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In this work we compare three multiscale measures for their ability to discriminate between participants having cardiac autonomic neuropathy (CAN) and aged controls. CAN is a disease that involves nerve damage leading to an abnormal control of heart rate, so one would expect disease progression to manifest in changes to heart rate variability (HRV). We applied multiscale entropy (MSE), multi fractal detrended fluctuation analysis (MFDFA), and Renyi entropy (RE) to recorded datasets of RR intervals. The latter measure provided the best separation (lowest p-value in Mann–Whitney tests) between classes of participants having CAN, early CAN or no CAN (controls). This comparison suggests the efficacy of RE as a measure for diagnosis of CAN and its progression, when compared to the other multiscale measures.

Cardiac rhythm is controlled by the membrane properties of the sino-atrial node, neuro-hormonal and autonomic nervous system (ANS) modulation [

A typical ECG signal is illustrated in

Examples of nonlinear measures include fractal and entropy measures. It is this group of measures which are examined in this work. In spite of a growing literature illustrating their use in HRV analysis, there is a shortage of empirical comparisons between different nonlinear/multiscale methods, particularly in relation to CAN and CAN progression. We examine how changing the value of scaling, length and parameters of the fractal and entropy algorithms provides a family of measures, and forms a multiscale series. The results of the different permutations of the multiscale measures are compared in terms of distinguishing CAN from controls in participants attending the diabetes complications screening clinic in Albury, Australia.

Multi-scale measures are those that provide a series of measures as a function of some scaling factor. The idea of multi-scale used in this paper requires some explanation. Not all of the measures considered here are multi-scale in the same sense. MSE examines RR intervals at a number of different scales in order to provide multiple measures based on the same time series. MFDFA and RE are multi-scale in the sense that a parameter is used to control the order or power that a quantity is raised to. This can take multiple values and so provides multiple measures. Previous work has compared several entropy measures in terms of their efficacy in distinguishing congestive heart failure and arrhythmia from normal sinus rhythm [

The aim of this work is to determine which, out of a variety of available multi-scale measures, is most suitable for detecting CAN, and to elucidate the reasons for the discriminatory power, or lack thereof, manifested by these methods. The work reported here used data from the Charles Sturt Diabetes Complications Screening Group (DiScRi), Australia [

The entropy _{1}_{n}

The probability _{i}_{m}(i)_{i}_{i+m−1}_{m}(i)_{i}_{m}(i)_{m}(j)

The sample entropy can be seen as the probability that a sequence of RR intervals of length _{m}_{m+1}_{i}_{j}

MFDFA is based on DFA, a fractal-like measure that examines the self-similarity properties of a time series, in this case a sequence of RR intervals. In the ensuing discussion, we follow the implementation of [_{1}_{n}

The transformed series _{ν}_{i}

As with the normal techniques for estimation of fractal dimension, these data are visualized as a graph where the values of _{m}

In order to derive a multifractal measure,

RE is naturally a multi-scale measure, as it generalizes the Shannon entropy and includes the Shannon entropy as a special case [_{i}

The density method estimates the probability of a sequence of RR intervals of length

The MSE was calculated using the default parameters of sequence length

Development of algorithms to characterize HRV of healthy and unhealthy subjects has been an ongoing endeavor, especially with relation to nonlinear methods as standard measures such as SDNN can lead to incorrect interpretation of pathology [

Results from the Mann–Whitney test using linear measures of HRV time and frequency domain analysis are shown in

Results from ROC analysis for time and frequency domain HRV measures are shown in

Summary statistics for the MSE are shown in

Results from the Mann–Whitney test are shown in

Results from ROC analysis are shown in

Summary statistics for MFDFA are shown in

Of interest is the observation that measures most successful in discriminating classes were all from the negative side of the spectrum. Examining

Results from ROC analysis are shown in

Results for RE, based on probabilities calculated using the density method, are shown in

