| | Froude number fractions to increase walking pattern dynamic similarities: Application to plantar pressure study in healthy subjectsReceived 20 July 2005; received in revised form 17 December 2005; accepted 22 December 2005. published online 23 January 2006. Abstract The purpose of this study was to determine if using similar walking velocities obtained from fractions of the Froude number (NFr) and leg length can lead to kinematic and kinetic similarities and lower variability. Fifteen male subjects walked on a treadmill at 0.83 (VS1) and 1.16 ms−1 (VS2) and then at two similar velocities (VSim27 and VSim37) determined from two fractions of the NFr (0.27 and 0.37) so that the average group velocity remained unchanged in both conditions ( ). NFr can theoretically be used to determine walking velocities proportional to leg lengths and to establish dynamic similarities between subjects. This study represents the first attempt at using this approach to examine plantar pressure. The ankle and knee joint angles were studied in the sagittal plane and the plantar pressure distribution was assessed with an in-shoe measurement device. The similarity ratios were computed from anthropometric parameters and plantar pressure peaks. Dynamically similar conditions caused a 25% reduction in leg joint angles variation and a 10% significant decrease in dimensionless pressure peak variability on average of five footprint locations. It also lead to heel and under-midfoot pressure peaks proportional to body mass and to an increase in the number of under-forefoot plantar pressure peaks proportional to body mass and/or leg length. The use of walking velocities derived from NFr allows kinematic and plantar pressure similarities between subjects to be observed and leads to a lower inter-subject variability. In-shoe pressure measurements have proven to be valuable for the understanding of lower extremity function. Set walking velocities used for clinical assessment mask the effects of body size and individual gait mechanics. The anthropometric scaling of walking velocities (fraction of NFr) should improve identification of unique walking strategies and pathological foot functions. 1. Introduction  Previous studies [1], [2] reported inter-subject dynamic similarities during the transition from walking to running. The pendulum model was applied to set similar velocities between subjects (Fig. 1). The mass of the body at the centre of gravity (Cg) oscillates at the end of a rigid segment length L representing the leg [3], [4]. At the time of a step, Cg of the subject describes an arc of a circle of radius L. At the linear velocity (V), Cg undergoes an acceleration (ä) equal to V2/L. When the leg is vertical, ä is directed upwards and perpendicular to the ground. This acceleration cannot be higher in intensity than gravitational acceleration (g) because the model would take-off. The take-off of the subject illustrates the disappearance of the double support phase and the transition from walking to running. The ratio (V2/gL) enabled the authors to estimate a theoretical Froude number  equal to l [3], [5] at the transition between walking and running. Using Thorstensson and Roberthson's [6] experimental data, Alexander [2] and Donelan and Kram [7] estimated that this transition occurred at a NFr of approximately 0.6. Wagenaar and Beek [8] and Zatsiorsky et al. [9] went even further by saying that fractions of NFr below 1 can be used to determine similar walking velocities between different subjects. A recent review by Vaughan and O’Malley [10] presented studies which used the Froude number in naval architecture, for locomotion analysis and advancements in robotic science. From Alexander [5] to Zijlstra et al. [11], many studies have used the Froude number to differentiate between the effects of walking velocity and anthropometry. However, many of these studies considered the Froude number (NFr) as the dimensionless walking speed and focussed on NFr as a means to normalise (set or spontaneously adopted) walking velocities. The use of the Froude number to determine similar dynamic conditions of walking has not been assessed. Similar dynamic conditions suppose geometric similarities between different subjects. The first idea is to consider that two subjects are anthropometrically proportional. Two subjects (S1 and S2) being proportional, the ratios computed from the leg length (l2/l1) and body mass (m2/m1) are defined as k and k3, respectively [9]. In a wider sense, these ratios defined the length (L) and mass (M) basic dimension ratios. Under similar dynamic conditions, the determining of similar velocities from NFr enables us to establish a time ratio (λ). Velocity is distance divided by time. The ratio of velocities is therefore equal to kλ−1. To express the ratio of similar velocities (VSim1 and VSim2) of the same two subjects, the constant NFr and g (gravitational acceleration) must be simplified. The velocity ratio is therefore equal to the square root