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Quantifying complexity and variability in phase portraits of gait
Clinical Biomechanics 25 (2010) 552–556
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Clinical Biomechanics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c l i n b i o m e c h
Quantifying complexity and variability in phase portraits of gait Louis A. DiBerardino III a, John D. Polk b, Karl S. Rosengren c, Jesse B. Spencer-Smith d, Elizabeth T. Hsiao-Wecksler a,⁎ a b c d
Department of Mechanical Science & Engineering, University of Illinois at Urbana-Champaign, Urbana IL, USA Department of Anthropology, University of Illinois at Urbana-Champaign, Urbana IL, USA Department of Psychology, Northwestern University, Evanston IL, USA Department of Psychology, University of Illinois at Urbana-Champaign, Urbana IL, USA
a r t i c l e
i n f o
Article history: Received 20 August 2009 Accepted 11 March 2010 Keywords: Asymmetric gait Treadmill walking Orthotic brace Elliptical Fourier Analysis
a b s t r a c t Background: Injuries to the lower extremity often cause limitations to joint motion and alter movement patterns of limb segments during gait. We hypothesized that complexity and variability of limb segment motion during gait would increase in both limbs due to unilateral injury. Using simulated injury to generate asymmetric gait, we developed new methods to quantify changes in the complexity and variability of limb segment angular phase portraits. Methods: To simulate reduced range of motion associated with knee injury, the right knee was constrained to full extension by an external brace. Thigh, shank and foot segment angular phase portraits were generated from 20 healthy male subjects walking for 3-minute trials with and without the brace. Using Fourier-based methods, complexity was quantified by the number of harmonic frequencies suitable for fitting the phaseportrait shape — with a larger number of harmonics indicating greater complexity. Variability was characterized by the drift and confidence area generated by the inter-cycle excursion of the phase-portrait centroid. Findings: Significant differences were found in complexity and variability measures due to bracing. Phaseportrait shape complexity and variability increased in the right (braced) limb, compared to the unbraced condition; while only variability increased for the left (contralateral) limb during bracing. Interpretation: These new methods proved successful at quantifying changes in the complexity and variability that have been visually observed in phase portraits during asymmetric gait. This work provides a method that can be incorporated into clinical assessments to provide quantifiable measures of more precise differences in gait dynamics. Published by Elsevier Ltd
1. Introduction Injuries to the lower extremity often cause limitations to joint motion and alter movement patterns of limb segments during gait. While considerable literature exists describing the effects of various injuries and pathologies on gait, relatively little work has focused on changes in the complexity and variability of motion patterns that are introduced by injury. Complexity and variability have been used as a way to combine gait analysis output with dynamic systems analysis; however existing methodologies are inconsistent and can be difficult to intuitively relate back to the physiological system. Our goals of this work were 1) to develop intuitive metrics for assessing the complexity and variability of specific gait-related motion patterns (e.g., phase portraits) that agree well with qualitative inspection, and ⁎ Corresponding author. Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, MC-244, 1206 W. Green Street, Urbana, IL 61801, USA. E-mail address: [email protected] (E.T. Hsiao-Wecksler). 0268-0033/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.clinbiomech.2010.03.007
2) to investigate whether gait cycle complexity and variability are affected by simulated injury. Previous studies have used various metrics for assessing the relationships between gait variability and lower limb injury. Researchers using nonlinear metrics such as Lyapunov exponents have shown that gait cycle variability decreases in individuals with injury and pain (Hamill et al., 1999; Lewek et al., 2003; Moraiti et al., 2007). Other researchers applied metrics derived from the statistical definition of variability, and have shown increases in variability in response to the presence of injury or pain (Madeleine et al., 2008; von Porat et al., 2006). Not surprisingly, these results lead to differing functional interpretations, with those finding decreased variability suggesting that it may limit an individual's ability to respond to different environmental conditions (Moraiti et al., 2007), and those finding increased variability suggesting a greater ability to avoid pain or recover from injury quicker. These contrasting results suggest that further investigations are necessary to characterize how injury and/or pain may alter the variability in limb segmental motion, and to explore alternative metrics for quantifying variability in limb motion during gait.
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Gait complexity has also been investigated using a variety of different techniques. The impetus to investigate measures of complexity derives from studies of heartbeat dynamics that show decreases in complexity with disease or age (e.g., Pool, 1989). Applications to human gait have revealed conflicting results with some finding decreased complexity in stride timing due to disease and aging using metrics like fractal analysis (e.g., Goldberger et al., 2002). Others have reported increased stride timing complexity due to disease and aging with metrics like correlation dimension and Hurst exponents (Munoz-Diosdado et al., 2003; Munoz-Diosdado et al., 2005). These results, like the variability results described above, are clearly affected by the metrics used to quantify complexity, and the functional interpretations of these results will obviously differ as well. Our goal in this study was to apply an alternative and intuitive method for quantifying the complexity of gait, and to use this method to assess the complexity of limb segmental motion patterns. In this study, we rely on phase-portrait representations of limb segment motion during gait. Phase portraits are cyclic, dynamic representations of motion obtained by plotting a position value against its velocity, obtained in our case by plotting angular position for a segment (e.g., thigh, shank or foot) against the angular velocity for that segment. Phase portraits have several advantages over traditional univariate gait metrics: they can be used to characterize multiple continuous gait cycles, and they represent motion patterns of individual body segments (rather than the resultant stride or step timing patterns that have been frequently used for complexity studies). Phase portraits and cyclograms (angle–angle plots) have previously been used to describe differences in motion patterns of various subject populations, but these analyses have mainly been limited to qualitative visual descriptions of the plots or relative phase angle differences between segments (Beuter and Garfinkel, 1985; Clark and Phillips, 1993; Clark et al., 1993; Stergiou et al., 2001). Others have attempted to quantify phase-portrait and cyclogram shapes using (perimeter and area) moment-based techniques or comparing normalized metrics like perimeter length divided by the square root of area (Hershler and Milner, 1980a; Hershler and Milner, 1980b; Goswami, 1998). These quantitative methods based on perimeter length and area break down when trajectories are not smooth and therefore are less appropriate for perturbed or abnormal gait, since perturbed gait has more irregularities in phase portraits, including abrupt direction changes. Elliptical Fourier Analysis (EFA) provides accurate descriptions of phase-portrait shape. Kuhl and Giardina (1982) used EFA to describe closed loop contours. Roughly elliptical shapes were approximated by Fourier series with various numbers of harmonics. These researchers found that with approximately 10 harmonics they could accurately describe most complex contours, such as the two-dimensional outline of a house or airplane. Previously, we adapted EFA to quantify changes in phase-portrait complexity and variability during infant gait development and found that the variability and complexity of lower limb segment movements are high in early walkers but some measures approach the lower adult levels after six months of walking experience (Polk et al., 2008). In another study, we applied EFA to analyze differences between children with and without developmental coordination disorder (DCD). We found significantly increased variability and complexity in the gait of children with DCD, compared to age and gender matched children with typical developmental patterns (Rosengren et al., 2009). In the current study, we utilized metrics to quantify the complexity and inter-cycle variability of phase-portrait shapes that correlate with visual differences seen in the portraits. Specifically, we defined intercycle variability using two metrics that quantified the fluctuation in phase-portrait location between gait cycles. These metrics were the statistical area and path length swept by the phase-portrait centroid over multiple gait cycles. We defined complexity with a single metric that quantified the amount of intra-cycle fluctuation, or smoothness,
