Musculoskeletal model

We adapted the musculoskeletal model used by Falisse et al. [8]–which is based on OpenSim’s gait2392 model [38,39] and the model proposed by Hamner et al. [40]–to include a midtarsal joint, plantar fascia, and a plantar intrinsic foot muscle. The resulting model has 33 skeletal degrees of freedom (dofs) (pelvis as floating base: 6 dofs, hip: 3 dofs, knee: 1 dof, ankle: 1 dof, subtalar: 1 dof, midtarsal: 1 dof, metatarsophalangeal (MTP): 1 dof, lumbar: 3 dofs, shoulder: 3 dofs, and elbow: 1 dof). We used Newtonian rigid body dynamics to model skeletal motion [41,42]. The lower limb and lumbar joints are actuated by 94 Hill-type muscle-tendon units (92 muscles according to the gait2392 model and the right and left plantar intrinsic foot muscles as described below) [32,43]. Muscle excitation-activation coupling is given by Raasch’s model [44]. The shoulder and elbow joints are actuated by ideal torque actuators [8]. Each joint has viscous friction (coefficient 0.1 Nm s rad^-1) and nonlinear stiffness to represent the net effects of ligaments [11,45].

A first series of model adaptations aimed at better capturing the properties of muscles spanning the ankle (Table A in S1 Text). During maximal isometric contraction of the triceps surae, Achilles tendon strain is reported to be 5.6–8% [46,47]. We reduced the normalised stiffness of the Achilles tendon such that maximal isometric contraction resulted in strains in the experimental range (6.8%). To better capture experimental peak isometric ankle moments, we increased the maximal isometric force of the triceps surae by 20% with respect to the force of the gait2392 model [38,39] (Fig E in S1 Text). We adapted the parameters related to the passive stiffness of the muscles spanning the ankle joint to better capture the experimental passive moment-angle relationship of the ankle. Passive stiffness of all muscles crossing the ankle joint was increased by shifting their passive force-length characteristic [32] by 10% of the optimal fibre length towards smaller fibre lengths. The resulting passive ankle moments are consistent with measurements [48] (see section 1 in S1 Text for additional information, validation, and sensitivity analysis).

We modelled the foot arch as a midtarsal joint with a single rotational degree of freedom, connecting the calcaneus and midfoot segments. Midfoot and forefoot segments are rigidly connected. Midtarsal joint centre, segment definitions, and segment mass properties were taken from Malaquias et al. [35]. The orientation of the midtarsal joint axis was set according to the mean finite helical axis calculated from the motion capture data during the stance phase of walking using the algorithm provided by Ancillao [49] (See section 2 in S1 Text for the sensitivity of midtarsal joint axis orientation). We defined the rotational stiffness of the midtarsal joint by combining the contributions of long and short plantar, and spring ligaments [35], considering a nonlinear stress-strain characteristic [50] (section 3 in in S1 Text).

We modelled the plantar fascia as a single elastic element with origin on the calcaneus and insertion on the toes [35], which wraps around a cylinder with radius 9.5 mm at the metatarsal head [51]. Based on in vivo ultrasound imaging in healthy adults, we set a nominal plantar fascia cross-section of 70 mm² [52]. We assumed a purely elastic plantar fascia with a nonlinear stress-strain characteristic presented by Natali et al. [53]. This stress-strain characteristic was based on uniaxial tension tests of plantar fascia samples [53,54]. We alternatively considered using the plantar fascia stress-strain characteristic presented by Gefen [50], however this stress-strain curve was more compliant than experimentally observed (Fig L in S1 Text). To test the effect of additional plantar fascia stiffnesses, we combined the stress-strain characteristic from Natali et al. with larger cross-sections (210 mm², 140 mm²), and with a smaller cross-section (24 mm², cfr. cadavers [55–57]) (Fig M in S1 Text). Plantar fascia slack length (146 mm) was chosen such that a standing foot, bearing only the weight of the foot and tibia, is in the anatomical position.

