Muscle-ligament interactions at the knee during walking J J Collins*, BA, MA, DPhil Oxford Orthopaedic Engineering Centre, Nuffield Orthopaedic Engineering Centre and the Department of Engineering Science, University of Oxford J J O’Connor, BE, MA, PhD Department of Engineering Science, University of Oxford A two-dimensional mathematical model of the knee is used with gait analysis to calculate muscle, cruciate ligament and tibioTfemora1 contact forces developed at the knee during normal level walking. Ten normal adult subjects-four females and six males-participated. The knee model is based upon a four-bar linkage comprising the femur, tibia and two cruciate ligaments. It takes account of the rolling and sliding of the femur on the tibia duringflexionlextension and the changes in direction of the ligaments and muscle tendons. W e considered forces transmitted by six elements: quadriceps, hamstrings, gastrocnemius, anterior and posterior cruciate ligaments, and tibio-femoral contact. The equations of mechanics can be used to determine the absolute values of only three of the knee forces simultaneously, so that twenty limiting solutions ofthree ofthe six forces were considered. A limiting solution was rejected ifany of the three forces were negative, corresponding to compressive muscle or ligament forces, or tensile contact forces. These constraints always reduced and at times removed the redundancy of the knee structures. The high incidence of predicted single muscle activity, supported by electromyography, suggested that the ligaments play a signijicant role in load transmission during gait. The temporal patterns of muscle and ligament activity and ligament force magnitudes were sensitive to the choice of model parameters. The analysis showed that each of four possible minimum principles of muscle selection-minimal muscle force, muscle stress, ligament force and contact force-was unlikely to be valid throughout the walking cycle.
1 INTRODUCTION
Although there have been numerous studies of the forces transmitted by the human knee during gait, very few of them have attempted to analyse ligament forces (1). This is an important omission in view of the frequency of ligament injury and the increased interest in its surgical treatment. The problem of determining the forces transmitted by the muscles, ligaments and articular surfaces at the knee is dynamically indeterminate. Considering only the resultant tensile forces transmitted by the patellar tendon, by the medial and lateral heads of hamstrings and gastrocnemius, by the iliotibial tract, by the two cruciate and the two collateral ligaments, and the resultant compressive forces transmitted by the medial and lateral compartments, there are twelve separate forces to be determined. By considering the dynamics of the lower leg, from the cleft of the knee joint down to the ground, only six independent dynamics equations are available for the determination of twelve forces at the knee. In theory, an infinite number of solutions that could satisfy the dynamic equations are possible. However, it is clear from electromyography (EMG) that not all muscles are active all the time during gait. Moreover, since ligaments can transmit only tensile force and since relative movements of the bones tend to stretch some ligaments and slacken others, it is likely that not all of the ligaments are loaded at every instant. If, in the limit, it is assumed that six of the twelve possible forces are zero, the values of the other six forces sufficient to satisfy the equations of dynamics can be The M S was received on 3 M a y 1990 and was accepted for publication on 1 1 February 1991. * Present address: NeuroMuscular Research Center, Boston University, Boston, Massachusetts, U S A , 02215, t o whom correspondence should be addressed. H02090 0 IMechE 1991
obtained. At every instant, there are 924 sets of such limiting solutions, lZC, .* The problem has usually been simplified by ignoring ligament forces and obtaining additional equations on the muscle forces by proposing some criterion to govern their selection. MacConnaill postulated in the ‘principle of minimal total muscular force’ that ‘no more total muscular force is used than is both necessary and sufficient for the task to be performed’ (2, p. 77). Variations of the minimal muscle force principle have been applied to models of the lower limb (35).Hardt (6)optimized a linear objective function based on the energy requirements of muscles. Others used inequality constraints which limited muscle stress (7-11). All of these studies, with the exception of Mikosz et al. (ll),considered the flexion axis of the knee to be fixed in relation to the bones, an assumption that could lead to substantial errors in the magnitude and sign of the moment of the knee (12). Mikosz et al. (11) took some account of the relative rolling and sliding movements of the articular surfaces of the knee but they assumed the directions of the flexor muscles to be fixed in relation to the tibia and did not attempt to evaluate ligament forces explicitly. The only knee model known to the authors that has been used in gait analysis to evaluate ligament forces was that of Morrison (1). From EMG, he chose the principal muscle group active at each instant and reduced the number of unknown forces to the number of available equations. However, he did not attempt to account for antagonistic or synergistic muscle action. * A system with m unknown forces and n available equations of mechanics, where m > n, is statically or dynamically indeterminate. If. in the limit, it is assumed that m - n of the m possible forces are zero, the absolute values of the n other forces sufficient to satisfy the equations of mechanics can be obtained. In the present paper, each combination of n of the m possible unknown forces will be referred to as a limiting solution.
