Computer Methods in Biomechanics and Biomedical Engineering
Vol. 12, No. 4, August 2009, 481–490
Computer simulation of the effects of shoe cushioning on internal and external loading during
running impacts
Ross H. Miller* and Joseph Hamill
Department of Kinesiology, University of Massachusetts, 110 Totman Building, 30 Eastman Lane, Amherst, MA 01003, USA
( Received 20 August 2008; final version received 16 December 2008 )
Biomechanical aspects of running injuries are often inferred from external loading measurements. However, previous
research has suggested that relationships between external loading and potential injury-inducing internal loads can be
complex and nonintuitive. Further, the loading response to training interventions can vary widely between subjects. In this
study, we use a subject-specific computer simulation approach to estimate internal and external loading of the distal tibia
during the impact phase for two runners when running in shoes with different midsole cushioning parameters. The results
suggest that: (1) changes in tibial loading induced by footwear are not reflected by changes in ground reaction force (GRF)
magnitudes; (2) the GRF loading rate is a better surrogate measure of tibial loading and stress fracture risk than the GRF
magnitude; and (3) averaging results across groups may potentially mask differential responses to training interventions
between individuals.
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Keywords: running injury; computer simulation; internal loading
1.
Introduction
External force variables associated with the rapid loading
of the body during the impact phase of running have
discriminated between runners with and without a history
of tibial stress fracture (Milner et al. 2006a,b). These
external loading variables, such as the ground reaction
force (GRF), are often used as surrogate measures for
internal bone loading and injury since direct measurement
of bone loading is difficult in human subjects (Crossley
et al. 1999; Bennell et al. 2004; Milner et al. 2006a,b).
However, most of these authors have recognised that the
GRF is only one of many factors that contribute to internal
loading. Along with external measures, a useful outcome
from mechanical analyses of human running is the
estimation of individual muscle forces during the stride
(Scott and Winter 1990; Glitsch and Baumann 1997;
Neptune et al. 2000a; Thelen et al. 2005). Muscle forces
provide information on the neuromuscular control of the
movement, and play a major role in determining the
internal contact forces at the joints. Joint contact forces are
often implicated in the aetiology of bone stress injuries to
runners (Crossley et al. 1999; Bennell et al. 2004;
Sasimontonkul et al. 2007).
Knowledge of both internal and external loading
would allow for an assessment of how factors that may
modify external loading influence internal loading.
Running injuries in general are thought to arise from the
internal loading of susceptible tissues at stress magnitudes
*Corresponding author. Email: rhmiller@kin.umass.edu
ISSN 1025-5842 print/ISSN 1476-8259 online
q 2009 Taylor & Francis
DOI: 10.1080/10255840802695437
http://www.informaworld.com
and frequencies beyond their capabilities to remodel
(Hreljac 2004). A drawback to investigating running
injuries with markers of external loading only is that
relationships between internal and external loading are
often complex and nonintuitive.
Scott and Winter (1990) reported that the peak vertical
impact force had little effect on peak forces estimated at
chronic running injury sites. Sasimontonkul et al. (2007)
reported that compressive bone contact forces at the distal
tibia estimated by static optimisation were primarily due to
muscle forces spanning the ankle, which are not mirrored
by GRFs or the resultant ankle joint forces. Wright et al.
(1998) suggested that changes in peak muscle forces with
simulated footwear conditions were not reflected by
changes in the peak impact force, and that responses to
footwear conditions varied widely between subjects.
Changes in peak muscle force magnitudes were less
variable, but the authors did not investigate in detail how
these changes may influence injury potential. Bates et al.
(1983) reported statistically significant subject – shoe
interactions when comparing GRF parameters between
five different running shoes. These findings suggest that
our understanding of running injuries may benefit by a
shift in focus from external to internal loading, and by a
greater emphasis on subject-specific investigations.
Therefore, the purpose of the study was to demonstrate
a computer simulation technique for investigating subjectspecific relationships between training interventions,
482
R.H. Miller and J. Hamill
internal loading and external loading during running. In
this example, we focus on the impact phase, shoe
cushioning, distal tibial loading and stress fracture
aetiology. The design of the study is similar to other
modelling studies of running (Gerritsen et al. 1995; Cole
et al. 1996; Wright et al. 1998), but is distinguished from
these studies in three regards: (1) the inclusion of a swing
leg in the musculoskeletal model, (2) an emphasis on
subject-specific simulations, results and conclusions and
(3) a focus on internal loading at a specific site of a
common running injury.