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of the phase portrait. The smoothness was characterized by the amount of harmonic content required to accurately describe the phase portrait with an EFA-based fit. In the current paper, we used a simulated injury to understand how controlled unilateral restrictions in joint motion can affect the complexity and variability in motion patterns of both limbs. We focused on how restrictive bracing affected the segmental motions of the braced and contralateral limbs. Bracing was used to provide a controlled experimental environment for simulating abnormal gait with known conditions. Artificially restricting knee range of motion (RoM) of one limb has been shown to affect the joint angular motion of both limbs (Shorter et al., 2008); therefore segmental complexity and variability should be significantly affected in both limbs as a result of the bracing. In addition, because we observed increased variability and complexity in our population of DCD children, we assume that other gait pathologies may result in increased complexity and variability. For this reason, we hypothesized that limb motion complexity and variability would be greater in both limbs of individuals while wearing the knee brace rather than without the brace. 2. Methods 2.1. Experimentation Twenty healthy male subjects, mean age 23 (SD 2) years, height 1.79 (SD 0.06) m, and mass 73 (SD 8) kg participated in the study. Based on self-report, subjects had no gait impairments, were experienced treadmill walkers and were all right-foot dominant (based on responding to which foot they would use to kick a ball). Subjects walked on a treadmill (Star Trac TR 4000; Irvine, CA, USA) at a self-selected comfortable pace (determined while wearing a knee brace) for 3 min unbraced, and for 3 min with a knee brace on their right limb (DonJoy 81099; Vista, CA, USA). The knee brace was locked in anatomical neutral position (0°, or full extension) to resist flexion. Subjects walked for at least 15 min on the treadmill with the brace before testing began, and were allowed at least 1 min to obtain steady-state walking before data were recorded. All procedures were approved by the University of Illinois at Urbana-Champaign Institutional Review Board and all participants gave informed consent. Kinematic data were captured at 120 Hz using a six-camera motion capture system (Vicon 460 Datastation; Oxford, UK). Relevant gait parameters were calculated from surface markers located over the sacrum, bilateral anterior superior iliac spine, mid-thigh, femoral lateral epicondyle, tibial tuberosity, lateral malleolus, 1st and 5th metatarsal heads, and heel. Gait cycles were defined from heel strike to the next heel strike separately for each limb. Missing marker data limited this study to using only 19 of the 20 subjects in the analysis and 20 consecutive cycles for each trial. Custom Matlab (R2008a, The MathWorks; Natick, MA, USA) code was used to calculate hip, knee, ankle and toe joint centers following the methodology of Vaughan et al. (1999). Two-dimensional thigh, shank, and foot segment angular positions and velocities were then calculated using sagittal plane projections of the joint centers. Phase portraits were obtained as segment angular position vs. angular velocity plots for each joint over the 20 gait cycles. Segment angles were measured counter-clockwise from a unit vector projecting anteriorly from the distal joint center of each segment. 2.2. Variability measures Variability was quantified by the inconsistency (or consistency) of the phase-portrait location between gait cycles throughout the trial, which we based on the fluctuation in phase-portrait centroid location from cycle to cycle. This methodology allows us to specifically quantify the inter-cycle variability throughout the trial, or the
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variability of the mean values for each cycle. The centroid was calculated as the mean of all (x, y) data points for each gait cycle. This inter-cycle variability was assessed by two measures, centroid area and centroid drift, which were adapted from traditional center of pressure stabilogram analyses used in postural steadiness studies (e.g., Prieto et al., 1996). Centroid area was defined as the bivariate 95% confidence ellipse area swept out by the centroids over 20 gait cycles. The total centroid drift was defined as the path length or total point-wise Cartesian distance that the centroid traveled on the phase plane over the 20 gait cycles of each trial (Fig. 1). The area metric gives a combined measure of the actual bivariate variability of the centroids around the overall mean centroid of the trial, and the drift quantifies the distance that the centroid traveled during the 20 gait cycles.
centroid). A reduced fit that eliminated 99.9% of the error between the full and zero-order fits was used because it represents the lowest (statistically motivated) value that characterized most of the full fit shape features. Calculating the number of harmonics that achieve a certain (high) percentage of the full fit allows us to better control for size differences between phase portraits, thereby making the resulting complexity measures more comparable across individuals and body segments. The mathematical details for this procedure are as follows. The maximum error between the full and zero-order fit was calculated using a point-wise sum of squared errors (SSE): n
SSEmax = ∑
i=1
2 2 xfull;i −xc + yfull;i −yc
ð1Þ
2.3. Complexity measure Complexity was quantified by the number of harmonics needed to describe the shape of the phase portrait (Fig. 1). EFA was performed on 20 gait cycles for each limb segment using a custom-modified version of an existing Matlab elliptical Fourier program (Thomas, 2006), based on the methodology of Kuhl and Giardina (1982). Complexity analyses were implemented using 200 points per cycle, i.e., 4000 points over the 20 gait cycles per trial. The complexity metric was defined as the minimum number of harmonics in a reduced-order fit, which was required to eliminate 99.9% of the error between a full-order fit and zero-order fit of the phase portrait. The full fit was defined as an over-parameterized 500 harmonic fit of the phase portrait (Polk et al., 2008; Rosengren et al., 2009). Therefore, a reduced-order fit will contain less than 500 harmonics. The zero-order fit is the worst possible Fourier fit of the data, and is equivalent to the mean value of the phase portrait (i.e.,
where (xfull,i, yfull,i) is the ith point in the full fit, (xc, yc) is the average centroid (zero-order fit) for the given trial, and n is the number of data points in the phase portrait (i.e., n = 4000). A reduced-order (j-harmonic) fit of the phase portrait was obtained and the error between it and the full fit was computed using: n
SSEj = ∑
i=1
2 2 xfull;i −xj;i + yfull;i −yj;i
ð2Þ
where (xj,i, yj,i) is the ith point in a reduced fit of j-harmonics. Then j was increased and Eq. (2) was recomputed until the error between the full and j-harmonic fit was less than 0.1% of the maximum error: SSEj ≤0:001*SSEmax :
ð3Þ
The minimum integer j satisfying Eq. (3) then defines our complexity metric. Since more harmonics (higher frequencies) are needed to accurately describe more complex shapes, a larger integer value suggests a more complex motion shape. 2.4. Statistical analysis Statistical tests (SPSS, v15; Chicago, IL, USA) on the three parameters (complexity, area, and drift) examined within-subject effects for limb (left and right) and bracing condition (no brace and knee brace) at each limb segment (thigh, shank, and foot). Pairwise comparisons of certain limb-brace combinations were also conducted. A 3 × 2 × 2 repeated measures multivariate ANOVA (MANOVA) confirmed that all main effects and interactions caused significant differences (P b 0.001). Therefore, to better address our hypothesis that restricted RoM in one joint affects complexity and variability of segment behaviors in both limbs, we performed paired t-tests on select knee-bracing conditions for each segment and measure (left limb: no brace vs. braced [LNB vs. LKB] and right limb: no brace vs. braced [RNB vs. RKB]). Bonferonni correction was used to account for the two tests performed on each measure (α = 0.025). These comparisons illustrate how restricting a single joint's RoM specifically affected each limb. While preliminary bilateral symmetry results (comparing LNB vs. RNB to LKB vs. RKB) were interesting, we felt symmetry between shape complexity and variability lacked physical meaning and did not add to any previous clinical results. Differences between LNB vs. RKB and LKB vs. RNB were also not investigated because they are not particularly interesting or useful in analyzing the differences between conditions. 3. Results
Fig. 1. Example of elliptical Fourier fitting method on segment angular position vs. velocity phase portraits for the right thigh segment — (a) without and (b) with knee brace. Solid line is full elliptical Fourier fit; dots indicate reduced fit (99.9% of full fit). In this figure, the original number of cycles (20) has been reduced to 4, and the dotted curves reduced from 200 to 50 points/cycle for clarity. The centroid for each cycle is identified with a “+” symbol.
