Journal of Biomechanics 47 (2014) 3776–3779
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
The effect of walking speed on local dynamic stability is sensitive to calculation methods Jan Stenum a, Sjoerd M. Bruijn b,c, Bente R. Jensen a,n a Biomechanics and Motor Control Laboratory, Section of Integrated Physiology, Department of Nutrition, Exercise and Sports, University of Copenhagen, Copenhagen, Denmark b Research Institute MOVE, Faculty of Human Movement Sciences, VU University Amsterdam, Amsterdam, The Netherlands c Department of Orthopedics, First Affiliated Hospital of Fujian Medical University, Fuzhou, Fujian, PR China
art ic l e i nf o
a b s t r a c t
Article history: Accepted 14 September 2014
Local dynamic stability has been assessed by the short-term local divergence exponent (λS), which quantifies the average rate of logarithmic divergence of infinitesimally close trajectories in state space. Both increased and decreased local dynamic stability at faster walking speeds have been reported. This might pertain to methodological differences in calculating λS. Therefore, the aim was to test if different calculation methods would induce different effects of walking speed on local dynamic stability. Ten young healthy participants walked on a treadmill at five speeds (60%, 80%, 100%, 120% and 140% of preferred walking speed) for 3 min each, while upper body accelerations in three directions were sampled. From these time-series, λS was calculated by three different methods using: (a) a fixed time interval and expressed as logarithmic divergence per stride-time (λS a), (b) a fixed number of strides and expressed as logarithmic divergence per time (λS b) and (c) a fixed number of strides and expressed as logarithmic divergence per stride-time (λS c). Mean preferred walking speed was 1.16 70.09 m/s. There was only a minor effect of walking speed on λS a. λS b increased with increasing walking speed indicating decreased local dynamic stability at faster walking speeds, whereas λS c decreased with increasing walking speed indicating increased local dynamic stability at faster walking speeds. Thus, the effect of walking speed on calculated local dynamic stability was significantly different between methods used to calculate local dynamic stability. Therefore, inferences and comparisons of studies employing λS should be made with careful consideration of the calculation method. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Local dynamic stability Walking speed Local divergence exponent Stability Gait
1. Introduction Local dynamic stability, quantified using local divergence exponents, is a promising measure of gait stability (Bruijn et al., 2013). Local divergence exponents quantify the average rate of logarithmic divergence of infinitesimally close trajectories in state space (Rosenstein et al., 1993) and is suggested to reflect the ability to attenuate the small perturbations that occur naturally during gait (van Schooten et al., 2014). A locally stable system is characterized by a negative local divergence exponent, whereas a locally unstable system is characterized by a positive exponent (Dingwell, 2006). Larger exponents indicate a more locally unstable system because of a more rapid expansion of the system’s principal axis. Growing evidence from both simulation and experimental studies suggests that the short-term local divergence exponent, λS, that quantifies the n Correspondence to: Biomechanics and Motor Control Laboratory, Section of Integrated Physiology, Department of Nutrition, Exercise and Sports, University of Copenhagen, Nørre Allé 51, 2200 Copenhagen N, Denmark. Tel.: þ45 3532 7303. E-mail address: [email protected] (B.R. Jensen).
http://dx.doi.org/10.1016/j.jbiomech.2014.09.020 0021-9290/& 2014 Elsevier Ltd. All rights reserved.
system’s response over a shorter time period may be related to stability impairments, the probability of falling in models and as an index for rehabilitation effect on fall risk (Manor et al., 2009; McAndrew et al., 2011; Roos and Dingwell, 2011; van Schooten et al., 2011; Bruijn et al., 2012; Hilfiker et al., 2013). Reported effects of walking speed on local dynamic stability have been inconclusive (Dingwell and Marin, 2006; England and Granata, 2007; Kang and Dingwell, 2008; Manor et al., 2008; Bruijn et al., 2009a, 2010; Yakhdani et al., 2010). For example, Dingwell and Marin (2006) and England and Granata (2007) found decreased local dynamic stability, i.e. higher λS, with faster walking, while Bruijn et al. (2009a) found different effects of walking speed on the movement directions studied. This might pertain to methodological differences in calculating λS. Specifically, Dingwell and Marin (2006) analyzed a fixed time interval for all walking speeds. This implies that more strides will be analyzed at faster walking speeds, which might affect λS (Bruijn et al., 2009b). In order to overcome this potential bias England and Granata (2007) analyzed a fixed number of strides for all speeds and time-normalized each time-series to 100 points per analyzed stride. However, instead of expressing λS as logarithmic
J. Stenum et al. / Journal of Biomechanics 47 (2014) 3776–3779
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Fig. 1. Schematic representation of short-term local divergence exponent (λS) calculation methods a, b and c, each quantifying local dynamic stability. Method a reconstructed state spaces from the original 3 min acceleration time-series for each walking speed and acceleration direction; normalized the temporal separation between data points to a percentage of the time it takes to complete a stride (stride-time); and expressed λS as mean logarithmic divergence of nearest neighbors in state space (〈ln(div)〉)/stride-time from 0–0.5 stride, i.e. λS has the unit g/stride (A). Method b reconstructed state spaces from time-normalized acceleration time-series for the first 115 strides for each walking speed and direction; normalized the temporal separation between data points to time in seconds; and expressed λS as 〈ln(div)〉/time from 0–0.5 stride, i.e. λS has the unit g/s (B). Method c reconstructed state spaces from time-normalized acceleration time-series for the first 115 strides for each walking speed and direction; and expressed λS as 〈ln(div)〉/stride-time from 0–0.5 stride, i.e. λS has the unit g/stride (C). Only accelerations for 20 s are shown and state spaces are reconstructed in 3 dimensions for graphical representation (stride duration ¼0.92 s).