of leg length ratio (Eq. (1)): Determining similar velocities between subjects means determining a time scale which is dependent on the distance scale. As the ratio of velocities (kλ−1) is equal to k0.5, consequently, the time ratio (λ) is equal to k0.5. Under similar conditions, the ratios of the basic dimensions, length (L), mass (M) and time (T) are k, k3 and k0.5, respectively. All the kinematic and kinetic ratios can be determined as a combination of the ratios of the basic dimensions. For example, the product of similarity ratios corresponding to each of the physical dimensions involved in pressure (ML−1T−2) expresses the pressure similarity ratio (Eq. (2)): Therefore, ratio “k” can be computed from the ratio of the pressure peaks (PP1 and PP2) of two different subjects (S1 and S2), i.e. k=PP2/PP1. If the dynamic conditions associated with walking velocity are similar, then the kinematic and kinetic parameters should be proportional from one subject to another. An exception exists, as the angles cannot be directly expressed from basic dimensions and the trigonometric functions are necessary to establish the link between lengths and angles. However, the effect of dynamic similarities on angles can be developed thus. According to the inverted pendulum model, linear velocity (VSim) of body mass at the centre of gravity swinging at the extremity of a leg of length (l) can be expressed in relation to the angular velocity (ω) (Eq. (3)). Angular velocity (ω) is the ratio of an angle (θ) and time (Δt): The ratio of similar velocities, VSim1 and VSim2, of the two subjects, S1 and S2, is equal to k0.5 and can be expressed in terms of leg lengths (l2 and l1), angles (θ2 and θ1) and times (Δt1 and Δt2) (Eq. (4)): Since the angle ratio (α) is equal to 1 if the angles θ2 and θ1 are identical (Eq. (4)), it follows that kinematic similarities should yield identical joint angles between one subject and another at “Similar” velocities. Being normalised with respect to L, M, T dimensions, the peak of pressure (corresponding to a force (kgms−2) distributed over a surface (m2)) can be expressed in a dimensionless form as: Dim(PP)=PPSL2/mg (PP, peak of pressure (kPa); SL, step length (m); m, body mass (kg) and g, acceleration due to gravity (ms−2)). The dimension of Dim(PP) is 1  . According to Stansfield et al. [12], the step length rather than leg length is used to introduce a ‘velocity-dependent’ parameter in the dimensionless expression of a pressure peak. Under similar dynamic conditions, the Dim(PP) of different subjects is theoretically unique. Under experimental conditions, a decrease in the inter-subject variability of Dim(PP) is expected as an effect of the determination of similar dynamic conditions using the Froude number fractions. Determining similar velocities should help to pinpoint similarities studied from the joint angles, dimensionless peaks of pressure and the similarity ratios of anthropometric parameters and pressure peaks. The purpose of the present study is to determine whether “Similar” velocities determined from fractions of the Froude number, lead to greater inter-subject walking pattern similarity. 2. Materials and methods 2.2. Determining “Similar” velocities To determine velocity, we devised two conditions that we called “Set” and “Similar”. In the former, two velocities were imposed on the subjects, the first of which was set arbitrarily at 0.83ms−1 (VS1) and then increased by 40% to define VS2 (1.16ms−1). From VS1 and VS2, two mean values of the Froude number were computed (Eqs. (5), (6)): In these equations, li represents the individual subject leg length, g the gravitational acceleration and n the number of subjects. The “Similar” velocity conditions (VSim27 and VSim37) for each subject were calculated using Eqs. (7), (8): Through this process, the means of the individual similar velocity conditions are kept equal to the set velocity conditions ( ). 2.3. Protocol The subjects performed the walking trials at “Set” (VS1 and VS2) and “Similar” (VSim27 and VSim37) velocities on a “Marquette Electronique MTC-1” treadmill. First, the subjects spent 20min familiarizing themselves with the different walking velocities. During the tests, the recordings of the video sequences and plantar pressure measurements of eight randomised consecutive steps were taken after 2min of walking, thus allowing for gait pattern stabilization [13]. The recordings were started using a remote control to synchronize both measurements. 