Krenz G S, Linehan J H, Dawson C A
Department of Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee 53233.
J Appl Physiol (1985). 1992 Jun;72(6):2225-37. doi: 10.1152/jappl.1992.72.6.2225.
The extant morphometric data from the intrapulmonary arteries of dog, human, and cat lungs produce graphs of the log of the vessel number, (N) or length (l) in each level vs. the log of the mean diameter (D) in each level that are sufficiently linear to suggest that a scale-independent self-similar or fractal structure may underlie the observed relationships. These data can be correlated by the following formulas: Nj = a1Dj-beta 1, and lj = a2Dj beta 2, where j denotes the level (order or generation) number measured from the largest vessel at the entrance to the arterial tree to the smallest vessel at the entrance to the capillary bed. With the hemodynamic resistance (R) represented by Rj = 128 microliterj/(Nj pi Dj4) and the vascular volume (Q) by Qj = Nj pi Dj2lj/4, the continuous cumulative distribution of vascular resistance (Rcum) vs. cumulative vascular volume (Qcum) (where Rcum and Qcum represent the total resistance or volume, respectively, upstream from the jth level) can be calculated from [formula: see text] where r = Dj/Dj+1 is a constant independent of j. Analogous equations are developed for the inertance and compliance distributions, providing simple formulas to represent the hemodynamic consequences of the pulmonary arterial tree structure.
来自狗、人类和猫肺内肺动脉的现有形态测量数据生成了各级血管数量（N）或长度（l）的对数与各级平均直径（D）的对数的关系图，这些图具有足够的线性，表明一种与尺度无关的自相似或分形结构可能是所观察到的关系的基础。这些数据可以用以下公式关联：Nj = a1Dj^-β1，以及lj = a2Dj^β2，其中j表示从动脉树入口处最大的血管到毛细血管床入口处最小的血管所测量的级别（顺序或代）数。用Rj = 128μl j/(NjπDj^4)表示血流动力学阻力（R），用Qj = NjπDj^2lj/4表示血管容积（Q），血管阻力（Rcum）与累积血管容积（Qcum）（其中Rcum和Qcum分别代表第j级上游的总阻力或容积）的连续累积分布可以根据[公式：见原文]计算得出，其中r = Dj/Dj + 1是一个与j无关的常数。针对惯性和顺应性分布推导出了类似的方程，提供了表示肺动脉树结构血流动力学后果的简单公式。
