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Yuan S, He Z, Zhao J, Yuan Z. Fusing depth local dual-view features and dual-input transformer framework for improving the recognition ability of motion artifact-contaminated electrocardiogram. COMPLEX INTELL SYST 2022. [DOI: 10.1007/s40747-022-00861-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
AbstractHeart health monitoring based on wearable devices is often contaminated by various noises to varying degrees. Using signal quality indicators (SQIs) to achieve signal quality assessment (SQA) is among the most promising ways to solve this problem, but the performance of SQIs in expressing ECG quality features contaminated by motion artifact (MA) noise remains disappointing. Here, we present a novel SQA method that fuses the proposed depth local dual-view (DLDV) features and the dual-input transformer (DI-Transformer) framework to improve the recognition ability of MA-contaminated ECGs. The proposed DLDV features are to identify subtle differences between MA and ECG through depth local amplitude and phase angle features. When it fuses with the temporal relationship features extracted by DI-Transformer, its accuracy is significantly improved compared to the SQIs-based methods. In addition, we also verify the robustness and the accuracy of DLDV features on four traditional classifiers. Finally, we conduct our experiments on the two datasets. On the PhysioNet/Computing in Cardiology Challenge dataset, the DLDV features (Acc = 95.49%) outperform the combination of six SQIs features (Acc = 91.26%). When combined with our DI-Transformer, it delivered an accuracy of 99.62%, outperforming the state-of-the-art SQA methods. On the artificial testset constructed by MA noise, our DI-Transformer outperforms four traditional methods and also delivered an accuracy of 97.69%.
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Cardiovascular Signal Entropy Predicts All-Cause Mortality: Evidence from The Irish Longitudinal Study on Ageing (TILDA). ENTROPY 2022; 24:e24050676. [PMID: 35626560 PMCID: PMC9142113 DOI: 10.3390/e24050676] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/30/2022] [Revised: 05/04/2022] [Accepted: 05/09/2022] [Indexed: 11/17/2022]
Abstract
In this study, the relationship between cardiovascular signal entropy and the risk of seven-year all-cause mortality was explored in a large sample of community-dwelling older adults from The Irish Longitudinal Study on Ageing (TILDA). The hypothesis under investigation was that physiological dysregulation might be quantifiable by the level of sample entropy (SampEn) in continuously noninvasively measured resting-state systolic (sBP) and diastolic (dBP) blood pressure (BP) data, and that this SampEn measure might be independently predictive of mortality. Participants’ date of death up to 2017 was identified from official death registration data and linked to their TILDA baseline survey and health assessment data (2010). BP was continuously monitored during supine rest at baseline, and SampEn values were calculated for one-minute and five-minute sections of this data. In total, 4543 participants were included (mean (SD) age: 61.9 (8.4) years; 54.1% female), of whom 214 died. Cox proportional hazards regression models were used to estimate the hazard ratios (HRs) with 95% confidence intervals (CIs) for the associations between BP SampEn and all-cause mortality. Results revealed that higher SampEn in BP signals was significantly predictive of mortality risk, with an increase of one standard deviation in sBP SampEn and dBP SampEn corresponding to HRs of 1.19 and 1.17, respectively, in models comprehensively controlled for potential confounders. The quantification of SampEn in short length BP signals could provide a novel and clinically useful predictor of mortality risk in older adults.
