The effect of prosthetic mass properties on the gait of transtibial amputees--a mathematical model.

Abstract

Present models in the literature, predicting that prostheses should not be too lightweight, are not supported by empirical evidence. Recent studies suggest that these models are incorrectly based on the assumption that the swing phase is uninfluenced by muscle activity. The purpose of the present study was to introduce a new mathematical model to predict the effect of mass properties on the gait of transtibial amputees, based on experimental findings that subjects adapt to mass perturbations by maintaining the same joint kinematics.
Effect of mass perturbations on the lower leg was evaluated in terms of muscular cost and forces between stump and socket, using a linked-segment model of the swing phase. Gait analysis and anthropometric data from 10 transtibial amputees were used as model input.
Location of perturbation strongly influenced the muscular cost. Cost generally increased after distally adding mass but decreased after proximally adding mass to the lower leg. Stump-socket interface forces always increased after mass addition.
A new model was introduced, predicting that the weight of distally located components (e.g. foot, ankle, shoe) strongly influence the estimated muscular cost, in contrast to proximal components. A comparison with experimental literature suggests this new model better describes the experimental data than existing models.

Full text

Page 1

The effect of prosthetic mass properties on the
gait of transtibial amputees—a mathematical
model
RUUD W. SELLES{*, JOHANNES B. J. BUSSMANN{,
A. J. ‘KNOEK’ VAN SOEST{, and HENK J. STAM{
{ Department of Rehabilitation Medicine, Erasmus MC—University Medical Center
Rotterdam, The Netherlands
{ Faculty of Human Movement Sciences and Institute of Fundamental and Clinical Human
Movement Sciences, Free University, Amsterdam, The Netherlands
Abstract
Purpose: Present models in the literature, predicting that
prostheses should not be too lightweight, are not supported
by empirical evidence. Recent studies suggest that these models
are incorrectly based on the assumption that the swing phase is
uninfluenced by muscle activity. The purpose of the present
study was to introduce a new mathematical model to predict
the effect of mass properties on the gait of transtibial
amputees, based on experimental findings that subjects adapt
to mass perturbations by maintaining the same joint kine-
matics.
Method: Effect of mass perturbations on the lower leg was
evaluated in terms of muscular cost and forces between stump
and socket, using a linked-segment model of the swing phase.
Gait analysis and anthropometric data from 10 transtibial
amputees were used as model input.
Results: Location of perturbation strongly influenced the
muscular cost. Cost generally increased after distally adding
mass but decreased after proximally adding mass to the
lower leg. Stump–socket interface forces always increased
after mass addition.
Conclusions: A new model was introduced, predicting that the
weight of distally located components (e.g. foot, ankle, shoe)
strongly influence the estimated muscular cost, in contrast to
proximal components. A comparison with experimental
literature suggests this new model better describes the
experimental data than existing models.
Introduction
Since the introduction in 1560 by Pare ´ of the first
‘modern’ above-knee prostheses made of steel, leather
and wood and weighing about 7 kg,1prosthetic mass
has been sharply reduced. Although not many will
dispute the beneficial effects of reducing the weight of
Pare ´ ’s prosthesis, the optimal weight of lower limb pros-
theses is still a topic of discussion. The current practice
of minimizing prosthetic mass has been challenged by
studies claiming that too much weight reduction leads
to a lower walking speed, increased energetic cost and
asymmetry between the movements of both legs.2–5
The idea that too much weight reduction of the lower
limb may not be optimal is based on the theory that the
swing phase of gait is passive, that is, not influenced by
muscle activity.6, 7According to this passive or ‘ballistic’
theory, during the swing phase the leg moves only under
the influence of gravitational and inertial forces. In a
passive swing phase, inertial properties (mass, centre of
mass location, and moment of inertia) influence the kine-
maticsoftheleginthesamewaythatthenaturalfrequency
of a pendulum is influenced by its inertial properties.
The prediction of the ballistic walking theory that
changing leg inertial properties will change the kine-
matics of gait, however, is in conflict with some of the
experimental literature in lower-limb amputees as well
as healthy control subjects. Selles et al.8reported in a
recent review that none of the eight studies investigating
the effect of prosthetic mass perturbation reported
significant effects on walking speed, stride length or
stride frequency. Recently, Mattes et al.9matched the
prosthetic inertial properties of transtibial amputees to
their contralateral leg by adding weights to the lower
leg and found more change in the metabolic cost than
in the kinematics of walking. In healthy subjects,
* Author for correspondence; Erasmus MC—University
Medical Center Rotterdam, Department of Rehabilitation
Medicine, P.O. Box2040,
The Netherlands. e-mail: r.selles@erasmusmc.nl
3000 CARotterdam,
DISABILITY AND REHABILITATION, 2004; VOL. 26, NO. 12, 694–704
Disability and Rehabilitation ISSN 0963–8288 print/ISSN 1464–5165 online # 2004 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/09638280410001704296