2.5. Variables 2.5.1. Kinematic similarities The step time was defined by two consecutive left heel strikes. The data time was then reported for the shorter step time recorded at 50fps. This being 820ms, it induced a 41-point data normalisation. The knee and ankle patterns of eight consecutive steps were thus expressed as a function of step time percentage. The assumption underlying the “Similar” velocity approach implies a reduction of the difference among a group of subjects. Therefore, the inter-subject coefficient of variability of joint angles was chosen to test this assumption at both “Set” and “Similar” velocities. The joint angle variability was studied at each of the 41 percentages of the normalised step time. 2.5.2. Kinetic similarities The dimensionless peaks of pressure (Dim(PP)) were computed from the pressure peaks recorded under the eight footprint locations. The Dim(PP) standard deviations were chosen to test both “Set” and “Similar” velocities effects on the variability. The similarity ratio (k) was computed from each measured parameter: leg length (l), body mass (BM), and the pressure peaks (PP) of eight footprint areas. The 15 subjects were classified by order of increasing leg length. The similarity ratios were computed for each subject pair combination with reversed ratios avoided. Hence, the number of similarity ratios rose to 105  . Table 1 summarizes the unities, the dimensions, the similarity ratios and the “k” ratios expressed from the parameters assessed. 2.5.3. Statistical analysis The Fisher–Snedecor F-test enabled analyses of ankle and knee angle ratios of variance observed in the “Set” and “Similar” conditions. It was repeated at each of the 41 percentages describing the normalised step time. The same test enabled analyses of Dim(PP) variability observed in both experimental conditions. To verify the existence of dynamical similarities, we had to verify that the similarity ratios computed from leg length, body masses and plantar pressure peaks were equal. We considered that this equality was verified when the data variances and means did not significantly differ and that the anthropometric and kinetic similarity ratios were significantly correlated. The statistical analysis was conducted thus: (i) a χ2-test enabled verification of the distribution normality of the different data; (ii) the “parameter” factor enabled us to differentiate the similarity ratios computed from leg lengths, body masses and plantar pressure peaks. A repeated-measure factor analysis of variance (ANOVA) detected differences associated with the factor “parameter” at each velocity. The Scheffe test for paired variables more precisely identified the significant differences observed between each of the similarity ratios and the ratios of leg lengths and body masses; (iii) a Fisher's “z” coefficient of correlation between the different ratios themselves refined the analysis. Our conclusion derived from this three-step analysis was that dynamic similarities exist when the kinetic parameters were proportional to leg lengths and/or body masses. The significance threshold was set at P<0.05 for all the analyses. 2.6. Results 2.6.1. Kinematic variables Fig. 2, Fig. 3 present angular change in ankle and knee in terms of step percentage recorded at VS1 versus VSim27 and VS2 versus VSim37, respectively. At “Similar” velocities (VSim27 and VSim37), coefficients of variation of ankle and knee joint angles were significantly lower by 15–35% than those observed at the “Set” velocities (VS1 and VS2) (Fig. 2, Fig. 3). 2.6.2. Kinetic variables Fig. 4 presents the dimensionless pressure peaks (Dim(PP)) observed under the eight footprint areas at VS1 versus VSim27 and VS2 versus VSim37. The Dim(PP) standard deviations significantly decreased by 11.16% on average for five locations (VS1 versus VSim27) and by 10.12% on average for six locations (VS2 versus VSim37). 2.6.4. Plantar pressure distribution dynamic similarities Fig. 5 presents the “k” ratios expressed from pressure peaks of the eight footprint areas. Only the ratios that can be considered as equal to leg length and body mass ratios are retained. 2.6.5. Similarities observed at VS1 versus VSim27 (Fig. 5A) At a “Similar” velocity (VSim27) similarities were found under the medial and lateral heel and under midfoot. Pressure peaks were proportional to body mass. Pressure peaks recorded under the fourth and fifth metatarsal heads were proportional to leg length at VS1 and to body mass at VSim27. Under the second and third metatarsal heads, pressure peaks were proportional to mass and to leg length with no difference being observed between VS1 and VSim27. At VSim27, the pressure peak under the first metatarsal head was proportional only to leg length whereas it was proportional to mass and leg length at VS1. Pressure peaks recorded under the medial midfoot and the Hallux were not proportional to mass or to leg length. 