Mathematical models of developmental vascular remodelling: A review

Over the past 40 years, there has been a strong focus on the development of mathematical models of angiogenesis, while developmental remodelling has received little such attention from the mathematical community. Sprouting angiogenesis can be seen as a very crude way of laying out a primitive vessel network (the raw material), while remodelling (understood as pruning of redundant vessels, diameter control, and the establishment of vessel identity and hierarchy) is the key to turning that primitive network into a functional network. This multiscale problem is of prime importance in the development of a functional vasculature. In addition, defective remodelling (either during developmental remodelling or due to a reactivation of the remodelling programme caused by an injury) is associated with a significant number of diseases. In this review, we discuss existing mathematical models of developmental remodelling and explore the important contributions that these models have made to the field of vascular development. These mathematical models are effectively used to investigate and predict vascular development and are able to reproduce experimentally observable results. Moreover, these models provide a useful means of hypothesis generation and can explain the underlying mechanisms driving the observed structural and functional network development. However, developmental vascular remodelling is still a relatively new area in mathematical biology, and many biological questions remain unanswered. In this review, we present the existing modelling paradigms and define the key challenges for the field.Author summary: In this review article, we examine the existing models of developmental vascular remodelling and discuss the many contributions these models have made to the field of vascular development. Four individual modelling paradigms are identified, and future directions for each paradigm are addressed. The four modelling paradigms we identify are as follows: (1) logarithmic models of vascular calibre determination; (2) hybrid models of vascular regression and calibre determination; (3) cell-based models of vascular remodelling; and (4) flow-focused models of regression. Finally we identify the development of multiscale models, subcellular models, and experimental model validation to be the major challenges for the field moving forward.