- Email: [email protected]
A comparison and update of direct kinematic-kinetic models of leg stiffness in human running
Journal of Biomechanics 64 (2017) 253–257
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
A comparison and update of direct kinematic-kinetic models of leg stiffness in human running Bernard X.W. Liew ⇑, Susan Morris, Ashleigh Masters, Kevin Netto School of Physiotherapy and Exercise Sciences, Curtin University, GPO Box U1987, Perth, WA 6845, Australia
a r t i c l e
i n f o
Article history: Accepted 25 September 2017
Keywords: Running Stiffness Kinematics Kinetics
a b s t r a c t Direct kinematic-kinetic modelling currently represents the ‘‘Gold-standard” in leg stiffness quantification during three-dimensional (3D) motion capture experiments. However, the medial-lateral components of ground reaction force and leg length have been neglected in current leg stiffness formulations. It is unknown if accounting for all 3D would alter healthy biologic estimates of leg stiffness, compared to present direct modelling methods. This study compared running leg stiffness derived from a new method (multiplanar method) which includes all three Cartesian axes, against current methods which either only include the vertical axis (line method) or only the plane of progression (uniplanar method). Twenty healthy female runners performed shod overground running at 5.0 m/s. Threedimensional motion capture and synchronised in-ground force plates were used to track the change in length of the leg vector (hip joint centre to centre of pressure) and resultant projected ground reaction force. Leg stiffness was expressed as dimensionless units, as a percentage of an individual’s bodyweight divided by standing leg length (BW/LL). Leg stiffness using the line method was larger than the uniplanar method by 15.6%BW/LL (P < .001), and multiplanar method by 24.2%BW/LL (P < .001). Leg stiffness from the uniplanar method was larger than the multiplanar method by 8.5%BW/LL (6.5 kN/m) (P < .001). The inclusion of medial-lateral components significantly increased leg deformation magnitude, accounting for the reduction in leg stiffness estimate with the multiplanar method. Given that limb movements typically occur in 3D, the new multiplanar method provides the most complete accounting of all force and length components in leg stiffness calculation. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Leg stiffness is thought to be an important control parameter in locomotion (Seyfarth et al., 2002; Shen and Seipel, 2015a), and is defined by the ratio of peak ground reaction force (GRF) and the change in leg length in the stance phase (Coleman et al., 2012). Presently, there are many methods of calculating the constituent components of leg stiffness (i.e. force and length components), which may produce differences in estimates of healthy biologic leg stiffness by up to 80% (Coleman et al., 2012). Evidently, the choice of leg stiffness methods has implications for intervention design (Beck et al., 2017), and the development of control theories for locomotion (Seyfarth et al., 2002; Shen and Seipel, 2015a). The direct method of measuring leg stiffness during three dimensional (3D) motion capture represents the current ‘‘Goldstandard” (Coleman et al., 2012), as it minimizes assumptions made when modelling the force and length components of leg stiff⇑ Corresponding author. E-mail address: [email protected] (B.X.W. Liew). https://doi.org/10.1016/j.jbiomech.2017.09.028 0021-9290/Ó 2017 Elsevier Ltd. All rights reserved.
ness. Currently, only the magnitudes of the vertical components of leg length and GRF (Farley and Gonzalez, 1996), or the sagittal plane scalar magnitudes have been used (Coleman et al., 2012). It has been implicitly argued that only including the sagittal plane scalar magnitudes into stiffness calculation sufficiently provide the most valid estimate of leg stiffness in running (Coleman et al., 2012), although this has not been formally verified. The medio-lateral (ML) component of GRF can reach up to 12% of vertical GRF (Cavanagh and Lafortune, 1980), and the ML foot displacement can differ between laterality by up to 0.05 m in amputees running (Arellano et al., 2015). Given that human gait typically involve limb movements and GRF in 3D, accounting for the ML force and length components will provide the most complete method of leg stiffness calculation. However, it is unknown if a method which accounts for force and length components in all 3D would produce statistically and clinically relevant differences from currently employed direct stiffness methods (Coleman et al., 2012; Farley and Gonzalez, 1996). The primary aim of this study was to investigate how estimates of healthy biologic leg stiffness in human running differs between
254
B.X.W. Liew et al. / Journal of Biomechanics 64 (2017) 253–257
2. Methods
length was defined by the difference between length at initial contact and length at peak resultant projected GRF. The time of peak projected GRF was specific to each method. Leg stiffness was expressed as dimensionless units (%BW/LL) (dividing raw values to bodyweight over static standing leg length) (Liew et al., 2017). The group mean normalizing factor was 759.8 N/m. GRF was expressed as a %BW, whilst leg length was expressed as a %LL.