Introduction of the knee brace on the right limb visibly altered each subject's gait and the segment angle phase portraits of each limb. For example, the right thigh segment phase portrait appears to have reduced in size and increased in complexity and inter-cycle variability as a result of the brace (Fig. 1). The increased complexity and
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variability due to bracing can be qualitatively assessed, respectively, by the greater amount of fluctuation seen within the cycles (or reduction of smoothness) and the greater amount of phase-portrait trajectory movement seen between the cycles (directly relating to centroid movement) (Fig. 1b). As stated in the Introduction, the goal of this study was to quantify these qualitative descriptions, which is done by using a frequency content measure to assess complexity and centroid movement measures to assess variability. Knee bracing caused significant differences in the phase-portrait complexity and variability measures for most of the lower limb segments (Table 1). In accordance with our prediction, phase-portrait shape complexity and inter-cycle variability increased in the right (braced) limb during bracing, compared to the unbraced condition, while only variability increased for the left (contralateral) limb during bracing. More specifically, shape complexity increased for all three segments of the right limb during bracing (P ≤ 0.001, see [RNB vs. RKB] column — Table 1). Inter-cycle variability (drift and area) also increased in the right limb, but only for the thigh and foot segments (P b 0.002). All three contralateral limb segments experienced increased variability due to bracing (P b 0.011, see [LNB vs. LKB] column — Table 1); however complexity measures did not differ significantly. These changes within each limb demonstrate how bracing directly affects the braced limb and indirectly affects the contralateral limb, most likely through compensation or coupling, during gait. 4. Discussion By applying these new metrics to limb segment angle phase portraits, we were able to quantify changes in gait complexity and variability induced by a simulated injury. Unilateral restriction of knee motion had significant impact on the complexity and variability of segmental motions within the ipsilateral limb, and the inter-cycle variability of the contralateral limb (Table 1). Our results compared very well to our hypothesis that limb motion complexity and variability would be greater in both limbs of individuals while wearing the knee brace rather than without the brace. While both limbs were significantly affected in one manner or another, variability of the ipsilateral shank and complexity of the entire contralateral limb did not increase significantly between bracing conditions. Bracing the right knee resulted in significant changes in phase portraits of all ipsilateral segments (Table 1 — [RNB vs. RKB] column). We found increased complexity in the phase portrait of the right limb for all three segments, as seen by the increased number of harmonics needed to describe their motion shapes, compared to the unbraced condition. Increased inter-cycle variability, as measured by drift and area, occurred in the thigh and foot but not shank. The lack of change in right shank variability may be explained by considering the lower limb's configuration during bracing. Normally the shank angle is controlled by a combination of hip and knee joint motions. Adding the
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right knee brace directly restricts right knee motion, essentially making the right shank an extension of the thigh. According to our metrics, the shank had inherently higher variability than the thigh during natural gait. The bracing effectively coupled the shank to the thigh creating a “combined” segment. The motion of this “combined” segment exhibited a significant increase in variability when compared to the unbraced thigh but not the unbraced shank. Thus while the brace perturbed natural gait, it had more of an effect on the variability of the right thigh than the right shank, possibly due to the higher original variability of the shank during unbraced walking (Table 1). While the right limb was affected as predicted, changes in the left limb were limited to inter-cycle variability and not complexity (Table 1 — [LNB vs. LKB] column). Drift and area increased in the left limb for all three segments while the right limb was braced. Increased contralateral variability is quite interesting, in that it suggests some sort of bilateral coupling between the two limbs. The coupling is most likely from the contralateral limb attempting to compensate for the reduced RoM and increased variability of the ipsilateral limb. The complexity of the left limb phase portraits was not affected by bracing, implying that the smoothness of the contralateral phase portraits was not affected by the compensation or coupling. It is possible that the changes in variability were caused by artifacts of wearing the knee brace, in addition to the restricted RoM, which might not be present in actual injuries. Our finding that gait variability increased due to simulated injury agrees with literature based on statistical variability metrics (Madeleine et al., 2008; von Porat et al., 2006), yet contradicts literature based on nonlinear metrics dealing with actual injury and pain (Hamill et al., 1999; Lewek et al., 2003; Moraiti et al., 2007). Those finding decreased variability tended to adapt methodology used in dynamic stability analysis (e.g., maximum Lyapunov exponent) to quantify variability (Lewek et al., 2003; Moraiti et al., 2007); however, Dingwell et al. (2001) showed that traditional variability measures do not correlate with similar stability results. In addition to relating variability with stability, many past researchers base their methods on the loss of complexity hypothesis (Pool, 1989), which states that biological systems lose complexity with age or disease. As mentioned by Lipsitz (2002), it may not necessarily be correct to equate complexity and variability, as they describe different aspects of a signal or shape (complexity relates to signal fluctuation frequency and irregularity while variability relates to deviation from the mean value). Still, our method of simulated injury may not be producing similar results to actual knee injury, and our intuitive phase-portrait variability metric could possibly agree with the existing nonlinear/stability metrics if performed on the actual data collected from injured individuals. For the data studied, our variability results agree with qualitative inspection, thus we are confident in these inter-cycle variability metrics. Contradictions in complexity results have also been found (Goldberger et al., 2002; Munoz-Diosdado et al., 2003; Munoz-Diosdado et al.,
Table 1 Mean (and standard deviation) values of complexity and variability measures separated by segment, limb, and bracing condition. Check marks signify statistical difference between paired comparisons of bracing condition within each limb (α = 0.025). Body segment
Thigh
Shank
Foot
a
Measure (units)
Complexity (# harmonics) Drift (Cartesian dist.) Area (rad2/s) Complexity Drift Area Complexity Drift Area
Brace worn on right limb.