divergence (ln(div)) per stride-time, they expressed it as ln(div) per time in seconds (England and Granata, 2007; Fig. 3), which introduced a dependency upon stride duration. Manor et al. (2008), analyzing a fixed number of strides for all speeds and expressing λS as ln(div) per stride-time, found decreased local dynamic stability at faster speeds for people with peripheral neuropathy, however, for healthy controls there was no effect of walking speeds on local dynamic stability. Presently, we hypothezised that analyzing a fixed time interval or a fixed number of strides for all walking speeds and expressing λS as ln(div) per stride-time or ln(div) per time, constituted important methodological differences in calculating λS that could induce different effects of walking speed on λS. The aim of the present study was to test if these methodological differences in calculating λS would induce different effects of walking speed on this measure.
2. Methods 2.1. Participants 10 healthy young participants were recruited (6 men and 4 women, (mean7 SD): age 22.6 7 2.8 years, body mass 70.6 7 6.5 kg, body height
1.78 7 0.08 m). Participants gave informed written consent before participation. The study was registered by the regional ethics committee (H-1-2014-FSP-006). 2.2. Procedure Participants reported to the laboratory on two non-consecutive test days, with 1 to 8 weeks between test days (3.57 3.0 weeks). On the first test day, preferred walking speed (PWS) was determined prior to testing on a treadmill. PWS was calculated using the method by Dingwell and Marin (2006). On both test days, participants walked for 3 min at 60%, 80%, 100%, 120% and 140% PWS, in random order, on a treadmill, with 1 min rest between trials. An accelerometer (range 76 g, Marq-Medical, Copenhagen, Denmark) was mounted on the sternum. Accelerations in vertical (VT), medio-lateral (ML) and antero-posterior (AP) directions were sampled at 64 Hz (Bluetooth connection) when the treadmill reached a constant speed. 2.3. Calculations 2.3.1. Pre-processing Time-series were not filtered before further analysis (Mees and Judd, 1993). The first 115 strides for each time-series, determined from the VT acceleration, were identified. 2.3.2. Local dynamic stability State spaces were reconstructed from each acceleration direction using the method of delays (Takens, 1981): Sðt Þ ¼ xðt Þ; xðt þ τÞ; …; xðt þ τðdE 1ÞÞ ;
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J. Stenum et al. / Journal of Biomechanics 47 (2014) 3776–3779
where S(t) is the state vector, x(t) is the original time-series, τ is the time delay and dE is the embedding dimension. An embedding dimension of dE ¼ 5 was chosen (Dingwell and Cusumano, 2000). From the reconstructed state spaces Euclidean distances between nearest neighbors in state space were calculated as a function of time and averaged over all nearest neighbors to obtain the average rate of logarithmic divergence of the distance between nearest neighbors in state space (Rosenstein et al., 1993): yðtÞ ¼
2.3.3. Statistics
λS for methods a, b and c for each direction were taken as dependent measures in a two-way (3 methods 5 speeds) analysis of variance (ANOVA) to determine if there were statistically significant interaction effects. To determine the possible effect of walking speed on each method a one-way ANOVA was performed for each direction. ANOVAs were calculated using SPSS (IBM Software). A p-value o 0.05 was considered significant.
