The Foot’s Arch and the Energetics of Human Locomotion

The energy-sparing spring theory of the foot’s arch has become central to interpretations of the foot’s mechanical function and evolution. Using a novel insole technique that restricted compression of the foot’s longitudinal arch, this study provides the first direct evidence that arch compression/recoil during locomotion contributes to lowering energy cost. Restricting arch compression near maximally (~80%) during moderate-speed (2.7 ms−1) level running increased metabolic cost by + 6.0% (p < 0.001, d = 0.67; unaffected by foot strike technique). A simple model shows that the metabolic energy saved by the arch is largely explained by the passive-elastic work it supplies that would otherwise be done by active muscle. Both experimental and model data confirm that it is the end-range of arch compression that dictates the energy-saving role of the arch. Restricting arch compression had no effect on the cost of walking or incline running (3°), commensurate with the smaller role of passive-elastic mechanics in these gaits. These findings substantiate the elastic energy-saving role of the longitudinal arch during running, and suggest that arch supports used in some footwear and orthotics may increase the cost of running.


Online Supplementary Material
Supplementary Video S1. High-speed video (300 frames s -1 ) of the insole material testing. The 18 material testing rig (Instron Model 8874, Illinois Tool Works Inc.) was configured to compress the insole 4mm during a simulated running foot contact). 20

Foot Model and Arch Compression Estimates 22
Because the markers placed on the first and fifth metatarsal bases were elevated from the foot, we used static pointer trials to identify the anatomical medial and lateral aspects of these landmarks, 24 expressed in the rearfoot anatomical coordinate system. The midpoint of these virtual landmarks was used to define both a rearfoot-forefoot joint center (metatarsophalangeal (MTP) joint center) as 26 well as the base of the foot (Manuscript Figure 3). A forefoot segment was created using the virtual metatarsal markers and a third virtual hallux marker, defined using a static pointer trial (these virtual 28 landmarks were expressed relative to a coordinate system defined by the metatarsal and hallux markers). A foot sole plane was created by transposing the rearfoot coordinate system so that its 30 origin was the rearfoot-forefoot joint center (the x-z-plane of this coordinate system is oriented parallel with the ground in a neutral standing posture). A continuous trace of the vertical 32 displacement of the navicular marker relative to the new rearfoot coordinate system in the y-axis was used as a measure of arch compression (Supplementary Figure S2). This method minimized any 34 effects that ankle inversion/eversion has on predicting arch compression since these motions do not alter the location of the navicular marker in the rearfoot coordinate system. The model also allowed 36 us to compute the inversion/eversion angle, which was found to be unaffected by the insole (Supplementary Figure S3). The navicular height relative to the sole of the foot (x-z-plane of the 38 rear-foot coordinate system) at initial foot contact in the shod only condition was used as a reference value to standardize arch compression. Results were compared to digitized high speed 40 video footage (navicular marker relative to top of the shoe midsole); maximal navicular compression from the two methods correlated strongly (r = 0.88). 42 Sagittal plane joint angles and net moments of the forefoot segment were calculated about the MTP 44 joint center using Vicon BodyBuilder software (Oxford Metrics, Oxford, UK). The ground reaction force (GRF) was ascribed to the forefoot segment when the center of pressure was anterior to the 46 MTP joint center and to the rearfoot when it was distal. Inverse dynamic calculations of the forefoot segment assumed that the moments generated by the mass and inertia of the segment were 48 negligible as per Stefanyshyn and Nigg 2 .

