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Global existence for an age and spatially structured haptotaxis model with nonlinear age-boundary conditions

Published online by Cambridge University Press:  01 April 2008

CHRISTOPH WALKER
CHRISTOPH WALKER*
Affiliation:
Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, D-30167 Hannover, Germany email: [email protected]
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Abstract

A model focusing on key components involved in tumour invasion is studied. Tumour cell migration is based on cell motility and haptotaxis, i.e., the directed migratory response of tumour cells up gradients of cell-adhesion molecules. Individual cell processes are modelled according to cell age and several tumour phenotypes are incorporated. Global existence and uniqueness of nonnegative solutions to the corresponding coupled system of nonlinear partial differential equations are shown.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 2 , April 2008 , pp. 113 - 147
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Amann, H. (1990) Dynamic theory of quasilinear parabolic equations–-II. Reaction-diffusion systems. Diff. Int. Equ. 3, 13–75.Google Scholar
[2]Amann, H. (1991) Multiplication in Sobolev and Besov spaces. In: Nonlinear Analysis: A Tribute in Honour of Giovanni Prodi. Quaderni, Scuola Normale Superiore di Pisa, pp. 27–57.Google Scholar
[3]Amann, H. (1995) Linear and quasilinear parabolic problems, Vol. I: Abstract Linear Theory. Birkhäuser, Basel, Boston, Berlin.CrossRefGoogle Scholar
[4]Amann, H. & Walker, Ch. (2005) Local and global strong solutions to continuous coagulation–fragmentation equations with diffusion. J. Diff. Equ. 218, 159–186.CrossRefGoogle Scholar
[5]Anderson, A. R. A. (2005) A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion. Math. Med. Biol. IMA J. 22, 163–186.CrossRefGoogle ScholarPubMed
[6]Anderson, A. R. A. & Chaplain, M. A. J. (1998) Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull. Math. Biol. 60, 857–900.CrossRefGoogle Scholar
[7]Anderson, A. R. A., Weaver, A. M., Cummings, P. T. & Quaranta, V. (2006) Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127, 905–915.CrossRefGoogle ScholarPubMed
[8]Ayati, B. P., Anderson, A. R. A. & Webb, G. F. (2006) Computational methods and results for structured multiscale models of tumor invasion. SIAM Multiscale Model. and Simul. 5, 1–20.CrossRefGoogle Scholar
[9]Bellomo, N. & Preziosi, L. (2000) Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Model. 32, 413–452.Google Scholar
[10]Carter, S. C. (1965) Principles of cell motility: The direction of cell movement and cancer invasion. Nature 208, 1183–1187.CrossRefGoogle ScholarPubMed
[11]Carter, S. C. (1967) Haptotaxis and the mechanism of cell motility. Nature 213, 256–260.Google Scholar
[12]Cusulin, C., Iannelli, M. & Marinoschi, G. (2005) Age-structured diffusion in multi-layer environment. Nonlin. Anal. Real World Appl. 6 (6), 207–223.Google Scholar
[13]DiBlasio, G. & Lamberti, L. (1978) An initial-boundary value problem for age-dependent population diffusion. SIAM J. Appl. Math. 35 (3), 593–615.CrossRefGoogle Scholar
[14]Dyson, J., Sánchez, E.Villella-Bressan, R.Webb, G. F. (2007) An age and spatially structured model of tumor invasion with haptotaxis. Discr. Cont. Dyn. Sys. B 8 (1), 45–60.Google Scholar
[15]Dyson, J., Sánchez, E.Villella-Bressan, R. & Webb, G. F. (2007) A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes. To appear in: Math. Model. Nat. Phenom.CrossRefGoogle Scholar
[16]Goldstein, J. A. (1985) Semigroups of Linear Operators and Applications. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York.Google Scholar
[17]Henry, D. (1981) Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840. Springer, Berlin, Heidelberg, New York.Google Scholar
[18]Kato, T. (1970) Linear evolution equations of “hyperbolic” type. J. Fac. Sc. Univ. Tokyo 25, 241–258.Google Scholar
[19]Kato, T. (1970) Linear evolution equations of “hyperbolic” type II. J. Math. Soc. Japan 25, 648–666.Google Scholar
[20]Kunisch, K., Schappacher, W. & Webb, G. F. (1985) Nonlinear age-dependent population dynamics with diffusion. Inter. J. Comput. Math. Appl. 11, 155–173.CrossRefGoogle Scholar
[21]Langlais, M. (1985) A nonlinear problem in age-dependent population diffusion. SIAM J. Math. Anal. 16 (3), 510–529.CrossRefGoogle Scholar
[22]Nickel, G. & Rhandi, A. (1995) On the essential spectral radius of semigroups generated by perturbations of Hille–Yosida operators. Tübinger Berichte zur Funktionalanalysis 4, 207–220.Google Scholar
[23]Orme, M. E. & Chaplain, M. A. J. (1996) A mathematical model of vascular tumour growth and invasion. Math. Comput. Model. 23, 43–60.CrossRefGoogle Scholar
[24]Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer Verlag, Berlin, New York, Heidelberg.Google Scholar
[25]Quaranta, V., Weaver, A. M., Cummings, P. T. & Anderson, A. R. A. (2005) Mathematical modeling of cancer: The future of prognosis and treatment. Clinica Chem. 357, 173–179.Google Scholar
[26]Rhandi, A. (1998) Positivity and stability for a population equation with diffusion on L 1. Positivity 2, 101–113.Google Scholar
[27]Rhandi, A. & Schnaubelt, R. (1999) Asymptotic behaviour of a non-autonomous population equation with diffusion in L 1. Disc. Cont. Dyn. Syst. 5, 663–683.CrossRefGoogle Scholar
[28]Triebel, H. (1995) Interpolation Theory, Function Spaces, Differential Operators. 2nd ed. Johann Ambrosius Barth, Heidelberg, Leipzig.Google Scholar
[29]Walker, Ch. (2007) Global well-posedness of a haptotaxis model with spatial and age structure. Diff. Int. Eq. 20 (9), 1053–1074.Google Scholar
[30]Walker, Ch. & Webb, G. F. (2007) Global existence of classical solutions for a haptotaxis model. SIAM J. Math. Anal. 38 (5), 1694–1713.CrossRefGoogle Scholar
[31]Webb, G. F. (1982) Diffusive age-dependent population models and an application to genetics. Math. Biosci. 61, 1–16.Google Scholar
[32]Webb, G. F. (2008) Population models structured by age, size, and spatial position. In: Magal, P. & Ruan, S. (editors). Structured Population Models in Biology and Epidemiology. To appear in: Lecture Notes in Mathematics 1936. Berlin. Springer.Google Scholar