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

64 Practical Applications in Biomedical Engineering

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

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].

effect upon ventilation-perfusion mismatch at the level of alveoli [68].

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.

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

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

scattergrams were constructed and coefficients of variation for both variables were determined for each 300 breath interval separately and for the whole 900 breaths. During CPAP trials, the Kolmogorov entropy indicating unpredictability [76] within the flowvolume loops was estimated, along with the dimension of the respiratory breathing pattern, defined as the number of clusters or clouds of trajectories in the flow-volume loops space, for each data interval. The authors found that the CV of tidal volume and PF, the Kolmogorov entropy and the fractal dimension of the spontaneous breathing pattern were all significantly smaller in the successfully weaning group compared with the failure weaning group.

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An Introduction to Complex Systems Theory Application in Pulmonary and Critical Care Medicine 67

the El-Khatib study [75] had a mean age of 50 years with predominantly underlying lung diseases and a longer duration of mechanical ventilation (11 days). Finally, in the Engoren study [72] cardiac surgery patients were included and indices of cardiac function were not presented, especially when low cardiac output states are associated with a Cheyne-Stokes

Recently, Wysocki and colleagues [78] examined breathing variability in 46 medical and surgical patients during 60 minutes spontaneous breathing trials (SBT) and during a 5 month period in four French tertiary university hospitals' ICUs. There were 32 successful and 14 failure weaning cases. SBT was performed with complete disconnection from the ventilator, meaning none form of ventilatory support. The time series analysis was performed by two independent investigators, one of whom was blinded to clinical information. Furthermore, a signal processing technique for checking stationarity of breath time series was performed and nonstationary data were excluded from analysis. The authors reported significantly increased CVs of tidal volume, Ti, Te, Ttot, Ti/Ttot and VT /Ti in the SW group, whereas the duration of autocorrelation (number of lags) was significantly shorter in the same group compared with patients who failed weaning trials, meaning that

low number of breaths were significantly autocorrelated in cases of weaning success.

limiting accuracy of comparisons with populations from the first two studies.

The different results from Engoren's and El-Khatib studies [72,75] were attributed to the prevailing conditions during respiratory variability analysis. According to Caminal and Brochard, different levels of pressure support may alter breath-to-breath variability in an inverse way [79,80], something that could have played a role in the two previous studies. Furthermore, it was postulated the low breathing variability could cause the deterioration of respiratory mechanics rather than result from it, by promoting microatelectasis. In conclusion, Wysocki's patients did not receive any ventilatory support during weaning trial,

In a recent study [81], we tried to investigate heart rate (HR) and respiratory rate complexity in patients with weaning failure or success, using both linear and nonlinear techniques. Forty-two surgical patients were enrolled in the study. There were 24 who passed and 18 who failed a weaning trial. Signals were analyzed for 10 minutes during two phases: 1. pressure support (PS) ventilation (15-20 cm H2O) and 2. weaning trials with PS: 5 cm H2O. Low and high frequency (LF, HF) components of HR signals, HR multiscale entropy (MSE), RR sample entropy, cross-sample entropy as an indication of coupling between cardiorespiratory signals and Poincaré plots were computed in all patients and during the two phases of PS. Multiscale entropy was recently introduced [82] for quantifying heart rate complexity. Briefly, for a given R-R time series, multiple 'coarsegraining' time series are constructed by averaging the data points within non-overlapping windows of increasing length τ, where τ represents the scale factor. Subsequently, sample entropy (SampEn) that represents the negative natural logarithm of the conditional probability that two sequences similar for m points remain similar at the next point with a tolerance r, is calculated for each time series and then plotted against the scale factor,

breathing pattern.

These results suggested that breath-to-breath variability during SIMV trials and complexity during CPAP trials were increased in patients who failed weaning trials. Patients from the two groups did not differ in terms of age, previous days on mechanical ventilation and blood gases at study entry, whereas the distribution of diagnosis was also similar between the two groups. Since the aim of this study was not to determine a cut-off threshold in terms of breathing pattern complexity for weaning prediction, no sensitivity or specificity were measured. Furthermore, the authors did not present any possible explanation of their results in terms of different pathophysiological mechanisms that may drive various levels of variability/ complexity.

