The purpose of this study was to test quantitatively the hypothesis that, as runners run along a more sharply curved track, greater torsional moments act on their tibiae. Six male participants were asked to run along a straight track and along counterclockwise curved tracks with turn radii of 15 m (gentle) and 5 m (sharp) at 3.5 m s−1. Data were collected using two high-speed cameras and force platforms. Each participant’s left (corresponding to the inside of the curves) foot and tibia were modeled as a system of coupled rigid bodies. For analysis, net axial moments acting on both ends of the tibia were calculated using free-body analysis. The torsional moment acting on the tibia was determined from the quasi-equilibrium balance of the tibial axial moments based on the assumption that the rate of change of the angular momentum about the tibial axis was negligible. The results showed that the torsional moments, which were in the direction of external rotational loading of the proximal tibiae, increased as the track curvature became sharper. Furthermore, the mean value of the maximum torsional moments, while running on a sharply curved track (28.5 Nm), was significantly higher than the values obtained while running on a straight track (11.0 Nm, p <.01) and on a gently curved track (12.2 Nm, p < .01). In conclusion, the present study has quantitatively confirmed that as runners run along a more sharply curved track, greater torsional moments act on their tibiae. The findings imply that athletes prone to tibial running injuries such as stress fractures should avoid repetitive running on sharply curved paths.
Key Words: load quantification, three-dimensional analysis, running injury
Introduction
In athletes, especially long-distance runners, stress fractures occur most frequently along the tibiae (e.g., Matheson, Clement, Mckenzie, et al., 1987). Repetitive overloading in a certain area of the bone can result in stress fractures, but the underlying mechanism for stress fractures is unclear. Therefore, a quantitative investigation of tibial loading during running may provide scientific information about both the risk and prevention of stress fractures in athletes. Lanyon, Hampson, Goodship, and Shah (1975) first measured human tibial strain during walking and running with strain gauges attached directly to the tibial shaft. They showed that during each cycle, the tibia was first deformed in a particular direction, released at least partially, and then deformed in another direction.
It is evident that in vivo measurements are essential for understanding the tibial loading pattern during human locomotion. However, conducting such measurements on multiple participants is difficult because of the invasive nature of the procedure. Scott and Winter (1990) estimated loads (compressive and shear forces and bending moment) acting on the tibiofibular complex during running using a two-dimensional musculoskeletal model of the lower extremities. Owing to its noninvasive nature, such a biomechanical estimation of tibial loading will be useful in both clinical practice and research. However, Scott and Winter did a two-dimensional analysis and therefore could not examine torsion. Since human long bones are most susceptible to torsional effects (Frankel & Nordin, 1980), a quantitative study of tibial torsional loading is relevant. Carter (1978) calculated torsional shear stresses on the tibia by an anisotropic analysis of the in vivo strain data from Lanyon et al. (1975). However, very little literature has been found concerning tibial torsional loading during human locomotion.
Not only long-distance runners but also most other athletes, including ballplayers, must include some kind of running exercise in their training. Furthermore, many athletes must run repeatedly on complex paths with varied curvatures as well as on straight paths. From this perspective, understanding how the sharpness of track curvatures affects tibial loading during running seems relevant. When runners run along sharply curved paths, it was speculated that their tibias were loaded in great torsion because runners are required to change direction while resisting a centrifugal force. However, such a speculation has not been quantitatively investigated. Therefore the purpose of the present study was to test quantitatively the hypothesis that, as runners run along a more sharply curved track, greater torsional moments act on their tibiae.
Methods
Six male physical education students (mean age 19.8 yrs ± 1.3; Height 1.73 m ± 0.05; Body mass 59.7 kg ± 7.9) took part in the experiment. Each one signed an informed consent form. Prior to the running trials, light-reflective spherical markers with a diameter of 20 mm were attached to the participants’ left limbs with adhesive tape. The markers were located at the medial tibial condyle, the fibular head, the medial and lateral malleoli, and the second and fifth metatarsal heads.
