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首页> 外文期刊>Journal of Theoretical Biology >On deriving Murray's law from constrained minimization of flow resistance
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On deriving Murray's law from constrained minimization of flow resistance

机译：从默里的最小化流动阻力中衍生默里法

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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. (C) 2020 Elsevier Ltd. All rights reserved.
机译：Murray定律指出，母血管半径的立方等于子血管半径的立方之和，该定律最初是通过最小化单个圆柱形管道中血液流动的操作成本得出的。谢尔曼（Sherman）提出的另一个广为引用的推导是基于在材料约束下最小化总流动阻力的优化问题，这项研究声称，“无论母血管是对称还是不对称分裂，无论它分裂成两个、三个、四个，或者可能是任意数量的子血管，半径立方之和的守恒是阻力最小的条件。”在本文中，我们证明Sherman的分析是有缺陷的，因为对于N个子血管，有2（N）-N-1组血管半径满足Murray定律，但不产生最小的总流动阻力。此外，我们还表明，当有N个子容器，每个子容器的半径相同时，当N>=1时，最小总流动阻力是N的递增函数。由于N=1对应于完全没有分支的退化情况，我们的结果表明，分支（N=2）实现了最小的总流动阻力。因此，我们的分析解释了在许多生物系统中，分叉（相对于三分叉或更高水平的分支）的优势。（C） 2020爱思唯尔有限公司版权所有。

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