  Angiogenesis is the process wherein endothelial cells (ECs) form sprouts that
elongate from the pre-existing vasculature to create new vascular networks. In
addition to its essential role in normal development, angiogenesis plays a
vital role in pathologies such as cancer, diabetes and atherosclerosis.
Mathematical and computational modelling has contributed to unravelling its
complexity. Many existing theoretical models of angiogenic sprouting are based
on the 'snail-trail' hypothesis. This framework assumes that leading ECs
positioned at sprout tips migrate towards low-oxygen regions while other ECs in
the sprout passively follow the leaders' trails and proliferate to maintain
sprout integrity. However, experimental results indicate that, contrary to the
snail-trail assumption, ECs exchange positions within developing vessels, and
the elongation of sprouts is primarily driven by directed migration of ECs. The
functional role of cell rearrangements remains unclear. This review of the
theoretical modelling of angiogenesis is the first to focus on the phenomenon
of cell mixing during early sprouting. We start by describing the biological
processes that occur during early angiogenesis, such as phenotype
specification, cell rearrangements and cell interactions with the
microenvironment. Next, we provide an overview of various theoretical
approaches that have been employed to model angiogenesis, with particular
emphasis on recent in silico models that account for the phenomenon of cell
mixing. Finally, we discuss when cell mixing should be incorporated into
theoretical models and what essential modelling components such models should
include in order to investigate its functional role.