2.1. Design
2.5. Statistical analyses
This is a secondary analysis of running data conducted on 20 healthy female recreational runners (25.1 (6.0) years, 1.66 (0.07) m, 61.3 (8.9) kg, 14 rearfoot strike and 6 forefoot strike patterns). These participants were originally recruited for an experiment on rigid hip taping and running kinematics. The study was approved by Curtin University’s Human Research Ethics Committee (PT022/2014), and all participants provided written informed consent.
A linear mixed model was used to analyse the effect of the independent variable (‘‘Method”) on leg stiffness, peak resultant GRF, and leg length change. Post-hoc analysis using Tukey’s pairwise comparison was used. This was performed in R software (v 3.2.5) within RStudio (v0.99.902, RStudio, Inc.) (Hothorn et al., 2008 ; Pinheiro et al., 2016).
2.2. Running protocol
3.1. Leg stiffness
Participants performed shod overground running at a controlled speed of 5.0 m/s (±10%), across three in-ground force platforms (3 m in total distance). Participants were given a 20 m run up to achieve the required speed, and a 10 m tail off for deceleration. Marker trajectories were collected using an 18 camera motion capture system at 250 Hz (Vicon T-series, Oxford Metrics, UK), whilst synchronized GRF were collected at 2000 Hz (AMTI, Watertown, MA). A 20 N force platform threshold was used to define initial and terminal contact.
All three methods differed in the magnitude of derived leg stiffness (F2,254 = 69.13, P < .001) (Table 1, Fig. 1). Leg stiffness using the line method was larger than the uniplanar method by 15.6%BW/LL (P < .001), and the multiplanar method by 24.2%BW/LL (P < .001) (Table 1, Fig. 1). Leg stiffness from the uniplanar method was larger than the multiplanar method by 8.5%BW/LL (P < .001) (Table 1, Fig. 1).
2.3. Biomechanical model
Resultant projected GRF did not differ between all three methods (F2,254 = 1.967, P = .1421) (Table 1, Fig. 2b). All three methods differed in the magnitude of resultant leg length change (F2,254 = 19.319, P < .001) (Table 1, Fig. 2f). Leg length change using the line method was smaller than the uniplanar method by 0.029%LL (P = .031), and the multiplanar method by 0.043%LL (P = .045) (Table 1, Fig. 2f). Leg length change from the uniplanar method was smaller than the multiplanar method by 0.014%LL (P = .014) (Table 1, Fig. 2f).
three different direct leg stiffness modelling methods, when different number of dimensions were accounted for in the constituent force and length components. In this paper we termed the method using only the vertical axis component as the ‘‘line method”, the vertical and anterior-posterior (AP) axes as the ‘‘uniplanar method” and all three axes as the ‘‘multiplanar method”.
A seven segment biomechanical model based on a previous study was used (Liew et al., 2016). The geometric and inertial characteristics of the biomechanical model was defined using Visual 3D (C-motion, Germantown, MD) default routines (Dempster, 1955; Hanavan, 1964). Marker trajectories and GRF were filtered at 15 Hz (fourth ordered, zero-lag, Butterworth).
3. Results
3.2. Ground reaction force and leg length
2.4. Leg stiffness methods 4. Discussion First, the leg was represented by a 3D (coordinates X – mediolateral, Y – anteriorposterior, Z – vertical) vector from the right hip joint centre (HJC) to the centre of pressure (COP) of the right foot. For the line method, the leg vector was defined by the vertical height of the HJC to the COP (Z-axis). For the uniplanar method, leg vector was defined by the YZ sagittal plane by setting the mediolateral component to zero for all data frames. For the multiplanar method, leg vector was defined by all three axes. For each method, the resultant length of the leg vector was used as the denominator for leg stiffness calculation. For the line method, the vertical GRF magnitude was used to calculate leg stiffness. For the uniplanar method, a 2D GRF vector was created by setting the mediolateral component of the GRF to zero for all data frames. For the multiplanar method, the original 3D GRF vector was used. For both uniplanar and multiplanar methods, the respective GRF vector was projected onto the respectively dimensioned leg vector by taking the dot product of the GRF vector by the unit vector of the leg. No projection of the GRF is needed for the line method. For all methods, leg stiffness was calculated by taking the ratio between the peak magnitude of the resultant projected GRF, and the resultant change in length of the leg vector. Change in leg
