Kamiya Akira, Takahashi Tatsuhisa
Nihon University General Research Center, Tokyo, Japan.
J Appl Physiol (1985). 2007 Jun;102(6):2315-23. doi: 10.1152/japplphysiol.00856.2006. Epub 2007 Mar 8.
The branching systems in our body (vascular and bronchial trees) and those in the environment (plant trees and river systems) are characterized by a fractal nature: the self-similarity in the bifurcation pattern. They increase their branch density toward terminals according to a power function with the exponent called fractal dimension (D). From a stochastic model based-on this feature, we formulated the fractal-based integrals to calculate such morphological parameters as aggregated branch length, surface area, and content volume for any given range of radius (r). It was followed by the derivation of branch number and cross-sectional area, by virtue of the logarithmic sectioning of the r axis and of the branch radius-length relation also given by a power function of r with an exponent (alpha). These derivatives allowed us to quantify various hydrodynamic parameters of vascular and bronchial trees as fluid conduit systems, including the individual branch flow rate, mean flow velocity, wall shear rate and stress, internal pressure, and circumferential tension. The validity of these expressions was verified by comparing the outcomes with actual data measured in vivo in the vascular beds. From additional analyses of the terminal branch number, we found a simple equation relating the exponent (m) of the empirical power law (Murray's so-called cube law) to the other exponents as (m=D+alpha). Finally, allometric studies of mammalian vascular trees revealed uniform and scale-independent distributions of terminal arterioles in organs, which afforded an infarct index, reflecting the severity of tissue damage following arterial infarction.
我们身体中的分支系统（血管和支气管树）以及环境中的分支系统（植物树和河流系统）具有分形性质：分支模式中的自相似性。它们根据幂函数朝着末梢增加分支密度，该幂函数的指数称为分形维数（D）。基于这一特征的随机模型，我们制定了基于分形的积分，以计算在任何给定半径（r）范围内的聚集分支长度、表面积和内容体积等形态参数。随后，通过对r轴进行对数划分以及利用r的幂函数（指数为α）给出的分支半径 - 长度关系，推导出分支数量和横截面积。这些推导使我们能够量化血管和支气管树作为流体管道系统的各种流体动力学参数，包括单个分支流速、平均流速、壁面剪切速率和应力、内部压力以及周向张力。通过将结果与在血管床中体内测量的实际数据进行比较，验证了这些表达式的有效性。通过对末梢分支数量的进一步分析，我们发现了一个简单的方程，将经验幂律（默里所谓的立方定律）的指数（m）与其他指数联系起来，即（m = D + α）。最后，对哺乳动物血管树的异速生长研究揭示了器官中末梢小动脉的均匀且与尺度无关的分布，这提供了一个梗死指数，反映动脉梗死后组织损伤的严重程度。
