Fluctuations and power laws in pulmonary physiology
Fluctuations and power laws in pulmonary physiology
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DOI：
10.1164/rccm.200202-152pp
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发表时间：
2002-07-15
影响因子：
24.7
通讯作者：
Suki, B
中科院分区：
文献类型：
--
作者：
Suki, B
变量的波动可以通过概率密度分布来表征，它指定在小范围内找到变量的特定值的可能性。为了估计分布，我们首先构建测量值的直方图，然后对直方图进行归一化，使其下方的面积统一。通常，变量 x 的分布 N (x) 遵循幂律形式：N (x) x d，这表示变量取 x 值的相对频率与 x 的 d 次方成正比。这种关系的对数是log(N)dlog(x)，log(N)和log(x)之间是线性关系。因此，为了估计指数 d，我们将 N (x) 绘制在双对数图上，d 是拟合 N 的直线的负斜率。幂律的一个重要特征是，与熟悉的分布（例如高斯分布）相比，它的尾部非常长。分布的尾部代表大事件发生的相对频率。由于幂律的尾部可能比高斯的尾部大几个数量级，因此幂律模型中发生大型或罕见事件的概率也高出几个数量级。大多数分布都有一个典型或“特征”值，例如对应于高斯峰值的值。然而，幂律分布不具有比其他尺度更受青睐的特征值或尺度。因此，幂律描述的过程或结构被称为“无标度”(3, 4)。其影响很重要。例如，图 1C 中分布的幂律尾部表明，长时间呼吸不足的统计风险明显大于高斯分布。幂律与 Mandelbrot 引入的分形密切相关 (3)。分形是自相似物体，因为在放大倍数增加的情况下，结构的一小部分看起来与整个物体相似 (4)。气道的三维结构是一个典型的例子，其中分支模式在多个长度尺度上重复出现 (5)。气管在厘米尺度上分支到主支气管，而周围气道在0.2毫米尺度上显示出类似的分支模式。结构尺寸的波动，例如气道直径的变化，取决于标尺的尺寸，在本例中是气道生成。此外，测量尺寸的分布是幂律，其指数描述了测量特征在连续放大倍数下如何变化。在分形结构上传播的动态过程（例如声波或电波）也表现出遵循幂律分布的时间波动（6）。此外，幂律出现在使用各种相关技术研究的空间和时间波动中 (4,7,8)。幂律行为的示例包括心率 (8)、呼吸频率 (9)、肺容量 (10)、通气量和灌注 (11)、潮气量、氧气和二氧化碳 (12)、血流量 (13、14) 或细胞力学 (15) 和肺组织 (16、17) 的波动。然而，分形并不是唯一的来源
Fluctuations of a variable can be characterized by the probability density distribution, which specifies the likelihood of finding a particular value of the variable within a small range. To estimate the distribution, we first construct a histogram of the measured values, and then we normalize the histogram so that the area under it is in unity. Often the distribution N (x) of a variable x follows a power law form: N (x) x d, which says that the relative frequency with which the variable takes a value x is proportional to x raised to the power d. The logarithm of this relationship is log (N) d log (x), a linear relationship between log (N) and log (x). Thus, to estimate the exponent d, we plot N (x) on a double-logarithmic graph, and d is the negative slope of a straight-line fit to N. An important feature of the power law is that its tail is very long compared with familiar distributions such as a Gaussian. The tail of a distribution is representative of the relative frequency of occurrence of large events. Because the tail of a power law can be orders of magnitude larger than the tail of a Gaussian, the probability that a large or rare event occurs in the power law model is also orders of magnitude higher. Most distributions have a typical or “characteristic” value, such as the value corresponding to the peak of the Gaussian. Power law distributions, however, do not have a characteristic value or scale that would be largely preferred over other scales. Thus, the process or structure that the power law describes is said to be “scale free”(3, 4). The implications are important. For example, the power law tail of the distribution in Figure 1C implies that the statistical risk for long periods of insufficient breathing is significantly larger than if the distribution were Gaussian. Power laws are closely related to fractals introduced by Mandelbrot (3). Fractals are self-similar objects because small parts of the structure at increasing magnifications appear similar to the entire object (4). The three-dimensional structure of the airways is a classic example in which the branching pattern repeats itself over multiple length scales (5). The trachea branches to the main bronchi at a scale of centimeters, whereas the peripheral airways show a similar branching pattern at a scale of 0.2 mm. The fluctuations in the size of a structure, for example, variability of airway diameter, depend on the size of the ruler, in this case airway generation. Furthermore, the distribution of measured sizes is a power law with an exponent that describes how the measured feature changes under successive magnifications. Dynamic processes (eg, acoustic or electric waves) propagating over fractal structures also exhibit fluctuations in time that follow power law distributions (6). In addition, power laws appear in spatial and temporal fluctuations studied using various correlation techniques (4, 7, 8). Examples of power law behavior include fluctuations in heart rate (8), respiratory rate (9), lung volume (10), ventilation and perfusion (11), tidal volume, oxygen and carbon dioxide (12), blood flow (13, 14), or the mechanics of cells (15) and lung tissues (16, 17). Nevertheless, fractals are not the only source of