The aim of this study was to investigate if accounting for all 3D within leg stiffness modelling could significantly alter healthy biologic estimates of leg stiffness, compared to current direct modelling methods. For a 60 kg adult running at 5.0 m/s, the new multiplanar method resulted in 6.5 kN/m smaller leg stiffness compared to the current ‘‘Gold-standard” uniplanar method. Given that a difference in leg stiffness by approximately 1 kN/m occurred after an exchaustive maximal run (Hayes and Caplan, 2014), a difference of 6.5 kN/m may have important clinical and scientific implications. Across this study and that of another (Liew et al., 2017), leg stiffness while running at 5 m/s varies between 34%BW/LL to 39% BW/LL (25–29 kN/m) using the ‘‘Gold-standard” uniplanar method. Surprisingly, the original paper which developed the current direct uniplanar method reported much lower leg stiffness values of 13.9 kN/m (Coleman et al., 2012). It may be that the mean velocity in Coleman et al. (2012) was much slower that of this study, although a range of velocities (2–6.5 m/s) was used. However, a separate study which had participants running at 3.3 m/s reported stiffness values using the uniplanar method of 35%BW/LL (Silder et al., 2015). Differences in stiffness values between studies may be
255
B.X.W. Liew et al. / Journal of Biomechanics 64 (2017) 253–257
Table 1 Mean, standard deviation (SD), standard error (SE), 95% confidence interval (CI) of leg stiffness (% BW/LL), resultant projected ground reaction force (% BW), resultant leg length change (% LL). Method
Observations
Mean
SD
SE
95% CI
Line Uniplanar Multiplanar
92 92 92
55.242 39.608 31.071
16.188 16.510 10.258
1.688 1.721 1.069
3.352 3.419 2.124
Line Uniplanar Multiplanar
92 92 92
2.657 2.653 2.678
2.657 2.653 2.678
2.657 2.653 2.678
2.657 2.653 2.678
Line Uniplanar Multiplanar
92 92 92
0.052 0.081 0.095
0.029 0.039 0.037
0.003 0.004 0.004
0.006 0.008 0.008
Kleg
GRF
Leg length
1% BW/LL = 759.80 kN/m 1% BW = 612.67 N 1% LL = 0.81 m
Projected GRF (N)
LINE UNIPLANAR MULTIPLANAR
Leg length (m) Fig. 1. Projected resultant force (N) versus leg vector length (m) curves in running.
due to differences in footstrike patterns (Sinclair et al., 2016), although this information was not presented by Coleman et al. (2012). The results of this study demonstrates that even in a healthy cohort, running in a straight line, accounting for all 3D in the force and length components, can significantly alter biologic leg stiffness estimates. The influence of different dimensions on leg stiffness calculation was due to the former’s influence only on leg length change. This was unsurprising given that peak GRF occurred close to mid-stance, where horizontal GRF components were close to zero (Fig. 2a). In contrast, the change in ML leg length between initial contact and peak GRF was 0.02 m (Fig. 2e). Given that the change in the vertical component of leg length was 0.03 m (Fig. 2c), the change in ML leg length reached 67% of the vertical length change. There is convergent evidence that the leg stiffness estimate using the new multiplanar method is closer to the true biologic leg function in running, compared to the uniplanar method. In a computer simulation of two dimensional, sagittal plane, human running at 5.0 m/s, dynamically stable running was achieved with a leg stiffness of 24 kN/m (Seyfarth et al., 2002). This is close to the stiffness value achieved by the new multiplanar method (23.5 kN/m). Given that running occurred only in 2D in the
simulation study (Seyfarth et al., 2002), stiffness using the uniplanar and multiplanar methods will converge. Across multiple human and mammalian running gait, dynamically stable and energetically optimal gait occurs with a dimensionless stiffness value between 7%BW/LL to 27%BW/LL (Shen and Seipel, 2015a, 2015b). The estimated leg stiffness value of 31%BW/LL provided by the multiplanar method is the closest to these stiffness values, amongst the methods investigated in this study. Experimentally, there is indirect evidence that healthy biologic leg stiffness in running has been over-estimated in the literature. This is evidenced by the metabolic improvements made by providing athletic amputees with a prosthesis that is less stiff than what is presently recommended by the prostheses’ manufactures (Beck et al., 2017). A reduction in overall leg stiffness (residual limb plus prosthetic stiffness) in an amputee running by 1 kN/m, reduced running cost of transport by 1.8% (Beck et al., 2017). The prescription of over-stiffed prosthesis could be a lack of accounting of the greater ML foot displacements inherent in amputees (Arellano et al., 2015). Alternatively, since mechanical properties of running specific prosthesis are manufactured to mimic healthy limb mechanical properties (Beck et al., 2016), this suggests that healthy leg stiffness may have been overestimated, as current methods do not account for all 3D.
256
B.X.W. Liew et al. / Journal of Biomechanics 64 (2017) 253–257
Fig. 2. (a) Mean (standard deviation [SD]) ground reaction force (GRF) vector; (b) Mean resultant projected GRF magnitude mean (SD); (c) Mean (SD) vertical component of virtual leg vector; (d) Mean (SD) anterior-posterior component of virtual leg vector; (e) Mean (SD) medial-lateral component of virtual leg vector; (f) Mean (SD) resultant length of leg vector.