Average (SD) values
Paired piecewise comparisons
Left lega: no brace
Left lega: knee brace
Right leg: no brace
Right leg: knee brace
194 0.42 0.0027 153 0.64 0.0057 159 0.96 0.0101
191 0.65 0.0056 153 0.92 0.0109 162 1.15 0.0167
195 0.51 0.0042 150 0.66 0.0062 171 0.86 0.0094
210 0.65 0.0062 181 0.70 0.0073 217 1.12 0.0156
(19) (0.12) (0.0015) (16) (0.23) (0.0043) (19) (0.34) (0.0054)
(15) (0.22) (0.0032) (14) (0.39) (0.0076) (14) (0.31) (0.0102)
(22) (0.19) (0.0030) (18) (0.31) (0.0047) (18) (0.26) (0.0052)
(21) (0.14) (0.0026) (25) (0.18) (0.0031) (21) (0.28) (0.0059)
LNB vs. LKB
RNB vs. RKB
√ √
√ √ √ √
√ √ √ √
√ √ √
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2005) with results changing depending on which metrics researchers choose to use. Our frequency based approach is unlike past approaches, but in line with the complexity description provided by Lipsitz (2002), where more fluctuation (produced by increased Fourier harmonics) leads to higher complexity. Similar to the differences in variability, this complexity measure should be executed across datasets before comparison with other methods used to quantify complexity. There are certain limitations of this study. First, our simulated RoM reduction is more drastic than what might be experienced with actual injury. These metrics might not detect subtle changes in gait due to minor or nearly healed injuries. Second, injured subjects also develop compensation strategies over time to manage their restrictions and pain. While our test subjects had approximately 15 min to walk with the brace before testing began, these subjects may not have developed long-term compensation strategies similar to an injured subject. Gait variability may decrease if our subjects walked with the brace for a long period of time. We found, however, for the 20 gait cycles in this study, that variability in terms of centroid drift was stationary. This suggests the subjects had at least developed a short-term movement strategy. (To assess stationarity of a phase portrait, we quantified centroid drift for each consecutive pair of centroids, i.e., distance between every two centroids. A straight-line fit, of centroid pair drift vs. cycle pair number, expressed the slope of the drift throughout the trial. No consistently increasing or decreasing trends due to bracing were found for data from three random subjects, suggesting stationarity of the phase portraits during each trial.). Future studies should examine the techniques presented in this paper on an actual injury population. Sensitivity of these metrics to a smaller gait perturbation could then be determined. Sensitivity to other controlled perturbations would also be useful. It would also be fruitful to apply previous methods (Hamill et al., 1999; Lewek et al., 2003; Moraiti et al., 2007) to measure gait of individuals wearing braces. These studies would hopefully provide converging evidence to aid in determining the best methods for quantifying complexity and variability across different populations. Other elliptical shapes found in gait dynamics, especially kinetic data, could be examined by this method as well.
5. Conclusions We have successfully developed new techniques that quantify changes in gait phase-portrait complexity and variability in agreement with qualitative inspection of the plots. We previously applied Elliptical Fourier Analysis to quantify motion changes due to gait development and pathology (Polk et al., 2008; Rosengren et al., 2009). In this paper, we have shown that this technique can be modified to successfully quantify the impact of bracing one limb. The physical restriction imposed in this study created large size (area, mean centroid, etc.) differences between braced and non-braced phase portraits, thus the metrics used were modified to factor out size (complexity) and more strictly follow intuitive definitions of variability (drift and area). These new metrics provided quantitative results in agreement with qualitative visual interpretation of the phase portraits while intuitively separating complexity and variability results. This approach was highly effective at distinguishing changes in motion patterns when range of motion was reduced due to a simulated knee injury. It also provides a tool that may be useful to clinicians for quantifying deviation from normal gait in terms of the complexity and variability changes of the resulting motion shapes.
Acknowledgments The authors thank Lauren Merry and K. Alex Shorter for their assistance with experimentation and data processing. They also thank Professors Harry Dankowicz, Sungjin Hong, and Michael Lague; and Nathaniel Helwig and Kiwon Park for their insight and advice. This study was funded by grants from the National Science Foundation (#0727083) and the Mary Jane Neer Disability Research Fund at the University of Illinois. References Beuter, A., Garfinkel, A., 1985. Phase plane analysis of limb trajectories in nonhandicapped and cerebral palsied subjects. Adapt Phys Activ Q. 2, 214–227. Clark, J.E., Phillips, S.J., 1993. A longitudinal-study of intralimb coordination in the 1st year of independent walking — a dynamical-systems analysis. Child Dev. 64, 1143–1157. Clark, J.E., Truly, T.L., Phillips, S.J., 1993. On the development of walking as a limit cycle system. In: Smith, L.B., Thelen, E. (Eds.), Dynamical Systems in Development: Applications. MIT Press, Cambridge. Dingwell, J.B., Cusumano, J.P., Cavanagh, P.R., Sternad, D., 2001. Local dynamic stability versus kinematic variability of continuous overground and treadmill walking. J Biomech Eng Trans ASME. 123, 27–32. Goldberger, A.L., Amaral, L.A.N., Hausdorff, J.M., Ivanov, P.C., Peng, C.K., Stanley, H.E., 2002. Fractal dynamics in physiology: alterations with disease and aging. Proc. Natl Acad. Sci. USA 99, 2466–2472. Goswami, A., 1998. A new gait parameterization technique by means of cyclogram moments: application to human slope walking. Gait Posture. 8, 15–36. Hamill, J., van Emmerik, R.E., Heiderscheit, B.C., Li, L., 1999. A dynamical systems approach to lower extremity running injuries. Clin Biomech. 14, 297–308. Hershler, C., Milner, M., 1980a. Angle–angle diagrams in above-knee amputee and cerebral-palsy gait. Am. J. Phys. Med. Rehabil. 59, 165–183. Hershler, C., Milner, M., 1980b. Angle–angle diagrams in the assessment of locomotion. Am. J. Phys. Med. Rehabil. 59, 109–125. Kuhl, F.P., Giardina, C.R., 1982. Elliptic Fourier features of a closed contour. Comput Graph Image Process. 18, 236–258. Lewek, M.D., Rudolph, K.S., Snyder-Mackler, L., 2003. Symptomatic knee osteoarthritis and knee pattern variability during gait. Med. Sci. Sports Exerc. 35, S252. Lipsitz, L.A., 2002. Dynamics of stability: the physiologic basis of functional health and frailty. J Gerontol A Biol Sci Med Sci. 57, B115–B125. Madeleine, P., Mathiassen, S.E., Arendt-Nielsen, L., 2008. Changes in the degree of motor variability associated with experimental and chronic neck-shoulder pain during a standardised repetitive arm movement. Exp. Brain Res. 185, 689–698. Moraiti, C., Stergiou, N., Ristanis, S., Georgoulis, A.D., 2007. Acl deficiency affects strideto-stride variability as measured using nonlinear methodology. Knee Surg. Sports Traumatol. Arthrosc. 