A fractal graph model of capillary type systems

TL;DR

This paper models blood flow in vessel-capillary systems using a fractal graph approach, establishing the existence and uniqueness of self-reproducing solutions and deriving boundary conditions that simplify flow analysis.

Contribution

It introduces a fractal graph model for capillary systems, demonstrating the existence and uniqueness of self-reproducing solutions based on scaling factors.

Findings

Existence and uniqueness of self-reproducing solutions

Derived a relation between pressure and flux at the junction

Established a Robin boundary condition simplifying flow analysis

Abstract

We consider blood flow in a vessel with an attached capillary system. The latter is modeled with the help of a corresponding fractal graph whose edges are supplied with ordinary differential equations obtained by the dimension-reduction procedure from a three-dimensional model of blood flow in thin vessels. The Kirchhoff transmission conditions must be satisfied at each interior vertex. The geometry and physical parameters of this system are described by a finite number of scaling factors which allow the system to have self-reproducing solutions. Namely, these solutions are determined by the factors' values on a certain fragment of the fractal graph and are extended to its rest part by virtue of these scaling factors. The main result is the existence and uniqueness of self-reproducing solutions, whose dependence on the scaling factors of the fractal graph is also studied. As a corollary we obtain a relation between the pressure and flux at the junction, where the capillary system is attached to the blood vessel. This relation leads to the Robin boundary condition at the junction and this condition allows us to solve the problem for the flow in the blood vessel without solving it for the attached capillary system.