Page 2

Donker et al.10found that adding weight to the right
ankle during walking did not influence stride frequency,
while Skinner and Barrack11found that weight added to
the ankles increased the oxygen cost per metre but did
not influence walking speed and stride length. In a study
on running, Sanderson and Martin12measured a kine-
matic pattern in transtibial amputees similar to that of
controls while finding different joint torques.
In addition to these studies suggesting that gait kine-
matics do not always change after mass perturbation,
the ballistic swing phase assumption itself has been chal-
lenged in recent studies on healthy subjects13as well as in
transtibial amputation (TTA) subjects,14reporting that
the swing phase cannot be understood as completely
ballistic and that muscle activity needs to be included
to accurately model the swing phase of gait.
Recently, Selles et al.15compared the prediction of
the ballistic walking model that kinematics change
after mass perturbation with an alternative hypoth-
esis that kinematics remain the same after mass
perturbation while kinetics (joint torques) change: a
kinematic invariancestrategy.
from ten TTA subjects indicated that the response
to mass perturbation was best characterized as a
kinematic invariancestrategy.
important implications for understanding the effect
of prosthetic inertial properties. It suggests that iner-
tial properties of lower limb prostheses should not
be optimized in terms of gait kinematics and walk-
ing speed, since these variables do not change after
mass perturbation. Instead, inertial properties should
be optimized in terms of the muscle activity required
to obtain ‘normal’ gait kinematics.
The aim of the present study was to develop a
new mathematical model to predict the effect of
prosthetic mass, centre of mass location and moment
of inertia on the swing phase kinetics. The model
evaluates the effects of a wide range of different
inertial properties based on the assumption that
TTA subjects adapt to mass perturbations by main-
taining the same swing phase kinematics (kinematic
invariance).Foreach inertial
model calculates the muscular
subjects need to perform their ‘normal’ (measured)
walking pattern. In addition, while optimal prosthe-
tic mass has mostly been studied from the perspec-
tive ofgait kinematics
perturbationwill also
stump–socket interface. Therefore, the same model
will be used to obtain a first estimate of these forces
by modelling the stump–socket interface as a rigid
joint and calculating the forces in this joint.
Gaitanalysisdata
Thisfinding has
configuration,
cost
the
the which
andenergetics,
forces
mass
theinfluencein
Methods
SUBJECTS
Ten unilateral transtibial amputees participated in the
study. Subjects were included if they could walk without
assistance for at least half an hour, had no cardiopul-
monary, neurological or orthopaedic disorders other
than their amputation, had no skin problems of the resi-
dual limb, and had been discharged at least 1 year earlier
from the rehabilitation programme. The hospital’s
Medical Ethics Commission approved the study, and
all subjects signed an informed consent.
MEASUREMENTS
The assessment was done in a gait analysis lab on a
15-metre straight track. Kinematics of the lower extre-
mities during eight trials were recorded with a 50 Hz,
three-cameraProReflexinfrared
Sweden). The camera system covered about three metres
in the middle of the track. Reflective markers were
placed on the greater trochanter, the lateral femoral
condyle and on the prosthesis on the equivalent of the
lateral malleolus location.16Subjects were instructed to
walk at the speed at which they felt most comfortable,
and all subjects wore their preferred walking shoes.
system (Qualisys,
?
Figure 1
definition of the joints, segments, and angles. In addition, the
definition of the net joint reaction forces between socket and
stump are shown.
Segment model of the prosthetic leg, indicating the
Effect of prosthetic mass properties on gait
695