15, 1406–1413. Munoz-Diosdado, A., Del Rio Correa, J.L., Angulo-Brown, F., Calleja Quevedo, E., 2005. Fractal analysis of human gait: old and young, healthy and ill subjects. Rev Mex Fis. 51, 14–21. Munoz-Diosdado, A., Del Rio Correa, J.L., Angulo Brown, F., 2003. Multifractality in time series of human gait. Annual International Conference of the IEEE Engineering in Medicine and Biology — Proceedings. Polk, J.D., Spencer-Smith, J., DiBerardino, L., Ellis, D., Downen, M., Rosengren, K.S., 2008. Quantifying variability in phase portraits: application to gait ontogeny. Infant Behav Dev. 31, 302–306. Pool, R., 1989. Is it healthy to be chaotic.3. Science 243, 604–607. Prieto, T.E., Myklebust, J.B., Hoffmann, R.G., Lovett, E.G., Myklebust, B.M., 1996. Measures of postural steadiness: differences between healthy young and elderly adults. IEEE Trans. Biomed. Eng. 43, 956–966. Rosengren, K.S., Deconinck, F.J., DiBerardino, 3rd, L.A., Polk, J.D., Spencer-Smith, J., De Clercq, D., Lenoir, M., 2009. Differences in gait complexity and variability between children with and without developmental coordination disorder. Gait Posture. 29, 225–229. Shorter, K.A., Polk, J.D., Rosengren, K.S., Hsiao-Wecksler, E.T., 2008. A new approach to detecting asymmetries in gait. Clin Biomech. 23, 459–467. Stergiou, N., Jensen, J.L., Bates, B.T., Scholten, S.D., Tzetzis, G., 2001. A dynamical systems investigation of lower extremity coordination during running over obstacles. Clin Biomech. 16, 213–221. Thomas, D., 2006. Elliptical Fourier shape descriptors. MATLAB Central, The MathWorks, Inc., http://www.mathworks.com/matlabcentral/fileexchange/12746. Vaughan, C.L., Davis, B.L., O'Connor, J.C., 1999. Dynamics of Human Gait. K. Publishers, Cape Town, South Africa. von Porat, A., Henriksson, M., Holmstrom, E., Thorstensson, C.A., Mattsson, L., Roos, E.M., 2006. Knee kinematics and kinetics during gait, step and hop in males with a 16 years old ACL injury compared with matched controls. Knee Surg. Sports Traumatol. Arthrosc. 14, 546–554.
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Clinical Biomechanics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c l i n b i o m e c h
Quantifying complexity and variability in phase portraits of gait Louis A. DiBerardino III a, John D. Polk b, Karl S. Rosengren c, Jesse B. Spencer-Smith d, Elizabeth T. Hsiao-Wecksler a,⁎ a b c d
Department of Mechanical Science & Engineering, University of Illinois at Urbana-Champaign, Urbana IL, USA Department of Anthropology, University of Illinois at Urbana-Champaign, Urbana IL, USA Department of Psychology, Northwestern University, Evanston IL, USA Department of Psychology, University of Illinois at Urbana-Champaign, Urbana IL, USA
a r t i c l e
i n f o
Article history: Received 20 August 2009 Accepted 11 March 2010 Keywords: Asymmetric gait Treadmill walking Orthotic brace Elliptical Fourier Analysis
a b s t r a c t Background: Injuries to the lower extremity often cause limitations to joint motion and alter movement patterns of limb segments during gait. We hypothesized that complexity and variability of limb segment motion during gait would increase in both limbs due to unilateral injury. Using simulated injury to generate asymmetric gait, we developed new methods to quantify changes in the complexity and variability of limb segment angular phase portraits. Methods: To simulate reduced range of motion associated with knee injury, the right knee was constrained to full extension by an external brace. Thigh, shank and foot segment angular phase portraits were generated from 20 healthy male subjects walking for 3-minute trials with and without the brace. Using Fourier-based methods, complexity was quantified by the number of harmonic frequencies suitable for fitting the phaseportrait shape — with a larger number of harmonics indicating greater complexity. Variability was characterized by the drift and confidence area generated by the inter-cycle excursion of the phase-portrait centroid. Findings: Significant differences were found in complexity and variability measures due to bracing. Phaseportrait shape complexity and variability increased in the right (braced) limb, compared to the unbraced condition; while only variability increased for the left (contralateral) limb during bracing. Interpretation: These new methods proved successful at quantifying changes in the complexity and variability that have been visually observed in phase portraits during asymmetric gait. This work provides a method that can be incorporated into clinical assessments to provide quantifiable measures of more precise differences in gait dynamics. Published by Elsevier Ltd
1. Introduction Injuries to the lower extremity often cause limitations to joint motion and alter movement patterns of limb segments during gait. While considerable literature exists describing the effects of various injuries and pathologies on gait, relatively little work has focused on changes in the complexity and variability of motion patterns that are introduced by injury. Complexity and variability have been used as a way to combine gait analysis output with dynamic systems analysis; however existing methodologies are inconsistent and can be difficult to intuitively relate back to the physiological system. Our goals of this work were 1) to develop intuitive metrics for assessing the complexity and variability of specific gait-related motion patterns (e.g., phase portraits) that agree well with qualitative inspection, and ⁎ Corresponding author. Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, MC-244, 1206 W. Green Street, Urbana, IL 61801, USA. E-mail address: [email protected] (E.T. Hsiao-Wecksler). 0268-0033/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.clinbiomech.2010.03.007
2) to investigate whether gait cycle complexity and variability are affected by simulated injury. Previous studies have used various metrics for assessing the relationships between gait variability and lower limb injury. Researchers using nonlinear metrics such as Lyapunov exponents have shown that gait cycle variability decreases in individuals with injury and pain (Hamill et al., 1999; Lewek et al., 2003; Moraiti et al., 2007). Other researchers applied metrics derived from the statistical definition of variability, and have shown increases in variability in response to the presence of injury or pain (Madeleine et al., 2008; von Porat et al., 2006). Not surprisingly, these results lead to differing functional interpretations, with those finding decreased variability suggesting that it may limit an individual's ability to respond to different environmental conditions (Moraiti et al., 2007), and those finding increased variability suggesting a greater ability to avoid pain or recover from injury quicker. These contrasting results suggest that further investigations are necessary to characterize how injury and/or pain may alter the variability in limb segmental motion, and to explore alternative metrics for quantifying variability in limb motion during gait.
