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Transactions on Biomedicine and Health vol 2, © 1995 WIT Press, www.witpress.com, ISSN 1743-3525
A method for the calculation of knee ligament
loads during rehabilitation following injury
or reconstruction
D.E.Toutoungi,* R.Giampaoli^ A.Leardini,^ F.Catani^
JJ. O'Connor*
"Oxford Orthopaedic Engineering Centre, University of Oxford,
*Movement Analysis Laboratory, Biomechanics Laboratory,
Istituto Ortopedici Rizzoli, Bologna, Italy
Abstract
A combination of non-invasive experimental measurements and theoretical
analysis, using a geometrical model of the knee joint, was used to calculate knee ligament forces during exercise. A "limiting solutions approach
[3] was used to resolve the indeterminacy problem. Three subjects were
evaluated. Muscle combinations tending to maximise ligament loads were
examined and found to be generally consistent with EMG. Calculated contact forces were found to be in line with previous studies. Peak ligament
loads were well below the ultimate loads of healthy ligaments, but close to
the limits of healing grafts.
1 Introduction
Injuries to the ligaments of the knee joint are very common. In a well-known
study carried out at the San Diego Kaiser Clinic in the late 1980s [8]. the
number of people suffering ligament injury was found to be approximately 1
per 1000 per year. Complete tears of the ligaments, particularly the anterior
cruciate ligament (ACL), are often treated by reconstructing the ligament.
Effective rehabilitation is crucial to the success of a reconstruction. It has
been shown that the probability of re-injury to damaged ligaments can be
significantly reduced by sound remedial treatment (Callahan 1980. quoted
in [10]) and so it is vital that the exercises used for rehabilitation do not
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damage the graft while it is healing. This means that we need to determine
the loads to which the ligaments are subjected during various movements,
so that potentially damaging exercises can be avoided.
From early on in a rehabilitation program, activities which require active muscle contraction are used. The most common types of exercise are
isometrics, isotonics (or progressive resistance exercises), isokinetics, controlled weightbearing and the so-called closed kinetic chain exercises [12]
such as squats. In this study, we have concentrated on isokinetic exercises.
An isokinetic movement occurs when the muscle contracts (or lengthens, in
the case of an eccentric contraction) at constant speed. This type of movement is produced by a machine which varies the external force it applies so
that the subject is forced to move the leg at a constant angular speed.
Determining the forces within structures of the musculoskeletal system
is a difficult task. Direct measurement in living subjects is not usually
possible due to the difficulty of instrumenting living tissues and to ethical
considerations. Studies with cadaver specimens are of limited use since it is
often impossible to reproduce accurately the movement of a living subject.
Several authors have used mathematical analysis of various movements
to calculate the joint forces. Smidt [13] and Yasuda and Sasaki [15] both
carried out a two-dimensional analysis of isometric extension and flexion
contractions. They calculated the cornpressive and shear forces between
the tibia and femur. The tibio-feinoral compressive and shear forces occurring during isokinetic movement were calculated by Nisell et al in 1989
[11]. They considered two speeds, 30°/sec and 180^/sec, and compared
the results of placing the external load proximally or distally on the tibia.
Most recently. Kaufmann et al. analysed forces in the tibiofemoral and
patellofemoral joints during isokinetic exercise [9]. However, none of these
studies have reported forces carried by the ligaments.
Ligament forces are difficult to calculate for two reasons. Firstly, the
position of the ligaments cannot be determined by X-ray or external examination, which means that the line of action of the ligament forces at a
particular flexion angle is unknown. Secondly, the knee joint is a mechanically indeterminate system; there are more potential force transmitting
structures than are needed for equilibrium. This means that, even if the
lines of action of all possible forces were known, the system could not be
solved exactly. In order to tackle these difficulties and arrive at predictions
of ligament force during movement, we have used a geometrical model of
the limb to give the lines of action of all forces and we have used the "limiting solutions approach" described by Collins and O'Connor [3] to resolve
the indeterminacy problem. The procedure is described in this paper and
some preliminary results are given.