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He also assumed that the joint flexes and extends about an axis fixed in the bones. Harrington (13) used a quasistatic version of Morrison's model to compare the gait of normal and pathological subjects during stance phase. In summary, there is not available in the literature an analysis of cruciate ligament forces in gait that takes account of the movements of the flexion axis relative to the bones and that, in that context, considers the possibility of antagonistic and synergistic muscle action. In the present paper, a two-dimensional mathematical model of the knee will be used to calculate the values of the muscle, cruciate ligament and tibio-femoral contact forces developed at the knee during normal level walking. Each of the limiting solutions, reduced in two dimensions to a maximum of twenty, is examined to see how many of them are consistent with the EMG evidence and how many of them imply the necessity of ligament forces. While a two-dimensional analysis is obviously a simplification, one can nonetheless gain some understanding of possible muscle-ligament interactions in walking without having to confront their full complexities. 2 METHODS
2.1 Experimental
Ten normal adult subjects, four female and six male (body weight: 49.C95.5 kg, mean 66.5 kg; height: 1.63-
(b)
1.88 m, mean 1.72 m), were studied in the gait laboratory of the Oxford Orthopaedic Engineering Centre. A description of the Vicon gait analysis system (Oxford Metrics, Oxford, UK OX2 OJB), Kistler force platforms (Kistler Instruments Limited, Hartley Wintney, UK RG27 8RN) and the methods of calculating limb position and joint angles from the Vicon data was given by Whittle (14). The present methods have also been used to study events around heelstrike (15). The kinematic and kinetic data were sampled at 50 Hz.
2.2 Model of the knee joint A mathematical model of the knee in the sagittal plane (16-18) (Fig. 1) was used to determine the lines of action of the tibio-femoral contact force and the cruciate and muscle forces at each position of the joint together with the lengths of the muscle lever-arms at the knee (Fig. 2). The model is based on a four-bar linkage comprising the femur, the tibia and the two cruciates. Knee and lower leg model parameters were estimated from anthropometrical studies (19). The model was validated by comparing its predictions with in uitro measurements of muscle and ligament forces (20).In order to assess the errors associated with using averaged anthropometrical measurements, a subject-specific set of parameters was obtained for one representative subject (one of the authors) from lateral X-rays of his right knee.
