**5. Fractal properties of the lung in disease: data from clinical and experimental studies**

Alterations to fractal properties are related to different pathologies and could have clinical implications for diagnosis and treatment. Physiologic time series, such as heart and respiratory signals, show similar alterations in their power law behavior in different disease states. Mackey and Glass [56] have introduced the term 'dynamic diseases' to describe states with loss of fractal properties of organs and power law dynamics of signals that are produced form the above structures. For example, loss of heart rate variability that is the variability of the R-R in the electrocardiogram, has been found in patients with heart failure [57], atrial fibrillation, septic shock and multiple organ dysfunction syndrome [6,26]. In respiratory disorders, a classical example is the highly periodic variation in respiratory frequency, seen in Cheyne-Stokes respiration. Penzel [58] has observed loss of fractal properties of heart rate signals during episodes of obstructive sleep apnea.

Macklem [59] was the first who raised the question of whether airway function can be studied using tools from chaos theory and the paradigm of complex systems. Que and coworkers [60] studied the distribution of forced oscillatory resistance in asthmatics and demonstrated that lung function exhibit loss of fractal properties during severe asthma. Frey and colleagues [61] applied fractal methods to twice-daily peak expiratory flow (PEF) in asthmatic patients and showed that the β slope was reduced, whereas it become more regular with standard long-acting β2-agonist treatment and more random with short-acting β2-agonist treatment, respectively. Moreover, the authors were able to demonstrate that the higher the β exponent when a patient was not under any treatment, the larger the improvement of his condition upon administration of long-acting β2-agonist therapy.

62 Practical Applications in Biomedical Engineering

can adapt more easily to their surroundings.

power law [51].

decreases with aging.

**experimental studies** 

probability distribution of airway opening during inspiration behaves also according to the

Another property of fractals and power laws in pulmonary physiology is error tolerance during development. In simulations of airway morphogenesis during lung development, West [52] compared a power law branching rule with an exponential decaying one and found that in the first case, the system was less susceptible to errors introduced into the branching process. These same properties suggest that living systems are capable to operate similarly at different scales, meaning that whenever environmental conditions change they

Aging has been proven by Lipsitz and Goldbereger [30] to be significantly associated with loss of complexity of physiological signals, leading to decreased ability to adapt to different physiological insults. Using different algorithms for estimating fractal properties and power law behavior, these authors found that the β slope of different signals in elderly was either reduced (decreased lower than 1) or augmented (increased higher than 1) compared to younger adults, indicating chaotic or periodic behavior, respectively. Peng and co-workers [53] showed that aging was associated with a breakdown of fractal dynamics of respiratory signals via a decrease in β slope towards 0.5 (randomness). Concerning early stages of development in humans, one study [54] has found that ultrasonographic patterns for assessment lung maturity showed fractal properties with a power law behavior. In addition, the β slope increased with gestational age from 28 to 38 weeks. Szeto and co-workers [55], calculated β slopes of different respiratory signals in human fetus and showed its movement from randomness towards fractal behavior with gestational age. In conclusion, it seems that there is great variability in complexity with age in early life, after which complexity

**5. Fractal properties of the lung in disease: data from clinical and** 

properties of heart rate signals during episodes of obstructive sleep apnea.

Alterations to fractal properties are related to different pathologies and could have clinical implications for diagnosis and treatment. Physiologic time series, such as heart and respiratory signals, show similar alterations in their power law behavior in different disease states. Mackey and Glass [56] have introduced the term 'dynamic diseases' to describe states with loss of fractal properties of organs and power law dynamics of signals that are produced form the above structures. For example, loss of heart rate variability that is the variability of the R-R in the electrocardiogram, has been found in patients with heart failure [57], atrial fibrillation, septic shock and multiple organ dysfunction syndrome [6,26]. In respiratory disorders, a classical example is the highly periodic variation in respiratory frequency, seen in Cheyne-Stokes respiration. Penzel [58] has observed loss of fractal

