# Detection of Atrial Fibrillation Episodes based on 3D Algebraic Relationships between Cardiac Intervals

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## Abstract

**:**

## 1. Introduction

#### 1.1. Existing Diagnostics Techniques for Atrial Fibrillation

#### 1.2. Perfect Matrices of Lagrange Differences as a Method for ECG Signal Analysis

## 2. Methods

#### 2.1. The Description of the Experimental Setup

#### 2.2. Participants

#### 2.3. Ethics Statement

#### 2.4. The Description of the Proposed Algorithm

#### 2.4.1. Preliminary Synopsis

#### 2.4.2. The Architecture of Third-Order Square Matrices of Lagrange Differences

- All elements of the matrix must be different.
- Zeroth-order differences are located on the main diagonal.
- First-order differences are located on the secondary diagonal.
- The matrix is balanced with respect to time (the sum of all time lags is equal to zero).
- The matrix is balanced with respect to lexicographic variables (the number of different symbols must be the same).

#### 2.4.3. The Sensitivity of the Proposed Algorithm

#### 2.4.4. Trials Computed upon the Dataset with Different Combination Patterns

#### 2.4.5. The Development of the Decision Support System

## 3. Results and Discussion

#### 3.1. Performing the Statistical Analysis

#### 3.2. Generation of the Variation Interval

#### 3.2.1. Condition 1

#### 3.2.2. Condition 2

#### 3.2.3. Condition 3

#### 3.3. First Test Candidate

#### 3.4. Second Test Candidate

## 4. Limitations

## 5. Conclusions

## 6. Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A schematic diagram illustrating the architecture of 2-by-2 and 3-by-3 perfect matrices of Lagrange differences. Zeroth-order differences (the diagonal elements of the matrices) are depicted in circles. First-order differences are depicted by arrows connecting respective elements.

**Figure 2.**The three recorded cardiac intervals are plotted as time series (the time is in minutes): (

**a**) the RR interval; (

**b**) the JT interval; (

**c**) the QRS interval.

**Figure 3.**The sensitivity of 3-by-3 matrices vs 2-by-2 matrices is demonstrated for one of the candidates. The x-axis denotes the time in minutes. The y-axis demonstrates the algebraic relationship for the 2-by-2 (

**a**–

**c**) and 3-by-3 (

**d**) matrix combination for the cardiac intervals JT, QRS, RR. (

**a**) The 2-by-2 matrix algebraic relation is computed for the combination of cardiac intervals JT, QRS. (

**b**) The 2-by-2 matrix algebraic relation is computed for the combination of cardiac intervals JT, RR. (

**c**) The 2-by-2 matrix algebraic relationship is computed for the cardiac intervals QRS, RR. (

**d**) The 3-by-3 matrix algebraic relationship is computed for the combination of cardiac intervals JT, QRS, RR.

**Figure 4.**The distribution of individuals in each cohort is approximated by the Gaussian distribut-ion. (

**a**) The Gaussian distribution plot computed for the cohort of healthy individuals. (

**b**) The Gaussian distribution plot computed for the cohort of unhealthy individuals.

**Figure 5.**(

**a**) The construction of the variation interval between the mean of the healthy distribution minus the standard deviation of the healthy distribution, and the mean of the unhealthy distribution plus the standard deviation of the unhealthy distribution. The variation interval is indicated by a double arrow and is used for the classification of a new incoming candidate. (

**b**) The decision support system is depicted as a probability distribution plot with a candidate’s likelihood (0.22) of having AF, indicated by an asterisk sign.

**Figure 6.**Test candidate #1. (

**a**) The algebraic relation is shown for the combination of JT, RR, and QRS intervals. (

**b**) The classification of the candidate is marked with an asterisk indicator. (

**c**) The decision support system recommends the probability of the AF equal to 0.07. (

**d**) The semi-gauge representation is exhibited for the candidate with the arrow of the gauge pointing towards green, indicating that the person is classified as a healthy individual.

**Figure 7.**Test candidate #2. (

**a**) The algebraic relation is shown for the combination of JT, QRS, and RR intervals. (

**b**) The classification of the candidate is marked with an asterisk indicator. (

**c**) The decision support system recommends the probability of the AF equal to 0.54. (

**d**) The semi-gauge representation is exhibited for the candidate with the arrow of the gauge pointing towards yellow, indicating that the person is classified as an unhealthy individual.