2 Methods and Materials
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2.1 Instruments
The exercise was controlled by an isokinetic dynamometer, the REV9000
(Technogym, Gambettola, Italy), sampling at lllHz. The dynamometer
motor varies the external resistance applied to the shank, via the cuff,
so that the rotational speed of the shank remains roughly constant. A
transducer measures the resisting torque. A passive stereophotogrammetric system, the ELITE (BTS, Milan, Italy), with two cameras, was used
for the acquisition of limb segment and dynamometer arm positions. The
sampling frequency was lOOHz. The dynamometer was positioned in the
stereophotogrammetricfieldof view, such that both the dynamometer arm
and the subject lower limb were visible at all times to both television cameras. EMG data was collected using a surface electromyograph (TELEMG
from BTS, Milan, Italy) with a sampling frequency of 500Hz. Pairs of surface electrodes were placed over the rectus femoris, vast us medialis, vast us
lateralis, biceps femoris, gastrocnemius and tibialis anterior.
2.2 Experimental Set-Up
The subject was seated in the dynamometer chair, with trunk and pelvis
strapped firmly in position. The hip flexion angle was set at 60°. The
dynamometer cuff was fastened to the shank at approximately 80% of the
shank-foot (s-f) segment length (measured from knee joint line to base of
calcaneous) below the tibial plateau. Rigid plates, mounted with three
retroreflective marker balls, were attached to the distal thigh and proximal
shank with wide elastic straps. Two further rigid clusters of markers were
fixed to the sliding cursor of the dynamometer and to the fixed dynamometer base. The positions of these arrays of markers were tracked by the
ELITE system during the exercises. In order to relate the movement of the
limb segments (anatomical frame) to the movement of the marker arrays
(technical frame), the positions of a number of anatomical landmarks were
recorded with the subject stationary, using the pointer technique (CAST
protocol) described by Cappozzo [2]. Knee flexion angle was calculated
from the positions of the anatomical frames, using the JCS convention of
Grood and Suntay [7]. The positions of several other points were recorded
using the same method in order to allow the calculation of the point of application of the external force, the orientation of the antero-posterior and
me di o-l at era! axes of the machine in the laboratory frame and the axis of
rotation of the dynamometer arm.
2.3 Experimental Protocol
Three healthy, adult, male volunteers were evaluated. The right leg of each
subject was tested during concentric isokinetic knee flexion and extension.
The subjects were instructed to perform a series of sequential extensionflexion movements at maximum voluntary contraction, through as much of
the set flexion range (0 to 90^) as possible. Four different nominal angular
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velocities were imposed by the dynamometer: 100, 180, 240 and 300°/sec.
At each velocity, after several warm-up cycles, the subject performed 5
repetitions, for which EMG data was gathered and the movements recorded
by the ELITE system. Finally, to allow the measurement of the anatomical
landmark locations, a few frames were recorded with the REV arm at 0
and 90° and then with the subject in a standing position with right leg
fully extended.
2.4 Calculation of resultant joint force and moment
The collected marker trajectories were filtered and smoothed and the EMG
output signals were passed through a high-pass filter. The sampling frequencies and data collection start-times of the REV and ELITE were different, so the two sets of data were synchronised and then the REV data was
resampled using linear interpolation to give data values at 0.01 sec intervals. This allowed the evaluation of the kinetic and kinematic data at the
same time instant. Using the reconstructed 3D positions of reference landmarks, combined with anthropornetric data, the following quantities were
calculated: dynamometer arm and tibial axis orientation relative to the
dynamometer base frame, knee flexion angle, line of action of the applied
force, moment arm of external force about the instant centre of the knee,
angular velocity of dynamometer arm and s-f segment, linear arid angular
accelerations of the centre of mass of the s-f segment. All the above quantities, expressed initially in the time domain, were recalculated as functions
of flexion angle, so that they could be combined with data from the theoretical knee model. The instant centre of rotation of the knee was given at
each flexion angle by the model. The resultant force and moment at the
instant centre were calculated from the equations of dynamic equilibrium.