Arch Elastic Energy and Total Limb Mechanical Work of Locomotion Estimates
First, the maximal experimental ankle compressive load during stance was estimated by summing 52 the computed joint reaction force with an estimate of the Achilles tendon force (the trial with the highest value was used). Achilles tendon force was estimated by dividing the net ankle joint moment 54 by the Achilles tendon moment arm taken from calliper measurements during standing. The participant's peak arch strain energy was then predicted from the estimated compressive ankle load 56 using the strain energy versus ankle load relationship from Ker et al. 3  The positive mechanical work during a single step ( , + ) was calculated for walking, level 74 running and incline running using the individual limb work force plate approach described by Donelan et al. 4 . Limb powers for the right and left limbs (P r , P l ) were calculated (from right heel strike 76 to left heel strike) as the dot product of the ground reaction force acting on the right and left limbs (F r , F l ), respectively, and the velocity of the center of mass ( ) (Eq. S1). The right and left limb 78 ground reaction forces were recorded from independent force plates under the right and left treadmill belts. where m is body mass and g is acceleration due to gravity (9.81 m s -1 ). During a single running step the GRF from the trailing limb (designated to be the left limb) equaled zero. Integration constants 90 (offsets) were applied to the calculated COM velocities: z was set so that the average COM vertical velocity over one step equaled 0; y was set so that the average fore-aft velocity over one step 92 equaled the treadmill speed; x was set so that the medio-lateral velocity at the start and end of the step were equal but opposite. Offsets were adjusted during the incline condition: z; treadmill speed 94 x sin 3°, y; treadmill speed x cos 3°. Limb powers were restricted to positive values and integrated with respect to time (t i and t f represent the right heel strike and left heel strike respectively) to 96 determine the positive mechanical work performed in the limbs over one step (Eq. S3). and + were summed as per the individual limbs methods described by Donelan et al. 4 .
In running, the left (trailing) limb generated no ground reaction force as it was in swing-phase and 104 thus , + was taken as ℎ + . The mechanical cost of transport ( + ; J kg -1 m -1 ) was computed by multiplying , + by two (assuming bilateral limb symmetry) and dividing by the 106 average distance traveled per stride and body mass.

Locomotor Mechanical Efficiency and Modelled Metabolic Energy Prediction
In addition to predicting increase in metabolic cost using a constant efficiency of 25%, we also 110 predicted costs using a computed mechanical locomotor efficiency (η + loc ). This was done by using the experimental gross energy cost (E tot ; J kg -1 m -1 ) and the positive limb mechanical work 112 (W limb + ; J kg -1 m -1 ) of the minimal shoe-only conditions: We subsequently computed a predicted increase in locomotor metabolic cost associated with restricting arch compression for each condition as: 118 where ∆W arch + and ∆W limb + are the differences in the amount of returned arch elastic energy and 122 positive limb mechanical work between the minimal shoe-only trial and the corresponding insole trial for level running, incline running and walking, respectively. 124

Maximal Oxygen Consumption and Lactate Threshold 126
In a separate session prior to the biomechanics testing, participant's maximum oxygen uptake (VȮ 2max ) and lactate threshold were determined by an incremental exercise test. In the 24 hours 128 prior to attending the laboratory, participants were asked not to exercise heavily or consume caffeinated food or beverages. In the two hours prior participant were instructed not to consume 130 any food or drink other than water. Before commencement of the test, a baseline heart rate (Polar F1 Heart Rate Monitor, Kempele, Finland) and fingertip capillary blood sample (Lactate-Pro, Arkray, 132 LT-1710, Kyoto, Japan) were collected. Participants were allowed to warm up at a self-selected pace and familiarize themselves with the motorized treadmill (VR 3000, NuryTech Inc, Germany) and 134 mouth piece.

136
A three minute exercise and one minute rest protocol was followed until volitional exhaustion. Given the weekly running distance inclusion criteria of the study, it was assumed that all participants were 138 of a good fitness level so all VȮ 2max tests commenced at a standard 10 km/hr. The treadmill belt speed was increased by 2 km/hr after each exercise bout until 16 km/hr after which speed was 140 increased by 1 km/hr. During the rest minute participants straddled the treadmill belt while a fingertip blood sample was collected and heart rate recorded. Throughout the test, expired gasses 142 were collected by a two-valve mouthpiece connected via two lightweight flexible tubes to a computerized oxygen and carbon dioxide gas analysis system [oxygen and carbon dioxide analysers 144 (Ametek SOV S-3A11 / Ametek COV CD-3A, Applied Electrochemistry, Ametek, Pittsburgh, PA)].
Ventilation was recorded at 15 second intervals using a turbine ventilometer (225A; Morgan, 146 Chatham Kent, UK). The ventilometer and gas analysers were calibrated before and immediately after each test using a one litre syringe pump and reference gas mixtures, respectively (BOC Gases, 148 Chatswood, Australia). Participant's peak oxygen consumption was determined by summing the four highest consecutive 15 second VO 2 values. The lactate threshold was determined using a Dmax 150 method 5 .