(C)
Fig. 1 Computer model of the knee. The positions of the bones on each other are controlled by the four-bar cruciate linkage ABCD. The perpendicular to the tibio-femoral surfaces at their point of contact passes through the instant centre I of the linkage. The patellar tendon and hamstrings tendons are attached to the tibia at R and H respectively. The gastrocnemius tendon is attached to the femur at G and to the calcaneal tendon (not shown). The quadriceps and patellar tendons intersect at P. In (a), at full extension, the gastrocnemius tendon intersects the tibia1 plateau at J. In (b), at 70" of flexion, the lever-arms of the muscle tendons are shown. In (c), at 140" of flexion, the quadriceps tendon wraps around the patellar facet of the femur @ IMechE 1991
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MUSCLE-LIGAMENT INTERACTIONS AT THE KNEE DURING WALKING
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Table 2 Self-equilibrating muscle, ligament, contact force systems. The notation is the same as that used in Table 1 Two muscles, one ligament, one contact Three muscles, one contact
E
s
E E 1.2
g E
E
QHAC, QHPC, QGAC, QGPC QHGC
1
40_-1---: 30
20
10
01 0
20
40
60
80
100
120
140
s
amstrings
0.6
Posterior cruciate
Flexion angle deg
Fig. 2 The calculated values of subject-specific muscle leverarms plotted against flexion angle. The symbols b, = quadriceps, b, = patellar tendon, b, = gastrocnemius, b, = hamstrings
;'77 0
20
40
' Patellar tendon
60
80
100
\ 120
140
Flexion angle
2.3 Limiting solutions of knee joint forces
deg
In the sagittal plane, the lower leg, from the joint cleft of the knee down t o the sole of the foot, has three degrees of freedom (ignoring the joints of the ankle and foot). The equations of mechanics can thus be used to determine the absolute values of only three of the knee joint forces simultaneously. In the modelling, forces transmitted by six elements were considered : quadriceps Q, hamstrings H, gastrocnemius G, anterior cruciate ligament A, posterior cruciate ligament P and tibio-femoral contact C. There are twenty limiting solutions of three of these forces, 6C3 (Table 1).
2.5 Analysis of forces at the knee
2.4 Self-equilibrating solutions In addition to the twenty limiting solutions of Table 1, there is the possibility that antagonistic muscles can contract simultaneously while keeping the limb position and the external loads unchanged. The muscle, ligament and contact forces generated at the joint by such isometric contractions are in addition to those required to balance the external loads and may be said to form selfequilibrating systems of forces, having no resultant (21). The five possible systems of four self-equilibrating forces are listed in Table 2. The three equations of mechanics are sufficient only to evaluate the magnitudes of three of the forces relative to a fourth (Fig. 3). Any of the selfequilibrating solutions of Table 2 can exist in isolation. However, superposition of such solutions upon each other or on the limiting solutions of Table 1 has to be
From gait analysis, the configuration of the subject's lower leg in space was known at every instant of the walking cycle. Data from the force platform defined the magnitude and direction of the resultant force applied by the ground to the foot and its relation in space to the lower leg. The inertial parameters for the lower leg were determined for each subject from Winter's (22) data according to weight and lower leg length. Quintic spline approximations were used to smooth and differentiate the time-displacement data t o determine the linear and angular accelerations of the lower leg (23). The values of the resultant force through the instant centre of the knee and the associated couple needed for dynamic equilibrium of the lower leg were calculated. The computer model of the knee was used to deduce the lines of action of the quadriceps, hamstrings and gastrocnemius tendons, the two cruciates and the tibio-
Fig. 3 Self-equilibrating forces. Loading of the cruciate liga-
ments by isometric quadriceps and hamstrings contractions. Patellar tendon, hamstrings, ACL and PCL forces per unit contact force plotted against flexion angle, using averaged parameters. The diagram shows that the combination QHAC is possible near extension, QHPC above 23" of flexion done with care because of the one-sided constraint that ligament and muscle forces must be tensile.