1 〈lnðdj ðiÞÞ〉; Δt
where dj(i) is the Euclidean distance between the jth pair of nearest neighbors after i discrete time steps, 〈 〉 denotes the average over all values of j and Δt is the interval where the rate is evaluated. The rate was evaluated at an interval from 0 to 0.5 stride providing an estimate of λS. Exponents were calculated using custom made MATLAB (The Mathworks, Inc., Natwick, MA) functions. Average values of λS across test days were used for further analysis. λS was calculated using three methods: (a) Accelerations for 3 min were analyzed. A time delay of 8 samples was chosen, determined as the average of the first minimum of the average mutual information function (Fraser, 1986). The temporal separation between data points was normalized to a percentage of the time it takes to complete a stride (stride-time), by dividing the time of each data point by the average stride duration. λS was expressed as 〈ln(div)〉/stride-time, i.e. the unit of λS was g/stride (Fig. 1A). (b) Accelerations for 115 strides were analyzed. Data were time-normalized to 11,500 data points using a shape-preserving spline interpolation. A time delay of 10 samples was chosen, as all time-series had the same frequency. From this time-normalized time-series, the divergence curve was calculated after which the temporal separation between data points was rescaled to time in seconds, by multiplying the stride-time of each data point with the average stride duration. λS was expressed as 〈ln(div)〉/time, i.e. the unit of λS was g/s (Fig. 1B). (c) Accelerations for 115 strides were analyzed. Data were time-normalized to 11,500 data points. A time delay of 10 samples was chosen. λS was expressed as 〈ln(div)〉/stride-time, i.e. the unit of λS was g/stride (Fig. 1C).
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The mean7SD PWS was 1.1670.09 m/s. Mean7SD stride durations were 1.4070.09 s, 1.2070.06 s, 1.0770.05 s, 1.0070.04 s and 0.9570.04 s for 60%, 80%, 100%, 120% and 140% PWS, respectively. For all directions, methods a, b and c showed different trends for the effect of walking speed on λS (Fig. 2). For method a, walking speed only had a significant effect in the AP direction. For method b, λS in VT and ML directions significantly increased with increasing walking speeds. For method c, λS in all directions showed a significant decrease with increasing walking speed. Thus, different effects of walking speed on local dynamic stability, as quantified by λS, were induced by methods a, b and c (Fig. 2).
4. Discussion The main finding of the current study was that the effect of walking speed on calculated local dynamic stability was significantly
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Fig. 2. The effect of walking speed (60%, 80%, 100%, 120% and 140% of preferred walking speed (PWS)) on the short-term local divergence exponent (λS) calculated using methods a, b and c for vertical (VT), medio-lateral (ML) and anterior posterior (AP) upper body acceleration directions. Method a calculated λS using accelerations during treadmill walking for 3 min at each walking speed and expressed λS as mean logarithmic divergence of nearest neighbors in state space (〈ln(div)〉) per stride-time for 0–0.5 stride. Method b calculated λS using accelerations for the first 115 strides during treadmill walking, time-normalized data to 11,500 data points and expressed λS as 〈ln(div)〉 per time for 0–0.5 stride. Method c calculated λS using accelerations for the first 115 strides during treadmill walking, time-normalized data to 11,500 data points and expressed λS as 〈ln(div)〉 per stride-time for 0–0.5 stride. Error bars represent standard deviations. nDenotes a significant effect of walking speed on the short-term local divergence exponent for a particular method.
J. Stenum et al. / Journal of Biomechanics 47 (2014) 3776–3779