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Gait complexity has also been investigated using a variety of different techniques. The impetus to investigate measures of complexity derives from studies of heartbeat dynamics that show decreases in complexity with disease or age (e.g., Pool, 1989). Applications to human gait have revealed conflicting results with some finding decreased complexity in stride timing due to disease and aging using metrics like fractal analysis (e.g., Goldberger et al., 2002). Others have reported increased stride timing complexity due to disease and aging with metrics like correlation dimension and Hurst exponents (Munoz-Diosdado et al., 2003; Munoz-Diosdado et al., 2005). These results, like the variability results described above, are clearly affected by the metrics used to quantify complexity, and the functional interpretations of these results will obviously differ as well. Our goal in this study was to apply an alternative and intuitive method for quantifying the complexity of gait, and to use this method to assess the complexity of limb segmental motion patterns. In this study, we rely on phase-portrait representations of limb segment motion during gait. Phase portraits are cyclic, dynamic representations of motion obtained by plotting a position value against its velocity, obtained in our case by plotting angular position for a segment (e.g., thigh, shank or foot) against the angular velocity for that segment. Phase portraits have several advantages over traditional univariate gait metrics: they can be used to characterize multiple continuous gait cycles, and they represent motion patterns of individual body segments (rather than the resultant stride or step timing patterns that have been frequently used for complexity studies). Phase portraits and cyclograms (angle–angle plots) have previously been used to describe differences in motion patterns of various subject populations, but these analyses have mainly been limited to qualitative visual descriptions of the plots or relative phase angle differences between segments (Beuter and Garfinkel, 1985; Clark and Phillips, 1993; Clark et al., 1993; Stergiou et al., 2001). Others have attempted to quantify phase-portrait and cyclogram shapes using (perimeter and area) moment-based techniques or comparing normalized metrics like perimeter length divided by the square root of area (Hershler and Milner, 1980a; Hershler and Milner, 1980b; Goswami, 1998). These quantitative methods based on perimeter length and area break down when trajectories are not smooth and therefore are less appropriate for perturbed or abnormal gait, since perturbed gait has more irregularities in phase portraits, including abrupt direction changes. Elliptical Fourier Analysis (EFA) provides accurate descriptions of phase-portrait shape. Kuhl and Giardina (1982) used EFA to describe closed loop contours. Roughly elliptical shapes were approximated by Fourier series with various numbers of harmonics. These researchers found that with approximately 10 harmonics they could accurately describe most complex contours, such as the two-dimensional outline of a house or airplane. Previously, we adapted EFA to quantify changes in phase-portrait complexity and variability during infant gait development and found that the variability and complexity of lower limb segment movements are high in early walkers but some measures approach the lower adult levels after six months of walking experience (Polk et al., 2008). In another study, we applied EFA to analyze differences between children with and without developmental coordination disorder (DCD). We found significantly increased variability and complexity in the gait of children with DCD, compared to age and gender matched children with typical developmental patterns (Rosengren et al., 2009). In the current study, we utilized metrics to quantify the complexity and inter-cycle variability of phase-portrait shapes that correlate with visual differences seen in the portraits. Specifically, we defined intercycle variability using two metrics that quantified the fluctuation in phase-portrait location between gait cycles. These metrics were the statistical area and path length swept by the phase-portrait centroid over multiple gait cycles. We defined complexity with a single metric that quantified the amount of intra-cycle fluctuation, or smoothness,
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of the phase portrait. The smoothness was characterized by the amount of harmonic content required to accurately describe the phase portrait with an EFA-based fit. In the current paper, we used a simulated injury to understand how controlled unilateral restrictions in joint motion can affect the complexity and variability in motion patterns of both limbs. We focused on how restrictive bracing affected the segmental motions of the braced and contralateral limbs. Bracing was used to provide a controlled experimental environment for simulating abnormal gait with known conditions. Artificially restricting knee range of motion (RoM) of one limb has been shown to affect the joint angular motion of both limbs (Shorter et al., 2008); therefore segmental complexity and variability should be significantly affected in both limbs as a result of the bracing. In addition, because we observed increased variability and complexity in our population of DCD children, we assume that other gait pathologies may result in increased complexity and variability. For this reason, we hypothesized that limb motion complexity and variability would be greater in both limbs of individuals while wearing the knee brace rather than without the brace. 2. Methods 2.1. Experimentation Twenty healthy male subjects, mean age 23 (SD 2) years, height 1.79 (SD 0.06) m, and mass 73 (SD 8) kg participated in the study. Based on self-report, subjects had no gait impairments, were experienced treadmill walkers and were all right-foot dominant (based on responding to which foot they would use to kick a ball). Subjects walked on a treadmill (Star Trac TR 4000; Irvine, CA, USA) at a self-selected comfortable pace (determined while wearing a knee brace) for 3 min unbraced, and for 3 min with a knee brace on their right limb (DonJoy 81099; Vista, CA, USA). The knee brace was locked in anatomical neutral position (0°, or full extension) to resist flexion. Subjects walked for at least 15 min on the treadmill with the brace before testing began, and were allowed at least 1 min to obtain steady-state walking before data were recorded. All procedures were approved by the University of Illinois at Urbana-Champaign Institutional Review Board and all participants gave informed consent. Kinematic data were captured at 120 Hz using a six-camera motion capture system (Vicon 460 Datastation; Oxford, UK). Relevant gait parameters were calculated from surface markers located over the sacrum, bilateral anterior superior iliac spine, mid-thigh, femoral lateral epicondyle, tibial tuberosity, lateral malleolus, 1st and 5th metatarsal heads, and heel. Gait cycles were defined from heel strike to the next heel strike separately for each limb. Missing marker data limited this study to using only 19 of the 20 subjects in the analysis and 20 consecutive cycles for each trial. Custom Matlab (R2008a, The MathWorks; Natick, MA, USA) code was used to calculate hip, knee, ankle and toe joint centers following the methodology of Vaughan et al. (1999). Two-dimensional thigh, shank, and foot segment angular positions and velocities were then calculated using sagittal plane projections of the joint centers. Phase portraits were obtained as segment angular position vs. angular velocity plots for each joint over the 20 gait cycles. Segment angles were measured counter-clockwise from a unit vector projecting anteriorly from the distal joint center of each segment. 2.2. Variability measures Variability was quantified by the inconsistency (or consistency) of the phase-portrait location between gait cycles throughout the trial, which we based on the fluctuation in phase-portrait centroid location from cycle to cycle. This methodology allows us to specifically quantify the inter-cycle variability throughout the trial, or the
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variability of the mean values for each cycle. The centroid was calculated as the mean of all (x, y) data points for each gait cycle. This inter-cycle variability was assessed by two measures, centroid area and centroid drift, which were adapted from traditional center of pressure stabilogram analyses used in postural steadiness studies (e.g., Prieto et al., 1996). Centroid area was defined as the bivariate 95% confidence ellipse area swept out by the centroids over 20 gait cycles. The total centroid drift was defined as the path length or total point-wise Cartesian distance that the centroid traveled on the phase plane over the 20 gait cycles of each trial (Fig. 1). The area metric gives a combined measure of the actual bivariate variability of the centroids around the overall mean centroid of the trial, and the drift quantifies the distance that the centroid traveled during the 20 gait cycles.