Table 1 Twenty limiting solutions of three out of six possible forces at the knee, where Q = quadriceps, H = hamstrings, G = gastrocnemius, A = anterior cruciate ligament, P = posterior cruciate ligament, C = tibio-femoral contact One muscle, distracting load One muscle, compressing load Two muscles, distracting load Two muscles, compressing load Three muscles, distracting load No muscle 0 IMechE
QAP, HAP, GAP QAC, QPC, HAC, HPC, GAC, GPC QHA, QHP, QGA, QGP, GHA, GHP QHC, QGC, HGC QHG APC Proc Instn Mech Engrs Vol 205
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- 20
3 RESULTS
Figure 4 shows the pattern of knee flexion angle during the gait cycle of the representative subject. The temporal patterns of the predicted limiting solutions and the value of the resultant flexion/extension moment of the loads about the instant centre of the knee for that subject using subject-specific and averaged parameters are presented in Fig. 5. The number of limiting solutions available at each data point for the subject-specific case is shown in Fig. 6. Figure 7 shows the corresponding EMG data. The temporal solution patterns consistent with the various minimum principles are presented in Fig. 8. Sample ligament force calculations for the representative subject are shown in Fig. 9, where limiting solutions are selected that most closely accord with Stance phase
To
To Swing phase
Stance phase
femoral contact at each sampling point of the walking cycle. For each data point on the walking cycle, the values of the forces in all twenty limiting solutions of Table 1 were calculated. A solution was rejected if any of the three forces were negative-corresponding to compressive muscle or ligament forces or to a tensile contact force. Four possible minimum principles of muscle selection-minimal muscle force, muscle stress, ligament force and contact force-were examined. The limiting solutions consistent with each principle were selected at each sampling point on the walking cycle. The effective cross-sectional areas of quadriceps, hamstrings and gastrocnemius needed to calculate muscle stress were taken from Alexander and Vernon (24).
J
Walking cycle %
Fig. 4 Knee flexion angle plotted over the walking cycle for the representative subject
EMG findings, both published (25) and the authors' own (Fig. 7). 3.1 Temporal solution patterns The following description applies to the temporal patterns for the representative subject using subject-specific parameters (Fig. 5a). For a brief period following heelstrike, a flexing moment was required at the knee. Three limiting solutions were found-two single muscle solutions, HPC (hamstrings, posterior cruciate ligament and tibio-femoral contact force) and GAC To
Stance phase
Swing phase
-
Swing phase
-
-
-
+xion
20
60
40
80
QAC QPC QHC HQC QGC GQC HAC HPC GAC GPC HGC QAP HAP GAP QHA QHP QGA QGP HGA HGP QHG 100
Extension
Extension Walking cycle
Walking cycle
%
%
(a) Subject-specific parameters
(b) Averaged parameters
Fig. 5 Temporal patterns of admissible limiting solutions for the representative subject. The notation is the same as that used for Table 1. Knee flexion/extension moment of the external loads about the instant centre of the knee is plotted at the bottom @
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MUSCLE-LIGAMENT INTERACTIONS AT THE KNEE DURING WALKING
Stance phase
.n
EMG temporal patterns Stance phase To
Swing phase
To
15
Swing phase
:
-t
Quadriceps
Hamstrings
Gastrocnemius
0
20
60
40
100
80
Walking cycle %
Fig. 7 Periods of muscle activity for the representative subject, as indicated by EMG
.
0
.
20
.
. . .
. 60
40
80
, 100
Walking cycle %
The number of admissible limiting solutions found at each data point of the walking cycle for the representative subject, using subject-specific parameters Stance phase
(gastrocnemius, anterior cruciate ligament and tibiofemoral contact force), and one synergistic solution, HGC. During the remainder of early stance, an extending moment was required at the knee. Only one limiting solution, QAC (quadriceps, anterior cruciate ligament and tibio-femoral contact force), was found to be physically realizable. For the bulk of mid-stance, two single muscle solutions, HPC and GAC, and one synergistic Stance phase
Swing phase
To
-
To
Swing phase
QAC HPC
-
GAC
QAP HAP
QAP HAP
i
.
20
.
40
60
80
1
100
40
20
60 Walking cycle
Walking cycle %
%
(b) Principle of minimal muscle stress
(a) Principle of minimal muscle force Stance phase
Swing phase
To
To
Stance phase
Swing phase
.
.)