3. Discussion Clinicians studying pathological walking are often confronted with the problem of high inter-subject variability when comparing a patient's walking pattern with a normal pattern. Using ankle and knee joint angles to detect spasticity-related differences in motor behaviour, Kelly et al. [19] were limited by the database's large standard deviation affecting the differentiation between spastic disorders of different origin. Likewise, McPoil and Cornwall [20] abandoned the use of the centre of pressure path as a means of gauging the efficacy of foot orthoses because of its high variability (ranging from 43.4% to 48.6%). As extensive research has shown, natural velocity is affected by numerous intrinsic and extrinsic factors [21], [22]. The use of natural velocity in studies may contribute to the variability of biomechanical parameters reported in the literature. The purpose and the originality of the present study were to test a method of decreasing the walking pattern inter-subject variability and verifying its influence on plantar pressure similarities and inter-subject variability. The study compared walking pattern similarities while subjects moved at the same velocity (Set conditions VS1 and VS2) and then at “Similar” velocities (VSim27 and VSim37). The latter was measured from each subject's leg length and from two Froude number fractions. In theory, the inter-segment angles of anthropometrically proportional subjects moving at “Similar” velocities should be identical. In practice, angle variance observed at “Similar” velocities (VSim27 and VSim37; Fig. 2, Fig. 3) is, on average, 25% lower than that observed in VS1 and VS2 “Set” conditions. It would seem that when Froude number fractions are used to determine velocities, inter-segment angles of subjects are closer and, therefore, kinematic similarities are better. These results can be compared with those Diedrich and Warren [23] who identified significant reductions in inter-segment angle variability during transition from walking to running at an NFr of 0.6. Our study on “Similar” walking velocities determined from Froude number fractions found similarities at velocities below this transition. As the kinematic findings demonstrated inter-subject similarities, plantar pressure distribution repercussions were expected. They would be contingent on anthropometric characteristics of the subjects and similar dynamic conditions linked to the way VSim27 and VSim37 velocities were determined. Ratios computed from leg length and body mass were correlated. However, the 0.5 correlation found between leg length and mass ratios highlights the limitations of the initial assumption that our subjects were anthropometrically proportional. The ratios computed from pressure peaks varying between 0.65 and 1.74, significant correlation coefficients from 0.21 to 0.49 (P<0.05) were considered to compensate for a low range of ratio values. These correlations between leg length (or mass) ratios and peak pressure ratios enabled us to check that anthropometric and kinetic ratios were equal and to put forward the idea of the existence of pressure peaks proportional to body masses and/or to leg lengths. This assumption is supported by the decrease in dimensionless peak pressure variability under the same footprint locations. However, the correlation between the anthropometric ratios suggests random relations between peak pressure and leg length or body mass. 3.1. Dynamic similarities under ‘heel’ No pressure peak was proportional to body mass or to leg length under lateral or medial heel at VS1 (Fig. 5), confirming Shorten et al. [24] who found no significant correlation between heel pressure peak and subject mass during a study where the same velocity was imposed on several subjects. Conversely, determining “Similar” velocities generates under-heel pressure peaks proportional to mass (VSim27; Fig. 5). These kinetic observations confirm kinematic findings. During the loading phase (0–30% of step time) coefficients of variation of angles recorded at ankle and knee were found to have decreased by 15% on average at VSim27 and VSim37 compared with “Set” conditions. The Dim(PP) variability decreases under these locations and the standard deviation difference reaches a significant −8.7% value under the lateral heel at VSim37 compared to VS2. Pressure peaks were mostly proportional to mass. According to Morag and Cavanagh [25], the main factors for determining under-heel pressure peaks were the mechanical characteristics of the heel pad soft tissue, age, the longitudinal structure of the plantar arch and the kinematics of the leg during heel strike. In our study, only the characteristics of the heel pad were not checked. Our subjects were all of similar age, subjects with flat or high arch feet were not selected, leg kinematics in similar conditions showed inter-subject similarities. The peak pressure similarities observed under the heel at VSim27 and VSim37 appeared to be due to the determination of similar walking velocities. 3.2. Dynamic similarities under ‘midfoot’ Results under midfoot were less straightforward. Where coefficients of variation of ankle and knee angles decreased by 35%, on average, at “Similar” velocities during the single support phase (30% and 60% of step time), the number of similarity ratios under lateral and medial midfoot areas increased, though only at the slower velocity (VSim27). At VSim37, the number (of similarity ratios) was identical to that observed at VS2. Peak pressures were mainly proportional to body mass in both cases (VS2 and VSim37). However, the standard deviation of Dim(PP) observed in similar conditions (VSim27 and VSim37) significantly decreased by 5.3% under the lateral midfoot and by 12% under the medial midfoot compared to “Set” conditions. The study of angle and pressure peaks variations showed good inter-subject similarities and confirmed that determining “Similar” velocities improved midfoot pressure similarities (Fig. 4, Fig. 5). 