We modelled the plantar intrinsic muscles of a foot as a single Hill-type muscle because they act as a functional unit [24]. Origin, insertion, and wrapping of the muscle are the same as the plantar fascia. Parameters describing the muscle were derived from cadaveric measurements of flexor digitorum brevis (FDB), abductor hallucis (ABDH), quadratus plantae (QP), and abductor digiti minimi (ABDM), which all span the longitudinal foot arch and MTP joint [20,21]. Based on the mean fibre lengths reported by [21], we set the optimal fibre length to 23 mm. Maximal contraction velocity is 10 optimal fibre lengths per second [11]. The pennation angle at optimal fibre length is 20°, equal to measurements of FDB and AH [20]. To determine the physiological cross-sectional area (PCSA) of the lumped muscle, we summed the PCSA of FDB, ABDH, QP, and ABDM for a total of 1831 mm² [21]. Since the PCSA is obtained from donors with an average age of 71 years, we compensated for age-related muscle atrophy. Older adults show a 24–40% reduction in strength in ankle and foot muscles, in comparison to young adults [58]. Assuming a 24% decrease in PCSA due to ageing, the lumped PCSA is 2400 mm² for a young adult. With a specific tension of 25 N cm^-2 [59], the maximal isometric force is 600 N. To model the stiffness provided by the internal tendon in highly pennate muscles, we shifted the passive force-length characteristic [32] with 10% of the optimal fibre length towards shorter fibre lengths. We determined tendon slack length based on the assumption that muscle fibres work close to their optimal length during walking. Since we modelled the muscle-tendon unit along the same path as the plantar fascia, it has the same length as the plantar fascia. The plantar fascia is expected to have 0–4% strain while walking [54,60] allowing us to estimate the expected muscle-tendon length based on the selected plantar fascia slack length. We then calculated normalized fibre lengths for different tendon slack lengths assuming an isometric contraction at an activation of 0.5. We selected the tendon slack length (123 mm) that resulted in fibre lengths around the optimal fibre length. To test the influence of parameter choices, we repeated the walking simulations with different parameter values: we tested maximal isometric force values of 120 N, 300 N, 456 N (i.e. not compensating for age-related atrophy), 720 N, 900 N and 1200 N. We tested optimal fibre lengths of 23±4 mm and 23±8 mm (4 mm standard deviation on reported FDB fibre length [21]). We tested tendon slack lengths of 121 mm, 122 mm, 124 mm and 125 mm (section 5 in S1 Text). We also tested a model without plantar intrinsic muscles.

The model was scaled to the anthropometry of the subject (see below) with the OpenSim scale tool [38,39]. Scale factors were calculated based on marker data of the static trial. For the foot (talus and distal segments), we included manual measurements to determine non-uniform scale factors. Anteroposterior scaling was based on the distance between the dorsal calcaneus marker and the hallux marker. Mediolateral scaling was based on the distance between the markers on the heads of the first and fifth metatarsal (section 10 in in S1 Text). In the vertical direction, the vertical distance between the centres of the medial malleolus and the first metatarsal head (based on manual measurements) determined the size of the foot. Because the scaling of the foot was non-uniform, insertion points of the tibialis anterior and posterior, and peroneus (brevis, longus, and tertius) were distorted. We substituted these insertion point locations from an alternative musculoskeletal model [35], after scaling it to match our subject-specific model. Foot proportions of this alternative model were close to the subject’s foot proportions; thus, scaling did not cause distortions. The optimal fibre length and tendon slack length of the adjusted muscles were linearly scaled (cfr. the OpenSim scaling method for these parameters [38,39]). To test the effects of this scaling method, we also created a model with uniformly scaled feet. The uniform scale factor is proportional to the distance between the dorsal calcaneus marker and the hallux marker. Uniform scaling resulted in a lower foot arch compared to non-uniform scaling.

