Arterial arcades and collaterals regress under hemodynamics-based diameteradaptation: a computational and mathematical analysis

Abstract

Segments in the arterial network have a >1000-fold span of radii. This is believed to result from adaptation of each segment to the wall shear stress (WSS), with outward respectively inward remodeling if WSS is higher or lower than some reference value. While this seems a straightforward mechanism for arterial tree design, the arterial network is not a tree but contains numerous arcades, collaterals and other looping structures. In this theoretical study, we analyzed stability of looping structures in arterial networks under WSS control. Simulation models were based on very simple network topologies as well as on published human coronary and mouse cerebral arterial networks. Adaptation was implemented as a rate of change of structural radius of each segment that is proportional to the deviation from its reference WSS. A more generalized model was based on adaptation to a large range of other local hemodynamic stimuli, including velocity, flow and power dissipation. For over 12,000 tested parameter sets, the simulations invariably predicted loss of loops due to regression of one or more of the segments. In the small networks, this was the case for both the WSS and the generalized model, and for a large range of initial conditions and model parameters. Loss of loopiness also was predicted by models that included direction-dependent adaptation rates, heterogeneous reference WSS or adaptation rates among the adapting segments, and adaptation under dynamic conditions. Loss of loops was also found in the coronary and cerebral artery networks subjected to adaptation to WSS. In a mathematical analysis we proved that loss of loops is a direct consequence of Kirchhoffs circuit law, which for each loop leads to a positive eigenvalue in the Jacobian matrix of partial derivatives in the adaptation model, and therefore to unstable equilibria in the presence of loops. Loss of loops is an inherent property of arterial networks that adapt to local hemodynamics. Additional mechanisms are therefore needed to explain their presence, including direct communication between connected segments.