**Table 1.**The nine different elements of the time series $x,y$, and $z$ with current, time-backward, and time-forward indexes.

${\mathit{x}}_{\mathit{n}-\mathit{\delta}}$ | ${\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}+\mathit{\delta}}$ |
---|---|---|

${\mathit{y}}_{\mathit{n}-\mathit{\delta}}$ | ${y}_{n}$ | ${y}_{n+\delta}$ |

${\mathit{z}}_{\mathit{n}-\mathit{\delta}}$ | ${z}_{n}$ | ${z}_{n+\delta}$ |

**Table 2.**The comparison between the sensitivity of 2-by-2 and 3-by-3 matrices. The variance value for each of the combination is tabulated for one of the candidates (see also Figure 2).

Candidate | Variance Values | |||
---|---|---|---|---|

Combination of JT–QRS Interval | Combination of JT–RR Interval | Combination of QRS–RR Interval | Combination of JT–QRS–RR Interval | |

0.0218 | 0.0059 | 0.0186 | 0.0244 |

**Table 3.**Statistical analysis outcome for the combination of parameters QRS interval, RR interval, and JT interval.

Healthy Candidates (H) | Unhealthy Candidates (UH) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{Variance}\text{}{\mathbf{\sigma}}_{\mathbf{H}}^{2}$ | $\mathbf{Mean}\text{}{\mathbf{\mu}}_{\mathbf{H}}$ | $\mathbf{Median}\text{}\mathbf{x}{~}_{\mathbf{H}}$ | $\mathbf{Standard}\text{}\mathbf{Deviation}\text{}{\mathbf{\sigma}}_{\mathbf{H}}$ | $\mathbf{Upper}\text{}\mathbf{Limit}\text{}(\mathbf{x}{~}_{\mathbf{H}}+{\mathbf{\sigma}}_{\mathbf{H}})$ | $\mathbf{Lower}\text{}\mathbf{Limit}\text{}\left(\mathbf{x}{~}_{\mathbf{H}}-{\mathbf{\sigma}}_{\mathbf{H}}\right)$ | $\mathbf{Area}\text{}{\mathbf{A}}_{\mathbf{H}}$ | $\mathbf{Variance}\text{}{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}}^{2}$ | $\mathbf{Mean}\text{}{\mathbf{\mu}}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Median}\text{}\mathbf{x}{~}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Standard}\text{}\mathbf{Deviation}\text{}{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Upper}\text{}\mathbf{Limit}\text{}(\mathbf{x}{~}_{\mathbf{U}\mathbf{H}}+{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}})$ | $\mathbf{Lower}\text{}\mathbf{Limit}\text{}(\mathbf{x}{~}_{\mathbf{U}\mathbf{H}}-{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}})$ | $\mathbf{Area}\text{}{\mathbf{A}}_{\mathbf{U}\mathbf{H}}$ |

0.00080583 | 0.736 | 0.734 | 0.028 | 0.763 | 0.707 | 0.359 | 0.0003 | 1.168 | 1.164 | 0.018 | 1.182 | 0.004 | 0.769 |

0.0016751 | 0.827 | 0.821 | 0.041 | 0.863 | 0.781 | 1.100 | 0.0002 | 1.281 | 1.279 | 0.015 | 1.294 | 1.264 | 0.875 |

0.0015197 | 0.917 | 0.916 | 0.039 | 0.956 | 0.878 | −0.376 | 0.0061 | 1.018 | 1.012 | 0.078 | 1.091 | 0.935 | 2.062 |

0.00073806 | 1.031 | 1.031 | 0.027 | 1.059 | 1.004 | 0.254 | 0.0006 | 0.932 | 0.936 | 0.025 | 0.962 | 0.912 | −1.217 |

0.0015323 | 1.024 | 1.027 | 0.039 | 1.067 | 0.989 | −1.307 | 0.0141 | 1.422 | 1.409 | 0.119 | 1.528 | 1.291 | 1.461 |

0.0010183 | 0.875 | 0.875 | 0.032 | 0.907 | 0.843 | 0.148 | 0.0027 | 0.822 | 0.811 | 0.052 | 0.864 | 0.760 | 2.422 |