3.3. Dynamic similarities under ‘forefoot’ During the push-off phase, determining “Similar” velocities brought about a 30% decrease in inter-leg angle variation. However, the method failed to increase the number of peak pressure similarity ratios under the forefoot. Pressure peaks under the metatarsal heads were closely linked to movements of the ankle and foot joints and to EMG activity in the gastrocnemius. Under-Hallux pressure peaks were associated with the structure and mobility of the first metatarsal joint [24]. These different observations indicate the complexity of the factors involved in determining pressure peak under this area. With the occurrence of a pressure peak proportional to body mass under M4-5 at VSim37, it becomes apparent that determining “Similar” velocities does indeed improve forefoot similarities but only at the higher velocity. The expected effect on Dim(PP) variability is stronger and similar conditions lead to a decrease in the standard deviation of 10.2% under the fourth to fifth metatarsal heads between VS1 and VSim27 and of 14.7% under the Hallux between VS2 and VSim37. Determining walking velocity using Froude number fractions does generate similar walking patterns that lead to lower kinetic and kinematic variations and greater similarities between different subjects. Studying peak pressure under eight footprint areas has shown that determining “Similar” velocities greatly improves under-heel similarities though somewhat less so under the forefoot. The complexity and mobility of the bone structures under the forefoot could explain the differences in plantar pressure distribution between subjects during the push-off phase. However, the variability of the dimensionless pressure peaks is significantly lower in “Similar” conditions. This observation concerns all the footprint areas and confirms that the inter-subject variability is partly linked to poor dynamic similarities induced by the “Set” conditions. 4. Conclusion This study demonstrated that the determination of walking speeds using Froude number fractions below one is a successful method to establish dynamic walking similarities between different subjects. It could be used, for example, when a low inter-subject variability is expected to define a data base of biomechanical parameters. Several studies demonstrate that the parameter variability induced by spontaneously chosen walking speed is a limit to the detection of patient characteristics [26], [27], [28]. Van der Linden et al. [29] or Wagenaar and Beek [8] compared the patients to healthy results at the same dimensionless walking speed to differentiate between effects caused by speed and underlying pathology. This study suggests a dimensionless database constructed for a healthy population walking at different Froude number fractions that should enable a decrease in the biomechanical parameter variability and allow easier identification of patient characteristics. The dimensionless results of a patient walking at self-selected walking velocity could thus be compared to those of the database obtained from healthy subjects walking at the same dimensionless walking speed, i.e. at the same Froude number fraction. This database should be constructed from healthy subjects selected on the basis of their anthropometric similarities to avoid the difficulty in observing proportionality to length and mass. Further studies should be conducted to explore the accuracy of a dimensionless database and to test its usefulness in differentiating ‘abnormalities’ due to anthropometric variation, such as obesity. Acknowledgements Financial support was given by the Conseil Regional Nord-Pas de Calais, the Delegation Regionale à Recherche et à la Technologie du CHRU and the Institut Regional de Recherche sur le Handicap of Lille. The authors would like to thank M. Holvoet and Prof. Pelayo for helpful comments about video analysis and drafts of the manuscripts, respectively. References [1]. [1]Novacheck TF. The biomechanics of running. Gait Posture. 1998;7:77–95. Abstract | Full Text |
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a Laboratoire d’Etudes de la Motricité Humaine (EA 3608), Faculté des Sciences du Sport et de l’Education Physique, 59790 Ronchin, France b Nike Sport Research Laboratory, Beaverton, OR, USA  Corresponding author at: Laboratoire d’Etudes de la Motricité Humaine (EA 3608), Faculté des Sciences du Sport et de l’Education Physique, 9, rue de l’Université, 59790 Ronchin, France. Tel.: +33 3 20 88 73 76; fax: +33 3 20 88 73 63.
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