Methods
2.1 Subjects
Data were collected from two subjects who participated in
accordance with the local university institutional review
board. The male subject (age 27 years, height 1.84 m, mass
77.1 kg) was an experienced recreational distance runner
with no history of major lower extremity injuries. The
female subject (24 years, 1.66 m, 58.2 kg) was a former
NCAA Division I track athlete with a history of tibial
stress fractures, but healthy at the time of data collection.
2.2
Experimental setup
Coordinates of retro-reflective markers (diameter 15 mm)
were sampled at 500 Hz using an eight-camera optical
motion capture system (Oqus 300, Qualisys, Gothenburg,
Sweden). GRFs were sampled synchronously at 5000 Hz
using a strain gage force platform (model OR6-5, AMTI,
Watertown, MA, USA).
2.3
2.5 Musculoskeletal model
A sagittal-plane musculoskeletal model (Figure 1) was
developed using Autolev 4.1 software (OnLine Dynamics,
Sunnyvale, CA, USA). The model’s equations of motion
were derived using Kane’s method of dynamics (Kane and
Wang 1965). The model had nine segments [bilateral toes,
foot, calf and thigh, and a combined head-arms-trunk
(HAT) segment] and 11 degrees of freedom. The segments
were connected by ideal hinge joints. Segment anthropometrics were defined according to de Leva (1996).
Wobbling masses representing soft tissue oscillations and
the dynamics of segments not explicitly modelled (e.g. the
arms) were attached to the calf, thigh and HAT segments
by linear spring-dampers (Appendix A).
The model was actuated bilaterally by 16 Hill-based
muscle models representing iliopsoas, rectus femoris,
glutei, hamstrings, vasti, gastrocnemius, soleus and tibialis
anterior. Details on the muscle model algorithm are given
in Appendix B. Mechanical properties of the muscle
models were defined for each subject using a dynamometry and optimisation procedure described by Gerritsen
Protocol
Clusters of retro-reflective markers were placed bilaterally
on the forefoot, rearfoot, calf, thigh, pelvis and trunk for
the measurement of three-dimensional (3D) segment
kinematics. Subjects performed a standing calibration trial
and eight trials of overground running while the camera
system and force platform sampled marker coordinates
and GRF. The subjects ran in their own shoes at their selfreported 5-km training paces (male: 4.5 m s21; female:
3.1 m s21). Each subject completed five acceptable trials.
HAT
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2.
a Cardan rotation technique (Davis et al. 1991) and a
flexion/extension – adduction/abduction – internal/external rotation sequence. The stance phase of each trial was
isolated based on the magnitude of the vertical GRF
component (threshold ¼ 15 N). Joint angles and GRF
from each trial were interpolated to 101 points and
averaged across trials. Between-trial standard deviations
were also calculated.
g
Th
igh
Calf
2.4
Data analysis
Marker coordinates and GRF were digitally filtered by a
fourth-order lowpass recursive Butterworth filter. Cut-off
frequencies were selected by residual analysis (Winter
1990). Bilateral sagittal angles of the trunk, hips, knees,
ankles and metatarsophalanges during the stance phase of
the right leg were calculated from the marker data using
N2
t
Foo
s
Toe
N1
Figure 1. Musculoskeletal model. HAT ¼ head, arms and
trunk. Swing leg muscles are not shown.
Computer Methods in Biomechanics and Biomedical Engineering
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et al. (1998). Equations for muscle lengths and moment
arms were polynomial functions of the appropriate joint
angles fit to data from the OpenSim lower extremity model
(Delp et al. 2007) after scaling the model’s segment
lengths to those of the subject. Passive joint moments were
defined at the hips, knees and ankles according to Silder
et al. (2007) and at the metatarsophalanges according to
Yamaguchi (2001).
Ground contact of the stance foot was modelled by
three elements located beneath the heel, toe and
metatarsophalange. The elements modelled vertical
contact force as a nonlinear spring-damper (Anderson
and Pandy 1999) and horizontal contact force as an
approximation of Coulomb friction (Song et al. 2001).
2.6 Simulation
Forward dynamic computer simulations of the stance
phase were formulated as an optimal control problem to
find the muscle excitation histories u(t) that minimised
errors between state variables XM(t) and target states XE(t).