0.010428 | 1.317 | 1.307 | 0.102 | 1.41 | 1.206 | 1.508 | 0.0015 | 1.011 | 1.006 | 0.039 | 1.045 | 0.967 | 1.466 |

0.0060534 | 0.899 | 0.881 | 0.078 | 0.959 | 0.804 | 4.437 |

**Table 4.**Statistical analysis outcome for the combination of parameters RR interval, JT interval, and QRS interval.

Healthy Candidates (H) | Unhealthy Candidates (UH) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{Variance}\text{}{\mathbf{\sigma}}_{\mathbf{H}}^{2}$ | $\mathbf{Mean}\text{}{\mathbf{\mu}}_{\mathbf{H}}$ | $\mathbf{Median}\text{}\mathbf{x}{~}_{\mathbf{H}}$ | $\mathbf{Standard}\text{}\mathbf{Deviation}\text{}{\mathbf{\sigma}}_{\mathbf{H}}$ | $\mathbf{Upper}\text{}\mathbf{Limit}\text{}(\mathbf{x}{~}_{\mathbf{U}\mathbf{H}}+{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}})$ | $\mathbf{Lower}\text{}\mathbf{Limit}\text{}(\mathbf{x}{~}_{\mathbf{U}\mathbf{H}}-{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}})$ | $\mathbf{Area}\text{}{\mathbf{A}}_{\mathbf{H}}$ | $\mathbf{Variance}\text{}{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}}^{2}$ | $\mathbf{Mean}\text{}{\mathbf{\mu}}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Median}\text{}\mathbf{x}{~}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Standard}\text{}\mathbf{Deviation}\text{}{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Upper}\text{}\mathbf{Limit}\text{}(\mathbf{x}{~}_{\mathbf{U}\mathbf{H}}+{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}})$ | $\mathbf{Lower}\text{}\mathbf{Limit}\text{}(\mathbf{x}{~}_{\mathbf{U}\mathbf{H}}-{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}})$ | $\mathbf{Area}\text{}{\mathbf{A}}_{\mathbf{U}\mathbf{H}}$ |

0.000792 | 0.736 | 0.736 | 0.028 | 0.764 | 0.708 | 0.242 | 0.0003 | 1.168 | 1.164 | 0.018 | 1.182 | 1.146 | 0.775 |

0.001677 | 0.827 | 0.821 | 0.041 | 0.862 | 0.780 | 1.147 | 0.0002 | 1.282 | 1.279 | 0.015 | 1.294 | 1.264 | 0.896 |

0.001713 | 0.921 | 0.922 | 0.041 | 0.963 | 0.880 | −0.494 | 0.0061 | 1.016 | 1.011 | 0.078 | 1.089 | 0.932 | 2.051 |

0.000767 | 1.033 | 1.033 | 0.026 | 1.061 | 1.005 | 0.310 | 0.0006 | 0.933 | 0.937 | 0.025 | 0.962 | 0.911 | −1.264 |

0.001689 | 1.031 | 1.034 | 0.041 | 1.075 | 0.999 | −1.086 | 0.0141 | 1.424 | 1.411 | 0.119 | 1.529 | 1.292 | 1.477 |

0.001000 | 0.881 | 0.879 | 0.032 | 0.911 | 0.848 | 0.231 | 0.0027 | 0.823 | 0.812 | 0.052 | 0.864 | 0.759 | 2.434 |

0.01042 | 1.319 | 1.310 | 0.102 | 1.412 | 1.208 | 1.525 | 0.0015 | 1.012 | 1.006 | 0.039 | 1.046 | 0.967 | 1.478 |

