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Glass SM, Ross SE. Direction-Specific Signatures of Sport Participation in Center of Pressure Profiles of Division I Athletes. Int J Sports Phys Ther 2021; 16:1260-1272. [PMID: 34631246 PMCID: PMC8486415 DOI: 10.26603/001c.28227] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2021] [Accepted: 08/05/2021] [Indexed: 12/04/2022] Open
Abstract
BACKGROUND Descriptive and comparative studies of human postural control generally report effects for component or resultant dimensions of a measured signal, which may obscure potentially important information related to off-cardinal directionality. Recent work has demonstrated highly specific balance behavior that is often not easily reconciled with conventional theories of postural control. PURPOSE The purpose of this study was to quantify the effects of sport-specific training history on directional profiles of center of pressure (COP) displacement and velocity among collegiate athletes. STUDY DESIGN Cross-Sectional Study. METHODS One-hundred sixty-seven NCAA Division-I varsity athletes (80 female: 19.12±1.08 years, 169.79±7.03 cm, 65.69±10.43 kg; 87 male: 19.59±1.33 years, 181.25±9.06 cm, 76.40±12.73 kg) representing four sports (basketball, soccer, tennis, and cross county) participated in this study. Participants balanced barefoot with eyes closed on a force plate for 10-s. in double leg and single leg stance. Effects of sport on mean COP velocity and total displacement were assessed within eight non-overlapping directions (i.e. heading bins). RESULTS Greater double leg COP displacement and velocity were observed within specific heading bins in cross country athletes when compared to soccer athletes. Greater double leg COP velocity was also observed in multiple heading bins in basketball athletes when compared to soccer athletes. Greater single leg (non-dominant limb) COP displacement was observed in the 135° heading bin in basketball athletes when compared to soccer athletes. CONCLUSIONS The observed effects are likely attributable to sport-specific sensorimotor adaptations, including lower extremity strength/power, proprioceptive acuity, and efficiency of integrating vestibular information. Other potential mechanism-namely the involvement of cutaneous feedback and/or muscle synergies-deserve consideration. Directional profiling of spontaneous COP motion may improve understanding of sport-related balance behavior, enhancing its application in therapeutic and performance monitoring contexts. LEVEL OF EVIDENCE 3b.
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Gill-Body KM, Hedman LD, Plummer L, Wolf L, Hanke T, Quinn L, Riley N, Kaufman R, Verma A, Quiben M, Scheets P. Movement System Diagnoses for Balance Dysfunction: Recommendations From the Academy of Neurologic Physical Therapy's Movement System Task Force. Phys Ther 2021; 101:6307337. [PMID: 34160028 DOI: 10.1093/ptj/pzab153] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 02/10/2021] [Revised: 04/11/2021] [Accepted: 05/30/2021] [Indexed: 11/13/2022]
Abstract
UNLABELLED The movement system was identified as the focus of our expertise as physical therapists in the revised vision statement for the profession adopted by the American Physical Therapy Association in 2013. Attaining success with the profession's vision requires the development of movement system diagnoses that will be useful in clinical practice, research, and education. To date, only a few movement system diagnoses have been identified and described, and none of these specifically address balance dysfunction. Over the past 2 years, a Balance Diagnosis Task Force, a subgroup of the Movement System Task Force of the Academy of Neurologic Physical Therapy, focused on developing diagnostic labels (or diagnoses) for individuals with balance problems. This paper presents the work of the task force that followed a systematic process to review available diagnostic frameworks related to balance, identify 10 distinct movement system diagnoses that reflect balance dysfunction, and develop complete descriptions of examination findings associated with each balance diagnosis. A standardized approach to movement analysis of core tasks, the Framework for Movement Analysis developed by the Academy of Neurologic Physical Therapy Movement Analysis Task Force, was integrated into the examination and diagnostic processes. The aims of this perspective paper are to (1) summarize the process followed by the Balance Diagnosis Task Force to develop an initial set of movement system (balance) diagnoses; (2) report the recommended diagnostic labels and associated descriptions; (3) demonstrate the clinical decision-making process used to determine a balance diagnosis and develop a plan of care; and (4) identify next steps to validate and implement the diagnoses into physical therapist practice, education, and research. IMPACT The development and use of diagnostic labels to classify distinct movement system problems is needed in physical therapy. The 10 balance diagnosis proposed can aid in clinical decision making regarding intervention.