Additional Statistical Analysis 152
A series of Wilcoxon's Signed Rank Tests (non-parametric) were used to assess statistical differences between RPE and questionnaire results during walking, level running and incline running conditions. 154

Temporal Parameters
The insoles had no effect on any temporal parameters (stance, swing or stride time).  Table S1 for additional temporal parameter data. 162

Incremental Exercise Test 164
The incremental exercise test identified that participant's lactate threshold was 6.1 ± 1.1 mmol/L which occurred at a speed of 4.5 ± 0.3 ms -1 . Given that the energy expenditure and lactate 166 concentrations when running at 2.7 ms -1 on both the level and incline were lower than that at which the lactate threshold occurred, (lactate concentrations; incline running shoe-only 2.5 ± 1.0 mmol/L, 168 incline running FAI 2.5 ± 1.2 mmol/L), it was concluded that all participants were exercising aerobically. The blood lactate concentration of one participant during the incline trial exceeded his 170 previously determined threshold. His incline and level running data were therefore removed from the analyses. 172

Insole Material Testing 174
Material testing on the insoles returned less than 0.4 Joules of energy during running ( Figure S1).
This value is equal to less than 3% of the elastic energy storage/return that was estimated to be lost 176 when wearing the insoles.

178
Supplementary Figure S1. Average insole load-displacement curve indicating minimal energy return from the insoles. 180 182