different between calculation methods. Method a calculated local dynamic stability using a fixed time interval and expressed λS as ln(div)/stride-time, method b used a fixed number of strides and expressed λS as ln(div)/time and method c used a fixed number of strides and expressed λS as ln(div)/stride-time. For method a, there was a minor effect of walking speed on λS. For method b, λS increased at faster walking speeds, and for method c, λS decreased at faster walking speeds. As hypothesized, analyzing a fixed time interval or a fixed number of strides for all walking speeds and expressing λS as ln(div)/stride-time or as ln(div)/time led to different effects of walking speed on λS. Both methods a and c expressed λS as ln(div)/stride-time. However, method a analyzed a fixed time interval, whereas method c analyzed a fixed number of strides for all speeds. Analyzing a fixed time interval for all speeds, thus analyzing more strides at faster speeds, did not increase λS at faster speeds as inferred from Bruijn et al. (2009b). State space reconstructions for methods b and c were similar, but method b expressed λS as ln(div)/time while method c expressed λS as ln(div)/stride-time. Faster walking leads to shorter stride duration, which will, for a given divergence, lead to an increased λS indicating decreased stability at faster speeds when λS is expressed as ln(div)/time. When expressing λS as ln(div)/stride-time, this is avoided. However, it is presently unknown which interpretation is more valid. The methodological choices of analyzing a fixed time interval or a fixed number of strides and expressing λS as ln(div)/stridetime or as ln(div)/time are important and significantly changes the effect of walking speed on λS as presently shown. However, the methodological choices presented here are by no means exhaustive; indeed, other important choices remain in calculating λS that similarly could cause changes in the effect of walking speed on λS. Specifically, the choice of time-series from upper body kinematics or lower body joint kinematics (Kang and Dingwell, 2009), the movement directions studied (Bruijn et al., 2009a), state space reconstruction (Gates and Dingwell, 2009; van Schooten et al., 2013), the calculation algorithm used (Cignetti et al., 2012; Rispens et al., 2014) and whether to calculate λS as the rate of divergence from 0–0.5 stride or 0–1 stride (Reynard and Terrier, 2014; Reynard et al., 2014) all constitute important methodological choices. Based on the present study, wherein local dynamic stability of upper body accelerations was calculated using three methods, we conclude that the effect of walking speed on local dynamic stability is sensitive to pertinent methodological choices in calculating λS. The present study does not allow specific support to any of the three calculation methods. Therefore, inferences of the effect of walking speed on local dynamic stability and comparisons across studies should be made with careful consideration of the methods applied. Additional studies need to be done to establish which of the three applied methodological approaches is more correct. Conflict of interest statement None of the authors have any financial or personal relationships with people or organizations that could inappropriately influence the present study. Acknowledgements Sjoerd M. Bruijn was funded by a grant from the Netherlands Organization for Scientific Research (NWO #451-12-041).
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References Bruijn, S.M., Bregman, D.J., Meijer, O.G., Beek, P.J., van Dieën, J.H., 2012. Maximum Lyapunov exponents as predictors of global gait stability: a modelling approach. Med. Eng. Phys. 34 (4), 428–436. Bruijn, S.M., Meijer, O.G., Beek, P.J., van Dieën, J.H., 2010. The effects of arm swing on human gait stability. J. Exp. Biol. 213 (23), 3945–3952. Bruijn, S.M., Meijer, O.G., Beek, P.J., van Dieën, J.H., 2013. Assessing the stability of human locomotion: a review of current measures. J. R. Soc. Interface 10 (83), 20120999. Bruijn, S.M., van Dieën, J.H., Meijer, O.G., Beek, P.J., 2009a. Is slow walking more stable? J. Biomech 42 (10), 1506–1512. Bruijn, S.M., van Dieën, J.H., Meijer, O.G., Beek, P.J., 2009b. Statistical precision and sensitivity of measures of dynamic gait stability. J. Neurosci. Meth. 178 (2), 327–333. Cignetti, F., Decker, L.M., Stergiou, N., 2012. Sensitivity of the Wolf’s and Rosenstein’s algorithms to evaluate local dynamic stability from small gait data sets. Ann. Biomed. Eng. 40 (5), 1122–1130. Dingwell, J.B., 2006. Lyapunov exponent. In: Akay, M. (Ed.), Wiley Encyclopedia of Biomedical Engineering. John Wiley & Sons Inc., New York, NY, pp. 2135–2146. Dingwell, J.B., Cusumano, J.P., 2000. Nonlinear time series analysis of normal and pathological human walking. Chaos 10 (4), 848–861. Dingwell, J.B., Marin, L.C., 2006. Kinematic variability and local dynamic stability of upper body motions when walking at different speeds. J. Biomech. 