×

zbMATH — the first resource for mathematics

Partial differential equations of chemotaxis and angiogenesis. (English) Zbl 0990.35014
In this paper the authors motivate the study of the PDE system \[ {\partial P\over \partial t} = D \nabla \cdot (P \nabla (\ln (P/\Phi(\omega)))), \] \[ {\partial\omega \over \partial t} = {\mathbf F}(P,\omega), \] and examine some of its properties. Here \(x \in \Omega \subset \mathbb{R}^n\), \(t > 0\), \(D>0\), and the system is supplemented with appropriate initial and no-flux boundary conditions. \(P\) is a population density and \(\omega\) is an \(m\)-dimensional vector of control parameters.
The main applications of the system above, described in Section 1 of the paper, are in the modelling of chemotaxis, in which case \(P(x,t)\) is typically the density of myxobacteria cells and \(\omega\) is the concentration of the chemotactic signal produced by and reacted to by the bacteria. Alternatively, in a model of tumour angiogenesis, \(P(x,t)\) stands for the density of endothelial cells, in which case \(\omega\) has at least two components, the concentration of tumour angiogenesis factor(s) and of fibronectin.
In Section 2 the authors review the theory of reinforced random walks and, by passing to a continuum limit, derive from the master equations the above PDEs. This section is best read in conjunction with the work of H. G. Othmer and A. Stevens [SIAM J. Appl. Math. 57, 1044-1081 (1997; Zbl 0990.35128)]. After a survey of numerical simulations in Section 3, the authors consider in Section 4 exact solutions of the system for the case of \(\Omega\) being one-dimensional, \(\omega\) scalar, and particular choices of \(\Phi(\omega)\) and \({\mathbf F}\). A wealth of phenomena including finite time collapse and blow-up is discovered. In Section 5 the authors survey results on aggregation following H. A. Levine and B. D. Sleeman [SIAM J. Appl. Math. 57, 683-730 (1997; Zbl 0874.35047)]. In Section 6 existence of travelling wave and other similarity solutions is discussed, while in Section 7 implications of inhomogeneity in the medium through which the cells diffuse for transition probability rates in the master equation is considered.

MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
92C17 Cell movement (chemotaxis, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
92C50 Medical applications (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Keller, Journal of Theoretical Biology 26 pp 339– (1970) · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5
[2] Molecular Biology of the Cell (3rd Edn). Garland Publishing Inc.: New York, 1994.
[3] Othmer, SIAM Journal on Applied Mathematics 51 pp 1044– (1997)
[4] Davis, Probability Theory and Related Fields 84 pp 203– (1990) · Zbl 0665.60077 · doi:10.1007/BF01197845
[5] Levine, SIAM Journal on Applied Mathematics 57 pp 683– (1997) · Zbl 0874.35047 · doi:10.1137/S0036139995291106
[6] Folkman, Advances in Cancer Research 43 pp 175– (1985) · doi:10.1016/S0065-230X(08)60946-X
[7] Folkman, Nature Medicine 1 pp 21– (1995) · doi:10.1038/nm0195-27
[8] Stokes, Journal of Theoretical Biology 152 pp 377– (1991) · doi:10.1016/S0022-5193(05)80201-2
[9] Modelling the Growth and Form of Capillary Networks in On Growth and Form, (eds). Wiley: New York, 1999; 225-249.
[10] Levine, Journal of Mathematical Biology
[11] A mathematical model for the roles of pericytes and macrophases in the onset of angiogenesis. I: The role of protease inhibitors in preventing angiogenesis. Mathematical Biosciences, in press. · Zbl 0986.92016
[12] Holmes, Journal of Theoretical Biology 202 pp 95– (2000) · doi:10.1006/jtbi.1999.1038
[13] Tumour induced angiogenesis as a reinforced random walk: modelling capillary network formation without endothelial cell proliferation. Mathematical and Computer Modelling, in press. · Zbl 1021.92016
[14] Mathematical Biology. Biomathematics Texts. Springer: Berlin, 1989.
[15] Anderson, Fundamentals of Applied Nematology 20 pp 165– (1997)
[16] Some results on reaction-diffusion systems modelling chemotaxis. Ph.D. Thesis, Wuhan-University, People’s Republic of China, 2000.
[17] Linear and Nonlinear Waves. Wiley: New York, 1974.
[18] Symmetry Methods for Differential Equations. Cambridge Texts in Applied Mathematics. Cambridge University Press: Cambridge, 2000. · doi:10.1017/CBO9780511623967
[19] On a system of nonlinear strongly coupled partial differential equations arising in biology. In Proceedings of Dundee Conference on Ordinary and Partial Differential Equations, (eds.), Lecture Notes in Mathematics, vol. 846. Springer: Berlin, 1980; 290-298.
[20] Rascle, Journal of Mathematical Biology 33 pp 388– (1995) · Zbl 0814.92014 · doi:10.1007/BF00176379
[21] Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society Translators, vol. 23. AMS: Providence RI, 1968.
[22] On existence and non-existence of global solutions to a system of reaction-diffusion equations modelling chemotaxis. Private communication, 2000.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.