Two Kistler force platforms (Type 9287A, Kistler, Switzerland) were mounted, one in front of the other, in the middle of a 20-m indoor runway (Figure 1). Two high-speed cameras (HSV-500C3, NAC, Japan) were placed near the platforms, with one camera located slightly to the right and anterior while the other was located to the left and slightly anterior. Two pairs of photocells (SpeedtrapII, Brower, USA) were placed on each side of the set of platforms. The distance between photocells was 2 m.
The volume was calibrated with a calibration frame located on the force platforms. The standard error for the global reference frame was 1.8 mm in the X-direction (posterior-anterior), 6.0 mm in the Y-direction (right-left), and 2.2 mm in the Z-direction (inferior-superior) (Figure 2). Ground reaction force (GRF) data were collected as the participants ran across the force platforms. The outputs of the charge amplifier (Type 9865, Kistler) were A-D converted (Type 5606A, Kistler) and recorded by a personal computer (FM-V, Fujitsu, Japan). The motions of the markers were recorded with the two high-speed cameras.
The GRF and kinematic data were synchronized and sampled at 250 Hz. The kinematic data were digitized and then converted into 3-D spatial coordinates by the direct linear transformation method (Frame-DIAS, DHK, Japan). The raw data were filtered with a fourth-order, zero-lag Butterworth digital low-pass filter with a cutoff frequency at 8 Hz. This cutoff frequency was determined by the residual analysis (Winter, 1990). The anthropometric data were collected by individual measurements of the participants. The location of the center of mass (COM) and inertia properties of each body segment were determined based on the relationships determined by Chandler, Clauser, McConville, Reynolds, and Young (1975).
To investigate how the sharpness of track curvature affects tibial torsional loading, we had the participants run along a straight track and along counterclockwise curved tracks with mm radii of 15 m (gentle) and 5 m (sharp) at 3.5 m s-1 (Figure 1). They ran without footwear in order to exclude external factors as much as possible. Running speed was determined using the photocells. Although a deviation of ± 10% from the required speed was permitted, the runners were required to repeat the trial if they were not within this range. Each was required to footstrike on either platform with his left foot, which was on the side corresponding to the inside of the counterclockwise curved tracks. The participants performed practice runs to become accustomed to the required speed and footstrike without adjustment of stride. To identify whether a participant was artificially accelerating or decelerating on the platforms, the anteroposterior force of ground reaction was examined after each trial. Five successful running trials were recorded for each condition. When running, all participants made contact with the heel first.
Each participant’s left foot and tibia were modeled as a system of coupled rigid bodies. Based on the markers attached, a right-hand Cartesian coordinate system was embedded in each segment (Figure 2). For the foot, the x′-axis was parallel to the line from the midpoint of the medial and lateral malleoli to the second metatarsal head. The y′-axis was in the plane defined by the midpoint of the malleoli, the second metatarsal head, and the fifth metatarsal head in an orientation perpendicular to the x′-axis and pointing to the fifth metatarsal head. The z′-axis was the cross-product of these axes.
For the tibia, the Z′-axis was parallel to the line from the midpoint of the medial and lateral malleoli to the midpoint of the medial tibial condyle and fibular head. The Y′-axis was in the plane defined by the midpoint of the malleoli, the midpoint of the medial tibial condyle and fibular head, and the fibular head in an orientation perpendicular to the Z′-axis and pointing to the fibular head. The X′-axis was the cross-product of these axes. The origin of each system was set at the COM of the segment. To specify the 3-D orientation and motion of each segment relative to the global system, Euler angles (σ, θ, and ψ) were calculated. The sequence of Euler rotations of the body-embedded systems was of the Z-Y-Z type (Appendix A).
To determine the tibial torsional moment, free-body analysis was conducted. The rate of change of angular momentum (RCAM) with respect to the COM of the segments was first calculated using Euler equations of motion (Appendix B). To calculate net intersegmental moments acting on both ends of the tibia (Md and Mp) by inverse dynamics, the RCAM were projected into the global reference frame. The intersegmental moments Md and Mp were calculated using Equations 1a and 1b, respectively. These are given with respect to the global reference frame.