centroid). A reduced fit that eliminated 99.9% of the error between the full and zero-order fits was used because it represents the lowest (statistically motivated) value that characterized most of the full fit shape features. Calculating the number of harmonics that achieve a certain (high) percentage of the full fit allows us to better control for size differences between phase portraits, thereby making the resulting complexity measures more comparable across individuals and body segments. The mathematical details for this procedure are as follows. The maximum error between the full and zero-order fit was calculated using a point-wise sum of squared errors (SSE): n
SSEmax = ∑
i=1
2 2 xfull;i −xc + yfull;i −yc
ð1Þ
2.3. Complexity measure Complexity was quantified by the number of harmonics needed to describe the shape of the phase portrait (Fig. 1). EFA was performed on 20 gait cycles for each limb segment using a custom-modified version of an existing Matlab elliptical Fourier program (Thomas, 2006), based on the methodology of Kuhl and Giardina (1982). Complexity analyses were implemented using 200 points per cycle, i.e., 4000 points over the 20 gait cycles per trial. The complexity metric was defined as the minimum number of harmonics in a reduced-order fit, which was required to eliminate 99.9% of the error between a full-order fit and zero-order fit of the phase portrait. The full fit was defined as an over-parameterized 500 harmonic fit of the phase portrait (Polk et al., 2008; Rosengren et al., 2009). Therefore, a reduced-order fit will contain less than 500 harmonics. The zero-order fit is the worst possible Fourier fit of the data, and is equivalent to the mean value of the phase portrait (i.e.,
where (xfull,i, yfull,i) is the ith point in the full fit, (xc, yc) is the average centroid (zero-order fit) for the given trial, and n is the number of data points in the phase portrait (i.e., n = 4000). A reduced-order (j-harmonic) fit of the phase portrait was obtained and the error between it and the full fit was computed using: n
SSEj = ∑
i=1
2 2 xfull;i −xj;i + yfull;i −yj;i
ð2Þ
where (xj,i, yj,i) is the ith point in a reduced fit of j-harmonics. Then j was increased and Eq. (2) was recomputed until the error between the full and j-harmonic fit was less than 0.1% of the maximum error: SSEj ≤0:001*SSEmax :
ð3Þ
The minimum integer j satisfying Eq. (3) then defines our complexity metric. Since more harmonics (higher frequencies) are needed to accurately describe more complex shapes, a larger integer value suggests a more complex motion shape. 2.4. Statistical analysis Statistical tests (SPSS, v15; Chicago, IL, USA) on the three parameters (complexity, area, and drift) examined within-subject effects for limb (left and right) and bracing condition (no brace and knee brace) at each limb segment (thigh, shank, and foot). Pairwise comparisons of certain limb-brace combinations were also conducted. A 3 × 2 × 2 repeated measures multivariate ANOVA (MANOVA) confirmed that all main effects and interactions caused significant differences (P b 0.001). Therefore, to better address our hypothesis that restricted RoM in one joint affects complexity and variability of segment behaviors in both limbs, we performed paired t-tests on select knee-bracing conditions for each segment and measure (left limb: no brace vs. braced [LNB vs. LKB] and right limb: no brace vs. braced [RNB vs. RKB]). Bonferonni correction was used to account for the two tests performed on each measure (α = 0.025). These comparisons illustrate how restricting a single joint's RoM specifically affected each limb. While preliminary bilateral symmetry results (comparing LNB vs. RNB to LKB vs. RKB) were interesting, we felt symmetry between shape complexity and variability lacked physical meaning and did not add to any previous clinical results. Differences between LNB vs. RKB and LKB vs. RNB were also not investigated because they are not particularly interesting or useful in analyzing the differences between conditions. 3. Results
Fig. 1. Example of elliptical Fourier fitting method on segment angular position vs. velocity phase portraits for the right thigh segment — (a) without and (b) with knee brace. Solid line is full elliptical Fourier fit; dots indicate reduced fit (99.9% of full fit). In this figure, the original number of cycles (20) has been reduced to 4, and the dotted curves reduced from 200 to 50 points/cycle for clarity. The centroid for each cycle is identified with a “+” symbol.
Introduction of the knee brace on the right limb visibly altered each subject's gait and the segment angle phase portraits of each limb. For example, the right thigh segment phase portrait appears to have reduced in size and increased in complexity and inter-cycle variability as a result of the brace (Fig. 1). The increased complexity and
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variability due to bracing can be qualitatively assessed, respectively, by the greater amount of fluctuation seen within the cycles (or reduction of smoothness) and the greater amount of phase-portrait trajectory movement seen between the cycles (directly relating to centroid movement) (Fig. 1b). As stated in the Introduction, the goal of this study was to quantify these qualitative descriptions, which is done by using a frequency content measure to assess complexity and centroid movement measures to assess variability. Knee bracing caused significant differences in the phase-portrait complexity and variability measures for most of the lower limb segments (Table 1). In accordance with our prediction, phase-portrait shape complexity and inter-cycle variability increased in the right (braced) limb during bracing, compared to the unbraced condition, while only variability increased for the left (contralateral) limb during bracing. More specifically, shape complexity increased for all three segments of the right limb during bracing (P ≤ 0.001, see [RNB vs. RKB] column — Table 1). Inter-cycle variability (drift and area) also increased in the right limb, but only for the thigh and foot segments (P b 0.002). All three contralateral limb segments experienced increased variability due to bracing (P b 0.011, see [LNB vs. LKB] column — Table 1); however complexity measures did not differ significantly. These changes within each limb demonstrate how bracing directly affects the braced limb and indirectly affects the contralateral limb, most likely through compensation or coupling, during gait. 4. Discussion By applying these new metrics to limb segment angle phase portraits, we were able to quantify changes in gait complexity and variability induced by a simulated injury. Unilateral restriction of knee motion had significant impact on the complexity and variability of segmental motions within the ipsilateral limb, and the inter-cycle variability of the contralateral limb (Table 1). Our results compared very well to our hypothesis that limb motion complexity and variability would be greater in both limbs of individuals while wearing the knee brace rather than without the brace. While both limbs were significantly affected in one manner or another, variability of the ipsilateral shank and complexity of the entire contralateral limb did not increase significantly between bracing conditions. Bracing the right knee resulted in significant changes in phase portraits of all ipsilateral segments (Table 1 — [RNB vs. RKB] column). We found increased complexity in the phase portrait of the right limb for all three segments, as seen by the increased number of harmonics needed to describe their motion shapes, compared to the unbraced condition. Increased inter-cycle variability, as measured by drift and area, occurred in the thigh and foot but not shank. The lack of change in right shank variability may be explained by considering the lower limb's configuration during bracing. Normally the shank angle is controlled by a combination of hip and knee joint motions. Adding the
