Mathematical analysis of a two-dimensional population model of metastatic growth including angiogenesis

Journal of Evolution Equations

Volumn 11, Issue 1, 2011, Pages 187-213

  Sébastien, Benzekry ^a,b  

^a AIX MARSEILLE UNIVERSITÉ   (France)

^b Laboratoire de Toxicocinétique et Pharmacocinétique UMR MD3   (France)

Author keywords

2D structured populations; Asymptotic behavior; Malthus parameter; Semigroup; Transport equation

Indexed keywords

EID: 79959725504     PISSN: 14243199     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00028-010-0088-5     Document Type: Article

References (27)

1. M. Adimy and F. Crauste. Un modèle non-linéaire de prolifération cellulaire: extinction des cellules et invariance. C. R. Math. Acad. Sci. Paris, 336(7):559-564, 2003.
2. H. T. Banks and F. Kappel. Transformation semigroups and L1-approximation for size structured population models. Semigroup Forum, 38(2):141-155, 1989. Semigroups and differential operators (Oberwolfach, 1988).
3. D. Barbolosi, A. Benabdallah, F. Hubert, and F. Verga. Mathematical and numerical analysis for a model of growing metastatic tumors. Math. Biosci., 218(1):1-14, 2009.
4. C. Bardos. Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport. Ann. Sci. École Norm. Sup. (4), 3:185-233, 1970.
5. F. Billy, B. Ribba, O. Saut, H. Morre-Trouilhet, T. Colin, D. Bresch, J-P. Boissel, E. Grenier, and J-P. Flandrois. A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy. J. Theor. Biol., 260(4):545-562, 2009.
6. M. Cessenat. Théorèmes de trace L p pour des espaces de fonctions de la neutronique. C. R. Acad. Sci. Paris Sér. I Math., 299(16):831-834, 1984.
7. M. Cessenat. Théorèmes de trace pour des espaces de fonctions de la neutronique. C. R. Acad. Sci. Paris Sér. I Math., 300(3):89-92, 1985.
8. J-P. Demailly. Numerical analysis and differential equations. (Analyse numérique et equations différentielles.) Nouvelle éd. Grenoble: Presses Univ. de Grenoble. 309 p., 1996.
9. A. Devys, T. Goudon, and P. Laffitte. A model describing the growth and the size distribution of multiple metastatic tumors. Discret. and contin. dyn. syst. series B, 12(4), 2009.
10. A. d'Onofrio and A. Gandolfi. Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999). Math. Biosci., 191(2):159-184, 2004.
11. M. Doumic. Analysis of a populationmodel structured by the cells molecular content. Math. Model. Nat. Phenom., 2(3):121-152, 2007.
12. J. Droniou. Quelques résultats sur les espaces de sobolev. http://www-gm3.univ-mrs.fr/polys/, 2001.
13. J. M.L. Ebos, C. R. Lee, W. Crus-Munoz, G. A. Bjarnason, and J. G. Christensen. Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis. Cancer Cell, 15:232-239, 2009.
14. N. Echenim, D. Monniaux, M. Sorine, and F. Clément. Multi-scale modeling of the follicle selection process in the ovary. Math. Biosci., 198(1):57-79, 2005.
15. K-J Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
16. P. Hahnfeldt, D. Panigraphy, J. Folkman, and L. Hlatky. Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment, response and postvascular dormancy. Cancer Research, 59:4770-4775, 1999.
17. K. Iwata, K. Kawasaki, and Shigesada N. A dynamical model for the growth and size distribution of multiple metastatic tumors. J. Theor. Biol., 203:177-186, 2000.
18. A.G. McKendrick. Applications of mathematics to medical problems. Proc. Edin. Math. Soc., 44:98-130, 1926.
19. P. Michel, S. Mischler, and B. Perthame. General relative entropy inequality: an illustration on growth models. J. Math. Pures Appl. (9), 84(9):1235-1260, 2005.
20. B. Perthame. Transport equations in biology. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.
21. B. Perthame and S. K. Tumuluri. Nonlinear renewal equations. In Selected topics in cancer modeling, Model. Simul. Sci. Eng. Technol., pages 65-96. Birkhäuser Boston, Boston, MA, 2008.
22. B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier, and J. P. Boissel. A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents. J. Theor. Biol., 243(4):532-541, 2006.
23. J. Serrin and D. E. Varberg. A general chain rule for derivatives and the change of variables formula for the Lebesgue integral. Amer. Math. Monthly, 76:514-520, 1969.
24. F.R. Sharpe and F.R. Lotka. A problem in age distribution. Phil. Mag., 21:435-438, 1911.
25. L. Tartar. An introduction to Sobolev spaces and interpolation spaces, volume 3 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin, 2007.
26. S. L. Tucker and S. O. Zimmerman. A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables. SIAM J. Appl. Math., 48(3):549-591, 1988.
27. G. F. Webb. Theory of nonlinear age-dependent population dynamics, volume 89 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York, 1985.