0.006422 | 0.906 | 0.891 | 0.080 | 0.976 | 0.810 | 4.392 |

**Table 5.**Statistical analysis outcome for the combination of parameters JT interval, RR interval, and QRS interval.

Healthy Candidates (H) | Unhealthy Candidates (UH) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{Variance}\text{}{\mathbf{\sigma}}_{\mathbf{H}}^{2}$ | $\mathbf{Mean}\text{}{\mathbf{\mu}}_{\mathbf{H}}$ | $\mathbf{Median}\text{}\mathbf{x}{~}_{\mathbf{H}}$ | $\mathbf{Standard}\text{}\mathbf{Deviation}\text{}{\mathbf{\sigma}}_{\mathbf{H}}$ | $\mathbf{Upper}\text{}\mathbf{Limit}\text{}(\mathbf{x}{~}_{\mathbf{H}}+{\mathbf{\sigma}}_{\mathbf{H}})$ | $\mathbf{Lower}\text{}\mathbf{Limit}\text{}\left(\mathbf{x}{~}_{\mathbf{H}}-{\mathbf{\sigma}}_{\mathbf{H}}\right)$ | $\mathbf{Area}\text{}{\mathbf{A}}_{\mathbf{H}}$ | $\mathbf{Variance}\text{}{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}}^{2}$ | $\mathbf{Mean}\text{}{\mathbf{\mu}}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Median}\text{}\mathbf{x}{~}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Standard}\text{}\mathbf{Deviation}\text{}{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Area}\text{}{\mathbf{A}}_{\mathbf{U}\mathbf{H}}$ | ||

0.0013 | 0.698 | 0.694 | 0.037 | 0.731 | 0.658 | 1.565 | 0.0006 | 1.397 | 1.393 | 0.025 | 1.418 | 1.367 | 0.9204 |

0.0030 | 0.882 | 0.880 | 0.055 | 0.934 | 0.825 | 0.527 | 0.0004 | 1.539 | 1.535 | 0.019 | 1.554 | 1.516 | 1.1994 |

0.0025 | 1.039 | 1.032 | 0.050 | 1.082 | 0.982 | 1.427 | 0.0090 | 1.051 | 1.043 | 0.095 | 1.138 | 0.948 | 2.2968 |

0.0014 | 1.165 | 1.168 | 0.037 | 1.206 | 1.131 | −0.806 | 0.0014 | 1.070 | 1.077 | 0.037 | 1.114 | 1.040 | −2.2801 |

0.0043 | 1.169 | 1.184 | 0.066 | 1.250 | 1.119 | −5.280 | 0.0067 | 1.293 | 1.283 | 0.082 | 1.365 | 1.201 | 1.1756 |

0.0029 | 0.905 | 0.909 | 0.054 | 0.963 | 0.856 | −0.469 | 0.0031 | 0.900 | 0.900 | 0.056 | 0.956 | 0.845 | 0.6442 |

0.0031 | 1.441 | 1.446 | 0.055 | 1.502 | 1.391 | −1.105 | 0.0014 | 1.140 | 1.139 | 0.038 | 1.177 | 1.101 | 0.5454 |

0.0021 | 1.003 | 0.998 | 0.046 | 1.044 | 0.953 | 1.300 |

**Table 6.**Statistical computations for the cohort of healthy and unhealthy individuals for the combination of parameters JT interval, QRS interval, and RR interval. The variance, mean, median, and standard deviation values are tabulated.

Healthy Candidates (H) | Unhealthy Candidates (UH) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{Variance}\text{}{\mathbf{\sigma}}_{\mathbf{H}}^{2}$ | $\mathbf{Mean}\text{}{\mathbf{\mu}}_{\mathbf{H}}$ | $\mathbf{Median}\text{}\mathbf{x}{~}_{\mathbf{H}}$ | $\mathbf{Standard}\text{}\mathbf{Deviation}\text{}{\mathbf{\sigma}}_{\mathbf{H}}$ | $\mathbf{Upper}\text{}\mathbf{Limit}\text{}(\mathbf{x}{~}_{\mathbf{H}}+{\mathbf{\sigma}}_{\mathbf{H}})$ | $\mathbf{Lower}\text{}\mathbf{Limit}\text{}\left(\mathbf{x}{~}_{\mathbf{H}}-{\mathbf{\sigma}}_{\mathbf{H}}\right)$ | $\mathbf{Area}\text{}{\mathbf{A}}_{\mathbf{H}}$ | $\mathbf{Variance}\text{}{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}}^{2}$ | $\mathbf{Mean}\text{}{\mathbf{\mu}}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Median}\text{}\mathbf{x}{~}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Standard}\text{}\mathbf{Deviation}\text{}{\mathbf{\sigma}}_{\mathbf{U}\mathbf{H}}$ | $\mathbf{Area}\text{}{\mathbf{A}}_{\mathbf{U}\mathbf{H}}$ | ||

