On the preservation of vessel bifurcations during flow-mediated angiogenic remodelling
Fig 3
Stability analysis of the weight parameter α in the mechanistic bifurcation rule.
The weight parameter α is used to scale the respective influence of the shear stress ratio and cell number ratio when calculating probability of a cell to enter each branch. We discretised α over its range. (A) On the left, the percentage of total regression events in all simulations and the percentage of those regression events that involved either the proximal or distal branch. On the right, the percentage of regression events over time. Setting α = 0.0 means only cell number was used to determine the probability of each branch; 91.9% of these simulations experienced regression and bifurcation loss, with 52% involving proximal regression vs. 48% distal (B) Setting α = 1.0 and using only shear stress to determine branch probability resulted in 100% of simulations losing the bifurcation, with 27% involving proximal regression vs. 73% distal. Additionally, these simulations were prone to lose the bifurcation much earlier, with near 100% regression reached after just 2 days of migration. (C) Contour plot of bifurcation loss over the whole range of α, which demonstrates a global minimum of stability for values of α ranging from 0.3 to 0.6. (D) The surface of bifurcation loss vs. α over time resembles and asymmetric saddle with similar rates of increase in bifurcation loss in both directions. (E) Bifurcation loss grouped within each day of migration. The majority of bifurcations were lost during day 2, while minimal loss occurred after that. (F) Mean diameter of the proximal (blue) and distal branch (red) with standard deviation for simulations within the stable region (α = 0.45). The high-flow proximal branch, on average, stabilised at a larger diameter than the low-flow distal branch. Note that both the proximal and distal branch experience some initial transients in cell number/diameter which eventually stabilise once cells have traversed each path completely. As a result, transients take longer to settle in the distal branch due to the longer path length.