**4. Atrial activity analysis**

6 Atrial Fibrillation

the time series [24].

after a certain *L*. Nonetheless, this algorithm requires a corrective term to estimate accurately the CE. The correction is thought to counteract the bias toward a reduction of the CE which occurs increasing the size of the conditioning vectors and depends strongly on the length of

It is interesting to remark that a slightly modified version of the CE, such as *Cross-CE* (CCE), is able to assess the coupling degree between two time series [24]. Synchronization occurs when interactive dynamics between two signals are repetitive. In this line, this index computes the amount of information included in the most recent sample of a times series when the past *L*-sample-length pattern of the other series is given. Given that CCE suffers from the same

Finally, other measure proposed to estimate coupling between time series is the *causal entropy* (CauEn) [25]. This index is an asymmetric, time-adaptive, event-based measure of the regularity of the phase- or time-lag with which point *i* fires after point *j*. It is calculated from two components: a non-parametric time-adaptive estimate of the probability density of spike time lag between two points *i* and *j* such that *i* follows *j* (and, independently, the distribution of *j* following *i*), and a cost function estimate of the spread and stability of the distribution. Although a variety of alternatives exists to compute this metric, CauEn can be easily estimated by choosing an event-normalized histogram as the time-adaptive density

A *Recurrence plot* (RP) is a visual representation of all the possible distances between the points constituting the phase space of a time series [21]. Whenever the distance between two points is below a certain threshold, there is a recurrence in the dynamics: i.e. the dynamical system visited multiple times a certain area of the phase space. From this transformation, well suited for the study of short non-stationary signals, many geometric features can be extracted. In this sense, there are four main elements characterizing a RP: isolated points (reflecting stochasticity in the signal), diagonal lines (index of determinism) and horizontal/vertical lines (reflecting local stationarity in the signal). The combination of these elements creates large-scale and small-scale patterns from which is possible to compute several features, mainly based on the count of number of points within each element.

On the other hand, the *Poincaré plots* (PPs) are a particular case of phase space representation created selecting *m* = 2 and *τ* = 1; that corresponds to displaying a generic sample *n* of the time series as a function of the sample *n* − 1 [21]. This is also known as a return map or a Lorenz plot. The main limitation of this technique is that assumes that a low dimensional representation of a dynamical attractor is enough to detect relevant features of the dynamics. Despite its simplicity, this transformation has been successfully employed also with high dimensional systems. The benefit is that, given the low dimensionality, it is possible to easily design and visualize several types of geometric features. These features are based on an ellipse fitted to the PP. These features can be seen as measures of nonlinear autocorrelation. If successive values in the time series are not linearly correlated, there will be a deviation from a line that is often properly modeled using an ellipse. The different features involve the centroid of the ellipse, the length of the two axes of the ellipse, the standard deviation in the direction of the identity line (called SD2) and the standard deviation in the direction

limitation as CE, it has to be corrected in the same way.

estimator and the ShEn as the cost function [25].

**3.5. Geometric structure quantification**

orthogonal to the identity line (called SD1).

Although the mechanisms of AF still are unclear, several studies have demonstrated that this arrhythmia is associated with the propagation, throughout the atrial tissue, of multiple activation wavelets, resulting in complex ever-changing patterns of electrical activity [5]. As a consequence, the morphology of the registered *f* waves during AF changes constantly both in time and space showing different levels of organization, according to a definition of organization as repetitive wave morphologies in the AF signals [19]. Given that various morphologies reflect different activation patterns such as slow conduction, wave collision, and conduction blocks [26], AF organization analysis plays an important role to understand the mechanisms responsible for its induction and maintenance. In addition, the analysis of the degree of complexity characterizing the shape of the activation waves could provide useful information to improve AF treatment, which still is unsatisfactory, and contribute to take the appropriate decisions on its management [27].

Since a rigorous definition of organization does not exist, a variety of nonlinear indices have been applied to the AA signal extracted from both surface ECG recordings and intraatrial EGMs to quantify AF pattern dynamic and morphology. In the next subsections, the state of the art related to the AF organization estimation by using nonlinear methods is summarized.

