# Spatio-temporal Organization During Ventricular Fibrillation in the Human Heart

## Abstract

In this paper, we present a novel approach to quantify the spatio-temporal organization of electrical activation during human ventricular fibrillation (VF). We propose three different methods based on correlation analysis, graph theoretical measures and hierarchical clustering. Using the proposed approach, we quantified the level of spatio-temporal organization during three episodes of VF in ten patients, recorded using multi-electrode epicardial recordings with 30 s coronary perfusion, 150 s global myocardial ischaemia and 30 s reflow. Our findings show a steady decline in spatio-temporal organization from the onset of VF with coronary perfusion. We observed transient increases in spatio-temporal organization during global myocardial ischaemia. However, the decline in spatio-temporal organization continued during reflow. Our results were consistent across all patients, and were consistent with the numbers of phase singularities. Our findings show that the complex spatio-temporal patterns can be studied using complex network analysis.

## Keywords

Global myocardial ischaemia Cross-correlation Complex network analysis Hierarchical clustering Spatio-temporal systems## Introduction

The heart is an electromechanical pump. During each normal heartbeat an electrical wave of activation is initiated by the heart’s natural pacemaker and spreads through entire heart to both trigger and synchronize mechanical contraction. Any disturbance to the normal pattern of generation and propagation of these electrical activation waves results in an abnormal heart rhythm or cardiac arrhythmia. The most critical cardiac arrhythmia is ventricular fibrillation (VF), during which synchronized and regular contractions of the heart are suppressed by rapid and self-sustaining waves of electrical activation in the ventricles. As a consequence the heart is not able to pump blood effectively and VF is quickly lethal unless halted by defibrillation.15

An unavoidable consequence of spontaneous VF is global myocardial ischaemia, resulting from ineffective mechanical function and reduced myocardial perfusion. Several studies in both animal4,14 and human hearts3,17 have established that as a result of progressive global myocardial ischaemia, VF exhibits a series of stages where the frequency and the complexity of electrical activation waves vary. Understanding the dynamics of electrical activation waves during episodes of VF with global myocardial ischaemia is of great importance because this knowledge could be used to develop novel therapeutic strategies that could be optimized for each stage of VF.2,24

Recent technological advances in recording techniques11,21 along with increasingly detailed models of cardiac electrophysiology5 have greatly added to our knowledge of the mechanisms that initiate and sustain VF in the human heart. However, a detailed understanding of these mechanisms remains elusive.

One aspect that is poorly understood is whether the spatio-temporal patterns of electrical activation during VF arise from random or organized processes.16 Early studies of VF emphasized the observation of turbulent electrical activation waves.19,30 In contrast, other studies have found spatial and temporal organization in the pattern of electrical activation.6,13,29 Spatial organization is of particular interest because it could arise from an underlying connectivity structure that drives complex activation patterns on the heart surface. Moreover, in human VF the pattern of electrical activation results from a small number of excitation waves,29 compared to animal hearts of similar size.13

A key tool in the analysis of activation patterns on the surface of the heart during VF has been phase analysis, which can identify both activation wavefronts and the phase singularities around which they rotate.7 Although the complexity of re-entrant activation patterns can be assessed from the numbers and lifetimes of phase singularities,29 this approach does not allow any underlying connectivity structure to be identified. The spatial characteristic length based on coherence and correlation functions has been used to quantify the extent of spatial organization in both animal and human VF.9 Correlation and coherence functions quantify the degree of functional association between electrical activation waves at different spatial locations in time and frequency domain respectively. These algorithms have shown promise in characterizing the extent of spatial organization in both atrial and ventricular activation sequences particularly in differentiating between normal sinus rhythm and cardiac arrhythmias.9,27 However, quantifying and monitoring the level of organization during different stages of ischaemic VF in the human heart has not to our knowledge been reported in the literature.

