Physiologic data measured at the bedside or in the laboratory invariably display fluctuations. These fluctuations often carry information about the underlying structure or process but are usually regarded as measurement or biologic noise and are neglected.
Consider the example in Figure 1
Figure 1. Breathing irregularities in a preterm infant. Interbreath intervals as a function of breath number at postconceptional ages of 39 (A) and 61 weeks (B). (C) Probability density distributions estimated from A and B and straight line fits on a log–log graph (from Frey and colleagues with permission [1]).
[More] [Minimize]The previous example illustrates three important points: (1) Irregular fluctuations can carry information, (2) fluctuations often follow a power law distribution, and (3) the exponent of the distribution may be sensitive to physiologic or pathologic changes. During the last decade, it has become evident that power laws are ubiquitous in nature (2, 3). In this review, power law distributions in pulmonary physiology are described, focusing on their interpretation. Following an overview, power laws through specific examples are discussed, including parenchymal structure in emphysema, dynamics of airway opening and crackle sound, and lung recruitment. Finally, some exciting new possibilities that use external fluctuations in life-support systems to improve lung function are discussed.
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 (e.g., 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 power laws. The data in Figure 1 were interpreted in terms of fluctuations propagating through the nonlinear neural network of the respiratory oscillator (1).
Computed tomography is a sensitive method for assessing alterations in lung structure induced by various diseases such as emphysema (18, 19). To characterize a computed tomography image, we generate low-attenuation area clusters as follows: pixels with density less than a threshold, for example, 950 Hounsfield units, are designated as air and assigned a value of 1, whereas pixels with a density of more than 950 are designated as tissue with a value of 0. A low-attenuation area cluster is defined as a contiguous region with values of 1. Summing the number of pixels in a cluster provides the cluster size. These binary maps are extremely inhomogeneous with many irregular clusters of widely varying sizes. Hence, the fractal framework appears suitable for their analysis. Indeed, the probability distribution of low attenuation cluster sizes follows a power law for both normal subjects and patients with chronic obstructive pulmonary disease with exponents characterizing the complexity of the parenchymal tissue structure (20). The chronic obstructive pulmonary disease patients had significantly smaller exponents than the healthy subjects, suggesting that the likelihood of finding a large low-attenuation cluster, where elastic recoil and gas exchange are compromised, is much higher in chronic obstructive pulmonary disease patients than in normal subjects. One interesting finding was that normal subjects and patients with early emphysema had a similar total low-attenuation area (sum of all clusters). However, because the exponent was smaller in the early emphysematous subjects, the spatial organization of the clusters is different in chronic obstructive pulmonary disease. The same total low attenuation can be distributed among many small clusters (normal subject) or fewer small and some larger clusters (chronic obstructive pulmonary disease), which decreases the exponent of the power law. Therefore, the exponent can be a clinically useful index sensitive to mild reorganization of the parenchyma in early emphysema that is not detectable by the conventional index of percent low-attenuation area.
Currently, the mechanism that generates a power law distribution of low-attenuation clusters in normal subjects is not well understood. However, a model has been offered to explain the process that alters the existing structure and the exponent in emphysema (20). The model is based on a large elastic network of the parenchyma in which mechanical forces can rupture alveolar walls. The model predicts that initially the clusters grow slowly around their perimeters, but soon significant stress concentration develops on alveolar walls separating neighboring clusters. When the wall separating two clusters breaks, the clusters coalesce, which results in a sudden rearrangement of the structure. Next, the forces are redistributed along the perimeter of the new cluster, which starts growing slowly until it coalesces with another cluster. This process is similar to crack propagation in plastic materials in which the distribution of crack sizes follows a power law (21). The coalescence of clusters decreases the exponent because essentially we take away two small clusters (decreasing probability) and add one larger cluster (increasing probability) so that the distribution becomes less steep, which may explain why the exponent is sensitive to small changes in cluster reorganization (20). It is noteworthy that experiments have recently been reported supporting the notion that mechanical forces during breathing are capable of rupturing alveolar walls (22).