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Table 1
location and radius of gyration are calculated relative to the knee. Lower leg mass (stump and prosthesis) is normalized to body mass, while centre of
mass and radius of gyration are normalized to leg length. Reference values for the inertial properties of non-amputees17are indicated
Subject demographics as well as walking speed, prosthetic mass and the lower leg (stump and prosthesis) inertial properties. Centre of mass
Transtibial
amputee
Age
(yr)
Height
(m)
Weight
(kg)
Walking
speed
(m/s)
Mass
prosthesis
(kg)
Lower
leg mass
Centre
of mass
Radius
of gyration
1
2
3
4
5
6
7
8
9
10
38
38
48
42
51
71
35
44
25
50
1.78
1.88
1.76
1.81
1.78
1.57
1.86
1.68
1.85
1.84
83
113
77
72
93
74
71
74
69
105
1.28
1.41
1.31
1.41
1.17
1.36
1.39
0.74
0.88
1.29
3.0
3.4
2.0
2.5
2.4
2.3
2.5
2.4
2.7
3.0
0.052
0.042
0.036
0.049
0.039
0.050
0.047
0.044
0.052
0.052
0.538
0.633
0.461
0.545
0.614
0.630
0.612
0.510
0.531
0.489
0.725
0.765
0.651
0.750
0.779
0.788
0.783
0.700
0.726
0.647
Mean (SD)44.2
(12)
1.781
(0.1)
83.05
(15)
1.22
(0.23)
2.62
(0.41)
0.046
(0.01)
0.56
(0.06)
0.73
(0.05)
Non-amputees0.061 0.6060.735
—
—
—
—
—
—
Figure2
conditions: (1)the measured (unperturbed)condition, (2) adding1 kg justbelowthe knee,(3) removing 1 kg from justbelow theknee,
(4) adding 1 kg to the heel, and (5) removing 1 kg from the heel. Torques are normalized to body weight and leg length.
Netjointtorqueinhip(A)andknee(B)duringtheswingphaseforsubject1inthemeasuredaswellasduringfoursimulated
R. W. Selles et al.
696