Arch Kinematics
A representative trace of arch compression over the stance phase of level running for one 184 participant is presented in Figure S2. Peak arch compression was significantly reduced in the HAI and FAI conditions compared to the minimal shoe-only (both p < 0.001; Figure S2). No statistically 186 significant difference in peak ankle inversion or eversion was identified between the minimal shoeonly and FAI level running conditions ( Figure S3  6.3 ± 1.2 13.0 ± 1.5 n/a n/a n/a n/a 9.5 ± 1.4 n/a n/a FFS n/a 6.6 ± 0.8 13.3 ± 1.1 n/a n/a n/a n/a 10.5 ± 2.4 n/a n/a Average n/a 6.5 ± 1.0 13.2 ± 1.3 n/a n/a n/a n/a n/a n/a n/a n/a 27.5 ± 3.4 n/a n/a FFS n/a 29.4 ± 2.5 36.1 ± 2.7 n/a n/a n/a n/a 30.3 ± 3.4 n/a n/a Average n/a 29.9 ± 2.5 36.5 ± 2.9 n/a n/a n/a n/a 28.9 ± 3.6 n/a n/a RFS n/a n/a n/a n/a 24.9 ± 3.4 n/a n/a FFS n/a 38.9 ± 2.2 45.6 ± 2.1 n/a n/a n/a n/a 27.5 ± 2.3 n/a n/a Average n/a 39.8 ± 2.6 46.5 ± 2.8 n/a n/a n/a n/a 26. this is one of the most commonly used methods for assessing arch compression (e.g. 6,10-12 ), it is not without errors and may have under-estimated bony midfoot motion 13 . 252 Our model of arch elastic energy storage/release depended on an estimate of arch compressive load 254 from inverse dynamic calculations of ankle joint moments and Achilles tendon force that are both subject to error. The Achilles tendon moment arm was measured in a static standing position as the 256 perpendicular distance from the mid-point of the lateral malleoli to the Achilles tendon 14 . While this measurement was performed by the same assessor it is understood that this method may differ 258 from imaging-based measurements and does not represent the active moment arm of the Achilles during locomotion 15 . Furthermore, we assumed the hysteresis of the arch spring to be 22% based 260 on the experimental data of Ker et al. 3 , although the actual hysteresis may differ between participants. Nevertheless, differences in the average hysteresis would only change the magnitude 262 of the predicted elastic energy return (and the subsequent predicted energy cost of locomotion), but the relative differences between conditions would remain similar. Thus, although it serves as a 264 simple estimation of energy storage in the arch, the modelled load-displacement curve may lead to errors in estimating arch elastic energy storage with a very high degree of accuracy. 266 Our modelled prediction of the metabolic effect of restricting arch compression and spring function 268 depended on assumptions regarding the efficiency of performing positive locomotor mechanical work. We used an efficiency of 25%, which represents a value close to the theoretical maximal 270 efficiency of muscle performing positive work 16 . These predictions therefore assume that the mechanical work performed to replace the lost elastic arch work is done by positive muscle fiber 272 work, and that this additional muscle fiber work dictates the changes observed in metabolic cost.
Although these assumptions are reasonable, it is important to note that the lost arch elastic work 274 may have been substituted, to an extent, with an increase in elastic work performed at other joints.
This would necessitate an increase in force in the muscle-tendon-units generating the elastic work, 276 which would exact a metabolic cost. In this case the increase in metabolic cost may not be dictated solely by increases in muscle fibers producing mechanical work at a constant efficiency. Therefore, 278 as a secondary approach to predicting the increase in metabolic cost of locomotion arising from restricting arch compression we used a computed locomotor mechanical efficiency of performing 280 the positive mechanical work in the minimal shoe-only conditions (Equations S4 and S5). This prediction does not necessitate that the additional metabolic energy expenditure is due strictly to 282 additional muscle fiber work alone, but rather results from a proportional increase in the combined mechanical costs that dictate the locomotor efficiency in the level running, incline running or 284 walking conditions (e.g. energy expended in isometric contractions, muscle fiber work, etc.).
Modelled metabolic cost predictions from the locomotor mechanical efficiency showed the same 286 overall findings as those using a constant efficiency of 25% (See Table S2). Finally, it is also possible that muscle fiber efficiency was different from the 25% used in this study. If average muscle fiber 288 efficiency was different this would alter the magnitude of the predicted increases in metabolic energy cost, although the relative differences between conditions would be expected to remain 290 similar.

292
We acknowledge the possibility that our observations reflect a more general effect of altering gait mechanics with an arch-restricting insole. However, notwithstanding the aforementioned 294 limitations, there are several factors that lead the authors to dispute this interpretation. First, while the difference in arch compression was nearly two-fold between the FAI and HAI, both insoles 296 resulted in a similar loss of arch elastic energy storage and exhibited a non-significant difference in metabolic cost (Figures 1 & 2 in the main article). These data suggest that the increase in metabolic 298 cost of running while wearing the insoles is not simply a systematic effect of altering arch kinematics per se, but instead an effect of the non-linear arch elastic energy storage/return. Secondly, the fact 300 that there was no change in energy cost during both the FAI incline running and walking conditions strengthens the interpretation that the energetic changes observed in level running are primarily 302 attributed to alterations in the arch spring mechanics. If other general modifications such as cocontraction, instability, intrinsic foot muscle activity 7 , cushioning 17 or discomfort 18 were the 304 primary factors leading to the increase in metabolic cost after restricting arch compression, it would be expected that they would also elevate metabolic costs during incline running and possibly also 306 during walking. Finally, peak ankle eversion during level running was unaffected by the insoles ( Figure S3) and there were small differences in total limb mechanical work between the insole and 308 minimal shoe only conditions (Table 1 main article).

Orthotic clause
The authors would like to note that the foot insoles used in this study do not represent conventional 312 prescription practices by health practitioners for symptomatic individuals in a clinical setting, but rather a tool for experimentally testing our hypotheses. 314