Results from ROC analysis are shown in ^{2}

Traditional HRV analysis is based on the assumptions of linearity, stationary and equilibrium nature of heart rate time signals. Nonlinear and multiscale studies applying MSE, MFDFA and RE have indicated a high level of complexity in HRV dynamics based on the observation of a power law relationship when viewing the power spectral density for varying time scales. Hence traditional HRV analysis using time and frequency domain are often not sufficient to characterize the complex dynamics inherent in the heart rate time series [

Multiscale measures of HRV have generated a great deal of interest but a systematic approach is required in order to identify appropriate measures that may be of benefit in the context early identification of disease is of clinical importance to the practitioner. In this work we compared three approaches that yield multiscale measures, and evaluated their potential for discriminating different classes of CAN. We found that MSE afforded no useful discrimination between disease categories. In contrast, MFDFA provided a set of measures that were clearly able to discriminate these classes, but only when using negative values of the parameter

Using RE we were able to demonstrate a very strong separation of disease classes across all values of the parameter

The authors would like to thank Bev de Jong for technical assistance and Roche Australia Pty for providing consumables for blood glucose measurements. The study was funded in part by a Compacts Funding from Charles Sturt University. Roche Australia Pty and Charles Sturt University played no role in the design, data collection, analysis and interpretation of the study.

Herbert F. Jelinek contributed the problem formulation, much of the background material, most of the literature searching, and collected and contributed the data. Mika Tarvainen contributed the processing of the data. David Cornforth performed the analysis and wrote most of the paper. All authors assisted in preparation of the manuscript for publishing. All authors have read and approved the final manuscript.

The authors declare no conflict of interest.

A typical ECG signal showing the RR interval.

Results of Mann–Whitney test for comparison between different classes of participants using time and frequency domain measures. All results are significant at the

Scale | SDNN | pNN50% | LF power (ms^{2}) |
HF power (ms^{2}) |
---|---|---|---|---|

N |
0.000 | 0.002 | 0.000 | 0.001 |

E |
0.011 | 0.002 | 0.017 | 0.024 |

D |
0.000 | 0.000 | 0.000 | 0.000 |

Area under the ROC curve for comparison between different classes of participants using common time and frequency domain measures. Results > 0.8 are shown in bold face. Comparisons shown are normal

Scale | SDNN | pNN50% | LF power (ms^{2}) |
HF power (ms^{2}) |
---|---|---|---|---|

N |
0.302 | 0.352 | 0.282 | 0.334 |

E |
0.261 | 0.237 | 0.275 | 0.287 |

D |

Summary statistics for different classes of participants (N—Normal, E—Early and D—Definite) using MSE. Headings in the row designated as “scale” provide the number of RR intervals used to derive the measure. Values are provided for the upper quartile, median, and lower quartile.

Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

| |||||||||||

Upper | 2.018 | 2.069 | 2.011 | 1.918 | 1.922 | 1.846 | 1.813 | 1.709 | 1.642 | 1.569 | |

Median | 1.855 | 1.983 | 1.889 | 1.791 | 1.764 | 1.648 | 1.631 | 1.519 | 1.439 | 1.339 | |

Lower | 1.700 | 1.770 | 1.610 | 1.621 | 1.513 | 1.428 | 1.352 | 1.277 | 1.199 | 1.066 | |

| |||||||||||

Upper | 2.068 | 2.128 | 2.035 | 1.961 | 1.937 | 1.930 | 1.902 | 1.781 | 1.722 | 1.625 | |

Median | 1.953 | 1.972 | 1.872 | 1.827 | 1.758 | 1.812 | 1.682 | 1.598 | 1.531 | 1.437 | |

Lower | 1.815 | 1.855 | 1.753 | 1.588 | 1.617 | 1.584 | 1.477 | 1.426 | 1.337 | 1.190 | |

| |||||||||||

Upper | 2.140 | 2.137 | 2.129 | 2.015 | 2.066 | 2.018 | 1.805 | 1.761 | 1.767 | 1.762 | |

