Murray’s law, which states that the cube of the radius of a parent vessel equals the sum of the cubes of the radii of the daughter vessels, was originally derived by minimizing the cost of operation of blood flow in a single cylindrical tube. An alternative widely cited derivation by Sherman is based upon the optimization problem of minimizing the total flow resistance subject to a material constraint, and that study claimed that “Conservation of the sum of the cubes of the radii is the condition for minimal resistance whether the parent vessel divides symmetrically or asymmetrically, and whether it divides into two, three, four, or, presumably, any number of daughter vessels.” In this paper we show that Sherman’s analysis is flawed, since with N daughter vessels there are $2^N-N-1$ sets of vessel radii which satisfy Murray’s law but which do not yield minimal total flow resistance. Moreover, we show that when there are N daughter vessels, each with the same radius, the minimal total flow resistance is an increasing function of N for $N⩾1$. Since $N=1$ corresponds to the degenerate case of no branching at all, our result implies that bifurcation ($N=2$) achieves the minimal total flow resistance. Our analysis thus offers an explanation for the preponderance of bifurcations (as opposed to trifurcations or higher level branchings) in many biological systems.

默里定律指出，父血管半径的立方等于子血管半径的立方之和，最初是通过最小化单个圆柱管中的血流操作成本来推导的。谢尔曼（Sherman）的另一种广为引用的推导基于最小化受材料约束的总流阻的优化问题，该研究声称“保持半径立方的总和是最小阻力的条件，无论母体血管对称或不对称地分裂，分为两个，三个，四个或大概任何数量的子代血管。” 在本文中，我们证明了谢尔曼（Sherman）的分析是有缺陷的，因为有N个子血管$2^{ñ}--ñ--\mathrm{1个}$满足穆雷定律但不会产生最小总流阻的一组容器半径。此外，我们表明，当有N个子容器，每个子容器具有相同的半径时，最小总流阻是N在$ñ⩾\mathrm{1个}$。以来$ñ=\mathrm{1个}$ 对应于根本没有分支的简并情况，我们的结果表明分叉（$ñ=2$）实现最小的总流阻。因此，我们的分析为许多生物系统中的分支（相对于分支或更高级别的分支）占优势提供了解释。