Md = Mfp + (Rfp − Rcmf)
× Ffp − (Rd −
Rcmf) × Fd − Mf, (1a)
Mp = −Md − (Rd − R[sup
cmt]) × Fd − (Rp −
Rcmt) × Fp + Mt, (1b)
where
Md and Mp = net intersegmental moments acting on the tibia at the distal and proximal ends of it, respectively;
Mf and Mt = rates of change of angular momentum with respect to the centers of mass of the foot and tibia, respectively;
Ffp = ground reaction force;
Mfp = free moment of ground reaction (Holden & Cavanagh, 1991);
Fd and Fp = net intersegmental forces acting on the tibia at the distal and proximal ends of it, respectively;
Rfp = position of the center of pressure;
Rcmf and Rcmt = position of the centers of mass of the foot and tibia, respectively;
Rd and Rp = positions of the distal and proximal ends of the tibia, respectively.
Next, net axial moments acting on both ends of the tibia (Md,sub,z, and Mp,sub z,) were calculated. These moments are the products of the direction cosine of the longitudinal axis (Z′ axis) of the tibia and the net intersegmental moments acting at both ends of it (Md and Mp, Equations 2a, 2b). Equations 2a and 2b are derived from the transformation matrix that transforms the global frame into the tibial frame (Appendix C).
Md, sub Z′ = cosφ sinθ Md, sub X
+ sinφ sinθ Md, sub Y
+ cos θ MP, sub Z
(2a)
Mp, sub Z′ = cosφ sinθ Mp, sub X
+ sinφ sinθ Mp, sub Y
-
cos θ Mp, sub Z
(2b)
The net tibial axial moments (Md, sub Z′, and Mp, sub Z′,) were determined as the torsional moments acting on the tibia from the quasi-equilibrium balance of them. This quasi-equilibrium balance of the moment was based on the assumption that the RCAM about the longitudinal axis of the tibia (Mcmt, sub Z′,) is negligible during the stance phase of running at 3.5 m s−1. For convenience, the tibial torsional moments are represented by the proximal ones (Mp, sub Z′) in the following section.
The values of the maximum torsional moments for all running trials were determined. The values were averaged for each runner and then averaged across all runners. Mean values were compared among the three running conditions. Repeated-measures one-way analysis of variance and Fisher-PLSD post hoc tests were used to test whether the maximum moment between running conditions was statistically significant. The level of significance was set at 0.05.
Results
Figure 3a shows that the rates of change of angular momentum about the tibial axis (Mcmt,sub Z′) were small enough to be neglected and the tibial axial moments (Md,sub Z′ and Mp,sub Z′) matched each other very well throughout the stance phase of running on a straight track. M[sup cmt,sub Z′, however, followed in a similar pattern of change in five trials, as shown in Figure 3b. These results were common for all running conditions investigated.
It should be noted that greater positive torsional moments acted on the tibiae in the latter portion of the stance phase of running along the sharply curved track compared to the straight and gently curved tracks, as shown in Figure 4. The positive moment reflected external rotational loading of the proximal tibia. Furthermore, the mean value ± one standard deviation for maximum torsional moment during running along the sharply curved track (28.5 ± 7.4 Nm) was significantly higher than the values for maximum torsional moments during running on the straight track (11.0 ± 9.6 Nm, p < .01) and the gently curved track (12.2 ± 10.1 Nm, p < .01, Figure 5). In contrast, as shown in Figure 5, the maximum torsional moment did not differ significantly between the straight track and the gently curved one, although slightly greater torsional moments acted on the tibiae during running on the gently curved track compared to running on the straight track (Figure 4).
The forward tilt angles were similar regardless of the type of running track used, as shown in Figure 6a. Figure 6b shows that the inward (i.e., to the centripetal direction) tilt angles of the shank became greater as the runners ran along a more sharply curved track.
Figure 7b shows that the medial force of ground reaction became greater as the runners ran along a more sharply curved track. Both the anteroposterior and vertical forces were similar regardless of the type of track used, as shown in Figures 7a and 7c. Analysis of the anteroposterior forces seen also showed that the runners were able to maintain a constant speed over the force platform (Figure 7a).
