Revisiting Murray's Law in Pulmonary Arteries: Exploring Branching Patterns and Principles

In 1926, Cecil D. Murray published a fundamental law of physiology relating the form and function of branched vessels. Murray's Law predicts that the diameter of a parent vessel branching into two child branches is mathematically related by a cube law based on parabolic flow and power minimization with vascular volume. This law is foundational for computational analyses of branching vascular structures. However, pulmonary arteries exhibit morphometric and hemodynamic characteristics that may deviate from classical predictions. This study investigates the morphometry of pulmonary arterial networks, examining relationships between parent and child vessel diameters across species. We analyzed three-dimensional segmentations of pulmonary arterial geometries from healthy subjects across four species: human (n = 7), canine (n = 5), swine (n = 4), and murine (n = 3). Our findings reveal an average exponent value of 2.31(±0.60) in human, 2.13(±0.54) in canine, 2.10(±0.49) in swine, and 2.59(±0.58) in murine, all lower than the predicted value of 3.0 from Murray's Law. Extending Murray's Law to fully developed pulsatile flow based on minimal impedance, we show that mean flow is proportional to radius raised to a power between 2.1 and 3, depending on the Womersley number. Our findings suggest that while Murray's Law provides a useful baseline, pulmonary artery (PA) branching follows a different optimization principle depending on Womersley number. This study contributes to a deeper understanding of pulmonary arterial structure-function relationships and implications for vascular disease modeling.