Optimal Geometry of Perfusion in Flow Networks

In organ capillaries, oxygen detaches from red blood cells and diffuses through a porous membrane into the nearby tissue, eventually getting absorbed by tissue cells. Motivated by this system, we study networks where flow is laden with nutrients able to diffuse out of the vessels and into the surrounding faces. In a uniform network with a single flow source and sink, the nutrients become depleted far from the source, forming a gradient of nutrient density that may leave some tissue undersupplied. Nonetheless, organs have developed capillary bed networks that avoid tissue starvation. In anticipation of engineering networks for artificial organs, we explore theoretical network designs for uniform perfusion over an extended area. Here, we first propose an algorithm with rules to induce local topological changes in a flow network, which serves to generate a wide range of network topologies. We then use vertex model dynamics to tune the vertex positions in a way that equalizes nutrient distribution. Ultimately, we hope to use this uniformly perfusing network ensemble to identify topological features that serve to dissipate an undesirable nutrient gradient.