The entire stance phase was simulated, but only the initial
impact phase (first 40 ms) was analysed. The state
variables entered into the objective function were the
model’s hip, knee and ankle joint angles and horizontal
and vertical GRF components, while the target states were
these same experimental means from the human subjects.
Simulations were performed on a subject-specific basis,
with the model’s segment lengths and inertial parameters,
wobbling mass parameters, and muscle model parameters
defined for the subject in question. The problem was
reduced to a parameter optimisation problem (Pandy et al.
1992) by defining each muscle excitation history as a
piecewise linear function of nine nodal values spaced
evenly across the stance phase. Each node could take on
any value from zero (no excitation) to one (full excitation).
The four stiffness and damping parameters of the HAT
wobbling mass and the initial muscle active states were
also allowed to vary as controls, for a total of 164 control
variables. The objective function minimised was:
OF ¼
"
tf !
8 X
X
X M;i ðtÞ 2 X E;i ðtÞ 2
i¼1 t¼o
SDi
;
ð1Þ
where SDi is the average between-trial standard deviation
of the ith target state and tf is the duration of stance.
Optimisations were performed using a simulated annealing algorithm (Goffe et al. 1994) on a 2.4 GHz Core 2
Duo CPU (Intel, Santa Clara, CA, USA). Optimisation
terminated when 16,000 consecutive function calls had not
reduced OF by 1% of its current lowest value. Initial
generalised coordinates were taken from the experimental
data at heel-strike. Initial wobbling mass coupling forces
483
were zero, and initial muscle model series elastic lengths
were set to their unloaded lengths.
After identification of the optimal controls, simulations
were repeated with the vertical stiffness and damping
parameters of the ground contact elements adjusted to
approximate harder or softer shoes (Aerts and de Clercq
1993). The parameters were adjusted such that simulated
impact tests on the heel element produced forcedeformation slopes of approximately 50 kN m21 for the
‘softer’ condition and 400 kN m21 for the ‘harder’
condition. These values were chosen to approximate the
‘soft’ and ‘hard’ force-deformation slopes from impact
tests of commercial running shoes (Aerts and de Clercq
1993). The initial ‘normal’ stiffness was 214 kN m21.
Simulations with the ‘softer’ and ‘harder’ parameters were
then performed using the controls from the optimised
simulations. An assumption inherent to these simulations
is that changes in footwear do not affect muscle activity
during impacts. Previous research has found no differences in lower extremity muscle activity (quantified via
electromyography) in the temporal domain when running
in different types of shoes (Komi et al. 1987; O’Connor
and Hamill 2004), although differences have been found in
the frequency domain (Wakeling et al. 2002).
Outcome variables were the peak vertical GRF,
vertical GRF loading rate and peak ankle joint contact
force in the calf reference frame (i.e. the contact force on
the distal tibia). Vertical loading rate was calculated as the
slope of the vertical GRF component between 20 and 80%
of the time from heel-strike to the impact peak (Milner
et al. 2006b). Tibial contact force was computed as 90% of
the difference between the resultant ankle joint force and
the vector sum of ankle muscle forces (Lambert 1971;
Funk et al. 2004; Sasimontonkul et al. 2007).
3.
Results
3.1 Computational performance
The optimisations met the termination criterion after
approximately 400,000 function calls per subject. Figure 2
compares the optimal simulated and experimental joint
angles and GRF for the male subject. The state variables
generally fell within two between-trial standard deviations
of the experimental means, with the exception of the
horizontal GRF and the stance leg ankle angle. Agreement
between the model and experimental data is not entirely
unexpected (recall the optimisation problems attempted to
match these data as closely as possible), but provides
confidence in the model’s ability to represent the salient
kinetic and kinematic characteristics of the impact phase.