0
20
40
60
100
80
80
100
0
20
60
40
Walking cycle %
Walking cycle
(c) Principle of minimal ligament force
(d) Principle of minimal contact force
80
. 100
%
Fig. 8 Temporal patterns of admissible limiting solutions consistent with the tested minimum principles of muscle selection Proc Instn Mech Engrs Vol 205
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Table 3 Subject-to-subject
4
force patterns
o
Anterior cruciate (QAC) Anterior cruciate (GAC)
A
Posterior cruciate (HPC)
A
Anterior cruciate (HAC)
3
F? L
variability:
temporal
Period of gait cycle
Admissible limiting solution(s)
Number of subjects (9 total)
Early stance Mid-stance Late stance Early swing Mid-swing Late swing
QAC HAC, GAC QAC QAC, QHC Distracting solutions HAC, GAC, HGC
8 7 7 9
9 9
number of admissible limiting solutions for any subject was six, occurring only at one data point in the swing phase. 0
0
20
40
60
80
100
3.4 Minimum principles of muscle selection
Walking cycle %
Fig. 9 Sample cruciate ligament forces for the representative subject over the gait cycle, where limiting solutions have been selected that most closely accord with EMG findings. Adjusted subject-specific parameters (solid lines), averaged parameters (dashed lines) solution, HGC, were again obtained. During the early and late portions of mid-stance, the antagonistic solution, HQC, was also found. Throughout most of late stance, only one solution, QAC, was obtained. However, for two frames preceding toe-off, the antagonistic solution, QHC, was also found. Following toe-off, two extensor solutions, QAC and QHC, were found. Mid-swing was briefly characterized by distracting solutions. During most of late swing, three flexor solutions were obtained-HPC, GAC and HGC. The antagonistic solution, HQC, was found only for a few data frames. It was not found peri-heelstrike where indicated by EMG.
3.2 Subject-specific versus averaged parameters The resultant flexion/extension moment at the knee (Figure 5) for the representative subject using subjectspecific and averaged parameters was nearly identical. The magnitudes of the subject-specific muscle tendon lever-arms (Fig. 1) were generally larger than those calculated using averaged parameters. There was similarity in the temporal patterns (Fig. 5) during early stance, late stance and early swing. Significant differences in temporal patterns occurred during mid-stance and late swing.
3.3 Subject-to-subject variability The patterns of limiting solutions at different phases of the walking cycle for the remaining nine subjects, using averaged parameters, are summarized in Table 3. There was remarkable unanimity in the predicted patterns of seven of the subjects during the stance phase and for all of them during swing. The ninth subject walked so as to eliminate the need for flexor activity during stance. In nearly all of the subjects, the single muscle solution, QAC, proved to be the only theoretical possibility over 35-60 per cent of the walking cycle, mainly in early and late stance. Of the twenty possible, the maximum
The validity of each minimum principle (Fig. 8) can be evaluated by comparing the obtained limiting solutions with temporal EMG patterns (Fig. 7). Three of the minimum theories-minimal muscle force, ligament force and contact force-selected gastrocnemius solutions during late swing and the period immediately following heelstrike, whereas EMG did not indicate such activity. The principles of minimal muscle stress, minimal ligament force and minimal contact force preferred hamstrings solutions during mid-stance, while EMG demonstrated only gastrocnemius action. 4 DISCUSSION
4.1 Number of admissible limiting solutions Although twenty limiting solutions could arise at each point of the walking cycle, the simple constraint that ligament and muscle forces should be tensile and contact forces compressive ruled out most of them (Figs 5 and 6). The locomotor system is less redundant than might appear at first sight. Possible solutions were found to be inappropriate for a number of reasons. When the resultant force transmitted across the joint had a compressive component, the distractive limiting solutions of Table 1 were inappropriate. This was the case over most of the gait cycle. Flexing loads ruled out the single muscle flexor solutions and vice versa. Since the ACL pulls the tibia backwards and the PCL pulls it forwards, one or other can always provide a shear force in the direction required for dynamic equilibrium so that a single muscle limiting solution involving only one cruciate ligament force was found at every instant. The same is not true of the muscle forces; at times in the walking cycle, some antagonistic and synergistic solutions were ruled out because the antagonistic or synergistic pull had a forwards component parallel to the tibia1 plateau when a backwards pull was required, or vice versa. Thus, many of the non-distractive solutions were ruled out because the directions of the soft tissue elements, muscles or ligaments, were inappropriate.