We propose that the level of spatial organization within each episode of VF can be quantified by determining patterns of connectivity underlying electrical activation recorded at different spatial locations. In theoretical neuroscience, analysis of cortical connectivity network, structural, functional and effective, has a long and rich history. Structural connectivity is built upon the anatomy of the brain while the functional and effective connectivity are based on analyzing neuroimaging data.10,28 In functional connectivity networks, network connections are derived based on correlation or coherence between different regions, whereas network connections in effective connectivity are obtained *via* causal modelling such as Granger causality,12 partial directed coherence (PDC)25 and more recently model-based frameworks which integrate anatomical information with multi-electrode electrophysiological recordings.1 Another important approach is graph theory analysis which explores properties of complex networks and has been extensively applied to brain connectivity data.28 Clustering techniques have been also applied to time series data or its corresponding functional network to group regions with similar functional activities.18 These techniques can be adopted to study the spatial and temporal activation patterns in the human heart. For example the effective connectivity network based on the PDC approach has been used to analyze intracardiac signals recorded during atrial fibrillation (AF).25 However, many of these methods have not yet been applied to human VF, and there is a sufficient volume of interesting results from theoretical neuroscience studies that warrants further investigation in this domain.

The aim of this study was therefore to evaluate techniques to quantify the level of spatial and temporal organization in electrical activation sequences during human VF. First, we used the cross-correlation function to develop a method for quantifying functional association between recordings from each epicardial electrode and hence to compute the global level of spatio-temporal organization. Second, we exploited the spatio-temporal network structure obtained based on cross-correlation coefficients to study the organization of electrical activation sequences as spatio-temporal functional networks. We also applied graph-theoretical measures to further quantify the properties of network-based representation. To examine and visualise the spatio-temporal patterns of similar functional association over different regions of the epicardial surface, a hierarchical clustering method based on the cross-correlation matrix was also employed. Temporal progressions of the underlying connectivity structures obtained from these methods were then quantified and monitored using a sliding window-based analysis.

## Materials and Methods

### Data Acquisition and Preprocessing

The recordings used in this study were from a group of ten patients (labelled as H055, H057–H060, H062–H066) described in a previous publication3 in which the patient details and methods used for data acquisition are covered in detail. Briefly, each recording was collected from patients undergoing cardiopulmonary bypass for routine cardiac surgery with cross clamp fibrillation, and the study was approved by the local hospital ethics committee (REC 01/0130).

Epicardial electrograms were pre-processed using two major steps. First, slow variations due to respiratory artefacts were removed from each channel with piecewise linear detrending. This also ensured that the mean of each preprocessed fibrillation electogram was approximately zero. The second step was to identify and exclude electrodes with poor quality signals. Only electrograms with dominant frequencies (DF) between 2 and 20 Hz were included in the analysis. Following this a visual examination was further performed to exclude electrograms with amplitudes close to zero. As a result, an average of 219 electrodes (range 186–240) with good quality signals throughout the recording were available for the analysis. The positions of epicardial electrodes in three-dimensional space were obtained by fitting the multi-electrode sock over a heart model. The three-dimensional epicardial electrode coordinates (Fig. 1a) were projected onto a cone-shaped surface enclosing both ventricles. This surface was then projected onto a two-dimensional flat disc (Fig. 1b).

### Linear Coupling Measure

*d*, was set to the maximum value of the average activation cycle which we estimated from the average DF of epicardial recordings. The DF of an electrode was defined as the modal frequency in its spectrum calculated using Welch’s power spectral density method with a window length of 3000 samples and 1500 samples overlap.

### Complex Network Analysis

In complex network analysis a network is defined as a collection of interrelated elements, which can be represented by a graph. The abstract mathematical formulation of a graph is defined as \(G=(V,E)\) where \(V=\left\{ V_i\right\} \) is a set of nodes or vertices and \(E=\left\{ V_i,V_j \right\} \) is a set of edges representing the connections between the nodes. The edges of a graph structure can be undirected or directed and binary or weighted.

We used a weighted undirected graph with no self or multiple connections is to represent the network structure. The nodes of the network were defined as epicardial electrodes and the weighted edges are the pairwise correlations obtained from the correlation matrix defined in Eq. (2). The edges can be interpreted as functional connectivity, i.e., a symmetrical association between regions of epicardial surface. For further analysis the network structure of weighted graph was represented by its connectivity matrix where rows and columns indicate nodes and matrix elements indicate weights. This is equivalent to the correlation matrix, *R*, with its main diagonal set to zero to exclude self-connections.