The discrepancies between the simulated and experimental ankle angle and horizontal GRF are likely due to
the limitations of the foot model and the weighting scheme
of the objective function (Equation (1)). The foot model
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R.H. Miller and J. Hamill
Hips
0
Knees
Ankles
200
40
– 10
10
20
– 20
0
0
–30
–10
–20
– 40
– 20
80
– 60
15
60
– 80
0
2000
10
0
–120
0
– 100
– 300
–100
20
100
– 200
5
40
300
20
AP GRF (N)
60
VE GRF (N)
Left joint angle (deg)
Right joint angle (deg)
484
–5
–140
–10
–20
0
25 50 75 100
0
25 50 75 100
0
25 50 75 100
Right Leg Stance (%)
Right Leg Stance (%)
Right Leg Stance (%)
1500
1000
500
0
0
25 50 75 100
Right Leg Stance (%)
Figure 2. Comparison of lower extremity joint trajectories (left six panels) and GRFs (right two panels) during right leg stance between
the optimised simulation and the experimental data for the male subject (77.1 kg) running at 4.5 m s21. Zero percentage stance denotes
heel-strike and 100% stance denotes toe-off. The shaded areas are two between-trial standard deviations around the experimental means.
Solid lines are the simulation results. Data prior to the vertical dashed lines denote the impact phase.
consisted of two rigid segments with no intrinsic muscles
and only three discrete contact points, while the real
human foot has at least four functional segments (Wolf
et al. 2008), many intrinsic muscles (Basmajian and
Stecko 1963) and a continuum of contact points. We have
found the weighting scheme (based on experimental
between-trial variability) tends to favour the vertical GRF,
which we felt was appropriate since the vertical impact
peak was a primary outcome variable.
3.2
Effects of shoe cushioning
Between the simulated ‘softer’ and ‘harder’ shoe
cushioning conditions, the magnitude of the impact peak
increased by 17 N (2% body weight) for the male subject
but decreased by 40 N (7% body weight) for the female
subject (Figure 3). Between the ‘softer’ and ‘harder’
conditions, the vertical GRF loading rate increased from
44 to 55 kN s21 for the male subject and from 47 to
58 kN s21 for the female subject. Table 1 shows the
simulated peaks of angular position and velocity for the
subjects’ right ankles. Between the ‘softer’ and ‘harder’
conditions, the peak ankle angle became more dorsiflexed,
and the peak ankle dorsiflexion velocity increased.
As shoe cushioning increased from ‘softer’ to ‘harder’,
the magnitude of the peak tibial compressive force
increased by 136 N (18% body weight) for the male
subject and by 120 N (21% body weight) for the female
subject (Figure 4, top panel). The peak tibial shear force
increased by 32 N (4% body weight) for the male subject
and by 35 N (6% body weight) for the female subject
(Figure 4, bottom panel).
4.
Discussion
The purpose of the study was to demonstrate the utility of
computer modelling and simulation for subject-specific
investigations of training interventions and internal/external
loading during running. The two subjects studied exhibited
different kinetic responses to the simulated footwear
adjustments. One subject showed a small increase (þ2%
body weight) in the impact peak from the ‘softer’ to ‘harder’
conditions, but the other subject showed a moderate
Figure 3. Simulated vertical GRF peaks during the impact
phase with different shoe cushioning levels. Filled bars are the
male subject (77.1 kg). Empty bars are the female subject
(58.2 kg).
Computer Methods in Biomechanics and Biomedical Engineering
Table 1. Peak simulated ankle angles (u) and angular velocities
(v) during the impact phase for the male and female subjects with
each shoe cushioning condition.
Shoes
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Male
Softer
Normal
Harder
Female
Softer
Normal
Harder
Ankle u (8)
Ankle v (8s21)
8.5
9.9
12.0
299.1
323.2
360.0
7.6
9.2
10.8
211.5
237.5
279.4
decrease (27% body weight). However, both subjects
exhibited increases in peak tibial compression on the order
of 20% body weight between the softer and harder
conditions. Peak tibial shear and the vertical GRF loading
rate also increased for both subjects.
These findings support the conclusion of Wright et al.
(1998) that the effects of shoes on internal loading are not
reflected by changes in external loading. While Wright
et al. (1998) concluded that shoe cushioning had no effect
on peak impact forces based on statistical analysis of nine
subjects, inspection of their results suggests substantial
Figure 4. Peak compressive (top) and shear (bottom) tibial
contact forces predicted by the simulations with different
simulated shoe cushioning levels. Filled bars are the male subject
(77.1 kg). Empty bars are the female subject (58.2 kg).
485
variability in the impact peak responses of different
subjects, with apparent changes on the order of 2 100 to
þ 200 N depending on the subject. Similar betweensubjects variability in impact peak responses to midsole
hardness, particularly at faster running speeds, appeared in
Nigg et al. (1987), although they did not report the
consistency of the responses between subjects. Bates et al.