39 (3), 444–452. England, S.A., Granata, K.P., 2007. The influence of gait speed on local dynamic stability of walking. Gait Posture 25 (2), 172–178. Fraser, A.M., 1986. Using mutual information to estimate metric entropy. In: MeyerKress, G. (Ed.), Dimensions and Entropies in Chaotic Systems. Springer-Verlag, Berlin, pp. 82–91. Gates, D.H., Dingwell, J.B., 2009. Comparison of different state space definitions for local dynamic stability analyses. J. Biomech. 42 (9), 1345–1349. Hilfiker, R., Vaney, C., Gattlen, B., Meichtry, A., Deriaz, O., Lugon-Moulin, V., AnchisiBellwald, A., Palici, C., Foinant, D., Terrier, P., 2013. Local dynamic stability as a responsive index for the evaluation of rehabilitation effect on fall risk in patients with multiple sclerosis: a longitudinal study. BMC Res. Notes 6, 260. Kang, H.G., Dingwell, J.B., 2008. Effects of walking speed, strength and range of motion on gait stability in healthy older adults. J. Biomech. 41 (14), 2899–2905. Kang, H.G., Dingwell, J.B., 2009. Dynamic stability of superior vs. inferior segments during walking in young and older adults. Gait Posture 30 (2), 260–263. Manor, B., Wolenski, P., Guevaro, A., Li, L., 2009. Differential effects of plantar desensitization on locomotion dynamics. J. Electromyogr. Kinesiol. 19 (5), e320–e328. Manor, B., Wolenski, P., Li, L., 2008. Faster walking speeds increase local instability among people with peripheral neuropahty. J. Biomech. 41 (13), 2787–2792. McAndrew, P.M., Wilken, J.M., Dingwell, J.B., 2011. Dynamic stability of human walking in visually and mechanically destabilizing environments. J. Biomech. 44 (4), 644–649. Mees, A.I., Judd, K., 1993. Dangers of Geometric Filtering. Physica D 68, 427–436. Reynard, F., Terrier, P., 2014. Local dynamic stability of treadmill walking: intrasession and week-to-week repeatability. J. Biomech. 47 (1), 74–80. Reynard, F., Vuadens, P., Deriaz, O., Terrier, P., 2014. Could local dynamic stability serve as an early predictor of falls in patients with moderate neurological gait disorders? A reliability and comparison study in healthy individuals and in patients with paresis of the lower extremities. PLoS One 9 (6), e100550. Rispens, S.M., Pijnappels, M., van Dieën, J.H., van Schooten, K.S., Beek, P.J., Daffertshofer, A., 2014. A benchmark test of accuracy and precision in estimating dynamical systems characteristics from a time-series. J. Biomech. 47 (2), 470–475. Roos, P.E., Dingwell, J.B., 2011. Influence of simulated neuromuscular noise on the dynamic stability and fall risk of a 3D dynamic walking model. J. Biomech. 44 (8), 1514–1520. Rosenstein, M.T., Collins, J.J., De Luca, C.J., 1993. A Practical method for calculating largest Lyapunov exponents from small data sets. Physica D 65 (1), 117–134. Takens, F., 1981. Detecting strange attractors in turbulence. In: Rand, D., Young, L.S. (Eds.), Dynamical Systems and Turbulence, Warwick 1980, vol. 898. Springer, Berlin, pp. 366–381. van Schooten, K.S., Rispens, S.M., Elders, P.J.M., van Dieën, J.H., Pijnappels, M., 2014. Toward ambulatory balance assessment: estimating variability and stability from short bouts of gait. Gait Posture 39 (2), 695–699. van Schooten, K.S., Rispens, S.M., Pijnappels, M., Daffertshofer, A., van Dieën, J.H., 2013. Assessing gait stability: the influence of state space reconstruction on inter- and intra-day reliability of local dynamic stability during over-ground walking. J. Biomech. 46 (1), 137–141. van Schooten, K.S., Sloot, L.H., Bruijn, S.M., Kingma, H., Meijer, O.G., Pijnappels, M., van Dieën, J.H., 2011. Sensitivity of trunk variability and stability measures to balance impairments induced by galvanic vestibular stimulation during gait. Gait Posture 33 (4), 656–660. Yakhdani, H.R., Bafghi, H.A., Meijer, O.G., Bruijn, S.M., van den Dikkenberg, N., Stibbe, A.B., van Royen, B.J., van Dieën, J.H., 2010. Stability and variability of knee kinematics during gait in knee osteoarthritis before and after replacement surgery. Clin. Biomech. 25 (3), 230–236.
