Bifurcations in adaptive vascular networks: towards model calibration

Transport networks are crucial for the functioning of natural and technological systems. We study a mathematical model of vascular network adaptation, where the network structure dynamically adjusts to changes in blood flow and pressure. The model is based on local feedback mechanisms that occur on different time scales in the mammalian vasculature. The cost exponent $\gamma$ tunes the vessel growth in the adaptation rule, and we test the hypothesis that the cost exponent is $\gamma=1/2$ for vascular systems [Hu and Cai, Phys. Rev. Lett., Vol. 111(13) (2013)1 ]. We first perform a bifurcation analysis for a simple triangular network motif with fluctuating demand, and then conduct numerical simulations on network topologies extracted from perivascular networks of rodent brains. We compare the model predictions with experimental data and find that $\gamma$ is closer to 1 than to 1/2 for the model to be consistent with the data. Our study thus aims at addressing two questions: (i) Is a specific measured flow network consistent in terms of physical reality? (ii) Is the adaptive dynamic model consistent with measured network data? We conclude that the model can capture some aspects of vascular network formation and adaptation, but also suggest some limitations and directions for future research. Our findings contribute to a general understanding of the dynamics in adaptive transport networks, which is essential for studying mammalian vasculature and developing self-organizing piping systems.