(1983) reported that five runners with similar body weights
varied widely in their individual impact peak responses
when running in five different shoes. The present results
suggest further that relationships between internal and
external loading should be reported on a subject-specific
basis, since different subjects do not appear to make
identical external loading responses when faced with an
intervention, even if internal loading responses are similar
between subjects.
Recent studies of running have drawn inferences on
internal loading from measures of external loading, based
on ensemble results from groups of subjects (Barrett et al.
2008; Cheung and Ng 2008; O’Leary et al. 2008). We
suggest that additional insights on the aetiology of running
injuries and strategies for injury prevention can be gained
from subject-specific investigations and simulations of
internal loading.
Most simulation studies of running have used muscle
model mechanical properties referenced from literature
sources (Gerritsen et al. 1995; Cole et al. 1996; Wright
et al. 1998; Neptune et al. 2000a,b; Thelen et al. 2005;
Chumanov et al. 2007). Since simulated GRF magnitudes
can depend strongly on the input muscle model
mechanical properties (Scovil and Ronsky 2006), we
suggest the use of subject-specific mechanical properties is
justified, and perhaps critical, when simulating the
responses of individual runners to a training intervention.
A final conclusion relates to the investigation of tibial
loading during running. Because increases in tibial shear
and compression estimates for both subjects were
accompanied by increases in the vertical GRF’s loading
rate but not its impact magnitude, the results provide
evidence that the loading rate is a stronger surrogate
measure of the tibial contact force than the impact
magnitude. In these subjects, the increased loading rate of
the GRF appeared to increase the speed of ankle
dorsiflexion during impact (following the rapid plantarflexion immediately after heel-strike), which stretched
soleus eccentrically and shortened tibialis anterior, placing
these muscle models at locations on their force– velocity
and force– length relationships, respectively, where they
could generate larger forces, resulting in greater tibial
contact forces. Hardin et al. (2004) found similar results in
human runners: running in harder midsoles increased the
peak ankle dorsiflexion velocity for 12 male runners.
However, since we did not establish a direct causal
relationship between these variables and have only
simulated two specific individuals, the conclusion that
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486
R.H. Miller and J. Hamill
loading rate is a better measure of tibial loading is
suggested only on the basis of correlation, and may not be
generalisable to all runners.
Some interesting questions are motivated from the
present results: by what mechanism(s) are external impact
forces maintained across different footwear conditions?
Why does this maintenance happen, and what are its
physiological costs and benefits? This effect has been
demonstrated in previous simulations (Wright et al. 1998)
as well as experimental studies (Clarke et al. 1983; Nigg
et al. 1987), although it is unclear if this effect is consistent
across subjects (Bates 1989). The present results support
the findings of Wright et al. (1998) that impact force peaks
can be modulated by passive mechanisms, i.e. by changes
in muscle kinematics with no change in muscle activation.
Here, ‘passive’ refers to changes in muscle forces in the
absence of changes in muscle activation, rather than forces
generated by purely passive structures such as ligaments.
In simulations of these two subjects, the increases in tibial
contact forces with increasing shoe hardness were
primarily due to concurrent increases in the peak forces
of tibialis anterior and, to a lesser extent, soleus, as noted
earlier. The increased muscle forces in the ‘harder’
condition may contribute to the greater oxygen costs
observed when running in harder shoes (Frederick et al.
1986). Since these changes occurred with muscle
activations held constant across all conditions, they must
have been modulated by changes in ankle joint kinematics
(see Table 1).
An answer to the latter question (costs and benefits of
regulating impact forces) is less clear. Impact force
regulation is possibly related to shock attenuation and the
maintenance of head accelerations (Shorten and Winslow
1992; Derrick et al. 1998). For both subjects, a small
change in the impact peak was accompanied by a
relatively larger increase in the estimated tibial contact
force. Whether this increase represents an increased injury
risk is debatable. Greater bone contact forces could induce
a stronger osteogenic response resulting in improved bone
strength, assuming adequate rest. Bone failure is more
directly due to the resulting strain rather than the applied
force, and we cannot estimate bone strain in these subjects
since bone geometries and material properties were
unknown. It is therefore unknown how these estimated
loads compare to the strength of cortical bone in cyclical
loading (Carter et al. 1981; Pattin et al. 1996). However, an
increase in the contact force places the tibia closer to this
threshold and could contribute to a bone stress injury,
particularly if inadequate recovery time is allowed
between loading bouts. It is also notable that the peak
contact force during impact is much smaller than the peak
during mid-stance (Sasimontonkul et al. 2007). In this
study, we focused on the impact phase since previous tibial
stress fracture research has identified external loading
discriminators during impacts (Milner et al. 2006a,b).