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
The effect of walking speed on local dynamic stability is sensitive to calculation methods Jan Stenum a, Sjoerd M. Bruijn b,c, Bente R. Jensen a,n a Biomechanics and Motor Control Laboratory, Section of Integrated Physiology, Department of Nutrition, Exercise and Sports, University of Copenhagen, Copenhagen, Denmark b Research Institute MOVE, Faculty of Human Movement Sciences, VU University Amsterdam, Amsterdam, The Netherlands c Department of Orthopedics, First Affiliated Hospital of Fujian Medical University, Fuzhou, Fujian, PR China
art ic l e i nf o
a b s t r a c t
Article history: Accepted 14 September 2014
Local dynamic stability has been assessed by the short-term local divergence exponent (λS), which quantifies the average rate of logarithmic divergence of infinitesimally close trajectories in state space. Both increased and decreased local dynamic stability at faster walking speeds have been reported. This might pertain to methodological differences in calculating λS. Therefore, the aim was to test if different calculation methods would induce different effects of walking speed on local dynamic stability. Ten young healthy participants walked on a treadmill at five speeds (60%, 80%, 100%, 120% and 140% of preferred walking speed) for 3 min each, while upper body accelerations in three directions were sampled. From these time-series, λS was calculated by three different methods using: (a) a fixed time interval and expressed as logarithmic divergence per stride-time (λS a), (b) a fixed number of strides and expressed as logarithmic divergence per time (λS b) and (c) a fixed number of strides and expressed as logarithmic divergence per stride-time (λS c). Mean preferred walking speed was 1.16 70.09 m/s. There was only a minor effect of walking speed on λS a. λS b increased with increasing walking speed indicating decreased local dynamic stability at faster walking speeds, whereas λS c decreased with increasing walking speed indicating increased local dynamic stability at faster walking speeds. Thus, the effect of walking speed on calculated local dynamic stability was significantly different between methods used to calculate local dynamic stability. Therefore, inferences and comparisons of studies employing λS should be made with careful consideration of the calculation method. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Local dynamic stability Walking speed Local divergence exponent Stability Gait
1. Introduction Local dynamic stability, quantified using local divergence exponents, is a promising measure of gait stability (Bruijn et al., 2013). Local divergence exponents quantify the average rate of logarithmic divergence of infinitesimally close trajectories in state space (Rosenstein et al., 1993) and is suggested to reflect the ability to attenuate the small perturbations that occur naturally during gait (van Schooten et al., 2014). A locally stable system is characterized by a negative local divergence exponent, whereas a locally unstable system is characterized by a positive exponent (Dingwell, 2006). Larger exponents indicate a more locally unstable system because of a more rapid expansion of the system’s principal axis. Growing evidence from both simulation and experimental studies suggests that the short-term local divergence exponent, λS, that quantifies the n Correspondence to: Biomechanics and Motor Control Laboratory, Section of Integrated Physiology, Department of Nutrition, Exercise and Sports, University of Copenhagen, Nørre Allé 51, 2200 Copenhagen N, Denmark. Tel.: þ45 3532 7303. E-mail address: [email protected] (B.R. Jensen).
http://dx.doi.org/10.1016/j.jbiomech.2014.09.020 0021-9290/& 2014 Elsevier Ltd. All rights reserved.
system’s response over a shorter time period may be related to stability impairments, the probability of falling in models and as an index for rehabilitation effect on fall risk (Manor et al., 2009; McAndrew et al., 2011; Roos and Dingwell, 2011; van Schooten et al., 2011; Bruijn et al., 2012; Hilfiker et al., 2013). Reported effects of walking speed on local dynamic stability have been inconclusive (Dingwell and Marin, 2006; England and Granata, 2007; Kang and Dingwell, 2008; Manor et al., 2008; Bruijn et al., 2009a, 2010; Yakhdani et al., 2010). For example, Dingwell and Marin (2006) and England and Granata (2007) found decreased local dynamic stability, i.e. higher λS, with faster walking, while Bruijn et al. (2009a) found different effects of walking speed on the movement directions studied. This might pertain to methodological differences in calculating λS. Specifically, Dingwell and Marin (2006) analyzed a fixed time interval for all walking speeds. This implies that more strides will be analyzed at faster walking speeds, which might affect λS (Bruijn et al., 2009b). In order to overcome this potential bias England and Granata (2007) analyzed a fixed number of strides for all speeds and time-normalized each time-series to 100 points per analyzed stride. However, instead of expressing λS as logarithmic
J. Stenum et al. / Journal of Biomechanics 47 (2014) 3776–3779
λ
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Fig. 1. Schematic representation of short-term local divergence exponent (λS) calculation methods a, b and c, each quantifying local dynamic stability. Method a reconstructed state spaces from the original 3 min acceleration time-series for each walking speed and acceleration direction; normalized the temporal separation between data points to a percentage of the time it takes to complete a stride (stride-time); and expressed λS as mean logarithmic divergence of nearest neighbors in state space (〈ln(div)〉)/stride-time from 0–0.5 stride, i.e. λS has the unit g/stride (A). Method b reconstructed state spaces from time-normalized acceleration time-series for the first 115 strides for each walking speed and direction; normalized the temporal separation between data points to time in seconds; and expressed λS as 〈ln(div)〉/time from 0–0.5 stride, i.e. λS has the unit g/s (B). Method c reconstructed state spaces from time-normalized acceleration time-series for the first 115 strides for each walking speed and direction; and expressed λS as 〈ln(div)〉/stride-time from 0–0.5 stride, i.e. λS has the unit g/stride (C). Only accelerations for 20 s are shown and state spaces are reconstructed in 3 dimensions for graphical representation (stride duration ¼0.92 s).
divergence (ln(div)) per stride-time, they expressed it as ln(div) per time in seconds (England and Granata, 2007; Fig. 3), which introduced a dependency upon stride duration. Manor et al. (2008), analyzing a fixed number of strides for all speeds and expressing λS as ln(div) per stride-time, found decreased local dynamic stability at faster speeds for people with peripheral neuropathy, however, for healthy controls there was no effect of walking speeds on local dynamic stability. Presently, we hypothezised that analyzing a fixed time interval or a fixed number of strides for all walking speeds and expressing λS as ln(div) per stride-time or ln(div) per time, constituted important methodological differences in calculating λS that could induce different effects of walking speed on λS. The aim of the present study was to test if these methodological differences in calculating λS would induce different effects of walking speed on this measure.