### **4.1. Surface organization assessment**

From a clinical point of view, the assessment of AF organization from the standard surface ECG is very interesting, because it can be easily and cheaply obtained [10]. Previous works have shown that structural changes into surface *f* waves reflect the intraatrial activity organization variation [28, 29]. Thus, it has been observed that ECGs acquired during intraatrial organized rhythms present *f* waves with well-defined and repetitive morphology and ECGs recorded during highly disorganized AA with fragmented activations contain surface *f* waves with very dissimilar morphologies [30]. Taking advantage of this finding, several nonlinear indices have been applied to single-lead ECG recordings to estimate the amount of repetitive patterns existing in their extracted AA signal. Leads V1 and II have been most often selected for this purpose, because the atrial signal is larger in these recordings [10].

The first proposed method to estimate non-invasively temporal organization of AF is based on the application of SampEn to the fundamental waveform of the AA signal, which have been named as main atrial wave (MAW) in the literature [4]. Note that SampEn computation directly from the AA has also been investigated, but an unsuccessful AF organization assessment has been reported by several authors [4]. The presence of ventricular residua and other nuisance signals together with the SampEn sensitivity to noise have been considered the main reasons for this poor result [4]. In contrast, the MAW-SampEn strategy has provided ability to reliably reflect the intraatrial fibrillatory activity dynamics [29] and has been validated by predicting successfully a variety of AF organization-dependent events. In this respect, the method has shown a high diagnostic accuracy in the paroxysmal AF termination prediction, presenting more regular *f* waves for terminating than non-terminating episodes [4]. This result is in agreement with the decrease in the number of reentries prior to sinus rhythm (SR) restoration observed in previous invasive studies, where AF termination was achieved by using different therapies [6]. In a similar way, according with the invasive observation that self-sustained AF is associated with more circulating wavelets that non-sustained AF [6], the method has noticed higher organization levels for paroxysmal than persistent AF episodes [31].

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a multilayer perceptron neural network was applied to predict the AF termination with an

The Contribution of Nonlinear Methods in the Understanding of Atrial Fibrillation

As an alternative to the use of surface recordings, AF organization can be quantified from single-lead atrial EGMs by analysis of the whole signal aimed to infer measures related to the dynamical complexity of the signal itself. As for surface ECG recordings, the presence of undisturbed portions of the signal or the repetitiveness over time of similar patterns, are indicative of high regularity, or low dynamical complexity, related to the temporal organization of the arrhythmia. Within this context, Wells et al [8] distinguished three types of AF. In type I AF, the EGMs showed discrete complexes of variable morphology separated by a clear isoelectric baseline. Type II AF EGMs were characterized by discrete atrial beat-to-beat complexes of variable morphology but, in contrast to type I AF, the baseline showed continuous perturbations of varying degrees. During type III AF, highly fragmented atrial EGMs could be observed with no discrete complexes or isoelectric intervals. An analysis looking at these characteristics in AF EGMs has a peculiar electrophysiological relevance, as it may reflect the propagation patterns underlying the maintenance of AF [6]. Indeed, the Wells' approach has been used in several clinical and experimental studies to identify organization patterns in paroxysmal and chronic AF and to support the ablative treatment of AF [39]. In addition, many authors have proposed to quantify automatically single-lead EGMs organization by using analysis of fractal fluctuations and

In this line, the study of Hoekstra et al [40] was the first exhaustive nonlinear analysis of AF in man. The authors estimated the CD and the CorEn of unipolar epicardial EGMs. Both indices were exploited to discriminate among EGMs during induced AF, revealing the presence of nonlinear dynamics in type I AF. In contrast, type II and type III AF did not appear to exhibit features of low-dimensional chaos. Both previous indices were also used to investigate the anti-fibrillatory properties of the class Ic agent cibenzoline in instrumented conscious goats in which sustained AF had been electrically induced [41]. Results showed that during drug administration the nonlinear parameters were not significantly different from control. Nonetheless, scaling regions in the correlation sum were observed after infusion of cibenzoline suggesting that the drug introduced low-dimensional features in the dynamics of AF, whereas SR recorded shortly after cardioversion was very regular. Hence, authors concluded that nonlinear analysis revealed that cibenzoline does not significantly alter the dynamics of sustained AF during pharmacological conversion other than a slowing down of the atrial activation and a somewhat increasing global organization of the atrial

More recently, Mainardi et al [24] have developed a regularity index based on the corrected CE for single-site atrial EGMs and LAP series, which has provided ability to discriminate among different atrial rhythms and, particularly within different AF complexity classes according to Wells' criteria [24, 42]. In a similar way, the index has been able to capture subtle changes due to isoproterenol infusion both during SR and AF [43]. On the other hand, ShEn has been tested as a measure of EGMs complexity for distinguishing complex fractionated atrial electrograms (CFAE) from non-CFAE signals [44]. Given that CFAE have been identified as targets for AF ablation, the development of robust automatic algorithms to

accuracy higher than 95%.

entropy measures.

activation pattern.

