Murray's law states that vascular branching follows a universal exponent: the cube of the parent vessel's radius equals the sum of cubed daughter radii. The exponent α = 3 minimizes the total cost of pumping blood through the network, balancing flow resistance against the metabolic cost of maintaining blood volume. It has been a cornerstone of theoretical biology since 1926.
Empirical measurements of arterial branching consistently give α ≈ 2.70, not 3. The discrepancy has been attributed to measurement error, to biological variation, to the difficulty of imaging vessel radii precisely. The universal law was assumed correct; the data was assumed noisy.
The correction is to account for the metabolic cost of the vessel wall itself. Murray's original derivation treats blood volume as the metabolic cost — the tissue must oxygenate and maintain the blood within the vessel. But the vessel wall is also tissue. It has its own metabolic demand, which scales differently from blood volume because wall thickness is not proportional to radius.
When wall metabolic cost is included, the optimal exponent becomes scale-dependent. At small vessel sizes, where the wall-to-lumen ratio is large, the wall cost dominates and pulls the exponent below 3. At large vessel sizes, where the wall is relatively thin, the classical Murray exponent is approximately recovered. The resulting exponent profile closes roughly one-third of the gap between the theoretical α = 3 and the empirical α ≈ 2.70.
The universality was an artifact of an incomplete cost function. The exponent was never truly universal — it was always scale-dependent, but the missing term (wall metabolism) is small enough at large scales that the deviation was within experimental noise for the aorta and large arteries where Murray's law was first tested. At the capillary scale, where the deviation is largest, the measurements are hardest, masking the discrepancy.
The law was right about the principle (metabolic optimization). It was wrong about the accounting.