In another study, Bien and colleagues [77] investigated breathing pattern variability in 78 mechanically ventilated patients with systemic inflammatory response syndrome (SIRS) who had undergone abdominal surgery. The patients were divided in the successfully weaning group (n=57) and the failure weaning group (n=21). Within 1 hour before the measurement of breathing variability and under ventilation with PS with a pressure level between 10 and 20 cm H2O, tidal volume, total breath duration (Ttot), Ti, Te and peak inspiratory flow were calculated. After obtaining these data, the ventilatory mode was switched to PS of 5 cm H2O plus 5 cm H2O PEEP for 30 minutes. The coefficients of variation and the SD1 and SD2 from the constructed Poincaré plots for the above variables were determined for at least 300 breaths. Moreover, AUC were computed for ROC curves and compared with airway occlusion pressure, its ratio to the maximal inspiratory pressure (P0.1/Pimax), RR, RR/VT, and the product P0.1\*( RR/VT). The authors found that average values and the CVs of the 5 measured parameters were significantly lower in the failure weaning group than in the successful one. In addition, Poincaré plot analysis showed that both SD1 and SD2 of the 5 breathing parameters were significantly reduced in the weaning failure group whereas the SD1-SD2 ratio showed no statistical significance. Finally, AUCs of the CVs of the 5 measured variables and both SD1 and SD2 did not differ significantly from those of the 5 clinically used weaning predictors. All the AUC values were within the range of 0.73 to 0.80.

This was the first study reporting that WF patients exhibited a decreased breath-to-breath variability and complexity. According to the authors, the different results from the two previous studies may be due to different weaning protocols that were implemented and different patient characteristics. Their patients had a mean age of 68 years, no history of pulmonary disease and a short duration of ventilatory support (3 days) whereas patients in the El-Khatib study [75] had a mean age of 50 years with predominantly underlying lung diseases and a longer duration of mechanical ventilation (11 days). Finally, in the Engoren study [72] cardiac surgery patients were included and indices of cardiac function were not presented, especially when low cardiac output states are associated with a Cheyne-Stokes breathing pattern.

66 Practical Applications in Biomedical Engineering

weaning group.

variability/ complexity.

of 0.73 to 0.80.

scattergrams were constructed and coefficients of variation for both variables were determined for each 300 breath interval separately and for the whole 900 breaths. During CPAP trials, the Kolmogorov entropy indicating unpredictability [76] within the flowvolume loops was estimated, along with the dimension of the respiratory breathing pattern, defined as the number of clusters or clouds of trajectories in the flow-volume loops space, for each data interval. The authors found that the CV of tidal volume and PF, the Kolmogorov entropy and the fractal dimension of the spontaneous breathing pattern were all significantly smaller in the successfully weaning group compared with the failure

These results suggested that breath-to-breath variability during SIMV trials and complexity during CPAP trials were increased in patients who failed weaning trials. Patients from the two groups did not differ in terms of age, previous days on mechanical ventilation and blood gases at study entry, whereas the distribution of diagnosis was also similar between the two groups. Since the aim of this study was not to determine a cut-off threshold in terms of breathing pattern complexity for weaning prediction, no sensitivity or specificity were measured. Furthermore, the authors did not present any possible explanation of their results in terms of different pathophysiological mechanisms that may drive various levels of

In another study, Bien and colleagues [77] investigated breathing pattern variability in 78 mechanically ventilated patients with systemic inflammatory response syndrome (SIRS) who had undergone abdominal surgery. The patients were divided in the successfully weaning group (n=57) and the failure weaning group (n=21). Within 1 hour before the measurement of breathing variability and under ventilation with PS with a pressure level between 10 and 20 cm H2O, tidal volume, total breath duration (Ttot), Ti, Te and peak inspiratory flow were calculated. After obtaining these data, the ventilatory mode was switched to PS of 5 cm H2O plus 5 cm H2O PEEP for 30 minutes. The coefficients of variation and the SD1 and SD2 from the constructed Poincaré plots for the above variables were determined for at least 300 breaths. Moreover, AUC were computed for ROC curves and compared with airway occlusion pressure, its ratio to the maximal inspiratory pressure (P0.1/Pimax), RR, RR/VT, and the product P0.1\*( RR/VT). The authors found that average values and the CVs of the 5 measured parameters were significantly lower in the failure weaning group than in the successful one. In addition, Poincaré plot analysis showed that both SD1 and SD2 of the 5 breathing parameters were significantly reduced in the weaning failure group whereas the SD1-SD2 ratio showed no statistical significance. Finally, AUCs of the CVs of the 5 measured variables and both SD1 and SD2 did not differ significantly from those of the 5 clinically used weaning predictors. All the AUC values were within the range