Median | 1.929 | 1.928 | 1.904 | 1.762 | 1.706 | 1.748 | 1.645 | 1.659 | 1.581 | 1.551 | |

Lower | 1.610 | 1.737 | 1.636 | 1.504 | 1.668 | 1.441 | 1.516 | 1.242 | 1.199 | 1.154 |

Results of test for comparison between different classes of participants using MSE. Results at the

Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

N |
0.361 | 0.376 | 0.450 | 0.369 | 0.143 | 0.083 | 0.156 | |||

E |
0.746 | 0.509 | 0.872 | 0.782 | 0.936 | 0.819 | 0.647 | 0.652 | 0.882 | 0.661 |

D |
0.661 | 0.812 | 0.665 | 0.976 | 0.740 | 0.420 | 0.749 | 0.455 | 0.307 | 0.270 |

Area under the ROC curve for comparison between different classes of participants using MSE. Results > 0.6 are shown in bold face. Comparisons shown are Normal

Scale | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

N |
0.545 | 0.543 | 0.537 | 0.544 | 0.571 | 0.584 | 0.569 | |||

E |
0.529 | 0.559 | 0.515 | 0.525 | 0.507 | 0.520 | 0.542 | 0.540 | 0.513 | 0.540 |

D |
0.538 | 0.522 | 0.538 | 0.502 | 0.530 | 0.572 | 0.529 | 0.567 | 0.592 | 0.599 |

Summary statistics for different classes of participants (N—Normal, E—Early and D—Definite) using MFDFA. Headings in the row labelled “q-order” denote the exponent used in

q-order | −5 | −4 | −3 | −2 | −1 | 0 | +1 | +2 | +3 | +4 | +5 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

| ||||||||||||

Upper | 0.081 | 0.081 | 0.084 | 0.091 | 0.119 | 0.191 | 0.294 | 0.353 | 0.381 | 0.390 | 0.396 | |

Median | 0.040 | 0.043 | 0.047 | 0.062 | 0.103 | 0.166 | 0.234 | 0.272 | 0.291 | 0.285 | 0.289 | |

Lower | 0.010 | 0.016 | 0.027 | 0.046 | 0.086 | 0.140 | 0.186 | 0.209 | 0.215 | 0.218 | 0.218 | |

| ||||||||||||

Upper | 0.130 | 0.129 | 0.120 | 0.121 | 0.132 | 0.207 | 0.303 | 0.376 | 0.402 | 0.403 | 0.401 | |

Median | 0.073 | 0.074 | 0.078 | 0.081 | 0.112 | 0.176 | 0.258 | 0.303 | 0.293 | 0.302 | 0.295 | |

Lower | 0.025 | 0.029 | 0.037 | 0.054 | 0.088 | 0.135 | 0.199 | 0.243 | 0.240 | 0.231 | 0.217 | |

| ||||||||||||

Upper | 0.181 | 0.179 | 0.171 | 0.148 | 0.142 | 0.213 | 0.330 | 0.402 | 0.435 | 0.441 | 0.448 | |

Median | 0.132 | 0.132 | 0.130 | 0.128 | 0.122 | 0.184 | 0.257 | 0.316 | 0.336 | 0.339 | 0.335 | |

Lower | 0.053 | 0.057 | 0.063 | 0.071 | 0.109 | 0.143 | 0.180 | 0.206 | 0.209 | 0.165 | 0.130 |

Results of Mann–Whitney test for comparison between different classes of participants using MFDFA variables calculated from RR interval series. Results at the

q-order | −5 | −4 | −3 | −2 | −1 | 0 | +1 | +2 | +3 | +4 | +5 |
---|---|---|---|---|---|---|---|---|---|---|---|