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Borin AMS, Humeau-Heurtier A, Virgílio Silva LE, Murta LO. Multiscale Entropy Analysis of Short Signals: The Robustness of Fuzzy Entropy-Based Variants Compared to Full-Length Long Signals. ENTROPY (BASEL, SWITZERLAND) 2021; 23:1620. [PMID: 34945926 PMCID: PMC8700117 DOI: 10.3390/e23121620] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/08/2021] [Revised: 11/26/2021] [Accepted: 11/28/2021] [Indexed: 11/16/2022]
Abstract
Multiscale entropy (MSE) analysis is a fundamental approach to access the complexity of a time series by estimating its information creation over a range of temporal scales. However, MSE may not be accurate or valid for short time series. This is why previous studies applied different kinds of algorithm derivations to short-term time series. However, no study has systematically analyzed and compared their reliabilities. This study compares the MSE algorithm variations adapted to short time series on both human and rat heart rate variability (HRV) time series using long-term MSE as reference. The most used variations of MSE are studied: composite MSE (CMSE), refined composite MSE (RCMSE), modified MSE (MMSE), and their fuzzy versions. We also analyze the errors in MSE estimations for a range of incorporated fuzzy exponents. The results show that fuzzy MSE versions-as a function of time series length-present minimal errors compared to the non-fuzzy algorithms. The traditional multiscale entropy algorithm with fuzzy counting (MFE) has similar accuracy to alternative algorithms with better computing performance. For the best accuracy, the findings suggest different fuzzy exponents according to the time series length.
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Affiliation(s)
- Airton Monte Serrat Borin
- Federal Institute of Education, Science and Technology of Triangulo Mineiro, Uberaba 38064-790, Brazil;
| | - Anne Humeau-Heurtier
- LARIS—Laboratoire Angevin de Recherche en Ingénierie des Systèmes, University of Angers, 49035 Angers, France;
| | - Luiz Eduardo Virgílio Silva
- Department of Internal Medicine, Ribeirão Preto Medical School, University of São Paulo, Ribeirão Preto 14049-900, Brazil;
| | - Luiz Otávio Murta
- Department of Computing and Mathematics, School of Philosophy, Sciences and Languages of Ribeirão Preto, University of São Paulo, Ribeirão Preto 14040-901, Brazil
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Knight SP, Newman L, Scarlett S, O’Connor JD, Davis J, De Looze C, Kenny RA, Romero-Ortuno R. Associations between Cardiovascular Signal Entropy and Cognitive Performance over Eight Years. ENTROPY (BASEL, SWITZERLAND) 2021; 23:1337. [PMID: 34682061 PMCID: PMC8534418 DOI: 10.3390/e23101337] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/21/2021] [Revised: 09/15/2021] [Accepted: 10/12/2021] [Indexed: 12/27/2022]
Abstract
In this study, the relationship between non-invasively measured cardiovascular signal entropy and global cognitive performance was explored in a sample of community-dwelling older adults from The Irish Longitudinal Study on Ageing (TILDA), both cross-sectionally at baseline (n = 4525; mean (SD) age: 61.9 (8.4) years; 54.1% female) and longitudinally. We hypothesised that signal disorder in the cardiovascular system, as quantified by short-length signal entropy during rest, could provide a marker for cognitive function. Global cognitive function was assessed via Mini Mental State Examination (MMSE) across five longitudinal waves (8 year period; n = 4316; mean (SD) age: 61.9 (8.4) years; 54.4% female) and the Montreal Cognitive Assessment (MOCA) across two longitudinal waves (4 year period; n = 3600; mean (SD) age: 61.7 (8.2) years; 54.1% female). Blood pressure (BP) was continuously monitored during supine rest at baseline, and sample entropy values were calculated for one-minute and five-minute sections of this data, both for time-series data interpolated at 5 Hz and beat-to-beat data. Results revealed significant associations between BP signal entropy and cognitive performance, both cross-sectionally and longitudinally. Results also suggested that as regards associations with cognitive performance, the entropy analysis approach used herein potentially outperformed more traditional cardiovascular measures such as resting heart rate and heart rate variability. The quantification of entropy in short-length BP signals could provide a clinically useful marker of the cardiovascular dysregulations that potentially underlie cognitive decline.