This was the first study reporting that WF patients exhibited a decreased breath-to-breath variability and complexity. According to the authors, the different results from the two previous studies may be due to different weaning protocols that were implemented and different patient characteristics. Their patients had a mean age of 68 years, no history of pulmonary disease and a short duration of ventilatory support (3 days) whereas patients in Recently, Wysocki and colleagues [78] examined breathing variability in 46 medical and surgical patients during 60 minutes spontaneous breathing trials (SBT) and during a 5 month period in four French tertiary university hospitals' ICUs. There were 32 successful and 14 failure weaning cases. SBT was performed with complete disconnection from the ventilator, meaning none form of ventilatory support. The time series analysis was performed by two independent investigators, one of whom was blinded to clinical information. Furthermore, a signal processing technique for checking stationarity of breath time series was performed and nonstationary data were excluded from analysis. The authors reported significantly increased CVs of tidal volume, Ti, Te, Ttot, Ti/Ttot and VT /Ti in the SW group, whereas the duration of autocorrelation (number of lags) was significantly shorter in the same group compared with patients who failed weaning trials, meaning that low number of breaths were significantly autocorrelated in cases of weaning success.

The different results from Engoren's and El-Khatib studies [72,75] were attributed to the prevailing conditions during respiratory variability analysis. According to Caminal and Brochard, different levels of pressure support may alter breath-to-breath variability in an inverse way [79,80], something that could have played a role in the two previous studies. Furthermore, it was postulated the low breathing variability could cause the deterioration of respiratory mechanics rather than result from it, by promoting microatelectasis. In conclusion, Wysocki's patients did not receive any ventilatory support during weaning trial, limiting accuracy of comparisons with populations from the first two studies.

In a recent study [81], we tried to investigate heart rate (HR) and respiratory rate complexity in patients with weaning failure or success, using both linear and nonlinear techniques. Forty-two surgical patients were enrolled in the study. There were 24 who passed and 18 who failed a weaning trial. Signals were analyzed for 10 minutes during two phases: 1. pressure support (PS) ventilation (15-20 cm H2O) and 2. weaning trials with PS: 5 cm H2O. Low and high frequency (LF, HF) components of HR signals, HR multiscale entropy (MSE), RR sample entropy, cross-sample entropy as an indication of coupling between cardiorespiratory signals and Poincaré plots were computed in all patients and during the two phases of PS. Multiscale entropy was recently introduced [82] for quantifying heart rate complexity. Briefly, for a given R-R time series, multiple 'coarsegraining' time series are constructed by averaging the data points within non-overlapping windows of increasing length τ, where τ represents the scale factor. Subsequently, sample entropy (SampEn) that represents the negative natural logarithm of the conditional probability that two sequences similar for m points remain similar at the next point with a tolerance r, is calculated for each time series and then plotted against the scale factor, giving rise to the MSE curve. The sum of SampEn over all scaling factors represents the MSE of a signal. For our MSE analysis, the parameter r was set at 15% of standard deviation (SD) of the time series, after normalization (SD=1) and the parameter m (embedding dimension), that is the length of sequences to be compared was set at 2 (data length ranging from 100 to 5000 data points) [82]. Furthermore, we extracted two more features from MSE curves after log transformation of SampEn and scale factor, the fast slopes for small time scales, i.e., those defined by SampEn values between scale 1 and 5 and the long slopes for higher scale values (Figures 1 & 2).