4.2 EMG/theory agreement and discrepancy Single muscle limiting solutions consistent with EMG phasic patterns were found over much of the gait cycle: 0 IMechE 1991
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MUSCLE-LIGAMENT INTERACTIONS AT THE KNEE DURING WALKING
for quadriceps during early and late stance, hamstrings during late swing and gastrocnemius throughout midstance. However, during the interval immediately preceding and following heelstrike, EMG indicated antagonistic quadriceps-hamstrings activity not found by the calculations. 4.3 Bi-articular muscles Quadriceps and hamstrings have elements that span the hip as well as the knee. Gastrocnemius spans the ankle as well as the knee. It is possible that mechanical considerations at the hip or ankle could influence the activity of these muscles at the knee and could, for instance, explain the quadriceps-hamstrings activity periheelstrike detected by EMG but not predicted by the present model. The fact that hip or ankle mechanics might demand flexor activity at the knee whereas knee mechanics demands extensor activity need not present a dificulty since the self-equilibrating solutions of Table 2 can always be superimposed on the load-based solutions of Table 1. If account were taken of the contact forces at the hip and ankle, both in magnitude and direction, as well as tensile forces in soleus, tibialis anterior, iliopsoas and gluteus maximus together with the six forces of the present model, there would be 498 possible limiting solutions for a two-dimensional study (26). Taking account of the bi-articular nature of muscles obviously complicates the problem. 4.4 Parameter sensitivity
The main sensitivity to parameter choice proved to be for those elements, ligaments or muscles, that can lie nearly perpendicular to the tibial plateau : the anterior cruciate ligament near extension, the posterior cruciate ligament at high flexion angles (not encountered in level walking), the patellar tendon near 90" of flexion, the hamstrings tendon near extension, the gastrocnemius tendon over most of the flexion range. Differences of 10 per cent in parameter values, well within the error bands of these measurements, could make a difference of 4-5" in the values of the muscle or ligament angles relative to the tibial plateau. When the muscle tendon angles are nearly 90°, such differences could make uncertain whether the muscle applies a backwards or a forwards pull to the tibia, the factor which mainly determines whether a given limiting solution is likely to be admissible or not. For instance, for the representative subject in early stance, when the hamstrings tendon is nearly perpendicular to the plateau, the averaged parameters gave the HAC solution only whereas the subject-specific parameters gave H P C and HGC. The main area of uncertainty in the present study lies in the calculated values of the anterior cruciate ligament force near extension, that is during the stance phase. The value of the ligament force is determined by the need to provide a shear force parallel to the tibial plateau, the shear force being equal to the horizontal component, L cos(B,,,), of the ligament force L. When the inclination of the ligament, Olig, is near 90", small changes in O,,, can result in large differences in the calculated value of L. The similarity between the resultant flexion/extension moment at the knee (Fig. 5) for the subject-specific and
17
averaged parameters reflected the fact that the calculated position of the instant centre relative to the line of action of the external loads was nearly identical in the two cases. The differences in the values of the muscle tendon lever-arms had consequences for the calculated forces-the larger subject-specific lever-arms resulted in smaller muscle forces at any instant during the gait cycle. The knee-model parameters were estimated from anthropometrical studies of cadaveric specimens, using X-ray and dissection techniques (19). These formed the basis of the set of averaged parameters used in the calculations. For all but the representative subject, individual measurements on the subjects in this study were limited to weight, lower leg length and marker position. Magnetic resonance imaging methods are now being developed that can detect more readily the points of soft tissue insertion into the bones and may make parameter estimation easier and more accurate for individual subjects.