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Affiliation(s)
- Silvin P. Knight
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (S.S.); (J.D.O.); (J.D.); (C.D.L.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
| | - Louise Newman
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (S.S.); (J.D.O.); (J.D.); (C.D.L.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
| | - Siobhan Scarlett
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (S.S.); (J.D.O.); (J.D.); (C.D.L.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
| | - John D. O’Connor
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (S.S.); (J.D.O.); (J.D.); (C.D.L.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
- School of Medicine, Dentistry and Biomedical Sciences, The Patrick G Johnston Centre for Cancer Research, Queen’s University, Belfast BT9 7BL, UK
| | - James Davis
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (S.S.); (J.D.O.); (J.D.); (C.D.L.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
| | - Celine De Looze
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (S.S.); (J.D.O.); (J.D.); (C.D.L.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
| | - Rose Anne Kenny
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (S.S.); (J.D.O.); (J.D.); (C.D.L.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
- Mercer’s Institute for Successful Ageing (MISA), St. James’s Hospital, D08 E191 Dublin, Ireland
| | - Roman Romero-Ortuno
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (S.S.); (J.D.O.); (J.D.); (C.D.L.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
- Mercer’s Institute for Successful Ageing (MISA), St. James’s Hospital, D08 E191 Dublin, Ireland
- Global Brain Health Institute, Trinity College Dublin, D02 PN40 Dublin, Ireland
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Knight SP, Newman L, O’Connor JD, Davis J, Kenny RA, Romero-Ortuno R. Associations between Neurocardiovascular Signal Entropy and Physical Frailty. ENTROPY (BASEL, SWITZERLAND) 2020; 23:E4. [PMID: 33374999 PMCID: PMC7822043 DOI: 10.3390/e23010004] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/30/2020] [Revised: 12/16/2020] [Accepted: 12/19/2020] [Indexed: 12/13/2022]
Abstract
In this cross-sectional study, the relationship between noninvasively measured neurocardiovascular signal entropy and physical frailty was explored in a sample of community-dwelling older adults from The Irish Longitudinal Study on Ageing (TILDA). The hypothesis under investigation was that dysfunction in the neurovascular and cardiovascular systems, as quantified by short-length signal complexity during a lying-to-stand test (active stand), could provide a marker for frailty. Frailty status (i.e., "non-frail", "pre-frail", and "frail") was based on Fried's criteria (i.e., exhaustion, unexplained weight loss, weakness, slowness, and low physical activity). Approximate entropy (ApEn) and sample entropy (SampEn) were calculated during resting (lying down), active standing, and recovery phases. There was continuously measured blood pressure/heart rate data from 2645 individuals (53.0% female) and frontal lobe tissue oxygenation data from 2225 participants (52.3% female); both samples had a mean (SD) age of 64.3 (7.7) years. Results revealed statistically significant associations between neurocardiovascular signal entropy and frailty status. Entropy differences between non-frail and pre-frail/frail were greater during resting state compared with standing and recovery phases. Compared with ApEn, SampEn seemed to have better discriminating power between non-frail and pre-frail/frail individuals. The quantification of entropy in short length neurocardiovascular signals could provide a clinically useful marker of the multiple physiological dysregulations that underlie physical frailty.
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Affiliation(s)
- Silvin P. Knight
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (J.D.O.); (J.D.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
| | - Louise Newman
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (J.D.O.); (J.D.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
| | - John D. O’Connor
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (J.D.O.); (J.D.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
- School of Medicine, Dentistry and Biomedical Sciences, The Patrick G Johnston Centre for Cancer Research, Queen’s University, Belfast BT9 7BL, UK
| | - James Davis
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (J.D.O.); (J.D.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
| | - Rose Anne Kenny
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (J.D.O.); (J.D.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
- Mercer’s Institute for Successful Ageing (MISA), St. James’s Hospital, D08 NHY1 Dublin, Ireland
| | - Roman Romero-Ortuno
- The Irish Longitudinal Study on Ageing (TILDA), School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland; (L.N.); (J.D.O.); (J.D.); (R.A.K.); (R.R.-O.)