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Body height, body mass and the lengths of thigh,
shank and foot of each subject were measured and used
to calculate mass, centre of mass and moment of inertia
of all segments.17The anthropometric properties of the
residual limb were determined using a geometric
model.18–20In addition, the prosthetic socket, shank
and foot were weighed, balanced on a straight edge to
determine centre of mass location and swung in a
pendulum with a small amplitude to determine the
moment of inertia.18, 21Combined properties of the resi-
dual limb and prosthesis were then calculated.
DATA ANALYSIS
A spline-fitting interpolation algorithm (Qview soft-
ware, Qualisys, Gothenburg, Sweden) was used to fill
marker dropouts in the kinematic data. A maximum
gap of six samples was found, occurring in the greater
trochanter marker because of arm swing. Marker trajec-
tories were filtered in Matlab (The MathWorks, Inc.,
Natick, MA, USA) with a second order low pass zero-
phase-lag Butterworth filter with a cut-off frequency of
9 Hz. Of the eight trials measured, six were selected
for analysis on the basis of the quality of the kinematic
data. In the six trials, complete gait cycles were selected
by marking foot contact in the ankle kinematics.22
Marker trajectories of the selected cycles were normal-
ised to the average cycle length and averaged. In the
averaged data, toe-off was selected using the fifth meta-
tarsal head kinematics.22Although 3D kinematic data
were measured, all kinematic and kinetic variables were
calculated in the sagittal plane only.
SIMULATIONS
Simulations were performed to evaluate the effect of
mass perturbations of the lower leg on the swing phase
of TTA subjects. A linked segment model of the pros-
thetic leg was used with the prosthesis (foot and shank),
stump and thigh as separate segments (figure 1). Pros-
thesis and stump were modelled as separate segments
to allow calculation of the net joint reaction forces
and torque in their interface. Relative motion between
stump and prosthesis was assumed to be negligible.
At first, net joint forces and torques without mass
perturbation were calculated using a linked segment
model, with the measured kinematics and segment iner-
tial properties as model input. This analysis was started
at the prosthetic segment to calculate net joint reaction
force and net joint torque at the stump–socket inter-
face; then, the inverse dynamical equations of motion
of the stump segment were solved to obtain the net joint
torque at the knee joint, and finally the equations of the
thigh segment were solved to obtain the net joint torque
at the hip (e.g.,).23
The effect of mass perturbation on the swing phase
was then modelled based on the assumption that kine-
matics do not change when mass is perturbed. Mass
perturbation was parameterized in terms of adding and
removing point masses of varying magnitude, at varying
locations. The amount of mass was varied in 11 steps of
0.5 kg from minus 2.5 kg (removing mass) to plus 2.5 kg
(adding mass). The location was varied in 12 steps of
10% of the knee–ankle length from directly below the
knee to 110%, the latter position estimating the location
of the heel. The whole lower leg was studied because
changes in sockets and liners can affect the properties
of the proximal part of the lower leg. Combining all loca-
tions and masses led to 132 mass conditions. In the
analysis, however, only conditions were considered in
which the perturbed segments had physically realistic
properties (a positive mass and positive moment of iner-
tia relative to the segment centre of mass).
The effect of each mass perturbation on the swing
phase was modelled by first calculating the new inertial
properties of the prosthetic leg. These inertial properties
were calculated by combining the initial inertial proper-
ties of the subject with the effect of the mass perturba-
tion. Then, the inverse dynamical equations were
0.4
0.2
TEhip+knee
0
0.4
0.2
0 –3
0
location (m)
3
mass change (kg)
Figure 3
masses ranging from 72.5 to +2.5 kg from locations
between knee and heel on TEhip+knee. *, zero-mass conditions;
*, increase-; ., decrease in TEhip+kneecompared to the zero-
mass condition.
Effects for subject 1 of adding and removing point
Effect of prosthetic mass properties on gait
697

Page 5

solved to calculate all joint torques and forces based on
the initial (measured) swing phase kinematics and the
new inertial properties after mass perturbation. The
outcome of the simulation of each mass perturbation,
therefore, was the net joint reaction forces and torque
in the stump–socket interface as well as the net joint
torques in hip and knee.
DATA COMPARISON
To improve inter-subject comparability, all torques
were normalised to body weight and leg length24and
the joint reaction forces were normalised to body
weight. An estimate of the total muscular cost
during the swing phase in the hip and knee was
?
??
???
???
Figure 4
from 72.5 to +2.5 kg from locations between knee and heel on TEhip+knee. For visualisation, only half of the simulated locations
are shown. *, zero-mass conditions; *, increase-; . decrease in TEhip+kneecompared to the zero-mass conditions.
Graphs similar to figure 3 for the remaining nine subjects, i.e. the effect of adding and removing point masses ranging
R. W. Selles et al.
698