Although this study only included healthy runners, and running in a straight line, it is possible that the relative merits of the multiplanar method over other direct methods, may be even more apparent in tasks and populations where ML GRF and leg length change are more pronounced than in this study (Arellano et al., 2015; Luo and Stefanyshyn, 2012). For example, ML GRF can reach 80% of vertical GRF in maximal curve sprinting (Luo and Stefanyshyn, 2012). A limitation in all direct stiffness modelling methods is the need for accurate HJC localisation, and its dynamic tracking in gait. Improvements in these areas of biomechanical modelling may further improve leg stiffness modelling. 5. Conclusion This study demonstrates that the ML componenets of GRF and leg vector should also be included into leg stiffness modelling. The new stiffness modelling method does not require significantly greater computational burden, compared to the present uniplanar method, but does produce significantly different estimates of leg stiffness. Direct kinematic-kinetic stiffness modelling remains the ‘‘Gold-standard” for quantifying leg stiffness in gait motion capture (Coleman et al., 2012), albeit when all 3D of the GRF and leg vector are included. Acknowledgements The authors would like to thank Mr. Paul Davey for his help in verifying the equations used in this study.
Conflicts of interest and Source of Funding No funds were received in support of this work. No benefits in any form have been or will be received from a commercial party related directly or indirectly to the subject of this manuscript. Mr. Bernard Liew is currently supported by an institutional doctoral scholarship. References Arellano, C.J., McDermott, W.J., Kram, R., Grabowski, A.M., 2015. Effect of running speed and leg prostheses on mediolateral foot placement and its variability. PLoS ONE 10, e0115637. Beck, O.N., Taboga, P., Grabowski, A.M., 2016. Characterizing the mechanical properties of running-specific prostheses. PLoS ONE 11, e0168298. Beck, O.N., Taboga, P., Grabowski, A.M., 2017. Reduced prosthetic stiffness lowers the metabolic cost of running for athletes with bilateral transtibial amputations. J. Appl. Physiol. 1985 (122), 976–984. Cavanagh, P.R., Lafortune, M.A., 1980. Ground reaction forces in distance running. J. Biomech. 13, 397–406. Coleman, D.R., Cannavan, D., Horne, S., Blazevich, A.J., 2012. Leg stiffness in human running: comparison of estimates derived from previously published models to direct kinematic-kinetic measures. J. Biomech. 45, 1987–1991. Dempster, W., 1955. Space Requirements of the Seated Operator: Geometrical, Kinematic, and Mechanical Aspects of the Body With Special Reference To The Limbs. Wright-Patterson Air Force Based, OH. Farley, C.T., Gonzalez, O., 1996. Leg stiffness and stride frequency in human running. J. Biomech. 29, 181–186. Hanavan, E., 1964. A Mathematical Model of the HUMAN BODY: BEHAVIOURAL SCIENCES LABoratory. Write-Patterson Air Force Base, OH. Hayes, P.R., Caplan, N., 2014. Leg stiffness decreases during a run to exhaustion at the speed at VO2max. Eur. J. Sport. Sci. 14, 556–562. Hothorn, T., Bretz, F., Westfall, P., 2008. Simultaneous inference in general parametric models. Biometr. J. 50, 346–363.
B.X.W. Liew et al. / Journal of Biomechanics 64 (2017) 253–257 Liew, B., Morris, S., Netto, K., 2017. Increase in leg stiffness reduces joint work during backpack carriage running at slow velocities. J. Appl. Biomech. https:// doi.org/10.1123/jab.2016-0226. Liew, B.X., Morris, S., Netto, K., 2016. The effects of load carriage on joint work at different running velocities. J. Biomech. 49, 3275–3280. Luo, G., Stefanyshyn, D., 2012. Limb force and non-sagittal plane joint moments during maximum-effort curve sprint running in humans. J. Exp. Biol. 215, 4314– 4321. Pinheiro, J., Bates, D., DebRoy, S., Sarkar, D., R Core Team., 2016. nlme: Linear and nonlinear mixed effects models. Available from:.
257
Seyfarth, A., Geyer, H., Gunther, M., Blickhan, R., 2002. A movement criterion for running. J. Biomech. 35, 649–655. Shen, Z., Seipel, J., 2015a. Animals prefer leg stiffness values that may reduce the energetic cost of locomotion. J. Theor. Biol. 364, 433–438. Shen, Z., Seipel, J., 2015b. The leg stiffnesses animals use may improve the stability of locomotion. J. Theor. Biol. 377, 66–74. Silder, A., Besier, T., Delp, S.L., 2015. Running with a load increases leg stiffness. J. Biomech. 48, 1003–1008. Sinclair, J., Atkins, S., Taylor, P.J., 2016. The effects of barefoot and shod running on limb and joint stiffness characteristics in recreational runners. J. Mot. Behav. 48, 79–85.