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Affiliation(s)
| | - Lois D Hedman
- Department of Physical Therapy and Human Movement Sciences, Northwestern University Feinberg School of Medicine, Chicago, Illinois, USA
| | - Laura Plummer
- Physical Therapy Department, MGH Institute of Health Professions, Boston, Massachusetts, USA
| | - Leslie Wolf
- Outpatient Neurologic Rehabilitation, OhioHealth, Columbus, Ohio, USA
| | - Timothy Hanke
- Physical Therapy Program, Midwestern University, Downers Grove, Illinois, USA
| | - Lori Quinn
- Department of Biobehavioral Sciences, Teachers College, Columbia University, New York, New York, USA
| | - Nora Riley
- Physical Therapy Department, St. Ambrose University, Davenport, Iowa, USA
| | - Regina Kaufman
- Department of Physical Therapy, Springfield College, Springfield, Massachusetts, USA
| | - Akanshka Verma
- Inpatient Rehabilitation Unit & Acute Neurology Service, New York- Presbyterian Hospital, New York, New York, USA
| | - Myla Quiben
- Department of Physical Therapy, School of Health Professions, University of North Texas Health Science Center, Fort Worth, Texas, USA
| | - Patricia Scheets
- Quality & Clinical Outcomes, Infinity Rehab, Wilsonville, Oregon, USA
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Sápi M, Fehér-Kiss A, Csernák K, Domján A, Pintér S. The Effects of Exergaming on Sensory Reweighting and Mediolateral Stability of Women Aged Over 60: Usability Study. JMIR Serious Games 2021; 9:e27884. [PMID: 34287215 PMCID: PMC8339979 DOI: 10.2196/27884] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2021] [Revised: 04/30/2021] [Accepted: 05/29/2021] [Indexed: 12/19/2022] Open
Abstract
Background Older adults tend to experience difficulties in switching quickly between various reliable sensory inputs, which ultimately may contribute to an increased risk of falls and injuries. Sideward falls are the most frequent cause of hip fractures among older adults. Recently, exergame programs have been confirmed as beneficial tools for enhancing postural control, which can reduce the risk of falls. However, studies to explore more precisely which mechanism of exergaming directly influences older women’s ability to balance are still needed. Objective Our aim was to evaluate, in a single-group pretest/posttest/follow-up usability study, whether Kinect exergame balance training might have a beneficial impact on the sensory reweighting in women aged over 60. Methods A total of 14 healthy women (mean age 69.57 [SD 4.66] years, mean body mass index 26.21 [SD 2.6] kg/m2) participated in the study. The volunteers trained with the commercially available games of Kinect for Xbox 360 console 3 times (30 minutes/session) a week over a 6-week period (total of 18 visits). Participants’ postural sway in both the anteroposterior (AP) and mediolateral (ML) directions was recorded with NeuroCom Balance Master 6.0. To assess and measure postural sensory reweighting, the Modified Clinical Test of Sensory Interaction in Balance was used, where volunteers were exposed to various changes in visual (eyes open or eyes closed) and surface conditions (firm or foam surface). Results In the ML direction, the Kinect exergame training caused a significant decrease in the sway path on the firm surface with the eyes open (P<.001) and eyes closed (P=.001), and on the foam surface with the eyes open (P=.001) and eyes closed (P<.001) conditions compared with baseline data. The follow-up measurements when compared with the baseline data showed a significant change in the sway path on the firm surface with the eyes open (P<.001) and eyes closed (P<.001) conditions, as well as on the foam surface with the eyes open (P=.003) and eyes closed (P<.001) conditions. Besides, on the firm surface, there were no significant differences in sway path values in the AP direction between the baseline and the posttraining measurements (eyes open: P=.49; eyes closed: P=.18). Likewise, on the foam surface, there were no significant differences in sway path values in the AP direction under both eyes open (P=.24) and eyes closed (P=.84) conditions. Conclusions The improved posturography measurements of the sway path in the ML direction might suggest that the Kinect exergame balance training may have effects on sensory reweighting, and thus on the balance of women aged over 60. Based on these results, Kinect exergaming may provide a safe and potentially useful tool for improving postural stability in the crucial ML direction, and thus it may help reduce the risk of falling.
