# Granger Causality and Jensen–Shannon Divergence to Determine Dominant Atrial Area in Atrial Fibrillation

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## Abstract

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## 1. Introduction

## 2. Materials

## 3. Methods

#### 3.1. Principal Component Analysis

- Bipolar electrogram signals were pre-processed using several steps proposed by Botteron [30]. Initially, the signals were band-pass filtered between 40 and 250 Hz. Subsequently, they were rectified and, finally, the absolute value of the filtered waveforms were lowpass filtered with a 20-Hz cut-off filter. This filtering process extracts a time-varying waveform proportional to the amplitude of the high-frequency components of the original atrial electrograms, enhancing the atrial activations, simplifying their shape variations while reducing noise, as illustrated in Figure 2.
- PCA was applied to the pre-processed recordings on the RA and on the LA with Orbiter catheter and on the PVs with Lasso catheter before ablation procedure. We extract the first PC from the four electrograms recorded on the four PVs, simultaneous to the recordings from the LA and the RA, where the first PC was extracted in each area.
- PCA was applied to the recordings recorded on the LA and on the RA with the Orbiter catheter in four phases: in basal state, after right PVs’ isolation, after left PVs’ isolation and after the ablation procedure.

#### Suitability of PCA Decomposition

#### 3.2. Granger Causality

**X**= ${X}_{i}$: i≥ 1 and

**Y**= ${Y}_{i}$: i≥ 1, to determine whether

**X**causes

**Y**,

**Y**is first modelled as an univariate autoregressive series with error correction term ${V}_{i}$:

**Y**is modelled again, using the

**X**series as causal side information:

#### 3.3. Jensen–Shannon Divergence

**Z**of all variables, where X,Y ∈

**Z**. Evaluating the proximity of probability distributions ${P}_{1}$ and ${P}_{2}$, with ${X}_{i}$, $i=1,2,\cdots ,N$ and ${Y}_{i}$, $i=1,2,\cdots ,N$, which we denote as ${p}_{i}^{(1)}={P}_{1}({X}_{i})$ and ${p}_{i}^{(2)}={P}_{2}({Y}_{i})$, with $0\le {p}_{i}^{(k)}\le 1$ and ${\sum}_{i=1}^{N}{p}_{i}^{(k)}=1$ for all $i=1,2,\cdots ,n$ and $k=1,2$. If ${\pi}_{1}$ denotes the weight of ${P}_{1}$ and ${\pi}_{2}$ is the weight ${P}_{2}$, with the restrictions ${\pi}_{1}+{\pi}_{2}=1$ and ${\pi}_{1},{\pi}_{2}\ge 0$, the JSD is defined by:

#### 3.4. Statistical Analysis

## 4. Results

#### 4.1. Analysis Pre-Ablation Procedure

#### 4.1.1. Granger Causality

**X**, is causally driving

**Y**(first PC from another region), in the recurrent and non-recurrent AF patients (Figure 5).

#### 4.1.2. Jensen–Shannon Divergence

#### 4.2. Analysis during the Ablation Procedure

- Phase 1: Basal state;
- Phase 2: After right PVs’ isolation;
- Phase 3: After left PVs’ isolation;
- Phase 4: After the procedure.

#### 4.2.1. Granger Causality

#### 4.2.2. Jensen–Shannon Divergence

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

AH | Arterial Hypertension |

AF | Atrial Fibrillation |

CS | Coronary Sinus |

ECV | Electrical Cardioversion |

G-Causality | Granger Causality |

JSD | Jensen–Shannon Divergence |

IQR | Interquartile Range |

LA | Left Atrium |

LIPV | Left Inferior Pulmonar Vein |

LSPV | Left Superior Pulmonar Vein |

PCA | Principal Component Analysis |

PC | Principal Component |

PV | Pulmonar Vein |

RA | Right atrium |

RIPV | Right Inferior Pulmonar Vein |

RSPV | Right Superior Pulmonar Vein |

SC | Structural Cardiopathology |

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**Figure 1.**Diagrammatic representation illustrating Orbiter and Lasso catheters’ distribution of electric poles along both atria and a pulmonary vein. The lower right figure represents the anatomic relation of the four cardiac heart chambers, showing atria and ventricles and one pulmonary vein.

**Figure 2.**Four original RA electrogram signals (

**top**) and pre-processing signals using the Botteron pre-processing chain (

**bottom**).

**Figure 5.**G-Causality in relation to atrial area in AF recurrent patients (red) and patients that maintain sinus rhythm (blue).

**Figure 6.**Jensen–Shannon divergence in relation to atrial area in recurrent AF patients (red) and patients that maintain sinus rhythm (blue).

**Figure 7.**Granger causality in relation to all the phases in recurrent AF patients (red) and patients that maintain sinus rhythm (blue).

