Download Electronic Supplemental Material A simple planar computer model

Electronic Supplemental Material
A simple planar computer model with one degree of freedom was developed in Simulink
(The Mathworks; Natick, MA, USA) to investigate the influence of foot length proportions on
total muscular work performed during forward dynamic simulations of maximal-effort
plantarflexion. A foot segment represented by a 28 cm long rigid bar with negligible mass and
inertia (m = 10-10 kg, I = 0 kg m2) was attached by a revolute joint to a shank segment (which
was itself rigidly attached to the ground). The ‘toe’ of the foot segment was attached by a
revolute joint to a slider mounted on wall, permitting the toe to slide without friction along the
surface of the wall. This arrangement caused the wall reaction force to be aligned with the long
axis of the tibia in a manner similar to way in which a sprinter’s ground reaction force is aligned
with the tibia during push-off [1]. Three Hill-type muscle-tendon actuators, representative of the
triceps surae group, applied force to the ‘heel’ of the foot segment during simulations. The
soleus muscle attached to the shank segment 22 cm from the ankle joint and both heads of the
gastrocnemius were attached to the shank segment 0.40 m from the ankle joint. Muscle
pennation angles, optimal fiber lengths, and peak isometric forces for the muscles in the model
were specified according to recent cadaver measurements by Arnold et al. [2] and are given in
Table S1. Tendon slack lengths were determined geometrically by subtracting the optimal fiber
lengths of each muscle from the distance between the muscle origins and insertion point when
the ankle was positioned in 10° plantarflexion.
To represent variation in foot proportions, the ankle joint location with respect to the heel
(the most posterior point on the foot segment and the insertion of all three muscles) ranged from
45 mm to 70 mm, and was varied in 5 mm increments (these were the ranges for pfMA that we
measured for our subjects). Static optimization was performed at the beginning of each
simulation to find muscle fiber lengths that satisfied the constraint that tendon force and muscle
force must be equal for each actuator. During each simulation, the wall was prescribed to recede
at a constant speed (0.4 m s-1 to 4.0 m s-1), which corresponded to a range of ankle velocities
observed during push-off in walking, sprint starts, and maximal speed sprinting [3–5]. Each
muscle was maximally activated throughout the entire trial. Simulations started at 10°
dorsiflexion and stopped when the sum of the three active muscle forces became 0 N or when
plantarflexion exceeded 50°. Muscle work was derived by taking the integral with respect to
time of the product of the wall speed and the force component that was perpendicular to the wall.
Variable timestep integration was implemented using the ode45 function in MATLAB with a
maximum step size of 10-3.
Lateral
Gastrocnemius
Peak Isometric Force (N)
3585.9
606.4
Optimal Fiber Length (mm)
44
59
Pennation Angle (°)
28.3
12.0
Origin to Ankle Distance (cm)
22
40
Table S1. Muscle parameters adapted from Arnold et al. [2].
Soleus
Medial
Gastrocnemius
1308.0
51
9.9
40
REFERENCES
1 Kugler, F. & Janshen, L. 2010 Body position determines propulsive forces in accelerated
running. J. Biomech. 43, 343-348. (doi:10.1016/j.jbiomech.2009.07.041)
2 Arnold, E. M., Ward, S. R., Lieber, R. L. & Delp, S. L. 2010 A model of the lower limb for
analysis of human movement. Ann. Biomed. Eng. 38, 269-279. (doi:10.1007/s10439-0099852-5)
3 Winter, D. A. 1983 Energy generation and absorption at the ankle and knee during fast,
natural, and slow cadences. Clin. Orthop. Relat. Res. , 147-154.
4 Slawinski, J., Bonnefoy, A., Ontanon, G., Leveque, J. M., Miller, C., Riquet, A., Chèze, L. &
Dumas, R. 2010 Segment-interaction in sprint start: Analysis of 3D angular velocity and
kinetic energy in elite sprinters. J. Biomech. 43, 1494-1502.
(doi:10.1016/j.jbiomech.2010.01.044)
5 Bezodis, I. N., Kerwin, D. G. & Salo, A. I. T. 2008 Lower-limb mechanics during the support
phase of maximum-velocity sprint running. Med. Sci. Sports Exerc. 40, 707-715.
(doi:10.1249/MSS.0b013e318162d162)