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Affiliation(s)
- Mariann Sápi
- Doctoral School of Clinical Medicine, University of Szeged, Szeged, Hungary
| | - Anna Fehér-Kiss
- Physiotherapy Center, Albert Szent-Györgyi Health Center, University of Szeged, Szeged, Hungary
| | - Krisztina Csernák
- Department of Psychiatry and Psychiatric Clinic, Bács-Kiskun County Hospital Kecskemét, Kecskemét, Hungary
| | - Andrea Domján
- Department of Physiotherapy, Faculty of Health Sciences and Social Studies, University of Szeged, Szeged, Hungary
| | - Sándor Pintér
- Doctoral School of Clinical Medicine, University of Szeged, Szeged, Hungary
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Borzucka D, Kręcisz K, Rektor Z, Kuczyński M. Differences in static postural control between top level male volleyball players and non-athletes. Sci Rep 2020; 10:19334. [PMID: 33168913 PMCID: PMC7653955 DOI: 10.1038/s41598-020-76390-x] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2020] [Accepted: 10/28/2020] [Indexed: 12/30/2022] Open
Abstract
It is argued that elite athletes often demonstrate superior body balance. Despite the apparent significance of perfect balance ability in volleyball, little is known about the specific nature of postural control adjustments among first-rate volleyball competitors. This study compared postural performance and strategies in quiet stance between world vice-champions and young, healthy, physically active male subjects. The center-of-pressure (COP) signals recorded on a force plate were used to compute several measures of sway. In both axes of movement, athletes had lower COP range, but not its standard deviation and higher COP speed and frequency than controls. These findings indicate that postural regulation in athletes was more precise and less vulnerable to external disturbances which support optimal timing and precision of actions. Postural strategies in athletes standing quietly were similar to those exhibited by non-athletes performing dual tasks. It demonstrates a significant effect of sport practice on changes in postural control. In anterior–posterior axis, athletes displayed a much higher COP fractal dimension and surprisingly lower COP–COG frequency than controls. This accounts for their high capacity to use diversified postural strategies to maintain postural stability and significantly reduced the contribution of proprioception to save this function for carrying out more challenging posture-motor tasks.
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Affiliation(s)
- Dorota Borzucka
- Faculty of Physical Education and Physiotherapy, Opole University of Technology, ul. Prószkowska 76, 45-758, Opole, Poland
| | - Krzysztof Kręcisz
- Faculty of Physical Education and Physiotherapy, Opole University of Technology, ul. Prószkowska 76, 45-758, Opole, Poland.