However, bone loading during the remainder of stance
may also be an important injury factor.
Until noninvasive measurements of internal forces
during human movement become more practical, validating simulation-based estimates of muscle and bone contact
forces will remain challenging. However, the use of
subject-specific muscle mechanical properties, good
quantitative agreement between the modelled and
experimental kinematics and kinetics, and qualitative
agreement between the simulated muscle excitations and
typical electromyographic patterns during running (Novacheck 1998) provide some confidence in the estimates.
Peak simulated Achilles tendon forces (6.5 body weights)
were comparable to in vivo measurements (Komi 1990).
The fact that the simulations predicted greater relative
tibial contact forces for the subject with a stress fracture
history (240% body weight) than the subject with no
history (223% body weight), even when running at a
slower speed, is also promising.
The design of our model was similar to earlier running
models (Gerritsen et al. 1995; Cole et al. 1996; Wright
et al. 1998; Neptune et al. 2000a) but differed by the
inclusion of a swing leg. The previous studies modelled
only the stance leg segments and combined the swing leg
into the inertial properties of a ‘rest-of-body’ segment.
While we did not perform simulations of these two
subjects with a one-legged model, in developing the
present model we found that a one-legged model had
difficulty simulating accurate trunk kinematics and impact
forces. We encourage future locomotion simulations to
consider modelling both legs, even if data from one-legged
stances are of primary interest.
There are several limitations to this study. The model
was confined to the sagittal plane only. While running is
primarily a sagittal-plane motion, kinematics in secondary
planes has discriminated between healthy runners and
runners prone to injuries such as patellofemoral pain
(Heiderscheit et al. 2002) and iliotibial band syndrome
(Noehren et al. 2007). A suitable computer model for
investigating these injuries would likely need to be capable
of 3D motion. The assumption that footwear does not alter
muscle activity during impacts is justified in some
previous studies (Komi et al. 1987) but appears
questionable based on others (Wakeling et al. 2002). The
advantage of this assumption is that it allowed us to
investigate the effects of changing a single parameter
(shoe cushioning), which would be difficult to accomplish
experimentally. When running in different shoes, human
subjects may, for example, strike the ground with different
initial kinematics, which can alter lower extremity loading
(Gerritsen et al. 1995; Derrick et al. 2002). Finally,
although the simulations were limited to a single stance
phase, previous research has implicated stride-to-stride
fluctuations in running mechanics with overuse injuries
(Hamill et al. 1999; Heiderscheit et al. 2002; Miller et al.
Computer Methods in Biomechanics and Biomedical Engineering
2008). As computers continue to increase in power and
decrease in cost, perhaps simulations of many consecutive
strides will be common in the future.
Acknowledgements
The authors gratefully acknowledge Brian Umberger for advice
and assistance in developing the model and Brent Edwards for
comments on the manuscript. Funded by the American Society of
Biomechanics graduate student grant-in-aid program. Readers
interested in acquiring any of the codes, parameters, or data used
in these simulations are encouraged to contact the corresponding
author (RHM).
Downloaded By: [Miller, Ross] At: 15:46 13 August 2009
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Appendix A: Wobbling mass model
The wobbling masses of the HAT, thighs and calfs were assigned
50, 66 and 46% of the total segment mass, respectively
(Gruber et al. 1998). The rest of the segment mass was assigned
to the rigid body skeletal segment. We have found the simulated
vertical GRF impact forces and tibial forces to be relatively
insensitive (#10% change in peak magnitude) to moderate
changes in mass distribution (^20% change in wobbling mass).
However, with no wobbling masses the simulations predicted
very large, unrealistic impact forces. We caution against defining
these distributions from bone versus non-bone cadaver tissue
masses, since muscle activity influences what proportion of a
segment behaves dynamically as a wobbling mass (Karlsson and
Tranberg 1999; Wakeling and Nigg 2001), and since
physical activity and aging have profound effects on bone
density (Lane et al. 1990).