We modelled foot-ground contact as deformable spheres (foot) interacting with a rigid plane (ground). The force acting on each sphere was calculated from Hertzian stiffness and Hunt-Crossley dissipation [61,62]. Each foot has five contact spheres: heel pad, lateral longitudinal arch, ball of the foot (2 spheres), and toes (Fig 1). The location and radius of each sphere were chosen to represent areas with high plantar pressure during standing and walking [63,64], and based on anatomical reference [65]. To tune the vertical position of the contact spheres, we tracked experimental joint angles from the static trial with skeletal and contact dynamics. We solved the static force equilibrium and adjusted the vertical contact positions to reduce offset with respect to experimental pelvis-to-ground coordinates. We set the stiffness parameter of each contact sphere to 10 MPa, instead of the 1 MPa used previously [8,11]. Preliminary simulations showed that this improved the realism of the contact deformation during walking. We evaluated how the configuration and stiffness of the contact spheres influenced the predicted gait (section 6 in S1 Text).

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Fig 1. Musculoskeletal model of the ankle and foot.

Plantar intrinsic muscle is not visible, because it shares the same path as the plantar fascia. Visualised via OpenSim [38,39].

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To dissociate the effect of modelling the foot in more detail versus altering parameters, we created a nominal 3-segment foot model that is equivalent to our nominal 4-segment model with the midtarsal joint locked in anatomical position. To be consistent with the model from Falisse et al., the MTP joint of the 3-segment model was not actuated by muscles, but by a spring (25 Nm/rad) and damper (2 Nm s/rad) [8]. We tested an alternative 3-segment foot model with muscle-actuated MTP (section 7 in S1 Text).

Simulations of standing foot under vertical loading

We replicated a series of experiments that evaluated the deformation of the foot under external loading [2,18,19] in simulation and compared simulated and experimental outcomes to validate our model. To simulate these experiments, we considered only the right tibia and foot of our musculoskeletal model. The angle between toes and ground was fixed (cfr. [18,19]), and muscle activity was set to zero for cadaver experiments and to a small baseline activation (0.01) for in vivo experiments. We imposed a vertical force on the tibia and solved for static equilibrium. We repeated this for different force values within the experimental range to obtain force-deformation curves.

First, we simulated an experiment from Ker et al. where they measured longitudinal elongation of the foot under vertical loading in different conditions by subsequently removing more soft tissue [2]. To replicate this experiment, we used a model in which individual ligaments were represented (before lumping them for computational efficiency in gait simulations). Ker et al. [2] did not specify whether intrinsic muscles were cut. We considered removing intrinsic muscles as an additional step after removing plantar fascia, and before removing long plantar ligament. To compare simulation outcomes to experimental data, which was obtained on the foot of an 85 kg male, we scaled forces by the ratio of body weights (0.76) and elongation by the square root of this ratio (0.87).

Next, we simulated two experiments that evaluated the effect of flexing or extending the toes on arch compression. Yawar et al. tested how extending the toes (15°) influenced the vertical compression of a cadaver foot (defined as vertical displacement of the transected tibia), compared to having the toes in their neutral position [19]. Welte et al. applied a vertical load up to body weight on the thigh of a seated person (the force was aligned to be vertical to navicular) and assessed how flexing or extending the toes by 30° influenced arch compression [18]. Arch compression was defined as the change in the vertical position of a marker on the navicular normalised to the maximal height of that marker for the subject [18].

Simulations of walking

We used our previously developed framework [11,41] to simulate walking. We formulated gait as an optimal control problem. We solved for muscle excitations that result in a prescribed average forward velocity of the musculoskeletal model (1.33 m s^-1, self-selected speed of the subject), and minimise an objective function. We imposed symmetry between the left and right steps, which limits the time horizon to half a gait cycle. The objective function was a weighted sum of squared metabolic energy rate, squared muscle activations, squared joint accelerations, and squared joint limit torques, integrated over the time horizon and divided by the distance travelled. We calculated the metabolic energy of muscles with the model presented by Bhargava et al. [66], which we made continuously differentiable by approximating conditional statements with a hyperbolic tangent (supplement S8). This framework was implemented in MATLAB (The Mathworks Inc., USA). We used direct collocation with a third-order Radau polynomial basis. We used algorithmic differentiation (CasADi 3.5.5 [67]) to calculate the derivatives required for gradient-based optimisation. The optimisation problems were solved with IPOPT [68] and MUMPS [69]. For a detailed description of the simulation framework, we refer to [11].