Signals propagating over a fractal structure also lead to power law behavior, which is shown next in relation to airway reopening. In the normal lung, airway closure occurs when lung volume is lowered below closing volume (23). In a disease such as asthma, closure can develop during normal breathing (24). We examine airway closure and reopening using a simple model of the periphery of the airway tree (Model A, Figure 2A)
Figure 2. (A) A four-generation tree model of the airways. The numbers in brackets are segments opening in an avalanche (thick lines) when pressure P = 0.5. Thin lines are segments that have not opened. The numbers to the right of the segments are the normalized opening pressures. (B) Parallel model with the same segments arranged in an increasing order of their opening pressures. (C) Crackle events as a function of time from the tree model. Groups of crackles come from avalanches. (D) Crackle events from the parallel model. (E) Distribution of intercrackle intervals in the two models.
[More] [Minimize]The avalanche-like opening is a consequence of the hierarchical structure of the tree. We recall that Segment 2 could not open before the root. Thus, the connectivity of the tree imposes constraints on the possible sequence of openings. What would happen if the airways were not arranged in a tree structure? We examine this possibility in Model B, which has the same segments and opening pressures as in Model A but arranged in parallel (Figure 2B). We assume that Model B is connected to a large pump, which increases the pressure simultaneously in all segments. We observe that Segment 2 now opens earlier than Segment 1, which was the root in Model A. The recorded crackle time series is shown in Figure 2D. Comparing the two cases, we notice irregularities in both time series. We may say that the fluctuations are unimportant, and we could average them. Thus, we calculate the average time interval between consecutive crackles, the intercrackle interval. The particular example in Figure 2 was constructed so that this average is the same for the two models. Next, we calculate the SD of intercrackle intervals, which is larger in Model A. Given the large variability, the means are not statistically different. If we did not know how these time series were generated, we could say the data perhaps came from similar processes. However, even visually, the nature of irregularities in the two time series appears different. Careful examination of the fluctuations reveals that the two processes are fundamentally different. To see this quantitatively, we estimate the distributions of intercrackle intervals for both models with 14 generations (Figure 2E). For Model A, the distribution is a power law (31), as evidenced by the linear decrease on a log–log graph. For Model B, the distribution is exponential, decreasing much faster than the straight line for Model A. Often we have access only to a fluctuating variable but not to the underlying process. The interpretation of these distributions is thus critical and is examined next.
In Model B, the absolute arrival times are uniformly distributed. The reason is that the pressure increases linearly with time, and hence, the arrival times are proportional to the opening pressures, which were uniformly distributed. The number of crackles emitted per unit time is similar to a Poisson process, and hence, the intercrackle intervals follow an exponential distribution (32), which is thus a direct consequence of the distribution of opening pressures. Alternatively, the presence of the tree structure in Model A fundamentally alters the character of the distribution. When an avalanche is triggered, crackles arrive in a rapid succession from segments that were involved in that avalanche, and we observe many short intervals. We also observe a few large intervals that correspond to the waiting times between avalanches. Therefore, the distribution will be high for small time intervals, but it will also have a few unexpectedly large time intervals (not observed in Model B), resulting in a power law distribution (31).
Model A provides a simple example of how a self-similar structure generates fluctuations that follow a power law distribution. The situation is more complicated in the lung. The distribution is also influenced by the time required for an airway to open (31), the asymmetry of the airway tree, and the fact that the distribution of opening pressures may not be uniform and may depend on the generation (33). In addition, as crackles propagate up the tree, the sound amplitude is attenuated at successive bifurcations. Interestingly, the distribution of crackle amplitudes measured in the bronchi is also a power law with an exponent related to airway geometry at bifurcations (29). Thus, to interpret the fluctuations correctly, we need a reasonable model of crackle generation and propagation over a realistic structure. If the model parameters can be estimated from measured data, the characteristics of an individual lung may be studied. Such approach may prove useful to study airway structure and airway closure in situ.
Airway opening via avalanches has implications for the interpretation of the lower knee on the inspiratory pressure–volume curve. The lower knee has generally been associated with recruitment (34–39). In most studies, recruitment is assumed to occur at the level of alveoli involving sequential openings of individual lung units (34, 37). This is exactly our Model B in which the recruited volumes are the volumes of the units attached to each tube. The distribution of recruited volumes is therefore very narrow (variability of acinar volume). However, if airways open via avalanches, the number of alveoli recruited should be influenced by the airway tree, a notion less appreciated in the medical literature. In Model A, the recruited volumes are distributed according to a power law with an exponent 2 (33), which has been indirectly confirmed from measurements of regional lung impedance in dogs (40). As an example, consider the size of a collapsed alveolar region that is recruited during inspiration. If the distribution of recruited volumes were Gaussian with mean and SD of 20 ± 10 alveoli, the alveoli would tend to open in groups of 10 to 30. The likelihood of finding a rare event, for example, 100 alveoli opening simultaneously, would be extremely small, 10−15. However, if the volume distribution is a power law with exponent 2 (40), the probability that 100 alveoli open simultaneously is 10 orders higher, approximately 10−5. Thus, the measured volumes do not correspond to the size of any known physiologic structure (e.g., acinus). Instead, the volume distribution represents a process, airway opening via avalanches (26), that generates a scale-free distribution.