| | - Zbigniew Rektor
- Faculty of Physical Education and Physiotherapy, Opole University of Technology, ul. Prószkowska 76, 45-758, Opole, Poland
| | - Michał Kuczyński
- Faculty of Physical Education and Physiotherapy, Opole University of Technology, ul. Prószkowska 76, 45-758, Opole, Poland.,Faculty of Physiotherapy, University School of Physical Education in Wroclaw, Al. I.J. Paderewskiego 35, 51-612, Wrocław, Poland
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Borzucka D, Kręcisz K, Rektor Z, Kuczyński M. Postural control in top-level female volleyball players. BMC Sports Sci Med Rehabil 2020; 12:65. [PMID: 33101691 PMCID: PMC7576872 DOI: 10.1186/s13102-020-00213-9] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2020] [Accepted: 10/09/2020] [Indexed: 11/18/2022]
Abstract
Background The aim of this study was to compare the postural control of the Poland national women’s volleyball team players with a control group of non-training young women. It was hypothesized that volleyball players use a specific balance control strategy due to the high motor requirements of their team sport. Methods Static postural sway variables were measured in 31 athletes and 31 non-training women. Participants were standing on a force plate with eyes open, and their center of pressure signals were recorded for the 20s with the sampling rate of 20 Hz in the medial-lateral (ML) and anterior-posterior (AP) planes. Results In both AP and ML planes, athletes had lower range and higher fractal dimension of the COP. They had also higher peak frequency than control group in the ML plane only. The remaining COP indices including variability, mean velocity and mean frequency did not display any intergroup differences. Conclusion It can be assumed that due to the high motor requirements of their sport discipline Polish female volleyball players have developed a unique posture control. On the court they have to distribute their sensory resources optimally between balance control and actions resulting from the specifics of the volleyball game. There are no clearly defined criteria for optimal postural strategies for elite athletes, but they rather vary depending on a given sport. The results of our research confirm this claim. Trial registration The tests were previously approved by the Bioethical Commission of the Chamber of Physicians in Opole. (Resolution No. 151/13.12.2007). This study adheres to the CONSORT guidelines.
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Affiliation(s)
- Dorota Borzucka
- Faculty of Physical Education and Physiotherapy, Opole University of Technology, ul. Prószkowska 76, 45-758 Opole, Poland
| | - Krzysztof Kręcisz
- Faculty of Physical Education and Physiotherapy, Opole University of Technology, ul. Prószkowska 76, 45-758 Opole, Poland
| | - Zbigniew Rektor
- Faculty of Physical Education and Physiotherapy, Opole University of Technology, ul. Prószkowska 76, 45-758 Opole, Poland
| | - Michał Kuczyński
- Faculty of Physical Education and Physiotherapy, Opole University of Technology, ul. Prószkowska 76, 45-758 Opole, Poland.,Faculty of Physiotherapy, University School of Physical Education in Wroclaw, Al. I.J. Paderewskiego 35, 51-612 Wrocław, Poland
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The Effects of Active Self-correction on Postural Control in Girls with Adolescent Idiopathic Scoliosis: The Role of an Additional Mental Task. INTERNATIONAL JOURNAL OF ENVIRONMENTAL RESEARCH AND PUBLIC HEALTH 2020; 17:ijerph17051640. [PMID: 32138359 PMCID: PMC7084475 DOI: 10.3390/ijerph17051640] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 01/27/2020] [Revised: 02/24/2020] [Accepted: 02/29/2020] [Indexed: 11/18/2022]
Abstract
Due to balance deficits that accompany adolescent idiopathic scoliosis (AIS), the potential interaction between activities of daily living and active self-correction movements (ASC) on postural control deserves particular attention. Our purpose was to assess the effects of ASC movements with or without a secondary mental task on postural control in twenty-five girls with AIS. It is a quasi-experimental within-subject design with repeated measures ANOVA. They were measured in four 20-s quiet standing trials on a force plate: no task, ASC, Stroop test, and both. Based on the center-of-pressure (COP) recordings, the COP parameters were computed. The ASC alone had no effect on any of the postural sway measures. Stroop test alone decreased COP speed and increased COP entropy. Performing the ASC movements and Stroop test together increased the COP speed and decreased COP entropy as compared to the baseline data. In conclusion, our results indicate that AIS did not interfere with postural control. The effects of the Stroop test accounted for good capacity of subjects with AIS to take advantage of distracting attentional resources from the posture. However, performing both tasks together exhibited some deficits in postural control, which may suggest the need for therapeutic consultation while engaging in more demanding activities.