Based on convergence analysis (section 9 in in S1 Text) we selected 100 mesh intervals, cold-start initial guess and convergence tolerance of 1e-04. All results shown were obtained from simulations with these settings. To evaluate the sensitivity of predicted gait to differences in the musculoskeletal model, we performed simulations for a range of models and model parameters. All simulations were run on a laptop (allocating two threads of an intel core i7-11850 CPU to each simulation).

To evaluate the realism of our simulations, we calculated normalised cross-correlation coefficients between simulated and mean experimental joint angles, moments and powers. We considered the simulated and experimental curves to be in good agreement if the normalised cross-correlation coefficient was above 0.90 [31], and in moderate agreement if it was between 0.70 and 0.90 (cfr. threshold used in [29]). When comparing different simulation results, we used a difference of 0.05 as the threshold. We also calculated weighted root mean square errors between simulated and experimental joint angles, moments and powers as (1) where mean_i and SD_i are the mean and standard deviation of the experimental data for one time point, and simulated_i is the simulated value on the corresponding time point (note that simulations were deterministic and therefore had 0 standard deviation). Mesh points of the simulation were chosen as discrete time points (i.e. 200 points per gait cycle). We considered an RMSE below 3 as moderately accurate, and below 2 as accurate [29,31], and set a difference of 1 as threshold for comparisons. We made separate comparisons for the stance phase and swing phase.

To further validate our simulation results with the 4-segment foot model, we compared them to unified deformable segment analysis results published by Takahashi et al. [70]. In this analysis, deformation power distal to a reference segment is calculated based on the absolute kinematics of the reference segment and distal ground reaction forces [70]. We selected these results as reference data because they do not depend on rigid-body assumptions or predefined joints, thus better capturing the true magnitude of power [70]. To obtain simulated power distal to a segment, we summed joint powers and contact deformation powers that are distal to the segment in our model. Distal to hallux only consists of the contact sphere under the toes. Distal to forefoot includes distal to hallux and also MTP joint and contact spheres under the forefoot. Distal to hindfoot includes MTP and midtarsal joints and all contact spheres. Distal to shank is the sum of distal to hindfoot and ankle and subtalar joint powers. Work was calculated by integrating power over the stance phase.

To test whether simulations with our model can capture the effects of manipulating foot properties, we simulated walking and running with an insole that limits arch compression [6] and walking with local anaesthesia of intrinsic foot muscles [7]. To model the insole, we added a spring to the midtarsal joint, that only engages when midtarsal joint angles are positive. The properties of the insole were chosen such that they reduced arch compression by 80% during level running at 2.7 m/s in agreement with experimental data [6]. We simulated gait at 2.7 m/s for different insole stiffnesses and selected the stiffness (50 Nm/rad) that reduced arch compression by 80% relative to the same speed without the insole. We then simulated walking at 1.33 m/s with this insole. To mimic the effect of a nerve block inhibiting intrinsic foot muscle activation, we modelled that the activation should remain equal to its lower bound (0.05 cfr. [11]).

Experimental data

We compared simulation outcomes to previously published experimental data [8]. As our simulations do not take into account uncertainty, they yielded one solution per condition (i.e. combination of model parameters and imposed gait speed) whereas we used experimental data from 10 gait cycles. The data (marker trajectories, ground reaction forces, surface electromyography) was collected from a healthy female adult (mass: 62 kg, height: 1.70 m, age: 35) during 10 gait cycles of overground walking at self-selected speed (1.33 m/s) [8]. The subject was instrumented with 59 retro-reflective skin-mounted markers. There were 10 markers mounted on each foot (supplement S10). Additionally, we measured (with measuring tape) the normal distance from the medial malleolus centre and the first metatarsal head centre to the ground while the subject was in a seated position with the foot flat on the ground. We scaled a musculoskeletal model to the anthropometry of the subject, as described in the section on the musculoskeletal model. We performed inverse kinematic and inverse dynamic analysis [38] based on the model with 4-segment feet and selected midtarsal joint orientation (Fig 1), and calculated joint powers for each individual gait cycle. After time-normalising the processed data, we calculated the mean and standard deviation at 100 points equally spread over the gait cycle duration. For the inverse dynamic analysis, we applied experimental ground reaction forces on the calcaneus, thus we did not obtain moments nor powers for the midtarsal and MTP joints. Foot joint moments are sometimes estimated by applying the total ground reaction forces to a more distal segment once the centre of pressure moves anterior to the joint [71]. This method assumes that only one foot segment is in contact with the ground at each instance in time. We did not make this assumption in our model and therefore we did not compare simulated moments to moments computed based on experimental data.

