### Abstract

We propose a mathematical model that governs endothelial cell pattern formation on a biogel surface. The model accounts for diffusion and chemotactic motion of the cells, diffusion of the growth factor and effective biochemical reactions. The model admits a basic steady state that corresponds to a spatially uniform distribution of both the cells and the growth factor. We perform a weakly nonlinear stability analysis of the basic state in order to determine whether spatially nonuniform steady patterns can appear in the system when the basic state becomes unstable. The main results can be summarized as follows. No steady patterns can bifurcate from the basic state if the rate of decay of the growth factor is small. Increasing the rate of decay of the growth factor allows one to observe steady patterns, provided that diffusion of the growth factor is sufficiently slow. Specifically, the work focuses on the occurrence of hexagons and stripes. Most often hexagons are observed. In order for stripes to occur, the chemotactic sensitivity of the endothelial cells and/or their biochemical activity have to be reduced.

Original language | English (US) |
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Pages (from-to) | 163-169 |

Number of pages | 7 |

Journal | Applied Mathematics Letters |

Volume | 20 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2007 |

### Keywords

- Angiogenesis
- Endothelial cell pattern formation
- Modeling
- Nonlinear analysis

### ASJC Scopus subject areas

- Applied Mathematics

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## Cite this

*Applied Mathematics Letters*,

*20*(2), 163-169. https://doi.org/10.1016/j.aml.2006.03.011