A chemotaxis model motivated by angiogenesis

Abstract We consider a simple model arising in modeling angiogenesis and more specifically the development of capillary blood vessels due to an exogenous chemo-attractive signal (solid tumors for instance). It is given as coupled system of parabolic equations through a nonlinear transport term. We show that, by opposition to some classical chemotaxis model, this system admits a positive energy. This allows us to develop an existence theory for weak solutions. We also show that, in two dimensions, this system admits a family of self-similar waves. To cite this article: L. Corrias et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

[1]  Alexander R. A. Anderson,et al.  A mathematical analysis of a model for capillary network formation in the absence of endothelial cell proliferation , 1999 .

[2]  W. Jäger,et al.  On explosions of solutions to a system of partial differential equations modelling chemotaxis , 1992 .

[3]  Collapsing bacterial cylinders. , 1999, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  M. Brenner,et al.  Physical mechanisms for chemotactic pattern formation by bacteria. , 1998, Biophysical journal.

[5]  M. Chaplain.  Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development , 1996 .

[6]  Fordyce A. Davidson,et al.  Steady-state solutions of a generic model for the formation of capillary networks , 2000, Appl. Math. Lett..

[7]  B. Sleeman,et al.  Mathematical modeling of the onset of capillary formation initiating angiogenesis , 2001, Journal of mathematical biology.

[8]  Dirk Horstmann,et al.  Lyapunov functions and $L^{p}$-estimates for a class of reaction-diffusion systems , 2001 .

[9]  Leo P. Kadanoff,et al.  Diffusion, attraction and collapse , 1999 .

[10]  Howard A. Levine,et al.  A System of Reaction Diffusion Equations Arising in the Theory of Reinforced Random Walks , 1997, SIAM J. Appl. Math..

[11]  Alexander R. A. Anderson,et al.  A Mathematical Model for Capillary Network Formation in the Absence of Endothelial Cell Proliferation , 1998 .

[12]  Miguel A. Herrero,et al.  Finite-time aggregation into a single point in a reaction - diffusion system , 1997 .

[13]  Michel Rascle.  On a system of non linear strongly coupled partial differential equations arising in biology , 1981 .

[14]  M. Rascle,et al.  Finite time blow-up in some models of chemotaxis , 1995, Journal of mathematical biology.

[15]  L. Preziosi,et al.  Modelling and mathematical problems related to tumor evolution and its interaction with the immune system , 2000 .