Simulations of standing foot under vertical loading to validate model parameters

To evaluate the realism of our foot model, we simulated a series of experimental studies that applied controlled vertical loads to the foot.

Loading the proposed foot model by a vertical force up to 3 kN resulted in a maximal longitudinal elongation of 3.5 mm (Fig 2A), which is lower than the 6–8 mm observed for an intact cadaver foot [2]. When using a more compliant plantar fascia stress-strain characteristic (Gefen et al. [50]), the simulated elongation was 5.8 mm and thus in closer agreement with experimental data. The shape of the force-elongation curve agreed with experimental data. Overall, our model captured the contribution of the different ligaments and passive muscle forces to foot stiffness as can be seen by the agreement between simulated and experimental force-elongation curves after sequentially removing soft tissues (Fig 2A).

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Fig 2. Deformation of a standing foot under vertical compression.

Full lines indicate experimental data (digitised from (a) Ker et al. [2], (b) Yawar et al. [19], (c) Welte et al. [18]), and circles represent simulations. (a) Repeated loading of a cadaver foot after sequentially removing different soft tissue. The table gives an overview of structures that are present in each condition. Elongation was computed as the difference in length between calcaneus origin (defined as the most inferior, lateral point on the posterior surface of the calcaneus [73]) and MTP joint centre during the virtual experiment with respect to the resting condition, i.e. the unloaded intact foot. (b) Vertical force-displacement curves for different toe dorsiflexion (DF) positions. Displacement was calculated as the vertical position of the ankle joint centre, relative to its position when no external load is applied to the foot with toes at 0° DF. (c) Vertical loading of a standing foot of an in vivo subject, with toes passively dorsiflexed 30° and -30°. Including baseline muscle activation influenced the force-displacement curves and changed the effect of the toe position.

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The next set of simulations involved imposing different toe positions which have been previously shown to alter the relationship between the applied vertical load and vertical compression of the foot [18,19]. In agreement with experimental data obtained from a cadaver foot [19], extending the toes by 15° shifted the force-compression curve towards lower displacements, without affecting the shape of the curve (i.e. foot stiffness remained constant). We predicted a 1.0 mm shift in the force compression curve (Fig 2B), compared to 1.3 mm observed in experiments [19]. Although our simulations captured the observed shift, the magnitude of displacement is much larger than reported by Yawar et al. (24 mm vs. 12 mm) [19]. Further analysis of the results revealed that the heel pad of our model reached compressions up to 14 mm, which is greater than the total thickness of a human heel pad [72]. Since our model yielded realistic heel pad deformations during walking simulations, and increasing contact stiffness had only minor effects on the predicted walking pattern (see Methods and supplement S6), we did not further address the discrepancies between the experimental and simulation results for this specific cadaver experiment. In vivo, dorsiflexing the toes (30°) reduced the stiffness of the foot arch (i.e. reduced the slope of the force-compression curve) relative to when the toes are plantarflexed (30°) (Fig 2C) [18]. Our simulations captured the change in stiffness with toe flexion when we imposed a small baseline activity to the muscles (0.01)—consistent with people never fully relaxing their muscles—but not in the absence of any activity. Our simulations slightly overestimated compression, especially for smaller loads. Note that our model does not capture the observed hysteresis throughout all experiments, because we assumed static equilibrium at every discrete load.