2.5 Calculation of internal knee joint forces
The knee model adopted for this study was the two-dimensional model
developed by O'Connor et al. [3, 6] A four-bar linkage is formed by inextensible line segments representing the isometric fibres of the cruciate
ligaments and by similar line segments representing the tibia arid femur.
The flexion axis of the joint passes through the instant centre of rotation, which lies at the intersection of the two links representing the ligaments. The model is described in detail in Zavatsky and O'Connor [16].
In addition, the patellofemoral joint model of Gill and O'Connor [5] was
incorporated. Three muscle groups (the hamstrings, the quadriceps and
the gastrocnemius), represented as single lines, were also included. The
cruciate ligaments themselves were represented as bundles of fibres, using
the two-dimensional extensible ligament model of Zavatsky [16, 17]. This
representation meant that antero-posterior tibial translation during active
movements could be modelled.
The model included six joint structures which could potentially transmit
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a force to the tibia, giving 6 unknown forces: the hamstrings force (H).
the pat ell ar tendon force (Q). the gastrocnemius force (G). the ACL force
(A), the PCL force (P) and the tibio-femoral contact force (C). There
were three independent equilibrium equations describing the system (two
translational and one rotational). Thus an analytical solution could only
be obtained for three out of the six forces at any one time. In addition.
the tibial displacement relative to the passive flexion position of the tibia
was unknown. The Z a vat sky model gives a non-linear relationship between
tibial displacement and resultant ligament force.
In order to solve the equations, it was necessary to make certain assumptions so that the number of forces were reduced to three and then using an
iterative method to reach a solution: by assuming a certain value for the
displacement, calculating the joint element forces that would result from
this displacement, testing whether the resultant of the joint element forces
is equal to the actual resultant joint force and then, if necessary, adjusting
the value of the displacement and repeating the process until the resultants
calculated by the two methods are equal. It was assumed that there is a
non-zero tibial displacement, which implies that there is a force either in
the ACL or in the PCL, but not both. In addition, it was assumed that
there was no antagonistic or synergistic muscle action. These assumptions
mean that there are six possible combinations of three forces which could
be solutions: QAC, QPC, HAC, HPC, GAC and GPC. Solutions for each
combination were calculated in turn. Combinations which produced tensile contact forces or or compressive muscle forces were discarded as being
physiologically impossible. Finally the combinations were compared to the
EMG patterns and the solutions which were in closest agreement at each
particular instant were selected.
3 Results and Discussion
Of the 6 potential solution combinations, a maximum of 2 were found to be
admissible at any given instant. During extension, the valid combinations
were QAC, at lower flexion angles, or QPC. at above about 7(F. During
flexion, there were two valid combinations throughout most of the range.
These were HPC and GAC. While the net joint moment is negative, as
through most of the extension phase, flexor muscle solutions (HAC, HPC,
GAC, GPC) cannot produce equilibrium and vice-versa during the flexion
phase. Since the ACL pulls the tibia backwards and the PCL pulls it forwards, one or other can always provide a balancing shear force, depending
on the direction of pull of the active muscle. Interestingly, we found that
towards the end (last 1(F) of the flexion phase, combinations typical of
extension occurred and, similarly, towards the end of the extension phase.
flexion-type combinations were found. This is possibly due to the need for
rapid deceleration at the end of each phase, prior to beginning the opposite
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EXTENSION
0.4 - ACL
E 0.2
&
(D 0
£>
£ -0.2
I -0.4
O) -0.6 PCL
-0.8
0
20
40
60
80
flexion angle (degrees)
ligament force (xBW)
ro ro -^- ^ p
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398
FLEXION
ACL
\
/
r^/
3
J
^
1^
PCL
20
40
60
80
flexion angle (degrees)
Figure 1: The mean calculated ligament forces, expressed as fractions of
subject body-weight (BW), during extension (left) and flexion (right) at
each test velocity (
100°/sec, - - - 180°/sec, • • • 240°/sec, - . - 300°/sec).
