M. (2014) Mathematical modelling of cancer invasion: Implications of cell adhesion variability for tumour infiltrative growth patterns. J. Theor. Biol. 361, 41–60.Google Scholar
[11] Eftimie, R., de Vries, G. & Lewis, M. (2007) Complex spatial group patterns result from different animal communication mechanisms. Proc. National Acad. Sci. 104 (17), 6974–6979.CrossRefGoogle ScholarPubMed
[12] Eftimie, R., de Vries, G. & Lewis, M. (2009) Weakly nonlinear analysis of a hyperbolic model for animal group formation. J. Math. Biol. 59 (1), 37–74.CrossRefGoogle ScholarPubMed
[13] Eftimie, R., de Vries, G., Lewis, M. A. & Lutscher, F. (2007) Modeling group formation and activity patterns in self-organizing collectives of individuals. Bull. Math. Biol. 69 (5), 1537–1565.Google Scholar
[14] Engwer, C., Stinner, C. & Surulescu, C. (2017) On a structured multiscale model for acid-mediated tumor invasion: The effects of adhesion and proliferation, Math. Models Methods Appl. Sci. 27 (7) 1355–1390.Google Scholar
[15] Gerisch, A. (2010) On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion. IMA J. Numer. Anal. 30 (1), 173–194.CrossRefGoogle Scholar
[16] Gerisch, A. & Chaplain, M. A. J. (2008) Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion. J. Theor. Biol. 250 (4), 684–704.CrossRefGoogle ScholarPubMed
[17] Gerisch, A. & Painter, K. (2010) Mathematical modelling of cell adhesion and its applications to developmental biology and cancer invasion. In: Chauviere, A., Preziosi, L., Verdier, C., (editors), Cell Mechanics: From Single Scale-Based Models to Multiscale Modeling, Chapman and Hall/CRC Press, Boca Raton, pp. 319–350.CrossRefGoogle Scholar
[18] Hillen, T. (2007) A classification of spikes and plateaus. SIAM Rev. 49 (1), 35–51.Google Scholar
[19] Hillen, T. & Painter, K. (2009) A user's guide to PDE models for chemotaxis. J. Math. Biol. 58 (1–2), 183–217.Google Scholar
[20] Hillen, T., Painter, K. & Schmeiser, C. (2007) Global existence for chemotaxis with finite sampling radius. Discr. Cont. Dyn. Syst. B 7 (1), 125–144.Google Scholar
[21] Horstmann, D. (2003) From 1970 until present: The Keller–Segel model in chemotaxis and its consequences I. Jahresberichte der DMV 105, 103–165.Google Scholar
[22] Horstmann, D., Painter, K. J. & Othmer, H. G. (2004) Aggregation under local reinforcement: From lattice to continuum. Eur. J. Appl. Math. 15 (5), 545–576.CrossRefGoogle Scholar
[23] Johnston, S. T., Simpson, M. J. & Baker, R. E. (2012) Mean-field descriptions of collective migration with strong adhesion. Phys. Rev. E 85 (5), 051922.CrossRefGoogle ScholarPubMed
[24] Johnston, S. T., Simpson, M. J. & Plank, M. J. (2013) Lattice-free descriptions of collective motion with crowding and adhesion. Phys. Rev. E 88 (6), 062720.CrossRefGoogle ScholarPubMed
[25] Kim, Y., Lawler, S., Nowicki, M. O., Chiocca, E. A. & Friedman, A. (2009) A mathematical model for pattern formation of glioma cells outside the tumor spheroid core. J. Theor. Biol. 260 (3), 359–371.CrossRefGoogle ScholarPubMed
[26] Ladyžhenskaja, O., Solonnikov, V. & Ural'ceva, N. (1968) Linear and Quasilinear Equations of Parabolic Type, AMS Providence, Rhode Island.CrossRefGoogle Scholar
[27] Lee, C. T., Hoopes, M. F., Diehl, J., Gilliland, W., Huxel, G., Leaver, E. V., McCann, K., Umbanhowar, J. & Mogilner, A. (2001) Non-local concepts and models in biology. J. Theor. Biol. 210 (2), 201–219.Google Scholar
[28] Mogilner, A. & Edelstein-Keshet, L. (1999) A non-local model for a swarm. J. Math. Biol. 38 (6), 534–570.Google Scholar
[29] Murakawa, H., & Togashi, H. (2015) Continuous models for cell-cell adhesion. J. Theor. Biol. 374, 1–12.CrossRefGoogle ScholarPubMed
[30] Othmer, H. & Hillen, T. (2002) The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62 (4), 1122–1250.Google Scholar
[31] Painter, K., Armstrong, N. & Sherratt, J. (2010) The impact of adhesion on cellular invasion processes in cancer development. J. Theor. Biol. 264 (3), 1057–1067.Google Scholar
[32] Painter, K. J., Bloomfield, J. M., Sherratt, J. A. & Gerisch, A. (2015) A nonlocal model for contact attraction and repulsion in heterogeneous cell populations. Bull. Math. Biol. 77 (6), 1132–1165.Google Scholar
[33] Painter, K. J., Horstmann, D. & Othmer, H. G. (2003) Localization in lattice and continuum models of reinforced random walks. Appl. Math. Lett. 16 (3), 375–381.CrossRefGoogle Scholar
[34] Porzio, M. & Vespri, V. (1993) Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Diff. Equ. 103 (1), 146–178.CrossRefGoogle Scholar
[35] Quittner, P. & Souplet, P. (2007) Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhäuser, Basel/Boston/Berlin.Google Scholar
[36] Sekimura, T., Zhu, M., Cook, J., Maini, P. K. & Murray, J. D. (1999) Pattern formation of scale cells in lepidoptera by differential origin-dependent cell adhesion. Bull. Math. Biol. 61 (5), 807–828.Google Scholar
[37] Sherratt, J., Gourley, S. A., Armstrong, N. & Painter, J. (2009) Boundedness of solutions of a non-local reaction-diffusion model for adhesion in cell aggregation and cancer invasion. Eur. J. Appl. Math. 20 (1), 123–144.Google Scholar
[38] Steinberg, M. (2007) Differential adhesion in morphogenesis: A modern review. Curr. Opin. Genet. Devel. 17 (4), 281–286.CrossRefGoogle Scholar
[39] Topaz, C. M., Bertozzi, A. L. & Lewis, M. A. (2006) A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68 (7), 1601–1623.Google Scholar
[40] Xiang, T. (2013) A study on the positive non constant steady states of nonlocal chemotaxis systems. Disc. Cont. Dyn. Sys. B 18 (9), 2457–2485.Google Scholar