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

An Introduction to Complex Systems Theory Application in Pulmonary and Critical Care Medicine 69

**Figure 2.** Heart rate multiscale entropy (MSE) curves between phase H (high PSV) and spontaneous breathing trial (SBT) or phase L (low PSV) in a weaning success patient. The MSE curve during the SBT has increased SampEn values; moreover, the slope during small scales (fast slope) looks similar with the

Different profiles of MSE curves on small time scales between the two groups and according to fast slopes may be due to the impact of respiratory rhythm on heart rate dynamics (respiratory sinus arrhythmia). In general, the lower the amplitude of respiratory modulation, the higher the entropy values tend to be [82]. Thus, in both groups but especially in weaning failure patients, sum MSE was significantly increased between different PS levels, whereas reduced fast slopes could reflect different time constants of

In another similar study [83], we enrolled thirty-two mechanically ventilated patients undertaking a weaning trial. There were 22 who passed and 10 who failed a weaning trial. Tidal volume and mean inspiratory flow were analyzed for 10 minutes during two phases: 1. pressure support (PS) ventilation (15-20 cm H2O) and 2. weaning trials with PS: 5 cm H2O. Sample entropy (SampEn), fractal dimension (FD) and largest lyapunov exponents (LLE) of the two respiratory parameters were computed in all patients and during the two phases of PS. FD was computed according to the Higuchi method [84]. Briefly, FD was based on a measure of length L(k) of a time series, computed at different scales, by using a segment of k samples as a unit in each scale. The value of FD was calculated by a least-squares linear bestfitting procedure as the angular coefficient of the linear regression of the log-log graph of the

slope during the high PSV phase.

respiratory effect upon heart rate control mechanisms.

We found that weaning failure patients exhibited significantly decreased RR sample entropy, LF and HF components, compared with weaning success subjects (p<0.001). Their changes were opposite between the two phases, except for MSE that increased between and within groups (p<0.001). A new model including rapid shallow breathing index (RSBI), RR and cross sample entropies predicted better weaning outcome compared with RSBI, airway occlusion pressure at 0.1 sec (P0.1) and RSBI\* P0.1 (conventional model, R2 = 0.887 vs 0.463, p<0.001). Areas under the curve were 0.92 vs 0.86, respectively (p<0.005).

**Figure 1.** Heart rate multiscale entropy (MSE) curves between phase H (high PSV) and spontaneous breathing trial (SBT) or phase L (low PSV) in a weaning failure patient. SampEn and scale factor have been logarithmically trans-formed. The MSE curve during the SBT has increased SampEn values; however, the slope during small scales (fast slope) is decreased compared with the high PSV phase.

and the long slopes for higher scale values (Figures 1 & 2).

p<0.001). Areas under the curve were 0.92 vs 0.86, respectively (p<0.005).

giving rise to the MSE curve. The sum of SampEn over all scaling factors represents the MSE of a signal. For our MSE analysis, the parameter r was set at 15% of standard deviation (SD) of the time series, after normalization (SD=1) and the parameter m (embedding dimension), that is the length of sequences to be compared was set at 2 (data length ranging from 100 to 5000 data points) [82]. Furthermore, we extracted two more features from MSE curves after log transformation of SampEn and scale factor, the fast slopes for small time scales, i.e., those defined by SampEn values between scale 1 and 5

We found that weaning failure patients exhibited significantly decreased RR sample entropy, LF and HF components, compared with weaning success subjects (p<0.001). Their changes were opposite between the two phases, except for MSE that increased between and within groups (p<0.001). A new model including rapid shallow breathing index (RSBI), RR and cross sample entropies predicted better weaning outcome compared with RSBI, airway occlusion pressure at 0.1 sec (P0.1) and RSBI\* P0.1 (conventional model, R2 = 0.887 vs 0.463,

**Figure 1.** Heart rate multiscale entropy (MSE) curves between phase H (high PSV) and spontaneous breathing trial (SBT) or phase L (low PSV) in a weaning failure patient. SampEn and scale factor have been logarithmically trans-formed. The MSE curve during the SBT has increased SampEn values; however, the slope during small scales (fast slope) is decreased compared with the high PSV phase.

**Figure 2.** Heart rate multiscale entropy (MSE) curves between phase H (high PSV) and spontaneous breathing trial (SBT) or phase L (low PSV) in a weaning success patient. The MSE curve during the SBT has increased SampEn values; moreover, the slope during small scales (fast slope) looks similar with the slope during the high PSV phase.