Page 6

made using the ‘torque effort’ (TE) in the hip and
knee joint,
Zhs
TEhipþknee¼
to
ðjThipjÞdt þ
Zhs
to
ðjTkneejÞdt
ð1Þ
where Thipand Tkneeare the normalised net joint torques
in knee and hip and to and hs refer to the time of toe-off
and heel strike. This definition of ‘torque effort’ assumes
Table 2
2.5 kg
Maximum changes in the muscular cost (TEhip+knee) of the swing phase that can be obtained through mass perturbations of maximally
TEhip+knee
Maximum decrease (%)Maximum increase (%)
Transtibial amputeeAdding massRemoving mass Adding mass Removing mass
1
2
3
4
5
6
7
8
9
10
713
716
79
769
762
755
751
754
770
770
741
751
749
152
161
319
187
201
166
209
226
200
158
5
7
2
0
9
4
5
2
2
2
0
712
712
720
72
73
75
Mean
79.2
757 1984
—
—
0.5
0
–0.50
0.1 0.20.30.40.5
(A)
(B)
(C)
1
0.5
0
–0.5
–10
0.1
0.2 0.30.4 0.5
socket torque (Nm/kg/m)
asg. force (N/Kg)
long. force (N/kg)
1
0.5
0
–0.5
–10
0.10.2
0.30.4 0.5
time (sec)
Figure 5
sagittal- (B) and longitudinal (C) joint reaction force between
stump and socket during the swing phase for subject 1 in the
measured as well as in four simulated conditions: (1) the
measured (unperturbed) condition, (2) adding 1 kg just below
the knee, (3) removing 1 kg from just below the knee, (4)
adding 1 kg to the heel, and (5) removing 1 kg from the heel.
Typical examples of the net joint torque (A), and the
0.12
0.08
0.04
0
TEstump
0.6
0.4
0.2
0 –3
location (m)
0
3
mass change (kg)
Figure 6
ranging from 72.5 to +2.5 kg from locations between knee
and heel on TEstump. *, zero-mass conditions; *, increase-; .,
decrease in TEstumpcompared to the zero-mass condition.
Effects of adding and removing point masses
Effect of prosthetic mass properties on gait
699

Page 7

the cost of muscle contraction to depend on muscle force
only25and estimates the total muscular cost needed to
perform the swing phase.
Since the stump–socket interface was modelled as a
rigid joint, a torque in this joint is needed to maintain
prosthesis and stump parallel. An estimate of the total
torque between stump and socket was calculated as
Zhs
TEstump¼
to
ðjTstumpjÞdt
ð2Þ
in which Tstump was the torque (normalised to body
weight and leg length) in the interface between the sock-
et and stump. It should be noted, however, that in
reality this torque is established by a set of forces acting
between socket and stump. Therefore, the torque should
be interpreted as the net result of many forces between
socket and stump.
Results
The characteristics of the subjects are given in table 1,
as well as the initial mass of their prosthesis and the
anthropometric properties of the combined lower leg
(stump and prosthesis).
MUSCULAR COST OF THE SWING PHASE
Figure 2 shows a typical example of the normalised
hip and knee torque during the unperturbed condition,
as well as during four simulated conditions. Removing
mass at the heel decreased the maximum flexion and
extension torque in hip and knee compared to the
zero-mass condition, whereas adding mass at the heel
increased these variables. In contrast, removing mass
at the knee increased the flexion and extension torques
while adding mass decreased the torques needed to
maintain the same movement pattern. The magnitude
of the effect of mass perturbation around the knee was
much smaller than at the heel. The same effects were
found when evaluating all mass perturbations for the
same subject (figure 3): TEhip+kneedecreased when mass
was removed from the heel or added to the knee; in
contrast, it increased when mass was removed from
the knee or added to the heel. Again, it should be noted
that the effects of mass perturbations in the distal part
are much larger than in the proximal part of the lower
leg.
Comparing the effect of mass perturbation for all
individual subjects (figure 3 for subjects 1 and figure 4
for the other nine subjects), differences were found in
the number of possible mass perturbations without
obtaining unrealistic physical properties of the total
segments. This is the result of different initial inertial
properties of the stump and prosthesis. In terms of the
effect of the mass perturbation on TEhip+knee, most
subjects showed the same four regions in the graphs as
in figure 3. Only one subject (TTA subject 4) showed a
pattern in which mass addition always increased and
mass removal always decreased TEhip+knee. In all
subjects, larger effects on TEhip+kneecould be obtained
by removing mass than by adding mass (table 2).
FORCES BETWEEN STUMP AND SOCKET
Figure 5 shows a typical example of the net joint
torque at the stump–socket interface as well as the
resultant sagittal and longitudinal reaction forces. As
Table 3
socket during the swing phase that can be obtained through mass perturbations of maximally 2.5 kg
Maximum changes possible in the peak longitudinal force (Flong), peak sagittal force (Fsag), and muscular cost (TEstump) between stump and
Maximum decrease (%)Maximum increase (%)
Transtibial
amputee
Adding massRemoving massAdding mass Removing mass
Flong
Fsag
TEstump
Flong
Fsag
TEstump
Flong
Fsag
TEstump
Flong
Fsag
TEstump
1
2
3
4
5
6
7
8
9
10
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
788
781
775
780
784
788
786
785
777
768
792
791
785
781
788
789
791
790
779
769
770
794
755
751
752
755
787
768
755
726
103
91
161
150
124
130
124
135
131
118
117
102
245
157
140
142
145
186
160
136
116
118
255
129
122
128
146
175
141
132
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Mean000
781
786
761 127153146000
R. W. Selles et al.
700

