  Angiogenesis is the formation of new blood vessels from pre-existing ones in
response to chemical signals secreted by, for example, a wound or a tumour. In
this paper, we propose a mesoscopic lattice-based model of angiogenesis, in
which processes that include proliferation and cell movement are considered as
stochastic events. By studying the dependence of the model on the lattice
spacing and the number of cells involved, we are able to derive the
deterministic continuum limit of our equations and compare it to similar
existing models of angiogenesis. We further identify conditions under which the
use of continuum models is justified, and others for which stochastic or
discrete effects dominate. We also compare different stochastic models for the
movement of endothelial tip cells which have the same macroscopic,
deterministic behaviour, but lead to markedly different behaviour in terms of
production of new vessel cells.