N |
0.065 | 0.108 | 0.282 | 0.141 | 0.068 | 0.156 | 0.375 | 0.500 | |||

E |
0.281 | 0.201 | 0.118 | 0.080 | 0.196 | 0.618 | 0.777 | 0.861 | 1.000 | 0.957 | 0.989 |

D |
0.054 | 0.296 | 0.302 | 0.342 | 0.553 | 0.585 | 0.695 |

Area under the ROC curve for comparison between different classes of participants using MFDFA variables calculated from RR interval series. Results > 0.7 are shown in bold face. Comparisons shown are Normal

q-order | −5 | −4 | −3 | −2 | −1 | 0 | +1 | +2 | +3 | +4 | +5 |
---|---|---|---|---|---|---|---|---|---|---|---|

N |
0.589 | 0.614 | 0.640 | 0.622 | 0.577 | 0.552 | 0.571 | 0.588 | 0.568 | 0.543 | 0.532 |

E |
0.598 | 0.616 | 0.642 | 0.659 | 0.618 | 0.545 | 0.526 | 0.516 | 0.500 | 0.505 | 0.501 |

D |
0.673 | 0.594 | 0.593 | 0.585 | 0.553 | 0.549 | 0.535 |

Summary statistics for different classes of participants (N—Normal, E—Early and D—Definite) using RE. Headings in the row labelled “exponent” denote

Exponent | −5 | −4 | −3 | −2 | −1 | +1 | +2 | +3 | +4 | +5 | |
---|---|---|---|---|---|---|---|---|---|---|---|

Upper | 1.230 | 1.202 | 1.163 | 1.107 | 1.038 | 0.998 | 0.996 | 0.995 | 0.994 | 0.993 | |

Median | 1.087 | 1.063 | 1.043 | 1.021 | 1.007 | 0.998 | 0.992 | 0.989 | 0.986 | 0.985 | |

Lower | 1.032 | 1.023 | 1.013 | 1.005 | 1.001 | 0.997 | 0.983 | 0.977 | 0.972 | 0.969 | |

| |||||||||||

Upper | 1.138 | 1.109 | 1.072 | 1.034 | 1.008 | 0.998 | 0.997 | 0.997 | 0.996 | 0.996 | |

Median | 1.031 | 1.021 | 1.012 | 1.005 | 1.001 | 0.998 | 0.996 | 0.995 | 0.994 | 0.993 | |

Lower | 1.005 | 1.003 | 1.001 | 1.000 | 0.999 | 0.997 | 0.993 | 0.991 | 0.989 | 0.988 | |

| |||||||||||

Upper | 1.105 | 1.082 | 1.052 | 1.020 | 1.003 | 0.998 | 0.998 | 0.997 | 0.997 | 0.997 | |

Median | 1.002 | 1.001 | 1.000 | 0.999 | 0.999 | 0.998 | 0.997 | 0.997 | 0.997 | 0.996 | |

Lower | 1.000 | 0.999 | 0.999 | 0.998 | 0.998 | 0.998 | 0.996 | 0.996 | 0.995 | 0.995 |

Results of Mann–Whitney test for comparison between different classes of participants using RE variables calculated from RR interval series. Results at the

Exponent | −5 | −4 | −3 | −2 | −1 | +1 | +2 | +3 | +4 | +5 |
---|---|---|---|---|---|---|---|---|---|---|

N |
||||||||||

E |
0.063 | 0.059 | 0.061 | 0.051 | 0.056 | |||||

D |

Area under the ROC curve for comparison between different classes of participants using RE variables calculated from RR interval series. Results > 0.7 are shown in bold. Comparisons shown are Normal

Exponent | −5 | −4 | −3 | −2 | −1 | +1 | +2 | +3 | +4 | +5 |
---|---|---|---|---|---|---|---|---|---|---|

N |
0.657 | 0.658 | 0.660 | 0.667 | 0.685 | |||||

E |
0.669 | 0.672 | 0.670 | 0.678 | 0.691 | 0.674 | 0.689 | 0.686 | 0.690 | 0.692 |

D |