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
A comparison and update of direct kinematic-kinetic models of leg stiffness in human running Bernard X.W. Liew ⇑, Susan Morris, Ashleigh Masters, Kevin Netto School of Physiotherapy and Exercise Sciences, Curtin University, GPO Box U1987, Perth, WA 6845, Australia
a r t i c l e
i n f o
Article history: Accepted 25 September 2017
Keywords: Running Stiffness Kinematics Kinetics
a b s t r a c t Direct kinematic-kinetic modelling currently represents the ‘‘Gold-standard” in leg stiffness quantification during three-dimensional (3D) motion capture experiments. However, the medial-lateral components of ground reaction force and leg length have been neglected in current leg stiffness formulations. It is unknown if accounting for all 3D would alter healthy biologic estimates of leg stiffness, compared to present direct modelling methods. This study compared running leg stiffness derived from a new method (multiplanar method) which includes all three Cartesian axes, against current methods which either only include the vertical axis (line method) or only the plane of progression (uniplanar method). Twenty healthy female runners performed shod overground running at 5.0 m/s. Threedimensional motion capture and synchronised in-ground force plates were used to track the change in length of the leg vector (hip joint centre to centre of pressure) and resultant projected ground reaction force. Leg stiffness was expressed as dimensionless units, as a percentage of an individual’s bodyweight divided by standing leg length (BW/LL). Leg stiffness using the line method was larger than the uniplanar method by 15.6%BW/LL (P < .001), and multiplanar method by 24.2%BW/LL (P < .001). Leg stiffness from the uniplanar method was larger than the multiplanar method by 8.5%BW/LL (6.5 kN/m) (P < .001). The inclusion of medial-lateral components significantly increased leg deformation magnitude, accounting for the reduction in leg stiffness estimate with the multiplanar method. Given that limb movements typically occur in 3D, the new multiplanar method provides the most complete accounting of all force and length components in leg stiffness calculation. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Leg stiffness is thought to be an important control parameter in locomotion (Seyfarth et al., 2002; Shen and Seipel, 2015a), and is defined by the ratio of peak ground reaction force (GRF) and the change in leg length in the stance phase (Coleman et al., 2012). Presently, there are many methods of calculating the constituent components of leg stiffness (i.e. force and length components), which may produce differences in estimates of healthy biologic leg stiffness by up to 80% (Coleman et al., 2012). Evidently, the choice of leg stiffness methods has implications for intervention design (Beck et al., 2017), and the development of control theories for locomotion (Seyfarth et al., 2002; Shen and Seipel, 2015a). The direct method of measuring leg stiffness during three dimensional (3D) motion capture represents the current ‘‘Goldstandard” (Coleman et al., 2012), as it minimizes assumptions made when modelling the force and length components of leg stiff⇑ Corresponding author. E-mail address: [email protected] (B.X.W. Liew). https://doi.org/10.1016/j.jbiomech.2017.09.028 0021-9290/Ó 2017 Elsevier Ltd. All rights reserved.
ness. Currently, only the magnitudes of the vertical components of leg length and GRF (Farley and Gonzalez, 1996), or the sagittal plane scalar magnitudes have been used (Coleman et al., 2012). It has been implicitly argued that only including the sagittal plane scalar magnitudes into stiffness calculation sufficiently provide the most valid estimate of leg stiffness in running (Coleman et al., 2012), although this has not been formally verified. The medio-lateral (ML) component of GRF can reach up to 12% of vertical GRF (Cavanagh and Lafortune, 1980), and the ML foot displacement can differ between laterality by up to 0.05 m in amputees running (Arellano et al., 2015). Given that human gait typically involve limb movements and GRF in 3D, accounting for the ML force and length components will provide the most complete method of leg stiffness calculation. However, it is unknown if a method which accounts for force and length components in all 3D would produce statistically and clinically relevant differences from currently employed direct stiffness methods (Coleman et al., 2012; Farley and Gonzalez, 1996). The primary aim of this study was to investigate how estimates of healthy biologic leg stiffness in human running differs between
254
B.X.W. Liew et al. / Journal of Biomechanics 64 (2017) 253–257
2. Methods
length was defined by the difference between length at initial contact and length at peak resultant projected GRF. The time of peak projected GRF was specific to each method. Leg stiffness was expressed as dimensionless units (%BW/LL) (dividing raw values to bodyweight over static standing leg length) (Liew et al., 2017). The group mean normalizing factor was 759.8 N/m. GRF was expressed as a %BW, whilst leg length was expressed as a %LL.
