Journal of Mathematical Biology

, Volume 42, Issue 3, pp 195-238

First online:

Mathematical modeling of the onset of capillary formation initiating angiogenesis

  • Howard A. LevineAffiliated withDepartment of Mathematics, Iowa State University, Ames, IA, 50011, USA. e-mail:
  • , Brian D. SleemanAffiliated withSchool of Mathematics, University of Leeds, Leeds LS2 9JT, England, UK. e-mail:
  • , Marit Nilsen-HamiltonAffiliated withDepartment of Biochemistry, Biophysics and Molecular Biology, Iowa State University, Ames, Iowa, 50011, USA. e-mail:

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It is well accepted that neo-vascular formation can be divided into three main stages (which may be overlapping): (1) changes within the existing vessel, (2) formation of a new channel, (3) maturation of the new vessel.

In this paper we present a new approach to angiogenesis, based on the theory of reinforced random walks, coupled with a Michaelis-Menten type mechanism which views the endothelial cell receptors as the catalyst for transforming angiogenic factor into proteolytic enzyme in order to model the first stage. In this model, a single layer of endothelial cells is separated by a vascular wall from an extracellular tissue matrix. A coupled system of ordinary and partial differential equations is derived which, in the presence of an angiogenic agent, predicts the aggregation of the endothelial cells and the collapse of the vascular lamina, opening a passage into the extracellular matrix. We refer to this as the onset of vascular sprouting. Some biological evidence for the correctness of our model is indicated by the formation of teats in utero. Further evidence for the correctness of the model is given by its prediction that endothelial cells will line the nascent capillary at the onset of capillary angiogenesis.

Key words: Angiogenesis – Capillary formation – Growth factors – Reinforced random walk – Michealis-Menten kinetics