Model Microvascular Networks Can Have Many Equilibria

Key Structural Features of Microvascular Networks Leading to the Formation of Multiple Equilibria

We analyse mathematical models of blood flow in two simple vascular networks in order to identify structural features that lead to the formation of multiple equilibria. Our models are based on existing rules for blood rheology and haematocrit splitting. By performing bifurcation analysis on these simple network flow models, we identify a link between the changing flow direction in key vessels and the existence of multiple equilibria. We refer to these key vessels as redundant vessels, and relate the maximum number of equilibria with the number of redundant vessels. We vary geometric parameters of the two networks, such as vessel length ratios and vessel diameters, to demonstrate that equilibria are uniquely defined by the flow in the redundant vessels. Equilibria typically emerge in sets of three, each having a different flow characteristic in one of the network's redundant vessels. For one of the three equilibria, the flow within the relevant redundant vessel will be smaller in magnitude than the other two and the redundant vessel will contain few Red Blood Cells (RBCs), if any. For the other two equilibria, the redundant vessel contains RBCs and significant flow in the two available directions. These structural features of networks provide a useful geometric property when studying the equilibria of blood flow in microvascular networks.

Modeling transit time distributions in microvascular networks

The time a red blood cell (RBC) spends in the microvasculature is of prime importance for a number of physiological processes. In this work, we present a methodology for computing an approximation of the so-called transit time distribution (TTD), i.e., the probabilistic description of how long a RBC will reside within the network. As a proof of concept, we apply this methodology to three flavors of the mesh networks. We show that each network type supports multiple distinct steady-state configurations and we present tools for analyzing the associated collection of TTDs, ranging from standard measures like mean capillary transit time (MCTT) and capillary transit time heterogeneity (CTTH) to novel metrics.

A few upstream bifurcations drive the spatial distribution of red blood cells in model microfluidic networks

The physics of blood flow in small vessel networks is dominated by the interactions between Red Blood Cells (RBCs), plasma and blood vessel walls. The resulting couplings between the microvessel network architecture and the heterogeneous distribution of RBCs at network-scale are still poorly understood. The main goal of this paper is to elucidate how a local effect, such as RBC partitioning at individual bifurcations, interacts with the global structure of the flow field to induce specific preferential locations of RBCs in model microfluidic networks. First, using experimental results, we demonstrate that persistent perturbations to the established hematocrit profile after diverging bifurcations may bias RBC partitioning at the next bifurcations. By performing a sensitivity analysis based upon network models of RBC flow, we show that these perturbations may propagate from bifurcation to bifurcation, leading to an outsized impact of a few crucial upstream bifurcations on the distribution of RBCs at network-scale. Based on measured hematocrit profiles, we further construct a modified RBC partitioning model that accounts for the incomplete relaxation of RBCs at these bifurcations. This model allows us to explain how the flow field results in a single pattern of RBC preferential location in some networks, while it leads to the emergence of two different patterns of RBC preferential location in others. Our findings have important implications in understanding and modeling blood flow in physiological and pathological conditions.