Different profiles of MSE curves on small time scales between the two groups and according to fast slopes may be due to the impact of respiratory rhythm on heart rate dynamics (respiratory sinus arrhythmia). In general, the lower the amplitude of respiratory modulation, the higher the entropy values tend to be [82]. Thus, in both groups but especially in weaning failure patients, sum MSE was significantly increased between different PS levels, whereas reduced fast slopes could reflect different time constants of respiratory effect upon heart rate control mechanisms.

In another similar study [83], we enrolled thirty-two mechanically ventilated patients undertaking a weaning trial. There were 22 who passed and 10 who failed a weaning trial. Tidal volume and mean inspiratory flow were analyzed for 10 minutes during two phases: 1. pressure support (PS) ventilation (15-20 cm H2O) and 2. weaning trials with PS: 5 cm H2O. Sample entropy (SampEn), fractal dimension (FD) and largest lyapunov exponents (LLE) of the two respiratory parameters were computed in all patients and during the two phases of PS. FD was computed according to the Higuchi method [84]. Briefly, FD was based on a measure of length L(k) of a time series, computed at different scales, by using a segment of k samples as a unit in each scale. The value of FD was calculated by a least-squares linear bestfitting procedure as the angular coefficient of the linear regression of the log-log graph of the

mean of k values Lm(k) for m=1,2,3…k, with k being an interval time. The length Lm(k) originating from time m was calculated as the normalized sum of absolute differences between the values of point pairs that are 'k samples distant' and the length of curve of the time interval k, L(k) was calculated as the mean of the k values Lm(k). If the L(k) related to the scale used (k) linearly in a log-log plot with slope FD, then the curve was said to show fractal dimension.

LLE were computed as follows: Complex systems are considered sensitive to initial conditions and exhibit an exponential divergence in the phase space, which describes in a 3 dimensional axis their different states. Estimation of Lyapunov spectrum and largest Lyapunov exponents (LLE) can assess sensitivity to initial conditions. Briefly, if we consider two points in adjacent trajectories-states of the phase space with a distance between them d(0), after time t the average divergence (separation) will be:

$$\mathbf{Id}\left(\mathbf{t}\right) = \mathbf{d}\left(\mathbf{0}\right) \* \mathbf{e}^{\text{LLE\*(i\"alt)}}\text{[}\tag{2}$$

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

An Introduction to Complex Systems Theory Application in Pulmonary and Critical Care Medicine 71

**Figure 3.** Phase space of minute ventilation (MV) in a weaning success patient. Different values x(i) of minute ventilation were plotted against the following ones in tree-dimensional axes: x, x+t and x+2t, giving rise to the phase space. Red dots represent data during high PS ventilation, whereas blue dots

**Figure 4.** Phase space of minute ventilation in a weaning failure patient. Red dots represent data during high PS ventilation, whereas blue dots represent data during the performance of a SBT. Scattering of data seem highly reduced, compared with findings in Figure 3, whereas ventilation values are

represent data during the performance of a SBT.

positioned in different parts of the space.

whereas LLE is the largest Lyapunov exponent. In this study, we computed LLE of mean inspiratory flow and tidal volume signals, using the algorithm proposed by Rosenstein [85], which seems to be useful, particularly in small data sets. Values higher than 0 reflect an unstable and unpredictable system, where nearby points will diverge to any arbitrary separation. Increased LLEs reflect increased sensitivity to initial conditions and characterize unpredictable variations, whereas low values indicate regularity [85].

Weaning failure patients exhibited significantly decreased respiratory pattern complexity, reflected in reduced sample entropy and lyapunov exponents of respiratory flow time series, compared to weaning success subjects (p<0.001). In addition, their changes were opposite between the two phases of the weaning trials. A new model including rapid shallow breathing index (RSBI), its product with airway occlusion pressure at 0.1 sec (P0.1), SampEn and LLE predicted better weaning outcome compared with RSBI, P0.1 and RSBI\* P0.1 (conventional model, R2 = 0.874 vs 0.643, p<0.001). Areas under the curve were 0.916 vs 0.831, respectively (p<0.05).

