Foot-ground impact is mechanically challenging for all animals, but how do large animals mitigate increased mass during foot impact? We hypothesized that impact force amplitude scales according to isometry in animals of increasing size through allometric scaling of related impact parameters. To test this, we measured limb kinetics and kinematics in 11 species of hoofed mammals ranging from 18–3157 kg body mass. We found impact force amplitude to be maintained proportional to size in hoofed mammals, but that other features of foot impact exhibit differential scaling patterns depending on the limb; forelimb parameters typically exhibit higher intercepts with lower scaling exponents than hind limb parameters. Our explorations of the size-related consequences of foot impact advance understanding of how body size influences limb morphology and function, foot design and locomotor behaviour.
Citation: Warner SE, Pickering P, Panagiotopoulou O, Pfau T, Ren L, Hutchinson JR (2013) Size-Related Changes in Foot Impact Mechanics in Hoofed Mammals. PLoS ONE 8(1): e54784. https://doi.org/10.1371/journal.pone.0054784
Editor: Carol V. Ward, University of Missouri, United States of America
Received: March 8, 2012; Accepted: December 17, 2012; Published: January 30, 2013
Copyright: © 2013 Warner et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: Funding was provided by the Biotechnology and Biological Sciences Research Council (http://www.bbsrc.ac.uk), grants BB/F01169/1 and BB/H002782/1 to JRH. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Mitigating the mechanical consequences of foot-ground impact is critically important to musculoskeletal tissue health. The repeated, cyclical applications of transient forces have been implicated in fatigue accumulation leading to tissue failure –; it is the temporal nature of impact (i.e. rate and frequency of force application) that is thought to make loading during this phase particularly destructive , . As terrestrial cursors , hoofed mammals may be particularly vulnerable to fatigue damage because their relatively stiff, erect limbs coupled with presumably limited foot compliance (due to their small, rigid hooves) have to withstand long daily travel distances and rapid speeds. Although no real data exist to demonstrate the importance of fatigue damage in non-domesticated species, we wondered how large hoofed mammals are able to cope with highly repetitive impact loads particularly as ungulates span such an impressive body size range.
The scaling of geometric volumes suggest that impact loading poses a greater challenge to larger, heavier ungulates because forces (inertial and gravitational) are proportional to mass. Previous work however, has shown that peak bone stresses are maintained with size, through changes in limb posture and duty factor, by reducing locomotor performance and in very large animals, by increasing bone robusticity , . Furthermore, the frequency at which these forces (and thus stresses) are applied is reduced , . We therefore consider impact force amplitude alongside other features of foot impact in order to explore how increased body mass is mitigated during impact loading. We use the geometric similarity model as the basis of our hypotheses because many aspects of the model are supported over a wide range of species –. With insufficient evidence to indicate that larger ungulates are more susceptible to mechanical injury, we speculate that allometric relationships may exist to ensure that impact mechanics remain within tolerable limits.
At foot impact, the magnitude of force experienced by the limb is determined by the mass and acceleration of the limb. Whereas segmental mass is pre-determined by morphology (increasing isometrically in geometrically similar animals), effective foot mass (Meff); i.e. the amount of limb mass than collides with the ground prior to the limb being loaded with body mass (see Methods section ); may vary with neuromuscular control of limb dynamics. Although geometric similarity suggests that muscle force is proportional to physiological cross sectional area (expected to scale ∼Mb0.67), changes in limb posture ensure that peak muscle stresses remain constant with increasing body mass . Previous work by More et al.  suggests that despite having similar muscle force producing capacity, large and small animals may differ in their ability to respond to external stimuli, particularly to rapid events like foot impact. We expect Meff to remain isometric (scaling ∼Mb 0.84), with forelimb Meff being greater than hind limb Meff . (For more information on how this value (0.84) was derived, see equation 1 and the text in the following paragraphs explaining how impact impulse, velocity and duration are expected to scale).
Extending the time period over which a collision takes place decreases acceleration and therefore impact force magnitude . In the case of foot impact in humans, the latency period of muscle prevents the body from actively extending impact duration via changing limb geometry or limb stiffness . Although passive damping is likely to be limited by foot morphology (rigid hoof, coupled with a small digital cushion), ungulates may be able to use the relative movement of the digits, a sequential landing pattern and foot slip to prolong impact. We speculate that these features are unlikely to change with body mass; however, stance duration has previously been shown to increase with body mass , . On that basis, we expect impact duration to scale with positive allometry (scaling with a slope higher than Mb0.17 ) in both the fore- and hind limbs. This value is derived from , who proposed that time related variables scale proportionally to the square root of length variables.