Discussion
The present study quantitatively tested the hypothesis that, as runners run along a more sharply curved track, greater torsional moments act on their tibiae. The track curvatures investigated were quite sharp compared to the curvatures seen in typical 400-m athletic tracks, whose mm radii are approximately 35 m. However, many recreational runners and other athletes such as ballplayers run repeatedly on complex paths containing various curvatures rather than on typical tracks. Thus, the significance of our work is supported.
Shortcomings in our experimental procedure left two questions unanswered. First, how do banks affect tibial torsional loading? Although many curved paths may also be banked, we did not investigate banked curves because of the technical difficulty involved in mounting force platforms on a banked track. Second, what is the amount of torsional loading when runners turn with the foot on the side corresponding to the outside of the curve? The focus in this study was on the tibia of the side corresponding to the inside of the curve because the tibia on this side is considered more susceptible to torsional overloading due to excessive pronation of the foot. These questions may perhaps be answered with additional research.
When the runners ran along a more sharply curved track, their shanks and the GRF tended to tilt more inward (Figures 6b and 7b). From the viewpoint of physics, when a runner runs along a curve at a constant speed, a centrifugal force that is inversely proportional to the turn radius of the curve acts on the runner’s center of gravity. To turn against a greater centrifugal force, the runner is required to incline the body inward, in the centripetal direction, and press on the ground more strongly in the centrifugal direction with his/her planted foot. The present results regarding shank kinematics and GRF reflect these turning strategies.
Considerable torsional moment reflecting external rotational loading of the proximal tibia appeared in the latter portion of the stance phase of running along a straight track (Figure 4). This loading pattern did not necessarily agree with the pattern of torsional shear stress on the tibia reported by Carter (1978). The discrepancy between Carter’s results and our own may have been caused by several factors. First, Carter’s predictions were based on strains measured at the specific site of the tibia (Lanyon et al., 1975), so the results could not be generalized to the whole bone. Second, Carter’s results mainly reflected individual characteristics because they were based on data from a single participant. Finally, differences in running speed (2.2 m s−1 in Carter’s study and 3.5 m s−1 in our study) could also have resulted in a discrepancy. Therefore the discrepancy does not necessarily refute the validity of the present results.
Much greater torsional moments acted on the tibiae in the latter portion of the stance phase of running along a sharply curved track compared to straight and gently curved tracks (Figure 4). Furthermore, the mean value for the maximum torsional moment during running along the sharply curved track was significantly higher than the values during running along the straight track (p < .01) and along the gently curved track (p < .01, Figure 5). These results suggest that sharp curves with mm radii of less than 5 m have a high risk of increasing torsional loading along the tibiae of runners. Although both the shank and GRF tended to tilt considerably inward during the stance phase of running along the sharply curved track, the inward tilt of the shank was larger than that of the GRF (Figure 8). Consequently, the relative angle between the shank and the GRF in the frontal plane became greater during running along the sharply curved track compared to the other tracks. Such a configuration may load the tibia with great torsion as a result of great shearing strains in the joint structures (i.e., ligaments and articular surfaces) attached to the tibia.
Our results for the maximum torsional moments were well below the static fracture load of the human tibiae, which is approximately 100 Nm as reported in an in vitro study (Yamada, 1973). Yet it should be noted that the mean value for the maximum torsional moment during running along the sharply curved track (28.5 Nm) reached over 28% of ultimate tibial loading. Forwood and Parker (1989) reported that less than 16.3% of the ultimate loading could cause observable microdamage in rat tibiae, and that crack probability strongly depended on the level of loading. Therefore, it is possible that the risk of running injuries associated with tibial torsion increases in athletes who run repeatedly on sharply curved paths.
No considerable difference was found for the pattern of torsional moment or its maximum value between the straight track and the gentle curved one (Figures 4 and 5), although both the shank kinematics and the GRF were considerably different. These results suggest that gentle curves with turn radii of more than 15 m have little effect on torsional loading along the tibiae of runners. Although both the shank and GRF tended to tilt more inwardly during the stance phase of running along the gently curved track, the configuration between them was similar to that seen in the straight track (Figure 8). It may be that runners unknowingly avoid excessive torsional loading along their tibiae by keeping the configuration between the shank and the GRF within an appropriate range when running along gently curved paths.