2. Methods 2.1. Participants 10 healthy young participants were recruited (6 men and 4 women, (mean7 SD): age 22.6 7 2.8 years, body mass 70.6 7 6.5 kg, body height
1.78 7 0.08 m). Participants gave informed written consent before participation. The study was registered by the regional ethics committee (H-1-2014-FSP-006). 2.2. Procedure Participants reported to the laboratory on two non-consecutive test days, with 1 to 8 weeks between test days (3.57 3.0 weeks). On the first test day, preferred walking speed (PWS) was determined prior to testing on a treadmill. PWS was calculated using the method by Dingwell and Marin (2006). On both test days, participants walked for 3 min at 60%, 80%, 100%, 120% and 140% PWS, in random order, on a treadmill, with 1 min rest between trials. An accelerometer (range 76 g, Marq-Medical, Copenhagen, Denmark) was mounted on the sternum. Accelerations in vertical (VT), medio-lateral (ML) and antero-posterior (AP) directions were sampled at 64 Hz (Bluetooth connection) when the treadmill reached a constant speed. 2.3. Calculations 2.3.1. Pre-processing Time-series were not filtered before further analysis (Mees and Judd, 1993). The first 115 strides for each time-series, determined from the VT acceleration, were identified. 2.3.2. Local dynamic stability State spaces were reconstructed from each acceleration direction using the method of delays (Takens, 1981): Sðt Þ ¼ xðt Þ; xðt þ τÞ; …; xðt þ τðdE 1ÞÞ ;
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where S(t) is the state vector, x(t) is the original time-series, τ is the time delay and dE is the embedding dimension. An embedding dimension of dE ¼ 5 was chosen (Dingwell and Cusumano, 2000). From the reconstructed state spaces Euclidean distances between nearest neighbors in state space were calculated as a function of time and averaged over all nearest neighbors to obtain the average rate of logarithmic divergence of the distance between nearest neighbors in state space (Rosenstein et al., 1993): yðtÞ ¼
2.3.3. Statistics
λS for methods a, b and c for each direction were taken as dependent measures in a two-way (3 methods 5 speeds) analysis of variance (ANOVA) to determine if there were statistically significant interaction effects. To determine the possible effect of walking speed on each method a one-way ANOVA was performed for each direction. ANOVAs were calculated using SPSS (IBM Software). A p-value o 0.05 was considered significant.
1 〈lnðdj ðiÞÞ〉; Δt
where dj(i) is the Euclidean distance between the jth pair of nearest neighbors after i discrete time steps, 〈 〉 denotes the average over all values of j and Δt is the interval where the rate is evaluated. The rate was evaluated at an interval from 0 to 0.5 stride providing an estimate of λS. Exponents were calculated using custom made MATLAB (The Mathworks, Inc., Natwick, MA) functions. Average values of λS across test days were used for further analysis. λS was calculated using three methods: (a) Accelerations for 3 min were analyzed. A time delay of 8 samples was chosen, determined as the average of the first minimum of the average mutual information function (Fraser, 1986). The temporal separation between data points was normalized to a percentage of the time it takes to complete a stride (stride-time), by dividing the time of each data point by the average stride duration. λS was expressed as 〈ln(div)〉/stride-time, i.e. the unit of λS was g/stride (Fig. 1A). (b) Accelerations for 115 strides were analyzed. Data were time-normalized to 11,500 data points using a shape-preserving spline interpolation. A time delay of 10 samples was chosen, as all time-series had the same frequency. From this time-normalized time-series, the divergence curve was calculated after which the temporal separation between data points was rescaled to time in seconds, by multiplying the stride-time of each data point with the average stride duration. λS was expressed as 〈ln(div)〉/time, i.e. the unit of λS was g/s (Fig. 1B). (c) Accelerations for 115 strides were analyzed. Data were time-normalized to 11,500 data points. A time delay of 10 samples was chosen. λS was expressed as 〈ln(div)〉/stride-time, i.e. the unit of λS was g/stride (Fig. 1C).