Accuracy of predictive gait simulation with a 4-segment foot model

We compared simulated gait patterns using the nominal 4-segment foot model, and the previous 3-segment model (Falisse et al. [8]) to experimental data to evaluate the realism of our simulations. (Fig 3, black and orange traces). Normalised cross-correlation coefficients (NCC) and weighted root mean square errors (RMSE) between simulated and experimental gait variables (joint angles, moments, and powers) are reported in S1 Text.

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Fig 3. Predictive simulation results with the presented model compared to experimental data and state-of-the-art simulation.

Vertical lines indicate stance-to-swing transition. (a) Kinematics. (b) Kinetics. (c) Joint powers. (d) Muscle activation. Experimental data was obtained from one subject and therefore the grey band represents stride-to-stride variability. Kinematics, kinetics, and powers of all joints can be found in section 11 in S1 Text.

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Unified deformable segment analysis.

Foot and ankle energetics, simulated based on the 4-segment foot model, are mostly in agreement with experimental results obtained from the unified deformable segment analysis reported in the literature. Simulated power curves for different parts of the foot (i.e. distal to hallux, forefoot, hindfoot, and shank) captured the results obtained by Takahashi et al. [70], except for two features (Fig 4A). First, our simulations largely overestimated the negative power distal to shank and hindfoot after initial contact. This peak is attributed to the absorption of impact energy after heel strike [75] and coincides with the overestimation of the rise in ground reaction forces after initial contact (Fig AD in S1 Text). Second, we underestimated the negative power distal to the hindfoot, and overestimated the negative power distal to the forefoot during terminal stance (Fig 4A). Our simulation had less negative and net work distal to the hindfoot compared to experiments, but captured the observation that the net work distal to the shank is low (Fig 4B), i.e. the ankle-foot complex is close to energy-neutral during walking [70].

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Fig 4. Deformation power and work by soft tissue distal to ankle-foot segments.

(a) Power distal to segments. (b) Positive, negative, and net work. Experiment-based data, digitised from Takahashi et al. [70], is plotted from reference.

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Effects of modelling choices on the predicted gait pattern

We evaluated the effect of modelling choices on the predicted walking motion to explore how different foot structures shape walking mechanics and energetics. We compared the effect of each specific modelling choice versus the proposed model (“nominal 4-segment foot model”). We discuss only those gait features sensitive to the change. The results of additional parameter sweeps are available in S1 Text. In all figures, black lines represent results based on the nominal model and coloured lines represent results based on a model with alternative parameter values.

Achilles tendon stiffness.

Achilles tendon stiffness had a large influence on ankle dorsiflexion during mid-stance. In both the nominal 4-segment and 3-segment foot models, high Achilles tendon stiffness (i.e. normalised tendon stiffness of the generic Hill-type model [32]) resulted in little or no ankle dorsiflexion during midstance in contrast to experimental observations (Fig 5). The energy storage in the Achilles tendon and ankle push-off power is lower in the simulations with a stiff Achilles tendon than in the simulations with the lower tendon stiffness that we propose here (i.e. half of the normalised tendon stiffness for the generic Hill-type model [32]). Using a higher versus lower Achilles tendon stiffness also led to a worse agreement between simulated and experimental soleus activity (Fig 5). Simulations with different Achilles tendon stiffness values showed that ankle dorsiflexion, ankle power and soleus activation peaks all increased with decreasing stiffness, while mid-stance knee extension decreased (Fig A in S1 Text).

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Fig 5. Effect of Achilles tendon stiffness.

A stiff Achilles tendon results in less ankle dorsiflexion and push-off power, and a different soleus activation pattern. Experimental data was obtained from one subject and therefore the grey band represents stride-to-stride variability.

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Plantar fascia stiffness.