In order to explore the connectivity matrix quantitatively we calculated a set of graph-theoretical measures for all nodes (electrodes) using adjacency and weights matrices. The weights matrix was obtained by thresholding the connectivity matrix where entries under a certain value were set to zero. The adjacency matrix was then obtained by replacing non-zero values with one in the weights matrix. It should be noted that thresholding was performed based on the absolute value of correlation coefficient.

Graph theoretical measures provide a powerful tool for a systematic study of the epicardial functional network, providing a quantitative connectivity structure which characterizes the level of organization of fibrillation electrograms over VF episodes. There are several graph-theoretical measures for analysis of complex networks in the literature. In this study connection density, node strength and strength distribution, and mean clustering co-efficient were chosen and calculated using the Brain Connectivity toolbox in Matlab.28 A brief description of each measure based on adjacency and weights matrices (undirected and weighted) is provided below.

#### Connection Density

Connection density can be calculated for a set of adjacency matrices obtained from different threshold values ranging from 0 and 1. A plot of connection density vs. threshold values can be then produced. To calculate the overall level of sparseness or interconnectedness in a network, we calculated area under the curve (AUC) measure that is independent of correlation-threshold value. This measure can be then used to monitor the changes in the overall level of organization across different VF episodes.

#### Node Strength and Strength Distribution

#### Clustering Coefficient

Clustering co-efficient can be used to examine densely interconnected groups or clusters with similar functional association. For an undirected and binary graph, the local clustering co-efficient of a node was defined as the number of triangles within its neighbourhood (its immediately connected neighbours) to the maximum possible number of edges between them.28

*i*, is calculated by

*i*and \(l_{i}\) is the total intensity of triangles attached to node,

*i*.

### Hierarchical Clustering Based on Correlation Matrix

*R*(Eq. (2)),

*X*and cluster

*Y*respectively.

## Results

### Linear Coupling Measure

In order to define the total number of lags in correlation analysis, we calculated the maximum activation cycle length for control VF, global myocardial ischaemic VF and reflow VF separately. Within each VF episode the average value of activation cycle length was estimated as the inverse of mean DF across all electrodes. For each patient this gave a mean activation cycle length for each segment of the time series. The calculated mean activation cycle lengths for ten patients were then averaged for each VF episode. The resultant cycle length during control VF was \(180 \pm 19.9\) ms (mean ± 99% confidence interval (CI)). For ischaemic VF this increased from \(165 \pm 14.7\) ms to \(207 \pm 24.9\) ms and reduced rapidly to \(161\pm 22.3 \) ms during reflow. Based on the maximum value of activation cycle length the range of time delays was set to \(-\,116\) to 116 ms, giving a cycle length of 232 ms. Using the calculated cycle length the correlation matrix for each 30 s VF epoch was computed.

Piecewise linear fit of organization measures.

Measures | Mean intercept | Mean change in slope (%) | ||||||
---|---|---|---|---|---|---|---|---|

Control | Isch1 | Isch2 | Isch3 | Isch4 | Isch5 | Reflow | ||

Frobenius norm | 0.667 ± 0.06 | − 0.14 ± 0.13 | 0.051 ± 0.27 | 0.166 ± 0.22 | − 0.086 ± 0.40 | − 0.171 ± 0.61 | 0.058 ± 0.22 | − 0.53 ± 0.20 |

Connection density | 0.646 ± 0.07 | − 0.15 ± 0.14 | 0.058 ± 0.30 | 0.184 ± 0.25 | − 0.097 ± 0.44 | − 0.182 ± 0.67 | 0.065 ± 0.23 | − 0.035 ± 0.21 |

Clustering coeff | 0.571 ± 0.08 | − 0.16 ± 0.14 | 0.069 ± 0.33 | 0.184 ± 0.26 | − 0.099 ± 0.45 | − 0.203 ± 0.69 | 0.072 ± 0.261 | − 0.068 ± 0.21 |