Vallverdu et al. [86] in another study with similar design compared with our two previous investigations examined heart rate and respiratory pattern complexity in 78 patients, during weaning trials and using information flow analysis, which describes the regularity of signals by estimating the auto- and mutual information functions. The authors were able to find reduced complexity and a more coupled nonlinear oscillator behavior in weaning failure subjects.

These results parallel those from Schmidt and colleagues [87] who reported increased LLE and Kolmogorov-Sinai entropy values of mean inspiratory flow signals in mechanically ventilated patients, after switching the ventilator from the pressure support mode to neurally adjusted ventilatory assist mode (NAVA). According to these authors, successful spontaneous breathing trials unmask underlying variability and complexity of central neural output, since inspiratory pressure inhibits the respiratory drive. This effect is nicely reflected through the increased complexity indices of flow and is responsible for better neuro-mechanical coupling.

0.831, respectively (p<0.05).

neuro-mechanical coupling.

subjects.

mean of k values Lm(k) for m=1,2,3…k, with k being an interval time. The length Lm(k) originating from time m was calculated as the normalized sum of absolute differences between the values of point pairs that are 'k samples distant' and the length of curve of the time interval k, L(k) was calculated as the mean of the k values Lm(k). If the L(k) related to the scale used (k) linearly in a log-log plot with slope FD, then the curve was said to show fractal dimension.

LLE were computed as follows: Complex systems are considered sensitive to initial conditions and exhibit an exponential divergence in the phase space, which describes in a 3 dimensional axis their different states. Estimation of Lyapunov spectrum and largest Lyapunov exponents (LLE) can assess sensitivity to initial conditions. Briefly, if we consider two points in adjacent trajectories-states of the phase space with a distance between them

whereas LLE is the largest Lyapunov exponent. In this study, we computed LLE of mean inspiratory flow and tidal volume signals, using the algorithm proposed by Rosenstein [85], which seems to be useful, particularly in small data sets. Values higher than 0 reflect an unstable and unpredictable system, where nearby points will diverge to any arbitrary separation. Increased LLEs reflect increased sensitivity to initial conditions and characterize

Weaning failure patients exhibited significantly decreased respiratory pattern complexity, reflected in reduced sample entropy and lyapunov exponents of respiratory flow time series, compared to weaning success subjects (p<0.001). In addition, their changes were opposite between the two phases of the weaning trials. A new model including rapid shallow breathing index (RSBI), its product with airway occlusion pressure at 0.1 sec (P0.1), SampEn and LLE predicted better weaning outcome compared with RSBI, P0.1 and RSBI\* P0.1 (conventional model, R2 = 0.874 vs 0.643, p<0.001). Areas under the curve were 0.916 vs

Vallverdu et al. [86] in another study with similar design compared with our two previous investigations examined heart rate and respiratory pattern complexity in 78 patients, during weaning trials and using information flow analysis, which describes the regularity of signals by estimating the auto- and mutual information functions. The authors were able to find reduced complexity and a more coupled nonlinear oscillator behavior in weaning failure

These results parallel those from Schmidt and colleagues [87] who reported increased LLE and Kolmogorov-Sinai entropy values of mean inspiratory flow signals in mechanically ventilated patients, after switching the ventilator from the pressure support mode to neurally adjusted ventilatory assist mode (NAVA). According to these authors, successful spontaneous breathing trials unmask underlying variability and complexity of central neural output, since inspiratory pressure inhibits the respiratory drive. This effect is nicely reflected through the increased complexity indices of flow and is responsible for better

\*( ) [ \* ] **LLE iΔ<sup>t</sup> dt d0 e** (2)

d(0), after time t the average divergence (separation) will be:

unpredictable variations, whereas low values indicate regularity [85].

**Figure 3.** Phase space of minute ventilation (MV) in a weaning success patient. Different values x(i) of minute ventilation were plotted against the following ones in tree-dimensional axes: x, x+t and x+2t, giving rise to the phase space. Red dots represent data during high PS ventilation, whereas blue dots represent data during the performance of a SBT.

**Figure 4.** Phase space of minute ventilation in a weaning failure patient. Red dots represent data during high PS ventilation, whereas blue dots represent data during the performance of a SBT. Scattering of data seem highly reduced, compared with findings in Figure 3, whereas ventilation values are positioned in different parts of the space.

