# Short-Time Estimation of Fractionation in Atrial Fibrillation with Coarse-Grained Correlation Dimension for Mapping the Atrial Substrate

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Study Population and Data Acquisition

#### 2.2. Coarse-Grained Correlation Dimension

#### 2.3. Selection of Computational Parameters

#### 2.3.1. Embedded Dimension

#### 2.3.2. Time Lag

#### 2.3.3. Distance in Phase Space

#### 2.3.4. Reference Points

#### 2.4. Data Preprocessing and Analysis

#### 2.5. Surrogate Data Analysis

#### 2.6. Statistical Analysis

^{®}Classification Learner (MathWorks, Natick, MA, USA) performed a coarse-tree analysis with a maximum split of 2, using 10-fold cross-validation. Normality and homoscedasticity of the median values for the three AF Types were tested with Shapiro–Wilk [55] and Levene tests [56], respectively. According to the results of the above tests, statistical differences between the median values of the three AF Types of each group were verified with the Kruskal–Wallis test [57], whereas statistical differences between the median values in pairs of AF Types were also tested, using a Mann–Whitney U test [58] with Bonferroni correction.

## 3. Results

#### 3.1. Surrogate Data Analysis

#### 3.2. Statistical Analysis

## 4. Discussion

## 5. Limitations

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Phase-Space Reconstruction and Correlation Dimension Using Small Data Size

**Y**${}_{1}^{4}=\phantom{\rule{3.33333pt}{0ex}}({x}_{1},{x}_{9},{x}_{17},{x}_{25})$ [67]. The reconstructed signal, will then look like the $\left(ii\right)$ case of Figure 2.

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**Figure 1.**Example of bipolar atrial fibrillation (AF) electrograms (EGMs) of different Types. AF Type IV consists of alternating Type I/II and Type III segments.

**Figure 2.**Example of one second segment of (

**i**) original and (

**ii**–

**iv**) reconstructed AF electrograms via CGCD. (

**ii**) Reconstructed signal with time lag $\tau =8$ ms, embedded dimension $m=4$. (

**iii**) Reconstructed signal with time lag $\tau =8$ ms, embedded dimension $m=10$. (

**iv**) Reconstructed signal with time lag $\tau =35$ ms, embedded dimension $m=10$. (

**a**) AF Type I, (

**b**) AF Type II, and (

**c**) AF Type III. Length p of reconstructed signal decreases as $\tau $ and m increase, as can be seen from Equation (1).

**Figure 3.**Illustration of algorithm steps and decisions taken for AF Type IV detection on the pseudo-real recordings of Group 3 in the database.

**Figure 4.**Surrogate data analysis indicating coarse-grained CorDim (CGCD) values for the entire database. Values of original data are presented with a small circle, whereas surrogate values are depicted as boxplots, generated from the 40 surrogates corresponding to each time series.

**Figure 5.**Box plots illustrating the distribution CGCD values as a function of the AF Types, where (

**a**) is for the most representative EGMs in Group 1, (

**b**) for the whole database in Group 2, and (

**c**) for Type IV pseudo-real EGMs in Group 3.

**Figure 6.**Receiver operating characteristics (ROC) curve analysis of discrimination between AF Types by using CGCD as a fractionation index. (

**a**,

**b**) Curves for the 24 most representative EGMs in Group 1 and (

**c**,

**d**) curves for the whole dataset analyzed in Group 2. AUC: area under the ROC curve.

**Figure 7.**Confusion matrices for the most representative EGMs in Group 1 (

**a**) and the whole database in Group 2 (

**b**). All EGMs in Group 1 were correctly classified by their AF type, whereas 17 EGMs of Group 2 were wrongly classified.

**Figure 8.**Decision tree together with thresholds obtained to classify EGMs by their AF Type through the application of CGCD. Scheme for the most representative EGMs in Group 1 (

**a**) and for the whole database in Group 2 (

**b**).

**Figure 9.**Scatterplots of CGCD values for the three AF Types in the most representative EGMs of Group 1 (

**a**), in the whole database of Group 2 (

**b**), and in Group 2 combined with the pseudo-real Type VI EGMs of Group 3 (

**c**).

**Table 1.**Statistical differences between the median CGCD values to discriminate between the three AF Types as well as for pairs of AF Types of Groups 1 and 2.

AF Types | Group 1 | Group 2 |
---|---|---|

AF Types I-II-III | p = 0.00004 | p < 0.000010 |

AF Types I vs. II/III | p = 0.00010 | p < 0.000010 |

AF Types III vs. I/II | p = 0.00010 | p < 0.000010 |

**Table 2.**Classification accuracy by coarse decision tree for Groups 1 and 2 and the corresponding thresholds for the discrimination by different AF Types. ${T}_{1}$, ${T}_{2}$, and ${T}_{3}$ are the thresholds for discriminating AF Types I, II, and III, respectively.

Group | Nr of EGMs | Accuracy | Wrongly Classified | Threshold |
---|---|---|---|---|

1 | 24 | $100\%$ | 0 | ${T}_{1}:<1.4958$ ${T}_{2}:\ge 1.4958,<2.0680$ ${T}_{3}:\ge 2.0677$ |

2 | 119 | 84.00–85.70% | 17 | ${T}_{1}:<1.3880$ ${T}_{2}:\ge 1.3880,<2.0326$ ${T}_{3}:\ge 2.0326$ |

**Table 3.**Mean and standard deviation of CGCD values of Groups 1 and 2 for AF Types I, II, and III. Results are presented as mean ± standard deviation.

AF Type | Group 1 | Group 2 |
---|---|---|

Type I | $1.0220\pm 0.2499$ | $1.0608\pm 0.2413$ |

Type II | $1.7097\pm 0.0934$ | $1.7587\pm 0.2576$ |

Type III | $2.6406\pm 0.2366$ | $2.3884\pm 0.3524$ |

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**MDPI and ACS Style**

Vraka, A.; Hornero, F.; Bertomeu-González, V.; Osca, J.; Alcaraz, R.; Rieta, J.J.
Short-Time Estimation of Fractionation in Atrial Fibrillation with Coarse-Grained Correlation Dimension for Mapping the Atrial Substrate. *Entropy* **2020**, *22*, 232.
https://doi.org/10.3390/e22020232

**AMA Style**

Vraka A, Hornero F, Bertomeu-González V, Osca J, Alcaraz R, Rieta JJ.
Short-Time Estimation of Fractionation in Atrial Fibrillation with Coarse-Grained Correlation Dimension for Mapping the Atrial Substrate. *Entropy*. 2020; 22(2):232.
https://doi.org/10.3390/e22020232

**Chicago/Turabian Style**

Vraka, Aikaterini, Fernando Hornero, Vicente Bertomeu-González, Joaquín Osca, Raúl Alcaraz, and José J. Rieta.
2020. "Short-Time Estimation of Fractionation in Atrial Fibrillation with Coarse-Grained Correlation Dimension for Mapping the Atrial Substrate" *Entropy* 22, no. 2: 232.
https://doi.org/10.3390/e22020232