Investigating a simple model of cutaneous wound healing angiogenesis

Abstract

A simple model of wound healing angiogenesis is presented, and investigated using numerical and asymptotic techniques. The model captures many key qualitative features of the wound healing angiogenic response, such as the propagation of a structural unit into the wound centre. A detailed perturbative study is pursued, and is shown to capture all features of the model. This enables one to show that the level of the angiogenic response predicted by the model is governed to a good approximation by a small number of parameter groupings. Further investigation leads to predictions concerning how one should select between potential optimal means of stimulating cell proliferation in order to increase the level of the angiogenic response.

Mathematical modelling of angiogenesis using continuous cell-based models

“…Another interesting approach is to apply a full continuum model in terms of a system of partial differential equations like by Maggelakis (2003), Maggelakis (2004) or by Gaffney et al. (2002). The latter approach allows to consider angiogenesis on a larger scale such as on a tissue scale.…”

“…(2014) and Gaffney et al. (2002), Maggelakis (2003), Maggelakis (2004) in the context of angiogenesis. The list of examples is far from complete.…”

A Three Species Model to Simulate Application of Hyperbaric Oxygen Therapy to Chronic Wounds

“…In 2002, Gaffney et al used a two-species model to investigate cutaneous wound healing [24]. Here a travelling wave analysis was used to identify a lower bound on the wave speed of the wound healing unit in terms of two key model parameters, namely, the random motility of capillary tips and the rate of budding of tips.…”

“…Here a travelling wave analysis was used to identify a lower bound on the wave speed of the wound healing unit in terms of two key model parameters, namely, the random motility of capillary tips and the rate of budding of tips. Importantly, Gaffney et al chose not to use a snail-trail approach, opting rather to consider the EC density explicitly in addition to the capillary tip density [24]. The flux of the capillary tips is determined by random motion and directed motion.…”

“…The flux of the EC is then assumed to be proportional to the tip flux, where the rate constant is the number of EC that makes up an average capillary tip. More recently, Schugart et al developed a seven species model of acute wound healing angiogenesis [25], using a similar approach to Gaffney et al [24] in their treatment of the EC flux, while Addison-Smith et al used a simple mechanistic model for the sprouting of vessels during tumour-induced angiogenesis [26]. Mantzaris et al provides an excellent review of continuum models of angiogenesis and concludes that continuum models are important for providing significant insight into the relative importance of different processes [27].…”