An interesting implication is that if recruitment is influenced by airway structure, then the pressure–volume curve may carry information about the airway tree. Indeed, the pressure–volume curve of normal lungs during first inflation from the degassed state has been successfully modeled using symmetric (41) and asymmetric trees (42) providing a mechanistic interpretation of the lower knee. Whether airways open via avalanches in acute lung injury is unclear. Several factors suggest that this may not be the case. First, in the avalanche model, we assumed a uniform distribution of opening pressures. In acute lung injury, the distribution of opening pressures may be Gaussian (39), which alone should not exclude avalanche-like openings, provided the opening pressure distributions overlap from generation to generation (33). Unfortunately, such data are not available. Second, in acute lung injury, alveoli may be partially or fully flooded with fluid of high surface tension (43). A recent model of a single flooded alveolus has been shown to account for the lower knee of the pressure–volume curve (44). However, in acute lung injury, ventilation and recruitment are influenced by gravity (35), and hence, heterogeneity of ventilation could be important even at the level of alveoli. Indeed, it has been observed in a model of acute lung injury that alveoli near one another may inflate normally, pop open, or behave as the fluid-filled alveolus (45). Additional experiments are needed to clarify the nature of recruitment in the injured lung.
Before concluding, we examine a particularly interesting case where forcing fluctuations on a system may have clinical benefits. Although mechanical ventilators have been essential life-support systems for over 40 years, they can also injure the lungs or propagate injury. It has been argued that, although spontaneous physiologic fluctuations are essential features of living organisms, mechanical ventilators eliminate this inherent variability (46). Fluctuations were reintroduced in the form of “biologic variability” to mechanical ventilation, such that breathing rate was randomized and tidal volume was adjusted to keep minute ventilation constant. Compared with conventional ventilation, this approach improved arterial oxygenation in an oleic acid-induced injury model without increasing mean inspiratory pressures (46, 47). A mechanism has also been offered to explain how external fluctuations may improve blood oxygenation (48). The pressure–volume relationship of an atelectatic lung is highly nonlinear. When the peak airway pressure is fixed, the recruited lung volume is also stable and low (assuming a stable patient condition). When fluctuations in the form of symmetrically distributed random noise is added to peak airway pressure, called “noisy ventilation,” the mean does not change, but occasionally, peak airway pressures explore higher values resulting in additional recruitment (Figure 3)
Figure 3. Normalized pressure–volume curve in a model of an atelectatic region of the lung. With conventional mechanical ventilation, pressure is cycled between P1 = 0.3 and P2 = 0.7, and the recruited volume is V2. During noisy ventilation, end-inspiratory pressure is varied from inflation to inflation sampling the Gaussian around P2. In one inflation, pressure increases to 0.65 losing volume, whereas for the next inflation, pressure increases to 0.75 gaining recruited volume. Because of the nonlinearity of the pressure–volume curve, the “gain” is far greater than the “loss,” and the recruited volume available for gas exchange increases by ΔV from V2 to V3 (from Suki and colleagues with permission [48]).
[More] [Minimize]The concepts reviewed here may offer a unifying approach to deal with fluctuations that appear to invade medical research. Quantifying the fluctuations using distributions is important because they can help early detection of diseases or may improve prognosis of exacerbations. In some cases, it is also possible to understand these power laws in terms of a model that may shed light on the origins of the fluctuations and perhaps the mechanism of how a particular disease develops or propagates.
The author is grateful for helpful discussions with Adriano M. Alencar, Arnab Majumdar, Urs Frey, and Michael Lichtwarck-Aschoff.
Supported by grant BES-0114538 from the National Science Foundation and grant HL59215-04 from the National Institutes of Health.
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