2.1. Design
2.5. Statistical analyses
This is a secondary analysis of running data conducted on 20 healthy female recreational runners (25.1 (6.0) years, 1.66 (0.07) m, 61.3 (8.9) kg, 14 rearfoot strike and 6 forefoot strike patterns). These participants were originally recruited for an experiment on rigid hip taping and running kinematics. The study was approved by Curtin University’s Human Research Ethics Committee (PT022/2014), and all participants provided written informed consent.
A linear mixed model was used to analyse the effect of the independent variable (‘‘Method”) on leg stiffness, peak resultant GRF, and leg length change. Post-hoc analysis using Tukey’s pairwise comparison was used. This was performed in R software (v 3.2.5) within RStudio (v0.99.902, RStudio, Inc.) (Hothorn et al., 2008 ; Pinheiro et al., 2016).
2.2. Running protocol
3.1. Leg stiffness
Participants performed shod overground running at a controlled speed of 5.0 m/s (±10%), across three in-ground force platforms (3 m in total distance). Participants were given a 20 m run up to achieve the required speed, and a 10 m tail off for deceleration. Marker trajectories were collected using an 18 camera motion capture system at 250 Hz (Vicon T-series, Oxford Metrics, UK), whilst synchronized GRF were collected at 2000 Hz (AMTI, Watertown, MA). A 20 N force platform threshold was used to define initial and terminal contact.
All three methods differed in the magnitude of derived leg stiffness (F2,254 = 69.13, P < .001) (Table 1, Fig. 1). Leg stiffness using the line method was larger than the uniplanar method by 15.6%BW/LL (P < .001), and the multiplanar method by 24.2%BW/LL (P < .001) (Table 1, Fig. 1). Leg stiffness from the uniplanar method was larger than the multiplanar method by 8.5%BW/LL (P < .001) (Table 1, Fig. 1).
2.3. Biomechanical model
Resultant projected GRF did not differ between all three methods (F2,254 = 1.967, P = .1421) (Table 1, Fig. 2b). All three methods differed in the magnitude of resultant leg length change (F2,254 = 19.319, P < .001) (Table 1, Fig. 2f). Leg length change using the line method was smaller than the uniplanar method by 0.029%LL (P = .031), and the multiplanar method by 0.043%LL (P = .045) (Table 1, Fig. 2f). Leg length change from the uniplanar method was smaller than the multiplanar method by 0.014%LL (P = .014) (Table 1, Fig. 2f).
three different direct leg stiffness modelling methods, when different number of dimensions were accounted for in the constituent force and length components. In this paper we termed the method using only the vertical axis component as the ‘‘line method”, the vertical and anterior-posterior (AP) axes as the ‘‘uniplanar method” and all three axes as the ‘‘multiplanar method”.
A seven segment biomechanical model based on a previous study was used (Liew et al., 2016). The geometric and inertial characteristics of the biomechanical model was defined using Visual 3D (C-motion, Germantown, MD) default routines (Dempster, 1955; Hanavan, 1964). Marker trajectories and GRF were filtered at 15 Hz (fourth ordered, zero-lag, Butterworth).
3. Results
3.2. Ground reaction force and leg length
2.4. Leg stiffness methods 4. Discussion First, the leg was represented by a 3D (coordinates X – mediolateral, Y – anteriorposterior, Z – vertical) vector from the right hip joint centre (HJC) to the centre of pressure (COP) of the right foot. For the line method, the leg vector was defined by the vertical height of the HJC to the COP (Z-axis). For the uniplanar method, leg vector was defined by the YZ sagittal plane by setting the mediolateral component to zero for all data frames. For the multiplanar method, leg vector was defined by all three axes. For each method, the resultant length of the leg vector was used as the denominator for leg stiffness calculation. For the line method, the vertical GRF magnitude was used to calculate leg stiffness. For the uniplanar method, a 2D GRF vector was created by setting the mediolateral component of the GRF to zero for all data frames. For the multiplanar method, the original 3D GRF vector was used. For both uniplanar and multiplanar methods, the respective GRF vector was projected onto the respectively dimensioned leg vector by taking the dot product of the GRF vector by the unit vector of the leg. No projection of the GRF is needed for the line method. For all methods, leg stiffness was calculated by taking the ratio between the peak magnitude of the resultant projected GRF, and the resultant change in length of the leg vector. Change in leg