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right knee brace directly restricts right knee motion, essentially making the right shank an extension of the thigh. According to our metrics, the shank had inherently higher variability than the thigh during natural gait. The bracing effectively coupled the shank to the thigh creating a “combined” segment. The motion of this “combined” segment exhibited a significant increase in variability when compared to the unbraced thigh but not the unbraced shank. Thus while the brace perturbed natural gait, it had more of an effect on the variability of the right thigh than the right shank, possibly due to the higher original variability of the shank during unbraced walking (Table 1). While the right limb was affected as predicted, changes in the left limb were limited to inter-cycle variability and not complexity (Table 1 — [LNB vs. LKB] column). Drift and area increased in the left limb for all three segments while the right limb was braced. Increased contralateral variability is quite interesting, in that it suggests some sort of bilateral coupling between the two limbs. The coupling is most likely from the contralateral limb attempting to compensate for the reduced RoM and increased variability of the ipsilateral limb. The complexity of the left limb phase portraits was not affected by bracing, implying that the smoothness of the contralateral phase portraits was not affected by the compensation or coupling. It is possible that the changes in variability were caused by artifacts of wearing the knee brace, in addition to the restricted RoM, which might not be present in actual injuries. Our finding that gait variability increased due to simulated injury agrees with literature based on statistical variability metrics (Madeleine et al., 2008; von Porat et al., 2006), yet contradicts literature based on nonlinear metrics dealing with actual injury and pain (Hamill et al., 1999; Lewek et al., 2003; Moraiti et al., 2007). Those finding decreased variability tended to adapt methodology used in dynamic stability analysis (e.g., maximum Lyapunov exponent) to quantify variability (Lewek et al., 2003; Moraiti et al., 2007); however, Dingwell et al. (2001) showed that traditional variability measures do not correlate with similar stability results. In addition to relating variability with stability, many past researchers base their methods on the loss of complexity hypothesis (Pool, 1989), which states that biological systems lose complexity with age or disease. As mentioned by Lipsitz (2002), it may not necessarily be correct to equate complexity and variability, as they describe different aspects of a signal or shape (complexity relates to signal fluctuation frequency and irregularity while variability relates to deviation from the mean value). Still, our method of simulated injury may not be producing similar results to actual knee injury, and our intuitive phase-portrait variability metric could possibly agree with the existing nonlinear/stability metrics if performed on the actual data collected from injured individuals. For the data studied, our variability results agree with qualitative inspection, thus we are confident in these inter-cycle variability metrics. Contradictions in complexity results have also been found (Goldberger et al., 2002; Munoz-Diosdado et al., 2003; Munoz-Diosdado et al.,
Table 1 Mean (and standard deviation) values of complexity and variability measures separated by segment, limb, and bracing condition. Check marks signify statistical difference between paired comparisons of bracing condition within each limb (α = 0.025). Body segment
Thigh
Shank
Foot
a
Measure (units)
Complexity (# harmonics) Drift (Cartesian dist.) Area (rad2/s) Complexity Drift Area Complexity Drift Area
Brace worn on right limb.
Average (SD) values
Paired piecewise comparisons
Left lega: no brace
Left lega: knee brace
Right leg: no brace
Right leg: knee brace
194 0.42 0.0027 153 0.64 0.0057 159 0.96 0.0101
191 0.65 0.0056 153 0.92 0.0109 162 1.15 0.0167
195 0.51 0.0042 150 0.66 0.0062 171 0.86 0.0094
210 0.65 0.0062 181 0.70 0.0073 217 1.12 0.0156
(19) (0.12) (0.0015) (16) (0.23) (0.0043) (19) (0.34) (0.0054)
(15) (0.22) (0.0032) (14) (0.39) (0.0076) (14) (0.31) (0.0102)
(22) (0.19) (0.0030) (18) (0.31) (0.0047) (18) (0.26) (0.0052)
(21) (0.14) (0.0026) (25) (0.18) (0.0031) (21) (0.28) (0.0059)
LNB vs. LKB
RNB vs. RKB
√ √
√ √ √ √
√ √ √ √
√ √ √
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2005) with results changing depending on which metrics researchers choose to use. Our frequency based approach is unlike past approaches, but in line with the complexity description provided by Lipsitz (2002), where more fluctuation (produced by increased Fourier harmonics) leads to higher complexity. Similar to the differences in variability, this complexity measure should be executed across datasets before comparison with other methods used to quantify complexity. There are certain limitations of this study. First, our simulated RoM reduction is more drastic than what might be experienced with actual injury. These metrics might not detect subtle changes in gait due to minor or nearly healed injuries. Second, injured subjects also develop compensation strategies over time to manage their restrictions and pain. While our test subjects had approximately 15 min to walk with the brace before testing began, these subjects may not have developed long-term compensation strategies similar to an injured subject. Gait variability may decrease if our subjects walked with the brace for a long period of time. We found, however, for the 20 gait cycles in this study, that variability in terms of centroid drift was stationary. This suggests the subjects had at least developed a short-term movement strategy. (To assess stationarity of a phase portrait, we quantified centroid drift for each consecutive pair of centroids, i.e., distance between every two centroids. A straight-line fit, of centroid pair drift vs. cycle pair number, expressed the slope of the drift throughout the trial. No consistently increasing or decreasing trends due to bracing were found for data from three random subjects, suggesting stationarity of the phase portraits during each trial.). Future studies should examine the techniques presented in this paper on an actual injury population. Sensitivity of these metrics to a smaller gait perturbation could then be determined. Sensitivity to other controlled perturbations would also be useful. It would also be fruitful to apply previous methods (Hamill et al., 1999; Lewek et al., 2003; Moraiti et al., 2007) to measure gait of individuals wearing braces. These studies would hopefully provide converging evidence to aid in determining the best methods for quantifying complexity and variability across different populations. Other elliptical shapes found in gait dynamics, especially kinetic data, could be examined by this method as well.
5. Conclusions We have successfully developed new techniques that quantify changes in gait phase-portrait complexity and variability in agreement with qualitative inspection of the plots. We previously applied Elliptical Fourier Analysis to quantify motion changes due to gait development and pathology (Polk et al., 2008; Rosengren et al., 2009). In this paper, we have shown that this technique can be modified to successfully quantify the impact of bracing one limb. The physical restriction imposed in this study created large size (area, mean centroid, etc.) differences between braced and non-braced phase portraits, thus the metrics used were modified to factor out size (complexity) and more strictly follow intuitive definitions of variability (drift and area). These new metrics provided quantitative results in agreement with qualitative visual interpretation of the phase portraits while intuitively separating complexity and variability results. This approach was highly effective at distinguishing changes in motion patterns when range of motion was reduced due to a simulated knee injury. It also provides a tool that may be useful to clinicians for quantifying deviation from normal gait in terms of the complexity and variability changes of the resulting motion shapes.
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