Wobbling masses were attached to the skeleton by linear
translational spring-dampers located at the segment’s centre-of-mass.
Computer Methods in Biomechanics and Biomedical Engineering
The coupling force between a wobbling mass and its associated
segment was:
F c ¼ kDr þ cD_r;
ðA1Þ
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where Fc is the coupling force, Dr the vector between the wobbling
mass and the segment centre-of-mass, k the spring stiffness
coefficient and c the damping coefficient. Each mass was attached
to the segment by two spring-dampers, one oriented along the
longitudinal segment axis, and one perpendicular to this axis, which
allowed the wobbling masses to have different vibrational
characteristics in these two directions.
For the thigh and calf segments, values fork and c were computed
from data presented by Wakeling and Nigg (2001). For the thigh, the
damped natural frequency ( fDN) was assumed to be 16 Hz
(longitudinal direction) and 18 Hz (perpendicular direction) and the
damping rate (s) was assumed to be 34 Hz (longitudinal direction)
and 29 Hz (perpendicular direction). For the calf, fDN was assumed to
be 22 Hz (longitudinal direction) and 18 Hz (perpendicular direction),
and s was assumed to be 28 Hz (longitudinal direction) and 23 Hz
(perpendicular direction). The stiffness and damping coefficients
were then computed from under-damped vibration theory.
The damping ratio (z) was (Formenti 1999):
s
z ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
2
s þ ð2pf N Þ2
ðA2Þ
489
length Lm was the sum of the CC and SEC lengths:
Lm ¼ LCC þ LSEC ;
ðB1Þ
where LCC is the length of the CC, and LSEC is the length of the
SEC. Similarly, the total muscle velocity vm was the sum of the
CC and SEC velocities:
vm ¼ vCC þ vSEC ;
ðB2Þ
where vCC is the CC velocity, and vSEC is the SEC velocity. Force
production by the CC was a function of the active state and the
CC kinematics:
F CC ¼ f ðA; LCC ; vCC Þ;
ðB3Þ
where FCC is the force produced by the CC. A is the active state,
defined as the muscle’s intrinsic capacity to produce force
(Gasser and Hill 1924). Since the muscle model did not explicitly
consider muscle fibre pennation, the CC force, SEC force and the
total muscle force, were equivalent:
F m ¼ F CC ¼ F SEC ;
ðB4Þ
where Fm is the total muscle force and FSEC is the force sustained
by the SEC.
where fN is the natural frequency of the system:
f DN
f N ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi :
1 2 z2
ðA3Þ
B.1
Excitation-activation
By combining Equations (A2) and (A3), the damping ratio could be
approximated iteratively:
Neural input to the muscles was modelled as a single parameter
E that represented the total contributions of all excitation sources.
CC excitation-activation dynamics were described by a firstorder ordinary differential equation (He et al. 1991):
!
"
2pf DN 2
1
þ 2 < 2 þ z 2:
s
z
!
"
E
E
1
A_ ¼ ðE 2 AÞ
2 þ
;
tR tF tF
ðA4Þ
The spring stiffness (k) was then computed from the natural frequency
definition:
fN ¼
1
2p
rffiffiffi
k
) k ¼ mð2pf N Þ2 ;
m
ðA5Þ
where m is the mass of the wobbling mass. The damping coefficient
(c) was computed from the definition of the damping ratio:
pffiffiffiffiffiffi
c
z ¼ pffiffiffiffiffiffi ) c ¼ 2z km:
2 km
ðA6Þ
Stiffness and damping coefficients were assumed to be bilaterally
identical for the left and right leg. Since the HAT segment represented
all segments not explicitly included in the model, its vibrational
characteristics were unknown. Therefore the HAT stiffness and
damping coefficients were entered as control variables in the
optimisation.
Appendix B: Muscle model
The muscle model was a two-component Hill-based model (Hill
1970), with a contractile component (CC) in series with an elastic
component (SEC). Parallel elasticity was accounted for by the
passive joint moment functions. At any time, the total muscle
ðB5Þ
where E is the neural excitation. Both A and E varied from zero
(no activation/excitation) to one (full activation/excitation).
The parameters tR and tF are time constants that define the time
lags between activation (rise time), deactivation (fall time), and
excitation.