0.0012 | 0.692 | 0.689 | 0.035 | 0.724 | 0.654 | 1.370 | 0.0006424 | 1.397 | 1.393 | 0.025 | 1.418 | 1.367 | 0.904 |

0.0030 | 0.881 | 0.880 | 0.055 | 0.934 | 0.825 | 0.435 | 0.0003502 | 1.538 | 1.535 | 0.019 | 1.553 | 1.516 | 1.219 |

0.0023 | 1.032 | 1.025 | 0.048 | 1.073 | 0.978 | 1.565 | 0.0082 | 1.038 | 1.031 | 0.091 | 1.122 | 0.940 | 2.147 |

0.0014 | 1.161 | 1.164 | 0.037 | 1.201 | 1.127 | −0.773 | 0.0015 | 1.065 | 1.073 | 0.039 | 1.112 | 1.034 | −2.483 |

0.0040 | 1.162 | 1.176 | 0.064 | 1.240 | 1.113 | −5.041 | 0.0064 | 1.285 | 1.274 | 0.080 | 1.354 | 1.194 | 1.532 |

0.0025 | 0.894 | 0.895 | 0.050 | 0.945 | 0.845 | 0.042 | 0.0030 | 0.897 | 0.900 | 0.055 | 0.955 | 0.844 | 0.355 |

0.0031 | 1.431 | 1.437 | 0.056 | 1.493 | 1.382 | −1.224 | 0.0014 | 1.135 | 1.134 | 0.038 | 1.172 | 1.096 | 0.548 |

0.0018 | 0.991 | 0.987 | 0.042 | 1.029 | 0.945 | 1.159 |

Condition 1 | Condition 2 | Condition 3 | |
---|---|---|---|

If | C $\le {\mathsf{\mu}}_{\mathrm{h}}-{\mathsf{\sigma}}_{\mathrm{h}}$ | $\text{}\mathrm{C}\ge {\text{}\mathsf{\mu}}_{\mathrm{uh}}+{\mathsf{\sigma}}_{\mathrm{uh}}$ | ${\text{}\mathsf{\mu}}_{\mathrm{uh}}-{\mathsf{\sigma}}_{\mathrm{uh}}\le \mathrm{C}\le {\text{}\mathsf{\mu}}_{\mathrm{h}}+{\mathsf{\sigma}}_{\mathrm{h}}$ |

Then, indicator is | $\mathrm{IND}=0$ | $\mathrm{IND}=1$ | $\mathrm{IND}=\frac{\mathrm{C}-({\mathsf{\mu}}_{\mathrm{uh}}-{\mathsf{\sigma}}_{\mathrm{uh}})}{\left({\mathsf{\mu}}_{\mathrm{h}}+{\mathsf{\sigma}}_{\mathrm{h}}\right)-\left({\mathsf{\mu}}_{\mathrm{uh}}-{\mathsf{\sigma}}_{\mathrm{uh}}\right)}$ |

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## Share and Cite

**MDPI and ACS Style**

Qammar, N.W.; Šiaučiūnaitė, V.; Zabiela, V.; Vainoras, A.; Ragulskis, M.
Detection of Atrial Fibrillation Episodes based on 3D Algebraic Relationships between Cardiac Intervals. *Diagnostics* **2022**, *12*, 2919.
https://doi.org/10.3390/diagnostics12122919

**AMA Style**

Qammar NW, Šiaučiūnaitė V, Zabiela V, Vainoras A, Ragulskis M.
Detection of Atrial Fibrillation Episodes based on 3D Algebraic Relationships between Cardiac Intervals. *Diagnostics*. 2022; 12(12):2919.
https://doi.org/10.3390/diagnostics12122919

**Chicago/Turabian Style**

Qammar, Naseha Wafa, Vaiva Šiaučiūnaitė, Vytautas Zabiela, Alfonsas Vainoras, and Minvydas Ragulskis.
2022. "Detection of Atrial Fibrillation Episodes based on 3D Algebraic Relationships between Cardiac Intervals" *Diagnostics* 12, no. 12: 2919.
https://doi.org/10.3390/diagnostics12122919