4.5 The role of ligaments in force transmission during gait Despite uncertainties about force magnitudes arising from parameter sensitivity, these studies show that the cruciate ligaments play an essential mechanical role in normal gait. Whenever EMG shows that a single muscle group, extensor or flexor, is active, it is a certainty that some of the ligaments are under load. Single muscle group activity is found over most of the cycle for normal level walking so that it can be concluded that the muscles do not normally act to protect the ligaments. Figure 9 shows selected cruciate ligament force calculations for the representative subject. These results are consistent with the EMG data for that subject, except in late swing and early stance. The results for the ten subjects were similar: the ACL was shown to have a possible role in 65-100 per cent of the cycle (Table 3). It is interesting to compare these calculations with those of Morrison (1, Fig. 8). Both sets of solutions show three waves of ACL load transmission during stance, but the present calculations yielded much higher force values. Morrison found PCL action during stance in two of his three subjects whereas it was not found in nine subjects in the present study. The calculations for the representative subject, using subject-specific parameters, gave possible PCL action with hamstrings during mid-stance, but this limiting solution was not confirmed with EMG. Both models found PCL action during swing, persisting in Morrison's results and in the case of our representative subject into early stance. Using subject-specific parameters for the representative subject, a maximum ACL force of 3.5 x BW (2.35 kN) was calculated for the single muscle solution QAC during early stance. This value is high compared to the experimental results of Noyes and Grood (27), who found that human anterior cruciate ligaments failed in vitro at maximum tensile loads of 1.7 kN & 0.7 kN, decreasing rapidly with advancing age. Apart from parameter sensitivity discussed above, there are several factors neglected in the theoretical analysis which could further reduce the predicted ligament loading. For example, ligament elasticity would allow some anteroposterior translation of the tibia rela-
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tive to the femur and would decrease the ligament force required to resist an arbitrary shear force acting along the plateau. Moreover, allowing for participation by the collateral ligaments would further reduce the estimates of cruciate ligament forces (28). The assumption of inextensible ligaments therefore provides an upper bound solution to the value of the cruciate ligament forces. Finally, the maximum value of the ACL force was found for the single muscle solution QAC in early stance where EMG indicated that hamstrings and quadriceps were acting simultaneously at the knee, although the antagonistic limiting solution QHC was not obtained by the model. It is theoretically possible, however, to superimpose the self-equilibrating solution QHPC onto the limiting solution QAC to give an antagonistic solution with ligament loading, that is QHAC. The addition of hamstrings action at the knee could reduce the loading of the ACL. 4.6 Muscle selection criteria
These studies, taking account of possible ligamentous contributions to load bearing, appear to rule out many of the global criteria previously considered as possible bases for the selection of muscle activity. Any contribution from the self-equilibrating solutions of Table 2 to the load-related single muscle limiting solutions would increase the predicted muscle and contact forces but may increase or decrease ligament forces (Fig. 3). The single muscle limiting solutions therefore represent the minimal muscle force response of the knee joint to load. In the presence of flexing loads acting at the knee, single extensor muscle limiting solutions, that is QAC and QPC, also represent the minimal contact force response of the knee. In the presence of extending loads acting at the knee, the minimal contact force response of the joint is characterized by synergistic flexor muscle limiting solutions, that is HGC (Fig. 8d). The present geometric and mechanical analysis suggests that it may be unrealistic to seek a single optimality criterion for muscle selection that applies throughout the walking cycle. ACKNOWLEDGEMENTS
The authors would like to thank Professor M. W. Whittle and Dr R. J. Jefferson for their support and encouragement during this investigation. The authors also thank Professor H. J. Woltring for providing them with a FORTRAN package for spline smoothing and differentiation. Dr E. Biden and Mrs T. L. Shercliff contributed to the development of the knee model and associated programming and were supported by grants from the Arthritis and Rheumatism Council. JJC held a Rhodes Scholarship for the duration of the study. REFERENCES 1 Morrison, J. B. The mechanics of the knee joint in relation to normal walking. J . Biomech., 1970, 3, 51-61.