**4.2. Intraatrial organization assessment**

On the other hand, the MAW-SampEn method has also presented a high discriminant ability in the prediction of ECV result before the procedure is attempted. According with previous invasive findings [32], SampEn reported higher AF organization levels in those patients who maintained SR during the first month post-cardioversion [4]. In addition, analyzing SampEn after each needed electrical shock to restore SR, a relative entropy decrease was observed for the patients who finally reverted to SR, but the largest variation took place after the first attempt, thus indicating that this shock plays the most important role in the procedure [33]. Finally, remark that the method has also been used to assess the organization evolution along onward episodes of paroxysmal AF and within an specific episode. In the first case, the achieved results, in close agreement with previous findings obtained from invasive recordings [34], proved several relevant aspects of arial remodeling [35]. Thus, a progressive disorganization increase along onward episodes of AF was observed for 63% of the analyzed patients, whereas a stable AF organization degree was appreciated in the remaining 37%. Moreover, a positive correlation between episode duration and SampEn and a remarkable influence of the fibrillation-free interval, preceding each episode, on the corresponding level of AF organization at the onset of the subsequent AF episode were noticed. With respect to the application of the method to track organization variations within each specific episode [4], a decrease in the first minutes after AF onset and an increase within the last minute before spontaneous AF termination were revealed, in coherence with previous works [6].

It is interesting note that *f* waves regularity has also been assessed through the application of SampEn to the wavelet domain of the AA signal [4]. In this case, the proposed approach reached a slightly lower discriminant ability than the MAW-SampEn method both for paroxysmal AF termination and ECV outcome predictions. Nonetheless, both methodologies showed to provide complementary information, their combination allowing to improve the identification of AF organization time course [4]. A similar result has been recently observed when the variability of the wavelet coefficients computed from the AA signal has been quantified by the central tendency measure [36]. This nonlinear metric is the percentage of points which falls within a certain radius from the centre of the PP of the first difference of the original time series [21] and, in view of the provided results, can be considered as a successful non-invasive estimator of temporal organization of AF.

In addition to SampEn, other nonlinear indices have also been applied to the AA signal time domain. Thus, Kao et al [37] computed the CD, LLE and LZC from the AA signal extracted for the lead V1 in order to distinguish between atrial flutter and AF episodes. According to the expected AF disorganization levels, results showed that during AF, nonlinear parameters concentrated on higher values, which were lower at typical flutter and middle in atypical flutter. In addition, the combination of these parameters by using a neural network classification allowed the differentiation of these arrhythmias with a high diagnostic accuracy around 95%. On the other hand, Sun and Wang [38] have investigated the spontaneous termination of paroxysmal AF by quantifying the RP structure of the AA signal. More precisely, eleven features were extracted from the RP including, among others, the point recurrence rate, the patterns along the main diagonal, the patterns along the 135◦ diagonal and square-like patterns. Thereafter, a sequential forward search algorithm was utilized to select the feature subset which could predict the AF termination more effectively. Finally, a multilayer perceptron neural network was applied to predict the AF termination with an accuracy higher than 95%.

### **4.2. Intraatrial organization assessment**

8 Atrial Fibrillation

with more circulating wavelets that non-sustained AF [6], the method has noticed higher

On the other hand, the MAW-SampEn method has also presented a high discriminant ability in the prediction of ECV result before the procedure is attempted. According with previous invasive findings [32], SampEn reported higher AF organization levels in those patients who maintained SR during the first month post-cardioversion [4]. In addition, analyzing SampEn after each needed electrical shock to restore SR, a relative entropy decrease was observed for the patients who finally reverted to SR, but the largest variation took place after the first attempt, thus indicating that this shock plays the most important role in the procedure [33]. Finally, remark that the method has also been used to assess the organization evolution along onward episodes of paroxysmal AF and within an specific episode. In the first case, the achieved results, in close agreement with previous findings obtained from invasive recordings [34], proved several relevant aspects of arial remodeling [35]. Thus, a progressive disorganization increase along onward episodes of AF was observed for 63% of the analyzed patients, whereas a stable AF organization degree was appreciated in the remaining 37%. Moreover, a positive correlation between episode duration and SampEn and a remarkable influence of the fibrillation-free interval, preceding each episode, on the corresponding level of AF organization at the onset of the subsequent AF episode were noticed. With respect to the application of the method to track organization variations within each specific episode [4], a decrease in the first minutes after AF onset and an increase within the last minute before

spontaneous AF termination were revealed, in coherence with previous works [6].

successful non-invasive estimator of temporal organization of AF.

