Flow and anastomosis in vascular networks.

We analyze the effect that the geometrical place of anastomosis in the circulatory tree has on blood flow. We introduce an idealized model that consists of a symmetric network for the arterial and venous vascular trees. We consider that the network contains a viscoelastic fluid with the rheological characteristics of blood, and analyze the network hydrodynamic response to a time-dependent periodic pressure gradient. This response is a measurement of the resistance to flow: the larger the response, the smaller the resistance to flow. We find that for networks whose vessels have the same radius and length, the outer the level of the branching tree in which anastomosis occurs, the larger the network response. Moreover, when anastomosis is incorporated in the form of bypasses that bridge vessels at different bifurcation levels, the further apart are the levels bridged by the bypass, the larger the response is. Furthermore, we apply the model to the available information for the dog circulatory system and find that the effect that anastomosis causes at different bifurcation levels is strongly determined by the structure of the underlying network without anastomosis. We rationalize our results by introducing two idealized models and approximated analytical expressions that allow us to argue that, to a large extent, the response of the network with anastomosis is determined locally. We have also considered the influence of the myogenic effect. This one has a large quantitative impact on the network response. However, the qualitative behavior of the network response with anastomosis is the same with or without consideration of the myogenic effect. That is, it depends on the structure that the underlying vessel network has in a small neighborhood around the place where anastomosis occurs. This implies that whenever there is an underlying tree-like network in an in vivo vasculature, our model is able to interpret the anastomotic effect.

Obstructions in Vascular Networks: Relation Between Network Morphology and Blood Supply.

We relate vascular network structure to hemodynamics after vessel obstructions. We consider tree-like networks with a viscoelastic fluid with the rheological characteristics of blood. We analyze the network hemodynamic response, which is a function of the frequencies involved in the driving, and a measurement of the resistance to flow. This response function allows the study of the hemodynamics of the system, without the knowledge of a particular pressure gradient. We find analytical expressions for the network response, which explicitly show the roles played by the network structure, the degree of obstruction, and the geometrical place in which obstructions occur. Notably, we find that the sequence of resistances of the network without occlusions strongly determines the tendencies that the response function has with the anatomical place where obstructions are located. We identify anatomical sites in a network that are critical for its overall capacity to supply blood to a tissue after obstructions. We demonstrate that relatively small obstructions in such critical sites are able to cause a much larger decrease on flow than larger obstructions placed in non-critical sites. Our results indicate that, to a large extent, the response of the network is determined locally. That is, it depends on the structure that the vasculature has around the place where occlusions are found. This result is manifest in a network that follows Murray's law, which is in reasonable agreement with several mammalian vasculatures. For this one, occlusions in early generation vessels have a radically different effect than occlusions in late generation vessels occluding the same percentage of area available to flow. This locality implies that whenever there is a tissue irrigated by a tree-like in vivo vasculature, our model is able to interpret how important obstructions are for the irrigation of such tissue.