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Lubetzky AV, Hujsak BD, Kelly JL, Fu G, Perlin K. Control Mechanisms of Static and Dynamic Balance in Adults With and Without Vestibular Dysfunction in Oculus Virtual Environments. PM R 2019; 10:1223-1236.e2. [PMID: 30503230 DOI: 10.1016/j.pmrj.2018.07.001] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2018] [Revised: 07/02/2018] [Accepted: 07/08/2018] [Indexed: 11/28/2022]
Abstract
BACKGROUND Deficits in sensory integration and fear of falling in complex environments contribute to decreased participation of adults with vestibular disorders. With recent advances in virtual reality technology, head-mounted displays are affordable and allow manipulation of the environment to test postural responses to visual changes. OBJECTIVES To develop an assessment of static and dynamic balance with the Oculus Rift and (1) to assess test-retest reliability of each scene in adults with and without vestibular hypofunction; (2) to describe changes in directional path and sample entropy in response to changes in visuals and surface and compare between groups; and (3) to evaluate the relation between balance performance and self-reported disability and balance confidence. DESIGN Test-retest, blocked-randomized experimental design. SETTING Research laboratory. PARTICIPANTS Twenty-five adults with vestibular hypofunction and 16 age- and sex-matched adults. METHODS Participants stood on the floor or stability trainers while wearing the Oculus Rift. For 3 moving "stars" scenes, they stood naturally. For a "park" scene, they were asked to avoid a virtual ball. The protocol was repeated 1-4 weeks later. OUTCOME Anteroposterior and mediolateral center-of-pressure directional path and sample entropy were derived from a force plate. RESULTS We observed good to excellent reliability in the 2 groups, with most intraclass correlations above 0.8 and only 2 at approximately 0.4. The vestibular group had higher directional path for the stars scenes and lower directional path for the park scene compared with controls, with large variability in the 2 groups. Sample entropy decreased with more challenging environments. In the vestibular group, less balance confidence strongly correlated with more sway for the stars scenes and less sway for the park scene. CONCLUSION Virtual reality paradigms can shed light on the control mechanism of static and dynamic postural control. Clinical utility and implementation of our portable Oculus Rift assessment should be further studied. LEVEL OF EVIDENCE II.
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Affiliation(s)
- Anat V Lubetzky
- New York University, Department of Physical Therapy, Steinhardt School of Culture Education and Human Development, 380 2nd Ave, New York, NY 10010(∗).
| | - Bryan D Hujsak
- Vestibular Rehabilitation, The Ear Institute, Hearing and Balance Center, The New York Eye and Ear Infirmary of Mount Sinai, New York, NY(†)
| | - Jennifer L Kelly
- Vestibular Rehabilitation, The Ear Institute, Hearing and Balance Center, The New York Eye and Ear Infirmary of Mount Sinai, New York, NY(‡)
| | - Gene Fu
- New York University, Department of Physical Therapy, Steinhardt School of Culture Education and Human Development, New York, NY(§)
| | - Ken Perlin
- New York University, Computer Science Department, Courant Institute of Mathematical Sciences, New York, NY(¶)
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Piątek E, Kuczyński M, Ostrowska B. Postural control in girls with adolescent idiopathic scoliosis while wearing a Chêneau brace or performing active self-correction: a pilot study. PeerJ 2019; 7:e7513. [PMID: 31528504 PMCID: PMC6717654 DOI: 10.7717/peerj.7513] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2019] [Accepted: 07/18/2019] [Indexed: 01/11/2023] Open
Abstract
Background It is known that adolescent idiopathic scoliosis (AIS) is often accompanied by balance deficits. This reciprocal relationship must be taken into account when prescribing new therapeutic modalities because these may differently affect postural control, interacting with therapy and influencing its results. Objective The purpose was to compare postural control in girls with AIS while wearing the Chêneau brace (BRA) or performing active self-correction (ASC) with their postural control in a quiet comfortable stance. Methods Nine subjects were evaluated on a force plate in three series of two 20-s quiet standing trials with eyes open or closed; three blocks were randomly arranged: normal quiet stance (QST), quiet stance with BRA, and quiet stance with ASC. On the basis of centre-of-pressure (COP) recordings, the