- Discipline of Medical Gerontology, School of Medicine, Trinity College Dublin, D02 R590 Dublin, Ireland
- Mercer’s Institute for Successful Ageing (MISA), St. James’s Hospital, D08 NHY1 Dublin, Ireland
- Global Brain Health Institute, Trinity College Dublin, D02 DK07 Dublin, Ireland
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Silva LEV, Fazan R, Marin-Neto JA. PyBioS: A freeware computer software for analysis of cardiovascular signals. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2020; 197:105718. [PMID: 32866762 DOI: 10.1016/j.cmpb.2020.105718] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2020] [Accepted: 08/18/2020] [Indexed: 06/11/2023]
Abstract
BACKGROUND AND OBJECTIVE Several software applications have been proposed in the past years as computational tools for assessing biomedical signals. Many of them are focused on heart rate variability series only, with their strengths and limitations depending on the necessity of the user and the scope of the application. Here, we introduce new software, named PyBioS, intended for the analysis of cardiovascular signals, even though any type of biomedical signal can be used. PyBioS has some functionalities that differentiate it from the other software. METHODS PyBioS was developed in Python language with an intuitive, user-friendly graphical user interface. The basic steps for using PyBioS comprise the opening or creation (simulation) of signals, their visualization, preprocessing and analysis. Currently, PyBioS has 8 preprocessing tools and 15 analysis methods, the later providing more than 50 metrics for analysis of the signals' dynamics. RESULTS The possibility to create simulated signals and save the preprocessed signals is a strength of PyBioS. Besides, the software allows batch processing of files, making the analysis of a large amount of data easy and fast. Finally, PyBioS has plenty of analysis methods implemented, with the focus on nonlinear and complexity analysis of signals and time series. CONCLUSIONS Although PyBioS is not intended to overcome all the necessities from users, it has useful functionalities that may be helpful in many situations. Moreover, PyBioS is continuously under improvement and several simulated signals, tools and analysis methods are still to be implemented. Also, a new module is being implemented on it to provide machine learning algorithms for classification and regression of data extracted from the biomedical signals.
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Affiliation(s)
| | - Rubens Fazan
- Ribeirão Preto Medical School, University of São Paulo, Ribeirão Preto, SP, Brazil.
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Mjahad A, Rosado-Muñoz A, Bataller-Mompeán M, Francés-Víllora JV, Guerrero-Martínez JF. Ventricular Fibrillation and Tachycardia detection from surface ECG using time-frequency representation images as input dataset for machine learning. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2017; 141:119-127. [PMID: 28241963 DOI: 10.1016/j.cmpb.2017.02.010] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2016] [Revised: 12/23/2016] [Accepted: 02/09/2017] [Indexed: 06/06/2023]
Abstract
BACKGROUND AND OBJECTIVE To safely select the proper therapy for Ventricullar Fibrillation (VF) is essential to distinct it correctly from Ventricular Tachycardia (VT) and other rhythms. Provided that the required therapy would not be the same, an erroneous detection might lead to serious injuries to the patient or even cause Ventricular Fibrillation (VF). The main novelty of this paper is the use of time-frequency (t-f) representation images as the direct input to the classifier. We hypothesize that this method allow to improve classification results as it allows to eliminate the typical feature selection and extraction stage, and its corresponding loss of information. METHODS The standard AHA and MIT-BIH databases were used for evaluation and comparison with other authors. Previous to t-f Pseudo Wigner-Ville (PWV) calculation, only a basic preprocessing for denoising and signal alignment is necessary. In order to check the validity of the method independently of the classifier, four different classifiers are used: Logistic Regression with L2 Regularization (L2 RLR), Adaptive Neural Network Classifier (ANNC), Support Vector Machine (SSVM), and Bagging classifier (BAGG). RESULTS The main classification results for VF detection (including flutter episodes) are 95.56% sensitivity and 98.8% specificity, 88.80% sensitivity and 99.5% specificity for ventricular tachycardia (VT), 98.98% sensitivity and 97.7% specificity for normal sinus, and 96.87% sensitivity and 99.55% specificity for other rhythms. CONCLUSION Results shows that using t-f data representations to feed classifiers provide superior performance values than the feature selection strategies used in previous works. It opens the door to be used in any other detection applications.