The tibial torsional moments were determined from the quasi-equilibrium balance of the net axial moments acting on both ends of the tibia. The rate of change of angular momentum about the longitudinal axis of the tibia (Mcmt,sub Z′) was small enough to be neglected and the tibial axial moments (Md,sub Z′, and Mp,sub Z′) matched each other very well throughout the stance phase of running (Figure 3a). This suggests that the quasi-equilibrium balance of the moments can be valid for the stance phase of running at 3.5 m s−1. A skin movement artifact might affect the calculation of the angular acceleration of tibial axial rotation. However, such an artifact seems to be negligible: Reinschmidt, Bogert, Nigg, Lundberg, and Murphy (1997) reported that tibial axial rotation could be determined with reasonable accuracy (error < 3 deg) with skin markers.
The filtering procedures performed might also attenuate the peaks of tibial angular acceleration. However, the rate of change of angular momentum was less than 3% of the net axial moments throughout the stance phase, even when the cutoff frequency of the low-pass filter was set at 20 Hz. Additionally, the estimated moment of inertia about the tibial axis was negligible (up to 0.021 kg m−2 for our participants). All these results may also support the use of the quasi-equilibrium balance of the tibial axial moments for most human locomotion activities.
We disregarded the fibular contribution to torsional loading distribution. To our knowledge, no literature concerning the torsional distribution between the shinbones has been published, although some researchers (e.g., Skraba & Greenwald, 1984) have demonstrated that the tibia plays a major role in axial loading. Therefore, disregarding the fibular contribution seems acceptable. Yet it should be noted that disregarding the fibula might result in some overestimation of tibial loading. Furthermore, tibial torsional stresses were not determined because of the uncertain cross-sectional geometries of the tibiae. If information on tibial geometry becomes available through MRI imaging or other sources, noninvasive determination of torsional stresses along the whole tibia may be possible.
In conclusion, the present study has quantitatively confirmed the hypothesis that, as runners run along more sharply curved paths, greater torsional moments act on their tibiae. Our findings imply that athletes prone to tibial running injuries such as stress fractures should avoid repetitive running on sharply curved paths.
Acknowledgments
We would like to thank Dr. Gordon E. Robertson, University of Ottawa, and T. Yoshihisa, Yokohama Sports Medical Center, for their assistance during the course of this work.
GRAPHS: Figures 3 — (a) Net tibial axial moments acting on the distal (thin line) and proximal (thick) ends of the tibia and the rate of change of angular momentum (broken) during the stance phase of running along a straight track; and (b) enlargement of the rate of change of angular momentum (including SD bars). Each curve is ensemble-averaged one for 5 trials by one runner.
GRAPH: Figure 4 — Tibial torsional moment acting on the tibia during the stance phase of running along the straight (thin line), gently curved (broken), and sharply curved (thick) tracks. Each curve represents ensemble-averaged one for 5 trials by one typical runner. Positive moment reflected external rotational loading of the proximal tibia.
GRAPH: Figure 5 — Statistical comparisons of the maximum tibial torsional moments (mean ± SD for 6 runners) among the three types of running tracks. ** Significant difference at p < .01 by the Fisher-PLSD post hoc test.
GRAPHS: Figures 6 — (a) Forward tilt angle of the shank in the sagittal plane, and (b) inward tilt angle in the frontal plane during the stance phase of running along the straight (thin), gently curved (broken), and sharply curved (thick) tracks. Each curve is ensemble-averaged one for 5 trials by one runner presented in Figure 3. The sagittal and frontal planes are defined as the Z-X and Y-Z planes of the global system, respectively (see Figure 2). Forward tilt angle is defined as the angle between the X axis of the global system and the Z′ axis of the tibia. Inward (to the centripetal direction) tilt angle is defined as the angle between the Z axis of the global system and the Z′ axis of the tibia.
GRAPH: Figures 7 — (a) Anterior-posterior, (b) mediolateral, and © vertical forces of ground reaction during the stance phase of running along the straight (thin), gently curved (broken), and sharply curved (thick) tracks. Each curve is ensemble-averaged one for 5 trials by one runner presented in Figures 3 and 4.
