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We consider blood flow in a vessel with an attached capillary system. Thelatter is modeled with the help of a corresponding fractal graph whose edgesare supplied with ordinary differential equations obtained by thedimension-reduction procedure from a three-dimensional model of blood flow inthin vessels. The Kirchhoff transmission conditions must be satisfied at eachinterior vertex. The geometry and physical parameters of this system aredescribed by a finite number of scaling factors which allow the system to haveself-reproducing solutions. Namely, these solutions are determined by thefactors' values on a certain fragment of the fractal graph and are extended toits rest part by virtue of these scaling factors. The main result is theexistence and uniqueness of self-reproducing solutions, whose dependence on thescaling factors of the fractal graph is also studied. As a corollary we obtaina relation between the pressure and flux at the junction, where the capillarysystem is attached to the blood vessel. This relation leads to the Robinboundary condition at the junction and this condition allows us to solve theproblem for the flow in the blood vessel without solving it for the attachedcapillary system.

A fractal graph model of capillary type systems