VT
2.5
* *
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The mean7SD PWS was 1.1670.09 m/s. Mean7SD stride durations were 1.4070.09 s, 1.2070.06 s, 1.0770.05 s, 1.0070.04 s and 0.9570.04 s for 60%, 80%, 100%, 120% and 140% PWS, respectively. For all directions, methods a, b and c showed different trends for the effect of walking speed on λS (Fig. 2). For method a, walking speed only had a significant effect in the AP direction. For method b, λS in VT and ML directions significantly increased with increasing walking speeds. For method c, λS in all directions showed a significant decrease with increasing walking speed. Thus, different effects of walking speed on local dynamic stability, as quantified by λS, were induced by methods a, b and c (Fig. 2).
4. Discussion The main finding of the current study was that the effect of walking speed on calculated local dynamic stability was significantly
ML
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3. Results
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* *
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Fig. 2. The effect of walking speed (60%, 80%, 100%, 120% and 140% of preferred walking speed (PWS)) on the short-term local divergence exponent (λS) calculated using methods a, b and c for vertical (VT), medio-lateral (ML) and anterior posterior (AP) upper body acceleration directions. Method a calculated λS using accelerations during treadmill walking for 3 min at each walking speed and expressed λS as mean logarithmic divergence of nearest neighbors in state space (〈ln(div)〉) per stride-time for 0–0.5 stride. Method b calculated λS using accelerations for the first 115 strides during treadmill walking, time-normalized data to 11,500 data points and expressed λS as 〈ln(div)〉 per time for 0–0.5 stride. Method c calculated λS using accelerations for the first 115 strides during treadmill walking, time-normalized data to 11,500 data points and expressed λS as 〈ln(div)〉 per stride-time for 0–0.5 stride. Error bars represent standard deviations. nDenotes a significant effect of walking speed on the short-term local divergence exponent for a particular method.
J. Stenum et al. / Journal of Biomechanics 47 (2014) 3776–3779
different between calculation methods. Method a calculated local dynamic stability using a fixed time interval and expressed λS as ln(div)/stride-time, method b used a fixed number of strides and expressed λS as ln(div)/time and method c used a fixed number of strides and expressed λS as ln(div)/stride-time. For method a, there was a minor effect of walking speed on λS. For method b, λS increased at faster walking speeds, and for method c, λS decreased at faster walking speeds. As hypothesized, analyzing a fixed time interval or a fixed number of strides for all walking speeds and expressing λS as ln(div)/stride-time or as ln(div)/time led to different effects of walking speed on λS. Both methods a and c expressed λS as ln(div)/stride-time. However, method a analyzed a fixed time interval, whereas method c analyzed a fixed number of strides for all speeds. Analyzing a fixed time interval for all speeds, thus analyzing more strides at faster speeds, did not increase λS at faster speeds as inferred from Bruijn et al. (2009b). State space reconstructions for methods b and c were similar, but method b expressed λS as ln(div)/time while method c expressed λS as ln(div)/stride-time. Faster walking leads to shorter stride duration, which will, for a given divergence, lead to an increased λS indicating decreased stability at faster speeds when λS is expressed as ln(div)/time. When expressing λS as ln(div)/stride-time, this is avoided. However, it is presently unknown which interpretation is more valid. The methodological choices of analyzing a fixed time interval or a fixed number of strides and expressing λS as ln(div)/stridetime or as ln(div)/time are important and significantly changes the effect of walking speed on λS as presently shown. However, the methodological choices presented here are by no means exhaustive; indeed, other important choices remain in calculating λS that similarly could cause changes in the effect of walking speed on λS. Specifically, the choice of time-series from upper body kinematics or lower body joint kinematics (Kang and Dingwell, 2009), the movement directions studied (Bruijn et al., 2009a), state space reconstruction (Gates and Dingwell, 2009; van Schooten et al., 2013), the calculation algorithm used (Cignetti et al., 2012; Rispens et al., 2014) and whether to calculate λS as the rate of divergence from 0–0.5 stride or 0–1 stride (Reynard and Terrier, 2014; Reynard et al., 2014) all constitute important methodological choices. Based on the present study, wherein local dynamic stability of upper body accelerations was calculated using three methods, we conclude that the effect of walking speed on local dynamic stability is sensitive to pertinent methodological choices in calculating λS. The present study does not allow specific support to any of the three calculation methods. Therefore, inferences of the effect of walking speed on local dynamic stability and comparisons across studies should be made with careful consideration of the methods applied. Additional studies need to be done to establish which of the three applied methodological approaches is more correct. Conflict of interest statement None of the authors have any financial or personal relationships with people or organizations that could inappropriately influence the present study. Acknowledgements Sjoerd M. Bruijn was funded by a grant from the Netherlands Organization for Scientific Research (NWO #451-12-041).
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