It is interesting note that *f* waves regularity has also been assessed through the application of SampEn to the wavelet domain of the AA signal [4]. In this case, the proposed approach reached a slightly lower discriminant ability than the MAW-SampEn method both for paroxysmal AF termination and ECV outcome predictions. Nonetheless, both methodologies showed to provide complementary information, their combination allowing to improve the identification of AF organization time course [4]. A similar result has been recently observed when the variability of the wavelet coefficients computed from the AA signal has been quantified by the central tendency measure [36]. This nonlinear metric is the percentage of points which falls within a certain radius from the centre of the PP of the first difference of the original time series [21] and, in view of the provided results, can be considered as a

In addition to SampEn, other nonlinear indices have also been applied to the AA signal time domain. Thus, Kao et al [37] computed the CD, LLE and LZC from the AA signal extracted for the lead V1 in order to distinguish between atrial flutter and AF episodes. According to the expected AF disorganization levels, results showed that during AF, nonlinear parameters concentrated on higher values, which were lower at typical flutter and middle in atypical flutter. In addition, the combination of these parameters by using a neural network classification allowed the differentiation of these arrhythmias with a high diagnostic accuracy around 95%. On the other hand, Sun and Wang [38] have investigated the spontaneous termination of paroxysmal AF by quantifying the RP structure of the AA signal. More precisely, eleven features were extracted from the RP including, among others, the point recurrence rate, the patterns along the main diagonal, the patterns along the 135◦ diagonal and square-like patterns. Thereafter, a sequential forward search algorithm was utilized to select the feature subset which could predict the AF termination more effectively. Finally,

organization levels for paroxysmal than persistent AF episodes [31].

As an alternative to the use of surface recordings, AF organization can be quantified from single-lead atrial EGMs by analysis of the whole signal aimed to infer measures related to the dynamical complexity of the signal itself. As for surface ECG recordings, the presence of undisturbed portions of the signal or the repetitiveness over time of similar patterns, are indicative of high regularity, or low dynamical complexity, related to the temporal organization of the arrhythmia. Within this context, Wells et al [8] distinguished three types of AF. In type I AF, the EGMs showed discrete complexes of variable morphology separated by a clear isoelectric baseline. Type II AF EGMs were characterized by discrete atrial beat-to-beat complexes of variable morphology but, in contrast to type I AF, the baseline showed continuous perturbations of varying degrees. During type III AF, highly fragmented atrial EGMs could be observed with no discrete complexes or isoelectric intervals. An analysis looking at these characteristics in AF EGMs has a peculiar electrophysiological relevance, as it may reflect the propagation patterns underlying the maintenance of AF [6]. Indeed, the Wells' approach has been used in several clinical and experimental studies to identify organization patterns in paroxysmal and chronic AF and to support the ablative treatment of AF [39]. In addition, many authors have proposed to quantify automatically single-lead EGMs organization by using analysis of fractal fluctuations and entropy measures.

In this line, the study of Hoekstra et al [40] was the first exhaustive nonlinear analysis of AF in man. The authors estimated the CD and the CorEn of unipolar epicardial EGMs. Both indices were exploited to discriminate among EGMs during induced AF, revealing the presence of nonlinear dynamics in type I AF. In contrast, type II and type III AF did not appear to exhibit features of low-dimensional chaos. Both previous indices were also used to investigate the anti-fibrillatory properties of the class Ic agent cibenzoline in instrumented conscious goats in which sustained AF had been electrically induced [41]. Results showed that during drug administration the nonlinear parameters were not significantly different from control. Nonetheless, scaling regions in the correlation sum were observed after infusion of cibenzoline suggesting that the drug introduced low-dimensional features in the dynamics of AF, whereas SR recorded shortly after cardioversion was very regular. Hence, authors concluded that nonlinear analysis revealed that cibenzoline does not significantly alter the dynamics of sustained AF during pharmacological conversion other than a slowing down of the atrial activation and a somewhat increasing global organization of the atrial activation pattern.