B.2
CC force –length
The CC force – length relationship (Gordon et al. 1966) was
described by an inverted parabola (Woittiez et al. 1983):
!
"
1 LCC
FL ¼ 2 2
2 1 2 þ 1;
Lo
W
ðB6Þ
where Lo is the optimal CC length and W is a parameter
describing the parabola width. For example, if W ¼ 0.5, the CC
could produce force when it was at lengths between 0.5 and 1.5
times Lo.
B.3
CC force –velocity
The concentric CC force – velocity relationship was described by
a double-hyperboloid, modified from the Hill (1938) original
formulation to scale the maximum isometric force for the current
490
R.H. Miller and J. Hamill
force – length and activation:
!
"
F m 2 F ov
;
vCC ¼ b
Fm þ a
F ov ¼ F o · A · FL;
ðB7Þ
ðB8Þ
where Fo is the maximum isometric muscle force (i.e. the muscle
force when A ¼ 1, FL ¼ 1 and vCC ¼ 0). The parameters a and
b are Hill’s dynamic constants (Hill 1938), which define the
shape of the parabola. The eccentric force– velocity relationship
(Katz 1939) was described according to FitzHugh (1977):
!
"
F m 2 F ov
vCC ¼ S
;
ðB9Þ
F m 2 CS F ov
!
"
CS F ov 2 F ov
S¼b
;
ðB10Þ
F ov þ a
where CS is the eccentric force plateau.
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B.4
SEC force– extension
The SEC force – extention relationship was described by an
exponential function (Caldwell 1995):
$ !
"%
LSEC
2 1 2 CK ;
ðB11Þ
FDL ¼ CK exp K
Lu
F m ¼ F o FDL;
ðB12Þ
where CK is the SEC scaling coefficient, K the SEC stiffness,
LSEC the SEC length and Lu the unloaded SEC length.
B.5
Muscle mechanical properties
Each muscle model required 11 input parameters that defined its
excitation-activation, force– length, force – velocity and force –
extension relationships: tR and tF , the activation and deactivation
time constants; Fo, the maximum isometric force; Lo, the optimal
CC length; W, the CC force – length parabola width; a and b, the
Hill force – velocity constants; CS, the eccentric force plateau; Lu,
the unloaded SEC length, K, the SEC stiffness and CK, the SEC
scaling coefficient. Previous research has demonstrated that the
data tracking accuracy of forward dynamics simulations depends
strongly on the values assigned to some of these parameters
(Scovil and Ronsky 2006). If subject-specific simulations are
desired, it is therefore important that muscle model parameters be
defined on a subject-specific basis when possible.
The excitation-activation time constants were defined as
functions of the muscle fibre type composition:
tR ¼ 30 2 25 · FT;
ðB13Þ
tF ¼ 45 2 30 · FT;
ðB14Þ
where FT is the proportion of fast-twitch fibres, from zero to one
(Winters and Stark 1985). Equations (B13) and (B14) give the
time constants in units of milli-seconds. Data for FT were
referenced from Yamaguchi et al. (1990) and the same values of
FT were assumed for both subjects.
Hill’s dynamic constants a and b were also defined as
functions of FT (Winters and Stark 1985):
aR ¼ 0:1 þ 0:4 · FT;
ðB15Þ
bR ¼ aR ð4 þ 8 · FTÞ;
ðB16Þ
where aR and bR are the relative (dimensionless) dynamic
constants. The dimensional parameter b was the product of bR
and Lo. The dimensional parameter a was the product of aR and
Fo after scaling Fo for the current force– length and activation
levels:
a ¼ aR · F o · FL · A:
ðB17Þ
Finally, a was multiplied by A 20.3 to account for the activationdependency of the force – velocity relationship (Umberger et al.
2003). The eccentric force plateau was assumed to be CS ¼ 1.45
for all muscles, which is the mean value presented by Zajac
(1989).
The SEC stiffness and scaling coefficients were defined such
that each muscle model produced Fo when the SEC was extended
by 4% of its unloaded length (Ettema and Huijing, 1989). This
assumption corresponded to values of K ¼ 92.08 and
CK ¼ 0.0258 N.
The remaining four parameters (Fo, Lo, W and Lu) were
determined for each subject using the dynamometry and
optimisation procedure described by Gerritsen et al. (1998).
K, CK, CS and FT were also allowed to vary within a ^10% range
during the optimisation.