Macklem [59] was the first who raised the question of whether airway function can be studied using tools from chaos theory and the paradigm of complex systems. Que and coworkers [60] studied the distribution of forced oscillatory resistance in asthmatics and In another study, Suki and co-workers [51] studied the dynamics of airway opening and crackles, using a simple mathematic model of the periphery of airway tree. Suki found that the time series of crackles emitted during airway opening follows a power law distribution. Additionally, as the crackles propagate up the tree, the sound amplitude is attenuated at successive bifurcations, whereas its distribution follows the power law. The same has been found for the time intervals of the 'jumps' by which airway resistance decreases upon lung inflation by a constant flow. In a study of Boser and colleagues [62], the fractal dimension of airways was computed using autopsy material from three groups: fatal asthma, non-fatal asthma and non-asthma controls. The authors were able to show that the average FDs of both fatal (1.72) and non-fatal asthma groups (1.76) were significantly lower than that of the third control group (1.83, p<0.05), whereas the lower fractal dimension correlated with a decreased overall structural complexity and pathologic severity of disease.

Venegas and colleagues [63], using positron emission tomography (PET) imaging and computer modeling showed that in cases of bronchoconstriction and when smooth muscle activation reaches a critical level, localized clusters of poorly ventilated lung regions can develop abruptly in discrete steps. These steps are called *avalanches* and can lead to new stable conditions. Because of the fractal structure of the airways, small initial heterogeneities that are always present and particularly in the diseased lung, can be amplified, leading to sudden patches of poorly ventilated lung regions. Another implication is that since airways are organized into a fractal network embedded in the elastic parenchyma, the constriction of one airway can propagate and cause an avalanche-like constriction in large parts of the lung. The same holds true for the opposite process, where opening of airways during inhalation takes place in discrete steps [63-65].

Suki [51] has also demonstrated that airway opening upon inflation occur in avalanches with power law distribution of both the size and time intervals between them. The significance of these results is that the probability of finding a large avalanche is much higher than it would be if the distribution were Gaussian or exponential, so both the magnitude and timing of pressure excursions applied at the airways (i.e., using mechanical ventilation) may be critical in triggering the avalanche process of alveolar recruitment [24].

In conclusion, these studies in asthma show that when the airways are likely to approach their critical closing threshold pressure, a small stimulus can provoke a catastrophic cascade of airway closure and for that reason, there is such poor correlation between the trigger and the outcome in asthmatic patients. Moreover, the history of symptom fluctuations seems related to the structural changes of the airway tree (power law distribution of airway diameter) [65].

Airway recruitment may affect alveolar recruitment as well. Sujeer and co-workers [66] have found in mathematical models that the recruited volumes upon inflation with constant flow are distributed according to a power law with a β slope equal to 2. From the above findings it can be supposed that since alveolar recruitment is influenced by airway structure, then the pressure-volume curve may carry information about the airway tree [24]. Whether such models have any value in acute lung injury (ALI) is unclear. In this syndrome, it has been found that the opening pressure distribution does not seem to be always Gaussian, something that is assumed to be the case in the avalanche model [67]. More studies are needed to investigate the pattern of recruitment in ALI, particularly in the case of gravity effect upon ventilation-perfusion mismatch at the level of alveoli [68].

Fractal Physiology, Breath-to-Breath Variability and Respiratory Diseases:

An Introduction to Complex Systems Theory Application in Pulmonary and Critical Care Medicine 65

**6. Breathing variability and complexity indices as weaning predictors in** 

Engoren and colleagues [72] studied 10 control patients who had undergone cardiac surgery within an interval of 12 hours prior to this experiment and 21 patients who required prolonged (> 7 days) ventilatory support. The control group was studied during a weaning trial of 5 cm H2O continuous positive airway pressure (CPAP). The patient group was studied during 60-to-120 min trials of spontaneous ventilation with 5 cm H2O positive endexpiratory pressure (PEEP) with a constant (12.2 +/- 6.6 cm H2O) level of pressure support (PS) for each trial. These patients passed 59 and failed 14 weaning trials. During spontaneous ventilation each breath's instantaneous respiratory rate and tidal volume were recorded and their ApEn values were calculated for the terminal 1000 breaths, in a series of 100, 300 and 1000 breaths. Receiver operating characteristic (ROC) curves identified cutpoint values of continuous variables that predicted a failed weaning outcome. While mean VT did not vary between groups, mean RR and frequency-tidal volume ratio increased progressively from the control group to both successful weaning (SW) and failure weaning (FW) groups. Conversely, ApEn of RR did not vary between every pair groups, but entropy of VT increased significantly from the control to FW group and from SW to group FW, with no difference between control and SW groups. The ROC curves for ApEn-VT and frequencytidal volume ratio showed similar sensitivity and specificity for predicting weaning failure whereas ApEn-VT proved to be successful in separating SW from FW, with similar area under the curve (AUC) values of 0.74, 0.75 and 0.73 for 100, 300 and 1000 breaths respectively. Since 100 breaths during a weaning trial may take only 3 to 4 minutes compared with 30 to 40 minutes required for a 1000 breaths trial, ApEn-VT assessment can