**Figure 8.**Jensen–Shannon Divergence between LA and RA along the for phases in recurrent AF patients (red) and patients in sinus rhythm (blue).

Patient | Age | Sex | LA Size | SC | AH | AF > | ECV | Paroxysmal | Recurrence |
---|---|---|---|---|---|---|---|---|---|

(Years) | (mm) | 6 Months | AF | ||||||

Pat 1 | 67 | Male | 42 | 0 | 0 | 0 | 1 | 0 | 0 |

Pat 2 | 63 | Female | 48 | 0 | 1 | 0 | 1 | 1 | 1 |

Pat 3 | 32 | Male | 42 | 0 | 0 | 0 | 0 | 1 | 0 |

Pat 4 | 52 | Male | 45 | 0 | 1 | 0 | 0 | 0 | 0 |

Pat 5 | 65 | Female | 45 | 0 | 0 | 1 | 1 | 0 | 0 |

Pat 6 | 24 | Male | 36 | 0 | 0 | 0 | 1 | 1 | 1 |

Pat 7 | 51 | Male | 54 | 0 | 1 | 1 | 1 | 0 | 1 |

Pat 8 | 39 | Male | 35 | 0 | 0 | 0 | 1 | 1 | 0 |

Pat 9 | 57 | Female | 50 | 0 | 0 | 0 | 0 | 1 | 1 |

Pat 10 | 38 | Male | 41 | 0 | 0 | 1 | 1 | 0 | 1 |

**Table 2.**G-Causality along three different anatomical areas in patients that maintain sinus rhythm and in patients with recurrent AF.

G-Causality | Recurrent AF | Non Recurrent AF | p |
---|---|---|---|

RA → VP | $1.21\times {10}^{-4}$ ± $1.04\times {10}^{-4}$ | $1.50\times {10}^{-4}$ ± $7.58\times {10}^{-4}$ | 0.099 |

LA → VP | $0.76\times {10}^{-4}$ ± $1.04\times {10}^{-4}$ | $0.65\times {10}^{-4}$ ± $6.45\times {10}^{-4}$ | 0.433 |

PV → RA | $1.12\times {10}^{-4}$ ± $0.75\times {10}^{-4}$ | $1.11\times {10}^{-4}$ ± $2.57\times {10}^{-4}$ | 0.499 |

LA → RA | $1.39\times {10}^{-4}$ ± $1.51\times {10}^{-4}$ | $1.56\times {10}^{-4}$ ± $7.34\times {10}^{-4}$ | 0.499 |

PV → LA | $0.73\times {10}^{-4}$ ± $1.78\times {10}^{-4}$ | $1.39\times {10}^{-4}$ ± $3.25\times {10}^{-4}$ | 0.181 |

RA → LA | $0.62\times {10}^{-4}$ ± $1.87\times {10}^{-4}$ | $1.97\times {10}^{-4}$ ± $7.97\times {10}^{-4}$ | 0.047 |

**Table 3.**Jensen–Shannon Divergence along AF ablation procedure in non recurrent and recurrent AF patients.

Phases | Non Recurrent AF | Recurrent AF | p |
---|---|---|---|

RA-LA JSD | RA-LA JSD | ||

Phase 1 | 0.11 ± 0.17 | 0.04 ± 0.10 | 0.420 |

Phase 2 | 0.16 ± 0.09 | 0.012 ± 0.08 | 0.042 |

Phase 3 | 0.14 ± 0.13 | 0.04 ± 0.04 | 0.122 |

Phase 4 | 0.14 ± 0.26 | 0.02 ± 0.02 | 0.044 |

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## Share and Cite

**MDPI and ACS Style**

Cervigón, R.; Castells, F.; Gómez-Pulido, J.M.; Pérez-Villacastín, J.; Moreno, J. Granger Causality and Jensen–Shannon Divergence to Determine Dominant Atrial Area in Atrial Fibrillation. *Entropy* **2018**, *20*, 57.
https://doi.org/10.3390/e20010057

**AMA Style**

Cervigón R, Castells F, Gómez-Pulido JM, Pérez-Villacastín J, Moreno J. Granger Causality and Jensen–Shannon Divergence to Determine Dominant Atrial Area in Atrial Fibrillation. *Entropy*. 2018; 20(1):57.
https://doi.org/10.3390/e20010057

**Chicago/Turabian Style**

Cervigón, Raquel, Francisco Castells, José Manuel Gómez-Pulido, Julián Pérez-Villacastín, and Javier Moreno. 2018. "Granger Causality and Jensen–Shannon Divergence to Determine Dominant Atrial Area in Atrial Fibrillation" *Entropy* 20, no. 1: 57.
https://doi.org/10.3390/e20010057