More recently, Mainardi et al [24] have developed a regularity index based on the corrected CE for single-site atrial EGMs and LAP series, which has provided ability to discriminate among different atrial rhythms and, particularly within different AF complexity classes according to Wells' criteria [24, 42]. In a similar way, the index has been able to capture subtle changes due to isoproterenol infusion both during SR and AF [43]. On the other hand, ShEn has been tested as a measure of EGMs complexity for distinguishing complex fractionated atrial electrograms (CFAE) from non-CFAE signals [44]. Given that CFAE have been identified as targets for AF ablation, the development of robust automatic algorithms to objectively classify these signals is clinically relevant. An index of fractional intervals (FI) has been traditionally used and validated as a semiautomatic algorithm to identify CFAE [45]. This measure takes the average interval between deflections of an EGM signal during AF. In contrast, ShEn computation requires each EGM amplitude sample to be classified into bins of defined amplitude ranges. After quantifying EGMs with a bin with of 0.125 their with a bin width of 0.125 times their standard deviation, ShEn provided comparable results to the index of FI in distinguishing CFAE from non-CFAE without requiring user input for threshold levels. Hence, authors claimed that ShEn can be a useful tool in the study of AF pathophysiology as well as help in the classification of CFAE, although its use for EGM-guided approaches in AF ablation requires further validation.

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organization implies judging the electrical activity at one site in relation to the activity of another. Measures derived in such a way emphasize the concepts of relative temporal behavior and spatial coordination between electrical activations occurring at different sizes. With respect to approaches developed for single-site EGMs, the introduction of algorithms involving two (or more) signals provide complementary information. For instance, synchronization measures have been exploited to investigate the preferential directions of waveforms propagation during arrhythmias, or to reflect the spatial dispersion of electrophysiological parameters such as conduction velocity and refractory period.

The Contribution of Nonlinear Methods in the Understanding of Atrial Fibrillation

To capture and quantify nonlinear interactions among EGMs, some entropy measures have been adapted to the analysis of endocardial signals. Indeed, the studies of Censi et al [51] and Mainardi et al [24] estimated the degree of nonlinear coupling between pairs of bipolar EGMs acquired by decapolar catheters by performing specific multivariate embedding procedures. In particular, Censi et al [51] assessed the organization of the LAP series during AF by means of two indices, namely independence of complexity and independence of predictability. These indices were computed on the basis of a multivariate embedding procedure for the estimation of CD and CorEn. Significant degrees of nonlinear coupling were found in segments belonging to types I and II, while type III EGMs turned out to be only weakly coupled. On the other hand, Mainardi et al [24] estimated spatio-temporal organization in the atria by means of a synchronization index assessing the coupling level between EGMs by means of the corrected CCE. Although this index is sensitive to various signal coupling mechanisms (linear or not), it provides superior performance when compared to linear indices derived from the cross-correlation function, as evidenced in many applications [24]. Thus, it was found to be the best discriminator between organized (sinus rhythm and AF I, classified according to Wells' criteria) and non-organized (AF II and AF III) rhythms [24], showing sensitivity and positive predictability higher than 95%. The index also provided ability to capture subtle changes in atrial dynamics, thus improving the understand the effect of the sympathetic nervous system activity during SR and AF in patients suffering from paroxysmal and persistent AF [43]. In a similar way, the synchronization index showed to be able to underline the effect of the adrenergic stimulation, highlighting variations related to the distance between recording sites [7]. These variations were not detected with the same level of detail by any other linear and nonlinear parameter. Finally, a reduction in the synchronization among EGMs was evidenced by using this index during isoproterenol

Coupling between atrial EGMs can also be assessed by quantifying the temporal synchronism between activation times in two sites. In this context, researches have focused their attention to either the LAP series or the activation time sequences. Thus, Censi et al [53] exploited RPs to show that a certain degree of organization during AF can be detected as spatio-temporal recurrent patterns of the coupling between the atrial depolarization periods at two atrial sites. They demonstrated a deterministic mechanism underlying the apparently random activation processes during AF. Other approach for the same purpose was proposed by Masé et al [54] who characterized the synchronization between two atrial signals through a measure of the properties of the time delay distribution by the ShEn. Specifically, the values of the propagation delay were quantized into severals bins and the entropy of their distribution was estimated. After introducing a corrective term to reduce the systematic underestimation of ShEn due to the approximation of the probabilities with the corresponding sample frequency, the index was validated with a computer model of atrial arrhythmias. It was

infusion in both SR and paroxysmal AF episodes [52].