be of significant value, permitting a faster determination of weaning tolerance.

responsible for different effects of cardiac and pulmonary disease on regularity.

According to the authors [72], increased irregularity of the FW group indicated enhanced external inputs to the respiratory controller whereas increased regularity of the SW group suggested greater component autonomy. The first case implies that enhanced external inputs from peripheral receptors that measure pH, Pco2, Po2 or from cortical areas that represent dyspnea or anxiety are responsible for increasing breathing complexity [19,20]. Conversely, many studies estimating heart rate variability have found inverse relations between mortality after myocardial infarction or endotoxemia and HRV [73,74]. It seems that patient's voluntary control over breathing (which does not exist for heart rate) is

El-Khatib et al. [75] studied 52 patients diagnosed mostly with lung diseases and under mechanical ventilation in two phases: 1. under synchronized intermittent mandatory ventilation (SIMV) with RR ≤ 4 breaths/min and with no PS for 60 minutes and 2. during a CPAP trial of 5 cm H2O with no PS ventilation for other 60 minutes. Thirty-nine patients (75%) were successfully extubated and the remaining 13 patients failed weaning trial. In both patient groups, different breathing signals were collected (airway flow, volume, proximal airway pressure) and were divided into three intervals of 300 breaths. During SIMV trials, the spontaneous and mechanical peak flows (PF) versus tidal volume

**mechanically ventilated patients** 

The application of fractal analysis has also shed light into the morphology of the lung in cases of emphysema. Computed tomography (CT) is a sensitive method for assessing lung structure in different pathologies. In general, low attenuation area (LAA) clusters are depicted in pixels with density less than 950 Hounsfield units. These areas incorporate mostly air and assigned a value of 1, whereas pixels with a density higher than 950 include tissue with a value 0. Summing the number of pixels in a cluster gives the cluster size. In that way, a binary map of the lung can be constructed and a few studies have shown that in normal conditions, this map is highly heterogeneous [24]. Mishima and colleagues [69] have found that the probability distribution of LAA clusters follows a power law for both normal subjects and patients with chronic obstructive pulmonary disease (COPD). However, patients exhibited significantly smaller β slopes or exponents, which did not correlate with pulmonary function tests except for diffusion capacity of the lung. The authors suggested that the neighboring smaller LAA clusters tend to coalesce and form larger clusters as the weak elastic fibers separating them break under tension. This process does not change the % LAA but decreases the number of small clusters in favor of larger ones, which result in a reduction of the β slope. Another assumption derived from this study is that the likelihood of finding large LAA clusters is much higher in COPD patients than in normal controls [65].

Another possible application of fractals in pulmonary and critical care medicine seems to include the mechanical ventilation of critically ill patients. In an oleic acid injury animal model, Mutch and colleagues [70] introduced fluctuations according to an algorithm, to mechanical ventilation (biological variable tidal volume and respiratory frequency proportional to pre-defined minute ventilation values). Compared with conventional ventilation (with similar minute ventilation), this approach increased respiratory arrhythmia and oxygenation and decreased dead space. According to Suki [24,71], when fluctuations in the form of symmetrically distributed random noise is added to peak airway pressures (noisy ventilation), the mean does not change but isolated values can be augmented, leading to significant alveolar recruitment. In a mathematical model, the authors found that the recruited lung can be 200% larger in the case of biological variable ventilation than during conventional ventilation. Moreover, the standard deviation (SD) of the noise can be manipulated in order to achieve better oxygenation (system's output), a phenomenon called 'stochastic resonance', which has already been confirmed in animal models of ALI [71].