Errorbars are shown for 100°/sec and indicate i 1 SD. Positive values
represent an ACL force, negative values represent a PCL force.
movement. This deceleration can only be achieved by the use of the opposing muscle groups, thus yielding a flexor solution at the end of extension
and vice-versa.
Figure 1 shows the values calculated for ligament force. The combinations shown are QxC for extension, where x is A or P depending on
the flexion angle (positive values indicate an ACL force, negative values a
PCL force), and HxC for flexion. These were the combinations that caused
maximum ligament loading, whilst being reasonably consistent with EMG.
So far, too few subjects have been evaluated for us to detect any significant relationship between maximum ligament force and exercise speed,
though it does appear that during extension, at lower speeds the peak forces
occur at smaller flexion angles. During extension, the transition from PCL
loading to ACL loading occurs at around 75° at all speeds. However it is
seen that the PCL becomes re-loaded during the extension phase and that
the ACL becomes loaded towards the end of the flexion phase, presumably
due to the deceleration of the tibia. The effects of terminal deceleration
were more exaggerated at the higher speeds. This is important, as it shows
that even during active flexion alone, which is often assumed not to load
the ACL and therefore used early in the rehabilitation process [4], there
is the potential for damage to a graft. It can also be seen that the forces
carried by the PCL are up to 3 times higher than those carried by the ACL.
It is not possible to compare these predictions with experimental results,
since there have been no previous studies measuring ligament loads during isokinetic movement. However, Kaufmann et al. [9] and Nisell et al.
[11] both calculated tibio-femoral contact forces for 180°/sec. Kaufmann
calculated the mean peak contact force in 5 subjects to be 3.8±0.9xBW
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during extension and 1.6±l.lxBW during flexion. Nisell calculated the
force in 8 subjects to be 5.5±0.9xBW during extension. The corresponding
values for 180^ /sec for this study were 7.0±1.8xBW during extension and
2.8±0.6xBW during flexion. Our value for extension does not differ significantly from that of Nisell (Student's f-test. p > 0.05), but does from that
of Kaufmann (p < 0.02). Our value for flexion does not differ significantly
from that of Kaufmann (p > 0.05). This suggests that the values calculated
for ligament force, which we cannot directly compare with the literature.
are also reasonable, although more subjects need to be evaluated.
The ultimate load of the ACL depends very much on age. but its value is
reported to be around 1950N in young subjects [14]. For a subject weighing
78. 5kg (the average body-weight of our subjects), this would be 2.54 xB\V.
The maximum ACL forces calculated are well below this limit, so isokinetic exercise presents no risk to a healthy subject. However, the ultimate
loads of ligament grafts are much lower than this. For example. Butler
[1] measured the ultimate loads of bone-patellar tendon-bone autografts in
monkeys to be 16% of that of the control ACL at 7 weeks post-operatively.
rising to 39% by 53 weeks. The maximum force we predicted for the ACL
was 0.31:1:0.03 xBW, which is around 12±1%, of the ACL ultimate load.
close to the failure load of the graft in the early stages of healing.
4 Conclusions
The purpose of this study was to assess the feasibility of calculating ligament loads during exercise. The combination of experimental measurements, theoretical model and limiting solutions approach gives predictions which are credible, though quantitative validation is not possible at
this stage. There are some limitations to the method, in that the theoretical models used are currently only two-dimensional. However threedimensional models are being developed and will eventually be incorporated. In the meantime, we feel that, since the movement occurs principally
in the sagittal plane only, a two-dimensional analysis can offer valuable insights into loading patterns and reasonable estimates of the ligament forces.
In summary, the results suggest that this is a viable method for calculating ligament loads in situations where invasive measurements would not
be practical.
Acknowledgments The authors would like to thank Mr. T-W. Lu, who
kindly supplied the C routines describing the behaviour of the knee model.
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