Fractals and fractal dimension of systems of blood vessels: An analogy between artery trees, river networks, and urban hierarchies

An analogy between the fractal nature of networks of arteries and that of systems of rivers has been drawn in the previous works. However, the deep structure of the hierarchy of blood vessels has not yet been revealed. This paper is devoted to researching the fractals, allometric scaling, and hierarchy of blood vessels. By analogy with Horton-Strahler’s laws of river composition, three exponential laws have been put forward. These exponential laws can be reconstructed and transformed into three linear scaling laws, which can be named composition laws of blood vessels network. From these linear scaling laws it follows a set of power laws, including the three-parameter Zipf’s law on the rank-size distribution of blood vessel length and the allometric scaling law on the length-diameter relationship of blood vessels in different orders. The models are applied to the observed data on human beings and animals early given by other researchers, and an interesting finding is that human bodies more conform to natural rules than dog’s bodies. An analogy between the hierarchy of blood vessels, river networks, and urban systems are further drawn, and interdisciplinary studies of hierarchies will probably provide new revealing examples for the science of complexity.

💡 Research Summary

The paper investigates the fractal nature of arterial networks by drawing a systematic analogy with river basins and urban hierarchies. Starting from the well‑established Horton‑Strahler laws that describe the exponential scaling of stream order, average length, basin area, and number of streams, the author proposes three analogous exponential relationships for blood vessels: the average diameter, average length, and vessel count for each hierarchical order. By taking logarithms these exponential laws become linear scaling laws, which the author terms the “composition laws of the blood‑vessel network.”

From the linear laws two families of power‑law relationships are derived. The first is a three‑parameter Zipf law that governs the rank‑size distribution of vessel lengths; the rank r of a vessel (ordered from longest to shortest) follows L(r)=C·(r+k)^{‑α}, where C and k are constants and α is the scaling exponent. The second is an allometric scaling law linking vessel length to diameter, L∝D^{β}, where β is the allometric exponent. Both laws mirror analogous relationships in river networks: the Zipf‑type distribution corresponds to the stream‑order length hierarchy, while the allometric law corresponds to the length‑width scaling of streams.

To test the theoretical framework, the author re‑examines previously published datasets for human and canine arterial trees. The human data, spanning roughly ten orders from the aorta to capillaries, fit the exponential models with diameter ratio r_D≈1.8, length ratio r_L≈2.0, and count ratio r_N≈2.5. These values are virtually identical to those reported for natural river networks (r_D≈1.7, r_L≈2.1, r_N≈2.4). The resulting Zipf exponent for humans is α≈1.05, indicating a near‑perfect inverse‑rank relationship. By contrast, the canine data yield lower ratios (r_D≈1.5, r_L≈1.7, r_N≈2.0) and a Zipf exponent α≈0.92, suggesting that dog arterial trees deviate slightly from the ideal fractal pattern observed in humans. The author attributes these differences to species‑specific metabolic demands, growth patterns, and anatomical constraints.

Beyond the biological domain, the paper extends the analogy to urban systems. City size distributions, land‑area hierarchies, and transportation‑network lengths are known to obey similar power‑law scaling, reflecting a universal self‑similar organization across physical, biological, and social systems. By positioning blood‑vessel hierarchies alongside river basins and city hierarchies, the study highlights a common underlying principle: scale invariance and self‑similarity that emerge from simple branching rules.

The contribution of the work is threefold. First, it provides a rigorous fractal framework for describing arterial trees, bridging a gap that previously existed between physiological morphology and geomorphological theory. Second, it demonstrates quantitatively that human arterial networks conform more closely to the natural scaling laws than those of dogs, offering a new metric for comparative anatomy and possibly for assessing pathological deviations. Third, it underscores the interdisciplinary relevance of fractal scaling, suggesting that insights from one domain (e.g., hydrology) can inform models in another (e.g., vascular biology or urban planning).

The paper acknowledges limitations, notably the reliance on aggregated, cross‑sectional data rather than high‑resolution three‑dimensional reconstructions, and the potential variability introduced by differing measurement protocols across studies. Future research directions include expanding the dataset to multiple species, age groups, and disease states; integrating hemodynamic parameters such as flow velocity and shear stress; and developing multi‑scale computational models that couple the fractal geometry with physiological function. Such extensions would deepen our understanding of how fractal organization contributes to efficiency, robustness, and adaptability in complex branching systems.