The Optimal Radius Ratio in Naturally Furcated Vessels and Branches: Toward a Unifying Model

Abstract

Abstract. Bifurcation is a ubiquitous design principle in nature, shaping the architecture of arteries, veins, airways, and plant branches. For centuries, researchers have sought the “optimal” branching law. As early as 1708, Keill proposed a cross-sectional area ratio of 0.7984 for arteries; later, Hess and Murray, invoking the principle of minimum work, derived a diameter ratio of 2−1/3 = 0.7937. Although their work focused on blood vessels, the underlying principles were extended to other branching systems, including trees, establishing the conceptual foundation for a universal model. From metabolic scaling, Kleiber and Brody suggested ratios of 0.75 and 0.73, respectively. These values, however, remained irreconcilable, with no universally accepted framework. In contrast to existing models, which are based on averages over large datasets, here, we develop a new unifying theoretical model using dimensional analysis. By constructing similarity numbers that capture flow, gravity, viscosity, and energy constraints, we derive an optimal universal radius ratio of 0.7579, a value that bridges Hess and Murray's theoretical law with Kleiber's quarter-power scaling. A case study of poly-furcated branches in Ficus microcarpa shows that our dimensional analysis predicts measured radii with only 4.7% error, outperforming classical models. This work provides the first mathematically grounded reconciliation of competing bifurcation rules, offering a unified framework applicable to both vascular and botanical branching systems.