Number of clusters | 0.133 ± 0.04 | 0.08 ± 0.08 | − 0.029 ± 0.15 | − 0.090 ± 0.12 | 0.037 ± 0.14 | 0.067 ± 0.23 | − 0.006 ± 0.11 | 0.068 ± 0.09 |

### Complex Network Analysis

We further studied the properties of functional network using a window-based graph-theoretical analysis. The window size and the overlap were chosen similar to the previous analysis. Within each window the connection density (Eq. (4)) and the average clustering coefficient (Eqs. (6)–(8)) were calculated using different threshold values ranging from 0 to 1 with an increment of 0.05. It should be noted that connection density and average clustering co-efficient provide measures of the total number of connections and spatial connectivities in a network respectively. To quantify the overall level of organization at all the threshold values, the AUC of connection density and average clustering coefficient vs. threshold values was then computed. The temporal changes in the AUC of these measures over different VF episodes are shown in Fig. 4. Similarly, we then fitted a piecewise linear model to quantify the trend within each episode and the results from piecewise linear model are shown in Fig. 4 and Table 1. These findings show that both connection density and average clustering coefficient show similar behavior across VF epochs. During control VF, the mean intercept of connection density decreased from \(0.646 \pm 0.072\) to \(0.599 \pm 0.074\) whereas the mean of average clustering coefficient decreased from \(0.571 \pm 0.081\) to \(0.522 \pm 0.080\). In global ischaemia, the mean intercepts of the measures decreased to \(0.494 \pm 0.050\) and \(0.412\pm 0.64\) respectively which was then significantly decreased to \(0.44 \pm 0.05\) and \(0.36\pm 0.06\) during reflow. This shows that the level of organization quantified using these two measures was low during reflow compared to control VF. This is in accordance with results based on correlation matrix analysis (Fig. 3), indicating that both measures can be used to characterize the global level of organization in VF episodes.

### Hierarchical Clustering Based on Correlation Matrix

In the previous sections we identified nodes with a similar level of functional organization at different locations on the epicardium. We then used a hierarchical clustering approach based on cross-correlation matrix to group nodes into different clusters, to define regions that share similar characteristics. The hierarchical clustering in Algorithm 1 was applied to each of the seven segments VF recording. A dendrogram was then constructed to represent the arrangement of clusters (not shown here). A cut-off distance of 0.3 was then used to split the parent cluster into different sub-clusters. The resulting clusters with \(70\%\) similarity in cross-correlation coefficients were mapped onto the two-dimensional epicardium surface (Fig. 2d). This value was chosen to identify the clusters with higher level of functional organization and also to compare with the network structures shown in Fig. 2c. In Fig. 2d a high level of localised spatio-temporal organization can be observed in different regions across VF epochs. As ischaemic VF progresses, larger clusters with a high level of spatial organization were formed (Isch 2, 3 and 4). These clusters were then divided into smaller ones towards the end of the episode (Isch 5). One important observation is the higher number of electrodes which do not form a cluster (white circles) during reflow compared to control.

Comparison of number of clusters and phase singularities.

Patient index | Coefficient of determination |
---|---|

H055 | 0.60 |

H057 | 0.89 |

H058 | 0.54 |

H059 | 0.62 |

H060 | 0.60 |

H062 | 0.53 |

H063 | 0.67 |

H064 | 0.76 |

H065 | 0.80 |

H066 | 0.60 |

### Comparison with Phase Analysis

Here, the AUC measure of normalized number of clusters is compared with the number of phase singularities obtained from phase analysis.3 Both methods provide measures of the inherent complexity of electrical activation and so should provide consistent results. The temporal changes in the number of phase singularities are shown in Fig. 6. The detailed description of the phase analysis approach used to compute the phase singularities is presented in a previous study.3

In order to examine the relationship between the two measures, linear correlation between the two was calculated using coefficient of determination, R-squared, from a linear regression model of the data (Table 2). The R-squared values illustrate that the results obtained from clustering approach is consistent with the phase analysis.