The aim of this study was to investigate if accounting for all 3D within leg stiffness modelling could significantly alter healthy biologic estimates of leg stiffness, compared to current direct modelling methods. For a 60 kg adult running at 5.0 m/s, the new multiplanar method resulted in 6.5 kN/m smaller leg stiffness compared to the current ‘‘Gold-standard” uniplanar method. Given that a difference in leg stiffness by approximately 1 kN/m occurred after an exchaustive maximal run (Hayes and Caplan, 2014), a difference of 6.5 kN/m may have important clinical and scientific implications. Across this study and that of another (Liew et al., 2017), leg stiffness while running at 5 m/s varies between 34%BW/LL to 39% BW/LL (25–29 kN/m) using the ‘‘Gold-standard” uniplanar method. Surprisingly, the original paper which developed the current direct uniplanar method reported much lower leg stiffness values of 13.9 kN/m (Coleman et al., 2012). It may be that the mean velocity in Coleman et al. (2012) was much slower that of this study, although a range of velocities (2–6.5 m/s) was used. However, a separate study which had participants running at 3.3 m/s reported stiffness values using the uniplanar method of 35%BW/LL (Silder et al., 2015). Differences in stiffness values between studies may be
255
B.X.W. Liew et al. / Journal of Biomechanics 64 (2017) 253–257
Table 1 Mean, standard deviation (SD), standard error (SE), 95% confidence interval (CI) of leg stiffness (% BW/LL), resultant projected ground reaction force (% BW), resultant leg length change (% LL). Method
Observations
Mean
SD
SE
95% CI
Line Uniplanar Multiplanar
92 92 92
55.242 39.608 31.071
16.188 16.510 10.258
1.688 1.721 1.069
3.352 3.419 2.124
Line Uniplanar Multiplanar
92 92 92
2.657 2.653 2.678
2.657 2.653 2.678
2.657 2.653 2.678
2.657 2.653 2.678
Line Uniplanar Multiplanar
92 92 92
0.052 0.081 0.095
0.029 0.039 0.037
0.003 0.004 0.004
0.006 0.008 0.008
Kleg
GRF
Leg length
1% BW/LL = 759.80 kN/m 1% BW = 612.67 N 1% LL = 0.81 m
Projected GRF (N)
LINE UNIPLANAR MULTIPLANAR
Leg length (m) Fig. 1. Projected resultant force (N) versus leg vector length (m) curves in running.
due to differences in footstrike patterns (Sinclair et al., 2016), although this information was not presented by Coleman et al. (2012). The results of this study demonstrates that even in a healthy cohort, running in a straight line, accounting for all 3D in the force and length components, can significantly alter biologic leg stiffness estimates. The influence of different dimensions on leg stiffness calculation was due to the former’s influence only on leg length change. This was unsurprising given that peak GRF occurred close to mid-stance, where horizontal GRF components were close to zero (Fig. 2a). In contrast, the change in ML leg length between initial contact and peak GRF was 0.02 m (Fig. 2e). Given that the change in the vertical component of leg length was 0.03 m (Fig. 2c), the change in ML leg length reached 67% of the vertical length change. There is convergent evidence that the leg stiffness estimate using the new multiplanar method is closer to the true biologic leg function in running, compared to the uniplanar method. In a computer simulation of two dimensional, sagittal plane, human running at 5.0 m/s, dynamically stable running was achieved with a leg stiffness of 24 kN/m (Seyfarth et al., 2002). This is close to the stiffness value achieved by the new multiplanar method (23.5 kN/m). Given that running occurred only in 2D in the
simulation study (Seyfarth et al., 2002), stiffness using the uniplanar and multiplanar methods will converge. Across multiple human and mammalian running gait, dynamically stable and energetically optimal gait occurs with a dimensionless stiffness value between 7%BW/LL to 27%BW/LL (Shen and Seipel, 2015a, 2015b). The estimated leg stiffness value of 31%BW/LL provided by the multiplanar method is the closest to these stiffness values, amongst the methods investigated in this study. Experimentally, there is indirect evidence that healthy biologic leg stiffness in running has been over-estimated in the literature. This is evidenced by the metabolic improvements made by providing athletic amputees with a prosthesis that is less stiff than what is presently recommended by the prostheses’ manufactures (Beck et al., 2017). A reduction in overall leg stiffness (residual limb plus prosthetic stiffness) in an amputee running by 1 kN/m, reduced running cost of transport by 1.8% (Beck et al., 2017). The prescription of over-stiffed prosthesis could be a lack of accounting of the greater ML foot displacements inherent in amputees (Arellano et al., 2015). Alternatively, since mechanical properties of running specific prosthesis are manufactured to mimic healthy limb mechanical properties (Beck et al., 2016), this suggests that healthy leg stiffness may have been overestimated, as current methods do not account for all 3D.
256
B.X.W. Liew et al. / Journal of Biomechanics 64 (2017) 253–257
Fig. 2. (a) Mean (standard deviation [SD]) ground reaction force (GRF) vector; (b) Mean resultant projected GRF magnitude mean (SD); (c) Mean (SD) vertical component of virtual leg vector; (d) Mean (SD) anterior-posterior component of virtual leg vector; (e) Mean (SD) medial-lateral component of virtual leg vector; (f) Mean (SD) resultant length of leg vector.