Page 8

expected, adding weight to the knee did not affect the
forces and torque, since only the properties of the stump
are perturbed. On the prosthesis, adding mass increased
the amplitude of all time series while removing mass
decreased the amplitudes (figure 6). Averaged over all
subjects (table 3), the maximum decrease that could be
obtained by removing mass ranged from 61% to 86%
for the three variables. The increase that could be
obtained by adding mass ranged from 127% to 153%.
Discussion
The aim of the present study was to develop a new
mathematical model to predict the optimal prosthetic
???
?
??
Figure 7
integrals indicate the time integral from toe-off to heel strike. For visualisation, only half of the simulated locations are shown.
Abbreviations: Tknee, knee torque; Thip, hip torque; Pknee, knee power; Phip, hip power.
TEhip+knee(A) and three alternative estimates (B–D) of the total muscular cost of the swing phase for subject 1. The
Effect of prosthetic mass properties on gait
701

Page 9

inertial properties for TTA subjects. In contrast with
prior theoretical approaches, this model is not based
on the assumption of a ballistic swing phase. Assuming
invariant kinematics during the swing phase, we evalu-
ated the effects of a systematic set of mass perturbations
in terms of the net joint torques in hip and knee as well
as the net joint reaction forces and torque between sock-
et and stump needed to obtain the measured swing
phase kinematics. The model is focussed on the swing
phase of gait, assuming that inertial properties of a pros-
thesis will not importantly influence the stance phase.
METHODOLOGICAL ISSUES
The validity of this model is related to the assump-
tions made. One of the main assumptions was that
TTA subjects adapt to mass perturbations by maintain-
ing the same kinematic pattern: a kinematic invariance
strategy.15The data on which this assumption is based
have been discussed more extensively elsewhere.14
Because the predictions in this study are based on
inverse dynamical simulations, the well-reported limita-
tions of this technique (e.g.26, 27) also apply to our data.
Another limitation relates to the measure of muscular
cost. Ideally, the cost of the swing phase would be eval-
uated in terms of the total cost of all muscle action
involved. However, this is presently considered difficult
or impossible in normal subjects and may be even more
difficult in amputation subjects, in which a standard
musculoskeletal model may not apply. In the present
study, muscular cost was estimated using TEhip+knee.
A comparison with three alternative measures for quan-
tifying mechanical cost of the swing phase (figure 7)
revealed that the main conclusion was not affected by
the chosen measure.
MUSCULAR COST OF THE SWING PHASE
In contrast to the forces between socket and stump, in
the muscular cost of the swing phase, not only the size
but also the direction of the effect of mass perturbation
depends on the location of perturbation. In 9 out of 10
subjects, muscular cost decreased when removing mass
distally as well as when adding mass proximally. The
magnitude of the effects was much larger when mass
was perturbed distally (figure 4 and table 2). In one
subject (figure 4), adding mass always increased the
muscular cost of the swing phase.
In the present study, we did not experimentally test
the simulation predictions on muscular cost. However,
there is a large body of literature on mass perturbations
of lower limb prostheses with which to compare our
data. In a literature review on prosthetic mass perturba-
tions, Selles et al.15reported significant effects of mass
perturbations in two of the five studies on the economy
of gait, one reporting a positive effect of adding mass,
the other a negative effect. These conflicting results
may be explained by finding that the effect of mass
perturbation depends on the location of the perturba-
tion, as well as by differences in effects found between
Figure 8
muscular cost (TEhip+knee) of the swing phase. Alterations in foot and shoe mass were modelled as point mass perturbations at the
heel, while the effect of socket and liner alterations were modelled as point mass perturbations halfway the stump. The inertial
symmetry was modelled based on the estimated contralateral lower leg mass, centre of mass location and moment of inertia.
Transformation of the data of figure 3 to indicate the effects of typical alterations of prosthetic components on the
R. W. Selles et al.
702