Plantar fascia stiffness had a strong influence on ankle push-off. Reducing plantar fascia stiffness (i.e. using stress-strain curves from Gefen [50] instead of Natali et al. [53]) resulted in increased ankle plantarflexion and increased midtarsal extension during stance (Fig 6A). These kinematics show similarities with pathological gait observed in patients with a midfoot break foot deformity [77]. Interestingly, reducing plantar fascia stiffness resulted in lower activation of the intrinsic foot muscle, instead of higher activation to compensate for the reduced stiffness. The muscle fibres underwent an elongation of up to 35% of optimal fibre length, which is much larger than in the nominal simulation (Fig 6D). Sweeping plantar fascia stiffness showed that midtarsal extension decreased with increasing stiffness (Fig M in S1 Text). Ankle plantarflexion and ankle power increased with plantar fascia stiffness.

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Fig 6. Effect of foot arch stiffness.

We reduced foot arch stiffness, or the ability to stiffen the arch, by removing the plantar intrinsic foot muscle, by reducing plantar fascia stiffness, or by reducing foot arch height. All changes resulted in alterations in (a) kinematics and (b) reductions in peak ankle power. (c) Simulated activation of the plantar intrinsic foot muscle. (d) Fibre length of the plantar intrinsic foot muscle, normalised by optimal fibre length. (e) Comparison of nominal and reduced foot arch height, visualised via OpenSim [38,39]. Experimental data was obtained from one subject and therefore the grey band represents stride-to-stride variability.

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Plantar intrinsic foot muscle.

Removing intrinsic foot muscles mainly affected the stance knee kinematics and ankle push-off power, possibly because extrinsic foot muscles that also span the ankle partially compensated for the absence of the intrinsic foot muscles to keep foot stiffness high. Removing the plantar intrinsic muscle resulted in increased stance knee flexion and earlier ankle dorsiflexion (Fig 6A). Midtarsal and MTP kinematics remained largely similar. Ankle, midtarsal, and MTP moments were higher in mid-stance, and had a lower peak in terminal stance (Fig 7A). The subtle changes in midtarsal and MTP kinematics resulted in an increased plantar fascia strain and hence a slightly higher plantar fascia moment around both joints (Fig 7B). During mid and late stance, the midtarsal flexion moment (i.e. resisting arch compression) by extrinsic foot muscles increased. The increased contribution of the extrinsic foot muscles was also reflected in the increased ankle plantarflexion moment in mid-stance (Fig 7B). However, extrinsic muscles and plantar fascia did not fully compensate for the contributions of the intrinsic muscle to the peak midtarsal and MTP moments in the nominal simulation, and hence the net effect of removing the intrinsic foot muscles was a reduced stiffness of the foot at push off. Removing the plantar intrinsic muscle resulted in lower peak triceps surae activation (Fig AJ in S1 Text) and a lower peak ankle moment at push-off (Fig 6B). The reduction in triceps surae knee moment, combined with increased quadriceps and decreased hamstrings knee moments, resulted in a lower knee flexion moment in mid-stance, which was not in agreement with experimental results (Fig 7B). Remarkably, the simulated gait was not sensitive to parameters of the plantar intrinsic muscle. The maximal isometric force needed to be changed to at least double or half its nominal value for changes in gait kinematics to occur (Fig N in S1 Text). The optimal fibre length and tendon slack length of the plantar intrinsic muscle only influenced its normalised fibre length (Figs O and P in S1 Text).

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Fig 7. Effect of removing the plantar intrinsic muscle on joint moments.

(a) Total joint moment. Experimental data was obtained from one subject and therefore the grey band represents stride-to-stride variability. (b) Joint moment by different muscle groups and plantar fascia. Extrinsic muscles crossing MTP are flexor digitorum and hallucis longus and extensor digitorum and hallucis longus. Extrinsic muscles crossing MTJ (midtarsal joint) additionally include tibialis anterior and posterior, and peroneus (brevis, longus, and tertius). Joint moments by individual muscles are shown in Fig AK in S1 Text.

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Simulating experimental manipulations to foot properties

To test whether simulations with our model can capture the effects of manipulating foot properties, we simulated walking and running with an insole that limits arch compression [6] and walking with local anaesthesia of intrinsic foot muscles [7].