## Discussion

We have presented a complex network analysis approach to study the spatio-temporal organization in the complex spatio-temporal patterns of human VF. To quantify the level of organization, we have proposed metrics from correlation-analysis, network analysis and hierarchical clustering. Correlation analysis quantified the linear coupling between the epicardial recordings, where the global level of organization within VF episodes was characterized using the Frobenius norm. By representing the functional associations as network connections between the electrodes, the information extracted from network analysis can be used to identify the underlying network of highly interconnected fibrillation regions which drives complex spatio-temporal patterns of electrical activation. Similar to the Frobenius norm measure, connection density provided an overall measure of organization while the spatial organization within a network was characterized using average clustering coefficient. It can be seen that the degree of global clustering is dependent on the global functional organization in electrograms, as the episodes with higher levels of organization show higher levels of clustering and vice versa (Fig. 4). On the other hand, the number of clusters measure from hierarchical analysis characterizes the inherent complexity within VF episodes. Both clustering coefficient and the number of clusters illustrate the presence of localised spatial organization on the epicardial surface through different procedures. Hence the result from one method can be used to examine and validate results obtained from the other approach. The mean change in slope parameters obtained from the piecewise linear fit to proposed metrics (Table 1) show that the changes in organization are consistent throughout the VF episodes. This illustrates that the proposed methods can be used to study the level of organization in VF.

Although the normalized Frobenius norm is an easier approach to estimate the global organization level, an advantage of hierarchical clustering is its ability to detect and visualise clusters of electrodes which share functional properties (Fig. 2d) and to characterize the global complexity level in VF episodes. Moreover the patterns obtained using clustering approach are more robust to outliers as it uses the average linkage method to calculate the inter-cluster distances.8

It should be noted that the temporal information, i.e., time delays, between nodes were not investigated in correlation analysis. This is because, cross-correlation function can be sensitive to the inherent noise in the recordings, which significantly affect the estimation of time delays.

Our key findings from this study can be summarized as follows: (1) During control VF, there is a steady decrease in the level of organization; (2) During global ischaemia, there were transient increases within distinct epochs followed by a progressive decrease in the level of organization; (3) During reflow, the level of organization was lower compared to that during perfusion. Our results are in agreement with the findings presented in a previous study,3 where the changes in dynamics of complex electrical activation patterns were studied using traditional measures such as phase singularities and dominant frequency. Here we have also demonstrated that the proposed measures of organization are consistent with the number of phase singularities from phase analysis. Phase singularities are identified from the spatial distribution of phase, which requires robust estimation of phase angle from the phase plane trajectory of two state variables. For example, phase plane trajectory can be constructed using voltage and time-delayed voltage as state variables, where an appropriate delay parameter and a stable centre of rotation should be specified.13 This method was extended by reconstructing the phase plane trajectory using voltage and its rate of change as state variables.23 Although this method does not involve any additional parameters, the inherent differentiation of voltage has the tendency to amplify the noise in the signal. This can be addressed by considering the integral of voltage26 or its Hilbert transform21 as second state variable. However these approaches involve various pre-processing steps to obtain a stable center of rotation for the robust estimation of phase angle.7 The proposed methods in this study do not require these additional pre-processing steps which makes it algorithmically straightforward to implement compared to the phase analysis.

## Conclusion

This paper demonstrates that it is possible to quantify trends in spatio-temporal organization of epicardial electrograms during human VF that could be attributed to coronary perfusion, global myocardial ischaemia and reflow. Our findings show that the complex spatio-temporal patterns can be studied using complex network analysis and hierarchical clustering. These analyses yield results that are consistent with more traditional measures such as the number of phase singularities and dominant frequency, and so provide new tools for quantifying and understanding the dynamics of complex electrical activation patterns in cardiac arrhythmias. One distinct advantage is their simple implementation which do not require additional pre-processing steps. We have found that epicardial activity during human VF becomes progressively more disorganized and complex during a period of global myocardial ischaemia, and that a 30 s period of reperfusion does not reverse this decline.

## Notes

### Acknowledgements

Jinny Robson would like to acknowledge the support of Harry Nicholson Postgraduate Research Scholarship by University of Sheffield. Prof. Visakan Kadirkamanathan is partly supported by Engineering and Physical Sciences Research Council grant (EP/K03877X/1). The authors would like to thank the anonymous reviewers for their valuable suggestions.

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