As a consequence of longer impact duration, we expect loading rate to scale with negative allometry (scaling with a slope lower than Mb0.83; derived from loading rate or force divided by time; i.e. dividing Mb1.00 by Mb0.17). Considering that the forelimb functions primarily to brake centre of mass (CoM) motion in most mammalian quadrupeds , we anticipate that the forelimb will experience a higher loading rate than the hind limb . Additionally, we expect impact impulse (i.e. the integral of force and time) to remain isometric (scaling ∼Mb 1.17). Considering that the hind limbs function primarily to propel the CoM in most mammals , we predict that the hind limbs will experience greater impact impulses than the forelimbs. While segment length scales isometrically in geometrically similar animals, limb angle at impact seems to scale lower than what isometry predicts . On that basis, we expect (horizontal) impact velocity to scale with negative allometry (scaling with a slope lower than Mb0.16), with the (propulsive) hind limbs impacting at faster velocities than the forelimbs , . The expected value of 0.16 for geometric similarity is derived from dividing distance (Mb0.33) by time (Mb0.17).
Here, for the first time in a broad comparative context we determine how features of foot impact scale in hoofed mammals spanning over two orders of magnitude of adult body mass. Using the geometric similarity model, we hypothesise that impact force amplitude (Fig. 1a) will remain isometric with increasing size (scaling ∼Mb1.00), through allometric changes in impact velocity and impact duration (Fig. 1c). Larger animals are likely to experience lower impact velocities and longer impact durations, countering the increased mass associated with their larger size. Consequently, loading rate is likely to scale allometrically while Meff and impact impulse (Fig. 1d and 1e) remain isometric. Furthermore, due to differential limb function as noted above, there are likely to be disparities between how fore- and hindlimb impact dynamics scale .
a) peak vertical impact force amplitude; b) peak vertical ground reaction force (GRF) amplitude; c) impact duration; d) vertical horizontal impact impulse and e) horizontal impact impulse; f) total decelerative impulse over the entire stance; g) total accelerative impulse over the entire stance.
Here we present our results for each parameter of impact dynamics; the exponent predicted by isometry is in parentheses after each subheading. Our LMM (linear mixed effect model) analysis (see Methods) provided the mean scaling exponents ± standard errors. Note that this approach differs from standard bivariate scaling regression, but produces comparable patterns that allow more statistically rigorous inferences about scaling to be made. Detailed results are shown in Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12, Tables 1 and 2, and also Tables S1–S27. Some apparent patterns of scaling were non-significant trends, but still strikingly different for the fore- vs. hind limbs. Therefore, in addition to explicitly presenting significant scaling patterns here, we denote any non-significant pattern that is of potential biological significance as a “trend”, with the hope that this clarity inspires future scaling studies to more unambiguously test their significance.
a) peak vertical impact force amplitude; b) peak horizontal impact force amplitude. Black markers denote forelimb walk data; grey markers denote forelimb slow run data; dark blue markers denote hind limb walk data; light blue markers denote hind limb slow run data. The correspondingly coloured trend lines represent the scaling outcome generated by the LMM analysis.
a) vertical impact velocity; b) horizontal impact velocity. Black markers denote forelimb walk data; grey markers denote forelimb slow run data; dark blue markers denote hind limb walk data; light blue markers denote hind limb slow run data. The correspondingly coloured trend lines represent the scaling outcome generated by the LMM analysis. Dashed lines show non-significant scaling outcomes, i.e. the slope is not different from a slope of zero.
Black markers denote forelimb walk data; grey markers denote forelimb slow run data; dark blue markers denote hind limb walk data; light blue markers denote hind limb slow run data. The correspondingly coloured trendlines represent the scaling outcome generated by the LMM analysis. Dashed lines show non-significant scaling outcomes, i.e. the slope is not different from a slope of zero.
Black markers denote forelimb walk data; grey markers denote forelimb slow run data; dark blue markers denote hind limb walk data; light blue markers denote hind limb slow run data. The correspondingly coloured trendlines represent the scaling outcome generated by the LMM analysis.
a) walk; b) slow run. Black markers denote forelimb data; blue markers denote hind limb data. The corresponding shaded areas show two standard errors from the fitted model. Although the intersection suggests that smaller species (below ∼750 kg Mb) have greater forelimb Meff, whereas larger species appeared to have greater hind limb Meff, the standard errors associated with model fitting mean these limb differences are not statistically significant.
a) maximum average loading rate (calculated over 0.5% rolling window throughout the impact period); b) maximum instantaneous rate of force application. Black markers denote forelimb walk data; grey markers denote forelimb slow run data; dark blue markers denote hind limb walk data; light blue markers denote hind limb slow run data. The correspondingly coloured trendlines represent the scaling outcome generated by the LMM analysis.