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Affiliation(s)
- A Mjahad
- GDDP, Group for Digital Design and Processing, University of Valencia - ETSE - Electronic Eng. Dpt., Av. Universitat, s/n, 46100, Burjassot, Valencia, Spain.
| | - A Rosado-Muñoz
- GDDP, Group for Digital Design and Processing, University of Valencia - ETSE - Electronic Eng. Dpt., Av. Universitat, s/n, 46100, Burjassot, Valencia, Spain.
| | - M Bataller-Mompeán
- GDDP, Group for Digital Design and Processing, University of Valencia - ETSE - Electronic Eng. Dpt., Av. Universitat, s/n, 46100, Burjassot, Valencia, Spain
| | - J V Francés-Víllora
- GDDP, Group for Digital Design and Processing, University of Valencia - ETSE - Electronic Eng. Dpt., Av. Universitat, s/n, 46100, Burjassot, Valencia, Spain
| | - J F Guerrero-Martínez
- GDDP, Group for Digital Design and Processing, University of Valencia - ETSE - Electronic Eng. Dpt., Av. Universitat, s/n, 46100, Burjassot, Valencia, Spain
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Bolea J, Pueyo E, Orini M, Bailón R. Influence of Heart Rate in Non-linear HRV Indices as a Sampling Rate Effect Evaluated on Supine and Standing. Front Physiol 2016; 7:501. [PMID: 27895588 PMCID: PMC5108795 DOI: 10.3389/fphys.2016.00501] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2016] [Accepted: 10/13/2016] [Indexed: 11/21/2022] Open
Abstract
The purpose of this study is to characterize and attenuate the influence of mean heart rate (HR) on nonlinear heart rate variability (HRV) indices (correlation dimension, sample, and approximate entropy) as a consequence of being the HR the intrinsic sampling rate of HRV signal. This influence can notably alter nonlinear HRV indices and lead to biased information regarding autonomic nervous system (ANS) modulation. First, a simulation study was carried out to characterize the dependence of nonlinear HRV indices on HR assuming similar ANS modulation. Second, two HR-correction approaches were proposed: one based on regression formulas and another one based on interpolating RR time series. Finally, standard and HR-corrected HRV indices were studied in a body position change database. The simulation study showed the HR-dependence of non-linear indices as a sampling rate effect, as well as the ability of the proposed HR-corrections to attenuate mean HR influence. Analysis in a body position changes database shows that correlation dimension was reduced around 21% in median values in standing with respect to supine position (p < 0.05), concomitant with a 28% increase in mean HR (p < 0.05). After HR-correction, correlation dimension decreased around 18% in standing with respect to supine position, being the decrease still significant. Sample and approximate entropy showed similar trends. HR-corrected nonlinear HRV indices could represent an improvement in their applicability as markers of ANS modulation when mean HR changes.
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Affiliation(s)
- Juan Bolea
- Centro de Investigación Biomédica en Red Bioingeniería, Biomateriales y NanomedicinaZaragoza, Spain
- BSICoS Group, Aragón Institute of Engineering Research (I3A), ISS Aragón, Universidad de ZaragozaZaragoza, Spain
| | - Esther Pueyo
- Centro de Investigación Biomédica en Red Bioingeniería, Biomateriales y NanomedicinaZaragoza, Spain
- BSICoS Group, Aragón Institute of Engineering Research (I3A), ISS Aragón, Universidad de ZaragozaZaragoza, Spain
| | - Michele Orini
- Institute of Cardiovascular Science, University College LondonLondon, UK
| | - Raquel Bailón
- Centro de Investigación Biomédica en Red Bioingeniería, Biomateriales y NanomedicinaZaragoza, Spain
- BSICoS Group, Aragón Institute of Engineering Research (I3A), ISS Aragón, Universidad de ZaragozaZaragoza, Spain
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A Conjecture Regarding the Extremal Values of Graph Entropy Based on Degree Powers. ENTROPY 2016. [DOI: 10.3390/e18050183] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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