Although this study only included healthy runners, and running in a straight line, it is possible that the relative merits of the multiplanar method over other direct methods, may be even more apparent in tasks and populations where ML GRF and leg length change are more pronounced than in this study (Arellano et al., 2015; Luo and Stefanyshyn, 2012). For example, ML GRF can reach 80% of vertical GRF in maximal curve sprinting (Luo and Stefanyshyn, 2012). A limitation in all direct stiffness modelling methods is the need for accurate HJC localisation, and its dynamic tracking in gait. Improvements in these areas of biomechanical modelling may further improve leg stiffness modelling. 5. Conclusion This study demonstrates that the ML componenets of GRF and leg vector should also be included into leg stiffness modelling. The new stiffness modelling method does not require significantly greater computational burden, compared to the present uniplanar method, but does produce significantly different estimates of leg stiffness. Direct kinematic-kinetic stiffness modelling remains the ‘‘Gold-standard” for quantifying leg stiffness in gait motion capture (Coleman et al., 2012), albeit when all 3D of the GRF and leg vector are included. Acknowledgements The authors would like to thank Mr. Paul Davey for his help in verifying the equations used in this study.
Conflicts of interest and Source of Funding No funds were received in support of this work. No benefits in any form have been or will be received from a commercial party related directly or indirectly to the subject of this manuscript. Mr. Bernard Liew is currently supported by an institutional doctoral scholarship. References Arellano, C.J., McDermott, W.J., Kram, R., Grabowski, A.M., 2015. Effect of running speed and leg prostheses on mediolateral foot placement and its variability. PLoS ONE 10, e0115637. Beck, O.N., Taboga, P., Grabowski, A.M., 2016. Characterizing the mechanical properties of running-specific prostheses. PLoS ONE 11, e0168298. Beck, O.N., Taboga, P., Grabowski, A.M., 2017. Reduced prosthetic stiffness lowers the metabolic cost of running for athletes with bilateral transtibial amputations. J. Appl. Physiol. 1985 (122), 976–984. Cavanagh, P.R., Lafortune, M.A., 1980. Ground reaction forces in distance running. J. Biomech. 13, 397–406. Coleman, D.R., Cannavan, D., Horne, S., Blazevich, A.J., 2012. Leg stiffness in human running: comparison of estimates derived from previously published models to direct kinematic-kinetic measures. J. Biomech. 45, 1987–1991. Dempster, W., 1955. Space Requirements of the Seated Operator: Geometrical, Kinematic, and Mechanical Aspects of the Body With Special Reference To The Limbs. Wright-Patterson Air Force Based, OH. Farley, C.T., Gonzalez, O., 1996. Leg stiffness and stride frequency in human running. J. Biomech. 29, 181–186. Hanavan, E., 1964. A Mathematical Model of the HUMAN BODY: BEHAVIOURAL SCIENCES LABoratory. Write-Patterson Air Force Base, OH. Hayes, P.R., Caplan, N., 2014. Leg stiffness decreases during a run to exhaustion at the speed at VO2max. Eur. J. Sport. Sci. 14, 556–562. Hothorn, T., Bretz, F., Westfall, P., 2008. Simultaneous inference in general parametric models. Biometr. J. 50, 346–363.
B.X.W. Liew et al. / Journal of Biomechanics 64 (2017) 253–257 Liew, B., Morris, S., Netto, K., 2017. Increase in leg stiffness reduces joint work during backpack carriage running at slow velocities. J. Appl. Biomech. https:// doi.org/10.1123/jab.2016-0226. Liew, B.X., Morris, S., Netto, K., 2016. The effects of load carriage on joint work at different running velocities. J. Biomech. 49, 3275–3280. Luo, G., Stefanyshyn, D., 2012. Limb force and non-sagittal plane joint moments during maximum-effort curve sprint running in humans. J. Exp. Biol. 215, 4314– 4321. Pinheiro, J., Bates, D., DebRoy, S., Sarkar, D., R Core Team., 2016. nlme: Linear and nonlinear mixed effects models. Available from:
257
Seyfarth, A., Geyer, H., Gunther, M., Blickhan, R., 2002. A movement criterion for running. J. Biomech. 35, 649–655. Shen, Z., Seipel, J., 2015a. Animals prefer leg stiffness values that may reduce the energetic cost of locomotion. J. Theor. Biol. 364, 433–438. Shen, Z., Seipel, J., 2015b. The leg stiffnesses animals use may improve the stability of locomotion. J. Theor. Biol. 377, 66–74. Silder, A., Besier, T., Delp, S.L., 2015. Running with a load increases leg stiffness. J. Biomech. 48, 1003–1008. Sinclair, J., Atkins, S., Taylor, P.J., 2016. The effects of barefoot and shod running on limb and joint stiffness characteristics in recreational runners. J. Mot. Behav. 48, 79–85.