Page 10

subjects. Recently, Mattes et al.9investigated the effect
of matching the prosthetic inertial properties of TTA
subjects to the inertial properties of their contralateral
leg. On average, 1.7 kg was added relatively distal to
the lower leg. As a result, step length was not influenced,
while swing time significantly increased by about 4%
after mass addition, the latter finding in contrast with
the kinematic invariance strategy. In line with our
assumptions, metabolic cost per second significantly
increased by about 7%. Simulating lower leg inertial
symmetry in our data showed a similar outcome:
TEhip+knee increased in all subjects by, on average,
25% (SD 18; see also figure 8). It should be noted that
these percentages can not be compared directly because
the present study focuses on the swing phase only, while
Mattes et al.19studied the complete gait cycle.
FORCES BETWEEN STUMP AND SOCKET
The effects of the mass perturbations on the stump–
socket interface were relatively straightforward, that is,
forces and torque always decreased when mass was
removed and increased when mass was added. As in
the muscular cost, the magnitude of the effects was
larger when mass was perturbed more distally.
The analysis of the effects of mass perturbations on
the stump–socket interface should be considered a first
estimate, indicating only the combined effect of all shear
and pressure forces acting on many locations on the
stump. Currently, little is known about the forces
between socket and stump and how they relate to skin
problems such as blisters and sores. Most of the experi-
mental studies as well as the finite element models on
pressure and shear inside a socket have focused only
the stance phase. An estimate of the forces during the
swing phase can be found in a study by Appoldt et
al.,28showing that pressures measured at several loca-
tions at the stump during gait in above-knee amputees
were low compared to the stance phase. In contrast, a
more recent study by Williams et al.,29indicated that
the pressure and shear forces during swing phase are
significant, although lower than during the stance phase.
Inthepresent studywedid notmakean assumptionon
how the resultant forces and torques are distributed over
the stump, since this may strongly depend on the socket
type used. For example, a total surface bearing (TSB)
socket using a silicon liner may distribute the resultant
forces and torques very differently over the stump than
a patellar tendon bearing (PTB) prosthesis, loading
mainly the patellar tendon area. The aim of the present
study was to show that, in contrast to the effects on
muscular cost, the resultant forces always increased after
adding mass and decreased after removing mass. Future
studies should investigate how these forces are distribu-
ted and whether the magnitude of the effects of mass
perturbation is relevant for prosthetic design.
Conclusion
In this study, we developed a new mathematical
model to describe the relation between prosthetic iner-
tial properties on the one hand and the muscular cost
of the swing phase and the forces in the stump–socket
interface on the other hand. Comparing the prediction
of the present model with experimental literature
suggests this new model better describes the experimen-
tal data than existing models based on the hypothesis of
a ballistic swing phase.
Based on the model outcomes, we found that the
effects on the forces and torque in the stump–socket
interface are straightforward, that is, forces increase
when mass is added and decrease when mass is removed.
The effects are largest after distal perturbation. As noted
before, however, this analysis only provides a first esti-
mate of the stump–socket forces. In addition, because
the present literature suggests that forces in the
stump–socket interface during the swing phase are
small compared to the stance phase, drawing conclu-
sions for clinical practice from these results seems far
stretched. Overall, the present data show no reason to
change the current practice to make prostheses as light-
weight as possible from the point of view of stump–
socket interface forces.
In terms of the muscular cost of the swing phase, the
present results indicate that for components located in
the upper 15 to 20 cm of the lower leg, such as liners
and sockets, mass is not of primary importance (see
figure 8). In this region, prosthetic mass can be increased
without increasing the muscular cost of the swing phase.
For more distally located components, such as prosthe-
tic feet and shoes, the muscular cost increases with
increasing mass, suggesting that weight reduction is
beneficial.
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