It is interesting note that approaches of nonlinear analysis have also been applied to each one of the bipolar signals collected by basked catheters, thus providing estimates of spatial organization of AF. In this respect, Pitschner et al [46] calculated the CD of the depolarization wavefronts on signals measured during paroxysmal AF and found that the area anterior to the tricuspid valve showed the most pronounced chaotic activity. Later, Berkowitsch et al [47] proposed a combination of symbolic dynamics and adaptive power estimation to compute the normalized algorithmic complexity of single-site bipolar EGMs. The algorithm produces a measure of the "redundancies" in patterns of the AF EGM so that the complexity is inversely related to the number of redundancies found in the analyzed signal. The method was used to show heterogeneous complexity among different atrial regions and complexity changes after drug administration [48]. In a similar way, Cervigón et al [49] analyzed the regularity differences in EGMs captured both from right (RA) and left (LA) atria after propofol administration. Global regularity from each atrium was estimated by applying both MSE and ShEn to each registered single EGM and averaging all the recordings acquired from each atrium. Results revealed differences between the MSE profiles in basal and propofol states and that EGMs at basal condition were sightly less irregular in RA than in LA. In addition, an irregularity decrease in EGMs was noticed, through the MSE, for RA during the proposal infusion. Note that this behavior was observed for all time scales, although MSE decreased on small scales and gradually increases indicating the reduction of complexity on the larger scales. The application of ShEn showed the same upward trend in the LA during propofol infusion, and downward trend in the RA in the anaesthesic state.

In a similar way, both MSE and ShEn have also been used to assess regional organization differences between paroxysmal and persistent AF episodes [50]. In this case, both for paroxysmal and persistent AF patients, no significant differences were found in an intra-atrial analysis (i.e. between the EGMs within the same atrium) in any atria. However, in an inter-atrial analysis, entropy values were higher at the LA than at the RA; i.e. the atrial activations were generally more organized at the RA than at the LA. However, compared with persistent AF, results from the analysis of paroxysmal AF demonstrated larger differences between the atrial chambers. Therefore, a regional gradient from the LA to RA in the organization degree of the atrial electrical activity was found in paroxysmal AF patients, whereas no gradient was found in persistent AF patients.

### **4.3. Intraatrial synchronization assessment**

Spatio-temporal organization of AF has been investigated from mutual analysis of pairs of EGMs simultaneously collected during different atrial rhythms. In this case, measuring organization implies judging the electrical activity at one site in relation to the activity of another. Measures derived in such a way emphasize the concepts of relative temporal behavior and spatial coordination between electrical activations occurring at different sizes. With respect to approaches developed for single-site EGMs, the introduction of algorithms involving two (or more) signals provide complementary information. For instance, synchronization measures have been exploited to investigate the preferential directions of waveforms propagation during arrhythmias, or to reflect the spatial dispersion of electrophysiological parameters such as conduction velocity and refractory period.

10 Atrial Fibrillation

objectively classify these signals is clinically relevant. An index of fractional intervals (FI) has been traditionally used and validated as a semiautomatic algorithm to identify CFAE [45]. This measure takes the average interval between deflections of an EGM signal during AF. In contrast, ShEn computation requires each EGM amplitude sample to be classified into bins of defined amplitude ranges. After quantifying EGMs with a bin with of 0.125 their with a bin width of 0.125 times their standard deviation, ShEn provided comparable results to the index of FI in distinguishing CFAE from non-CFAE without requiring user input for threshold levels. Hence, authors claimed that ShEn can be a useful tool in the study of AF pathophysiology as well as help in the classification of CFAE, although its use for