DIAGRAM: Figure 1 — Experimental setup of force platforms, cameras, and photocells.
DIAGRAM: Figure 2 — Definitions of the global (O-XYZ), foot (g-xyz), and tibial (G-X′Y′Z′) coordinate systems. G and g represent the centers of mass of the tibia and foot, respectively. The force platforms are shown by dotted squares.
DIAGRAM: Figure 8 — Configurations of the foot, shank, and ground reaction force (GRF) in the frontal plane when tibial torsional moment reached the maximum level during the stance phase of running along the three types of tracks. Each figure is based on a single trial of one typical runner.
References
Carter, D.R. (1978). Anisotropic analysis of strain rosette information from cortical bone. Journal of Biomechanics, 11, 199–202.
Chandler, R.F., Clauser, C.E., McConville, J.T., Reynolds, H.M., & Young, J.W. (1975). Investigation of the inertial properties of the human body (Report DOT HS-801430). Springfield, VA: National Technical Information Service.
Forwood, M.R., & Parker, A.W. (1989). Microdamage in response to repetitive torsional loading in the rat tibia. Calcified Tissue International, 45, 47–53.
Frankel, V.H., & Nordin, M. (1980). Basic biomechanics of the skeletal system. Philadelphia: Lea & Febiger.
Holden, J.P., & Cavanagh, P.R. (1991). The free moment of ground reaction in distance running and its changes with pronation. Journal of Biomechanics, 24, 887–897.
Lanyon, L.E., Hampson, W.G., Goodship, A.E., & Shah, J.S. (1975). Bone deformation recorded in vivo from strain gauges attached to the human tibial shaft. Acta Orthopaedica Scandinavica, 46, 256–268.
Matheson, G.O., Clement, D.B., Mckenzie, D.C., Taunton, J.E., Lloyd-Smith, D.R., & Macintyre, J.G. (1987). Stress fractures in athletes—A study of 320 cases. The American Journal of Sports Medicine, 15, 46–58.
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Skraba, J., & Greenwald, A.S. (1984). The role of the interosseous membrane on tibiofibular weightbearing. Foot & Ankle, 4, 301–304.
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Appendix A — Sequence of Euler rotations used in this study
Assume that the respective axes of the segment G-X′Y′Z′ and the global system O-XYZ coincide.
- First, the segment was rotated through the angle φ about the Z′ = Z′ axis.
- Second, the segment was rotated through the angle θ about the Y′ axis.
- Third, the segment was rotated through the angle ψ about the Z′ axis.
Appendix B — Calculation of the rate of change of angular momentum of the body-embedded systems
First, angular velocities of the segment were determined using Euler angles as follows:
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where Ω = {Ωx′, Ωy′, Ωz′} indicates the angular velocity of the segment.
Second, angular accelerations of the segments were determined as the fist-time derivatives of the angular velocities.
Finally, the rate of change of angular momentum was determined by Euler equations of motion as follows:
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where
Mcms = {Mcms,sub X′, Mcms,sub Y′, Mcms,sub Z′} represents the rate of change of angular momentum with respect to the center of mass of the segment.
I = {IX′, IY′, IZ′} represents the moment of inertia of the segment.
Appendix C — Transformation matrix relating the global frame and the body-embedded frame
Equations 2a and 2b were derived from the transformation matrix that transformed the global frame (O-XYZ) into the tibial frame (G-X′ Y′Z′) as follows:
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By Ryuji Kawamoto, Keio University; Yusuke Ishige, Yokohama Sports Medical Center; Koji Watarai, University of Tokyo and Senshi Fukashiro, University of Tokyo
Ryuji Kawamoto is with the Faculty of Policy Management, Keio University, 5322, Endo, Fujisawa, Kanagawa, Japan.
Yusuke Ishige is with the Yokohama Sports Medical Center, Yokohama, Kanagawa, Japan.
Koji Watarai is with the Sports Sciences Lab, Dept. of Life Sciences, University of Tokyo, Meguro, Tokyo, Japan.
Senshi Fukashiro is with the Sports Sciences Lab, Dept. of Life Sciences, University of Tokyo, Meguro, Tokyo, Japan.