spatial and temporal COP parameters were computed. Results and Discussion Performing ASC was associated with a significant backward excursion of the COP mean position with eyes open and closed (ES = 0.56 and 0.65, respectively; p < 0.05). This excursion was accompanied by an increase in the COP fractal dimension (ES = 1.05 and 0.98; p < 0.05) and frequency (ES = 0.78; p = 0.10 and ES = 1.14; p < 0.05) in the mediolateral (ML) plane. Finally, both therapeutic modalities decreased COP sample entropy with eyes closed in the anteroposterior (AP) plane. Wearing BRA resulted in ES = 1.45 (p < 0.05) while performing ASC in ES = 0.76 (p = 0.13). Conclusion The observed changes in the fractal dimension (complexity) and frequency caused by ASC account for better adaptability of patients to environmental demands and for their adequate resources of available postural strategies in the ML plane. These changes in sway structure were accompanied by a significant (around 25 mm) backward excursion of the mean COP position. However, this improvement was achieved at the cost of lower automaticity, i.e. higher attentional involvement in postural control in the AP plane. Wearing BRA may have an undesirable effect on some aspects of body balance.
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Affiliation(s)
- Elżbieta Piątek
- Faculty of Physiotherapy, University School of Physical Education in Wroclaw, Wrocław, Poland
| | - Michał Kuczyński
- Faculty of Physical Education and Physiotherapy, Opole University of Technology, Opole, Poland
| | - Bożena Ostrowska
- Faculty of Physiotherapy, University School of Physical Education in Wroclaw, Wrocław, Poland
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Lubetzky AV, Harel D, Lubetzky E. On the effects of signal processing on sample entropy for postural control. PLoS One 2018; 13:e0193460. [PMID: 29494625 PMCID: PMC5832259 DOI: 10.1371/journal.pone.0193460] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2017] [Accepted: 02/12/2018] [Indexed: 11/29/2022] Open
Abstract
Sample entropy, a measure of time series regularity, has become increasingly popular in postural control research. We are developing a virtual reality assessment of sensory integration for postural control in people with vestibular dysfunction and wished to apply sample entropy as an outcome measure. However, despite the common use of sample entropy to quantify postural sway, we found lack of consistency in the literature regarding center-of-pressure signal manipulations prior to the computation of sample entropy. We therefore wished to investigate the effect of parameters choice and signal processing on participants’ sample entropy outcome. For that purpose, we compared center-of-pressure sample entropy data between patients with vestibular dysfunction and age-matched controls. Within our assessment, participants observed virtual reality scenes, while standing on floor or a compliant surface. We then analyzed the effect of: modification of the radius of similarity (r) and the embedding dimension (m); down-sampling or filtering and differencing or detrending. When analyzing the raw center-of-pressure data, we found a significant main effect of surface in medio-lateral and anterior-posterior directions across r’s and m’s. We also found a significant interaction group × surface in the medio-lateral direction when r was 0.05 or 0.1 with a monotonic increase in p value with increasing r in both m’s. These effects were maintained with down-sampling by 2, 3, and 4 and with detrending but not with filtering and differencing. Based on these findings, we suggest that for sample entropy to be compared across postural control studies, there needs to be increased consistency, particularly of signal handling prior to the calculation of sample entropy. Procedures such as filtering, differencing or detrending affect sample entropy values and could artificially alter the time series pattern. Therefore, if such procedures are performed they should be well justified.
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Affiliation(s)
- Anat V Lubetzky
- Department of Physical Therapy, Steinhardt School of Culture Education and Human Development, New York University, New York, NY, United States of America
| | - Daphna Harel
- Department of Applied Statistics, Social Science and Humanities, Steinhardt School of Culture Education and Human Development, New York University, New York, NY, United States of America
| | - Eyal Lubetzky
- Courant Institute of Mathematical Sciences, New York University, New York, NY, United States of America
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