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

An Introduction to Complex Systems Theory Application in Pulmonary and Critical Care Medicine 73

their environment (open systems): from near to equilibrium, like crystals to far from equilibrium systems, like the weather. 'The amount of energy dissipated determines where the system is situated along the continuum. Between the crystals and weather a sudden phase transition occurs over a small range of energy consumption. It is only there where life can flourish'. From the above statement it is concluded that either a decrease in energy consumption, i.e., during myocardial ischemia, or an increase may both contribute to a shift away from stable state. It has been shown by Que [60] that during asthma, the local increase in metabolic rate is associated with increased variability of respiratory impedance to flow. The same author has proposed the term 'homeokinesis' instead of homeostasis [60], as a basic property of living systems that 'describes the ability of an organism to utilize external energy sources to maintain a highly organized internal environment fluctuating within

There is good basic science describing the physiological and fractal variability observed in respiratory dynamics and substantial evidence that conditions prevalent in the ICU impacts breath-to-breath variability. Different clinical studies have assessed the ability of markers of respiratory variability as predictors of successful separation from mechanical ventilation. However, these studies are not comparable since the authors implemented different weaning protocols in heterogeneous groups of patients and examined various indices of both breathing pattern variability (linear) and complexity (nonlinear) that by their nature capture different aspects of signal dynamics. It seems that the study of Wysocki [78] performed better, since it was conducted in 5 medical centres, stationarity tests were done and possible bias in the time series analysis was significantly controlled. Furthermore, lack of ventilatory support during weaning trial could have eliminated any possible impact of positive pressure ventilation upon breathing variability and could be responsible for the different findings between this and the first two studies. Unfortunately, the authors did not perform complexity nonlinear analysis. Since neither ApEn, nor any other complexity index was calculated, we cannot make assumptions regarding inherent breathing dynamics during discontinuation from mechanical ventilation, because variability and complexity

We suggest that new studies are needed, for estimating both linear and non-linear indices of breathing variability and complexity for weaning outcome assessment, in different and homogeneous groups of patients, and comparing their diagnostic accuracy, along with other traditional and already used physiological predictors. Their association, in terms of 'coupling' or 'uncoupling' with other biosignals such as the electroencephalogram (EEG), with the aid of robust methods from signal processing theory (i.e., cross-ApEn, coherence etc) could uncover possible inter-relations between breathing and central nervous system

However, we think that the development of a consensus between clinicians and modelers is urgently needed for the assessment of different methods used for breathing pattern variability prior to the performance of other studies. In the cardiology literature, the Task Force on HRV has standardized the use of a set of signal processing techniques, including target variables, frequency and duration of measurements and signal quality assessment,

acceptable limits in a far from equilibrium state'.

may change in opposite directions.

dynamics during weaning trials.

**Figure 5.** Distribution of LLE between the two groups of patients (S: successful weaning and U: unsuccessful), during high (H) and low (L) pressure support ventilation. It seems that SW group demonstrated an increase in LLE after shifting to spontaneous breathing, contrary to what have occurred to the FW group.

In another study, Mangin and colleagues [88] investigated ventilatory chaotic dynamics in 17 mechanically ventilated patients during switching the ventilator from the assist-control mode to pressure support mode. They were able to show that both fractal dimension and LLE were increased, particularly in 5 patients who were successfully extubated. Furthermore, the authors supposed that increased breathing complexity may also be attributed to higher vagal afferent feedback during unassisted breathing, as has already been shown by Sammon and Bruce [40].

These studies support our findings that transition between mechanical and spontaneous ventilation is associated with increased complexity of respiratory signals in weaning success patients, since duration of ventilation before the SBTs was similar between groups with different weaning outcome. Moreover, in a study of Burykin and Buchman [89] investigating cardiorespiratory dynamics and synchronization during controlled and unassisted breathing in 13 surgical patients, it was demonstrated that mechanical ventilation reduces significantly both heart and respiratory rate complexity whereas spontaneous respiration is more irregular with increased uncoupling of cardiorespiratory rhythms in weaning success patients.
