Mechanobiological Modelling of Angiogenesis: Impact on Tissue Engineering and Bone Regeneration - Computational Modeling in Tissue Engineering - page 374

of the chemotactic response or the rate of capillary tip formation influence
angiogenesis.
Models of the continuum type are usually derived from mass conservation
equations and chemical kinetics. This results in a system of nonlinear partial
differential equations, modelling macroscopic quantities such as cell density and
chemical concentrations. They have been extensively studied in tumor induced
angiogenesis and wound healing.
Tumour-induced angiogenesis has received a lot of attention since 1985, when
Balding and McElwain [ 5 ] proposed a simple one dimensional model to describe
the vascular network through the introduction of a chemotactic stimulus. There-
fore, they concentrated only on the chemotactic response of the ECs to angiogenic
factors. Later, models were extended to two dimensions and included the hapto-
tatic response of ECs to adhesive gradients, usually of fibronectin, as well as the
role of the inhibitors in the angiogenic process [ 2 , 89 ]. Following Orme and
Chaplain [ 90 ], Levine et al. [ 67 , 68 ] have developed highly complex biochemi-
cally-based models that account for specific angiogenic factors, interactions
between ECs, and cells such as pericytes and macrophages that are also involved
in angiogenesis. Whereas in these previous models, vascular remodelling has been
neglected, it has been considered in recent studies [ 1 , 20 , 71 , 77 , 78 , 91 , 114 , 115 ].
The models of angiogenesis mentioned above do not account for the effect of
blood flow, which has stimulated the development of hybrid models.
Given the recognized biological similarities between tumour and wound
angiogenesis [ 18 , 37 ], the models for tumor induced angiogenesis may also be
relevant to wound healing, wherein growth factors released at the wound site by
inflammatory cells perform analogous roles to the tumour angiogenesis factor [ 88 ].
There are several continuum mathematical models of wound healing which
incorporate the effect of angiogenesis [ 15 , 35 , 41 , 88 , 98 , 100 , 108 , 124 ]. As in
tumour angiogenesis these models consist on a system of nonlinear partial dif-
ferential equations. Olsen et al. [ 88 ] and Gaffney et al. [ 41 ] proposed a simple
2-species model. The former aimed to investigate the role of the extracellular
matrix on the proliferative and migratory response of endothelial cells whilst the
latter described the passive motion of capillary sprouts following their leading tip
cell. In 1996, Pettet et al. [ 98 , 100 ] developed two models of wound healing
angiogenesis where they incorporated the chemotactic response of endothelial
cells to angiogenic factors. Later, Byrne et al. [ 15 ] with a three species-model also
incorporated chemotaxis and compared their results with experimental data. More
recently, Schugart et al. [ 108 ] developed a seven species model of acute wound
healing angiogenesis, using a similar approach to Gaffney et al. [ 41 ] and recog-
nizing the role of tissue oxygenation in wound care. Xue et al. [ 124 ] developed an
8-variables model which involves the concentration of oxygen, PDGF and VEGF,
the densities of macrophages, fibroblasts, capillary tips and sprouts, and the density
and velocity of the ECM. Some models additionally incorporated the oxygen [ 35 ].
The angiogenic process involved during the formation of the gap vasculature
network during fracture healing has been less well studied. Geris et al. [ 44 ]
proposed a continuous mathematical model that describes healing in terms of