It is interesting note that approaches of nonlinear analysis have also been applied to each one of the bipolar signals collected by basked catheters, thus providing estimates of spatial organization of AF. In this respect, Pitschner et al [46] calculated the CD of the depolarization wavefronts on signals measured during paroxysmal AF and found that the area anterior to the tricuspid valve showed the most pronounced chaotic activity. Later, Berkowitsch et al [47] proposed a combination of symbolic dynamics and adaptive power estimation to compute the normalized algorithmic complexity of single-site bipolar EGMs. The algorithm produces a measure of the "redundancies" in patterns of the AF EGM so that the complexity is inversely related to the number of redundancies found in the analyzed signal. The method was used to show heterogeneous complexity among different atrial regions and complexity changes after drug administration [48]. In a similar way, Cervigón et al [49] analyzed the regularity differences in EGMs captured both from right (RA) and left (LA) atria after propofol administration. Global regularity from each atrium was estimated by applying both MSE and ShEn to each registered single EGM and averaging all the recordings acquired from each atrium. Results revealed differences between the MSE profiles in basal and propofol states and that EGMs at basal condition were sightly less irregular in RA than in LA. In addition, an irregularity decrease in EGMs was noticed, through the MSE, for RA during the proposal infusion. Note that this behavior was observed for all time scales, although MSE decreased on small scales and gradually increases indicating the reduction of complexity on the larger scales. The application of ShEn showed the same upward trend in the LA during

EGM-guided approaches in AF ablation requires further validation.

propofol infusion, and downward trend in the RA in the anaesthesic state.

whereas no gradient was found in persistent AF patients.

**4.3. Intraatrial synchronization assessment**

In a similar way, both MSE and ShEn have also been used to assess regional organization differences between paroxysmal and persistent AF episodes [50]. In this case, both for paroxysmal and persistent AF patients, no significant differences were found in an intra-atrial analysis (i.e. between the EGMs within the same atrium) in any atria. However, in an inter-atrial analysis, entropy values were higher at the LA than at the RA; i.e. the atrial activations were generally more organized at the RA than at the LA. However, compared with persistent AF, results from the analysis of paroxysmal AF demonstrated larger differences between the atrial chambers. Therefore, a regional gradient from the LA to RA in the organization degree of the atrial electrical activity was found in paroxysmal AF patients,

Spatio-temporal organization of AF has been investigated from mutual analysis of pairs of EGMs simultaneously collected during different atrial rhythms. In this case, measuring To capture and quantify nonlinear interactions among EGMs, some entropy measures have been adapted to the analysis of endocardial signals. Indeed, the studies of Censi et al [51] and Mainardi et al [24] estimated the degree of nonlinear coupling between pairs of bipolar EGMs acquired by decapolar catheters by performing specific multivariate embedding procedures. In particular, Censi et al [51] assessed the organization of the LAP series during AF by means of two indices, namely independence of complexity and independence of predictability. These indices were computed on the basis of a multivariate embedding procedure for the estimation of CD and CorEn. Significant degrees of nonlinear coupling were found in segments belonging to types I and II, while type III EGMs turned out to be only weakly coupled. On the other hand, Mainardi et al [24] estimated spatio-temporal organization in the atria by means of a synchronization index assessing the coupling level between EGMs by means of the corrected CCE. Although this index is sensitive to various signal coupling mechanisms (linear or not), it provides superior performance when compared to linear indices derived from the cross-correlation function, as evidenced in many applications [24]. Thus, it was found to be the best discriminator between organized (sinus rhythm and AF I, classified according to Wells' criteria) and non-organized (AF II and AF III) rhythms [24], showing sensitivity and positive predictability higher than 95%. The index also provided ability to capture subtle changes in atrial dynamics, thus improving the understand the effect of the sympathetic nervous system activity during SR and AF in patients suffering from paroxysmal and persistent AF [43]. In a similar way, the synchronization index showed to be able to underline the effect of the adrenergic stimulation, highlighting variations related to the distance between recording sites [7]. These variations were not detected with the same level of detail by any other linear and nonlinear parameter. Finally, a reduction in the synchronization among EGMs was evidenced by using this index during isoproterenol infusion in both SR and paroxysmal AF episodes [52].

Coupling between atrial EGMs can also be assessed by quantifying the temporal synchronism between activation times in two sites. In this context, researches have focused their attention to either the LAP series or the activation time sequences. Thus, Censi et al [53] exploited RPs to show that a certain degree of organization during AF can be detected as spatio-temporal recurrent patterns of the coupling between the atrial depolarization periods at two atrial sites. They demonstrated a deterministic mechanism underlying the apparently random activation processes during AF. Other approach for the same purpose was proposed by Masé et al [54] who characterized the synchronization between two atrial signals through a measure of the properties of the time delay distribution by the ShEn. Specifically, the values of the propagation delay were quantized into severals bins and the entropy of their distribution was estimated. After introducing a corrective term to reduce the systematic underestimation of ShEn due to the approximation of the probabilities with the corresponding sample frequency, the index was validated with a computer model of atrial arrhythmias. It was shown to discriminate among different AF types and to elicit spatial heterogeneities in the synchronization between different atrial sites. Moreover, a comparison of the real data with simulation results linked the different shapes of the time delay distribution, and thus the proposed index, to different underlying electrophysiological propagation patterns.

