A multi-fractal vascular tree model for characterizing blood flow in cardiovascular system.

The cardiovascular system's hemodynamics is pivotal for tissue oxygen and nutrient delivery. However, accurately characterizing its blood flow characteristics remains a major challenge due to the complex distribution and geometry of the cardiovascular system. This study introduces a novel multi - fractal vascular tree (MFVT) model. By integrating heterogeneous fractal dimensions and scaling laws, considering vessel branching and tortuosity, the model depicts blood flow across various vessels. Formulas for blood flow rates in single vessels, fractal vessel trees, and bundles of fractal vessel trees were derived, along with the analytical expression for cardiovascular system permeability. The model's reliability was verified via four case studies, with a maximum relative error of less than 5% between theoretical and experimental values. Sensitivity analyses showed that parameters significantly influenced blood flow. For instance, as the maximum diameter of zero-level vessels increased from 0.2 to 4 mm, the flow rate rose notably. When the diameter ratio increased from 0.5 to 0.8, the flow rate also increased, especially at higher pressure differences. An increase in the initial length of zero-level vessels from 5 to 50 mm led to a decrease in the flow rate. These phenomena indicate that blood flow resistance is negatively correlated with vessel diameter and positively correlated with vessel length. This work advances the mechanistic understanding of blood flow distribution and provides a computational tool for studying vascular pathologies.