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http://dx.doi.org/10.5772/53407

preconditions AF onset. Additionally, the segments preceding AF onset were divided into three successive 10 minutes periods and analyzed with SampEn in order to show the presence of a possible trend. A decreasing complexity trend towards the onset of AF was observed independently on the presence or absence of ectopics, although in the later case the tendency was less pronounced. According to the authors, the decrease in complexity via SampEn before the onset of AF resulted mainly from atrial ectopy. Moreover, the decrease was in consistent agreement with the observed ectopic firing significance, that serves as a trigger of paroxysmal AF, in subjects without evidence of other structural cardiac abnormalities [60]. On the other hand, because of paroxysmal AF has been classified into vagally-mediated and sympathetically-mediated types, based on the autonomic profile and the clinical history, Shin et al [61] analyzed the HRV complexity in these types of AF. In this study, for 44 episodes, divided in three subgroups (vagal, sympathetic and non-related types), the 60 minutes segment of normal sinus rhythm preceding AF onset was divided into 6 periods of 10 minutes. The DFA showed a poor tendency to decrease before the onset of AF and the change of this parameter was divergent according to the AF type. In contrast, ApEn and SampEn revealed a linear decrease of complexity irrespective of AF type. In addition, this result in both ApEn and SampEn before AF onset was not affected whether excluding the ectopic beats or not. In the authors' opinion, the meaning of this progressive entropy reduction before the start of AF was that the heart rate became more orderly before AF; that is, there is a loss of normal "healthy" complexity, thus leading to a cardiac environment

The Contribution of Nonlinear Methods in the Understanding of Atrial Fibrillation

It is interesting note that although nonlinear indices, especially SampEn, have provided to be better predictors than standard HRV measures, their diagnostic ability in paroxysmal AF prediction is far from clinically optimal. Thus, in order to improve their discriminant capability, these nonlinear indices have been combined with other HRV metrics making use of different classification approaches. In this respect, Chesnokov [62] analyzed the combination of spectral features, SampEn and MSE of the HRV by using different artificial intelligent methods. More recently, Mohebbi and Ghassemian have proposed two different combinations of parameters to reach a diagnostic accuracy higher than 95%. Thus, in a first way, they computed a RP of the RR-interval series together with five statistically significant features: recurrence rate, length of longest diagonal segments, average length of the diagonal lines, entropy and trapping time. These parameters were combined making use of a support vector machine (SVM)-based classifier [63]. In the second alternative, a SVM-based classifier was also used to combine spectrum and bispectrum features with SampEn and PP-extracted

Finally, several studies have applied nonlinear indices to the HRV after coronary artery bypass graft surgery, i.e. before the onset of AF. Thus, Hogue et al [65] showed that patients who developed AF presented reduced heart rate complexity through ApEn and that standard measures of HRV did not distinguish between these two groups. Logistic regression analysis indicated that only lower complexity via ApEn and higher heart rate were independently associated with AF. In addition, ApEn did not correlate with any other HRV variable, so that the data provide little evidence for a direct relationship between the magnitude of ApEn and the level of autonomic modulation of heart rate. In a similar way, Chamchad et al showed that ApEn provides little complexity in predicting AF after off-pump coronary artery bypass graft surgery [66] and that the CD was independently associated with AF after

vulnerable to the occurrence of AF.

parameters from the HRV [64].

Finally, CauEn has been recently used to monitor coupling between temporal variations from two atrial EGMs for paroxysmal and persistent AF episodes [50]. Results showed differences between both atrial chambers with a higher disorganization in the LA than RA in paroxysmal AF patients and a more homogenous behavior along the atria in persistent AF patients. These findings were in strong agreement with the hypothesis that high-frequency periodic sources located in the LA drive AF [55]. Nonetheless, the result may also support the multiple wavelet hypothesis, which have a random movement throughout the atria [5].