2 MacConnaill, M. A. The ergonomic aspects of articular mechanics. In Studies ofthe anatomy andfunction ofbones and joints (Ed. F. G. Evans), 1967, pp. 69-80 (Springer, Berlin). 3 Seireg, A. and Arvikar, R. J. The prediction of muscular load sharing and joint forces in the lower extremities during walking. J . Biomech., 1975,8, 89-102. 4 Pedotti, A., Krishnan, V. V. and Stark, L. Optimization of muscle force sequencing in human locomotion. Math. Biosci., 1978, 38, 57-76. 5 Rohrle, H., Scholten, R., Sigolotto, C. and Sollbach, W. Join! forces
in the human pelvis-leg skeleton during walking. J. Biomech., 1984,17,409424. 6 Hardt, D. E. Determining muscle forces in the leg during normal human walking-an application and evaluation of optimization methods. Trans. ASME, J. Biomech. Engng, 1978,100,72-78. 7 Crowninshield, R. D. and Brand, R. A. A physiologically based criterion of muscle force prediction in locomotion. J. Biomech., 1981, 14,793-801. 8 Patriarco, A. G., Mann, R. W., Simon, S. R. and Mansour, J. M. An evaluation of the approaches of optimization models in the prediction of muscle forces during human gait. J. Biomech., 1981, 14,513-525. 9 An, K. N., Kwak, B. M., Chao, E. Y. and Morrey, B. F. Determination of muscle and joint forces: a new technique to solve the indeterminate problem. Trans. ASME, J. Biomech. Engng, 1984, 106,364367. 10 Calderdale, P. M. and Scelfo, G. A mathematical model of the locomotor apparatus. Engng Med., 1987,16, 147-161. 11 Mikosz, R. P., Andriacchi, T. P. and Andersson, G. B. J. Model analysis of factors influencing the prediction of muscle forces at the knee. J . Orthop. Res., 1988,6,205-214. I2 Nissan, M. Review of some basic assumptions in knee biomechanics. J . Biomech., 1980, 13, 37S381. 13 Harrington, I. J. A bioengineering analysis of force actions at the knee in normal and pathological gait. Biomed. Engng, 1976, 11(5), 167-172. 14 Whittle, M. W. Calibration and performance of a threedimensional television system for kinematic analysis. J . Biomech., 1982,15, 185-196. 15 Jefferson, R. J., Collins, J. J., Whittle, M. W., Radin, E. L. and O’Connor, J. J. The role of quadriceps in controlling impulsive forces around heelstrike. Proc: Instn h e c h . Engrs, Part H; 1990, 204(H1),21-28. 16 O’Connor, J., Shercliff, T. and Goodfellow, J. The mechanics of the knee in the sagittal plane. In Surgery and arthroscopy of the knee (Eds W. Muller and W. Hackenbruck), Proceedings of the Second Congress of the European Society for Surgery and Arthroscopy of the Knee, 1988, pp. 12-30 (Springer-Verlag, Berlin). 17 O’Connor, J. J., Shercliff, T. L., Biden, E. and Goodfellow, J. W. The geometry of the knee in the sagittal plane. Proc-. Instn Mech. Engrs, Part H, 1989,203(H4), 223-233. 18 O’Connor, J., Shercliff, T., FitzPatrick, D., Bradley, J., Daniel, D., Biden, E. and Goodfellow, H. Geometry of the knee. In Knee ligaments-structure, Junction, injury, and repair (Eds D. M. Daniel, W. H. Akeson and J. J. O’Connor), 1990, pp. 163-200 (Raven Press, New York). 19 Bradley, J., FitzPatrick, D., Daniel, D., Shercliff, T. and O’Connor, J. Orientation of the cruciate ligament in the sagittal plane: a method of predicting length change vs. knee flexion. J . Bone Jt Surg., 1988,70B, 9499. 20 O’Connor, J., Biden, E., Bradley, J., FitzPatrick, D., Young, S., Kershaw, C., Daniel, D. and Goodfellow, J. The muscle-stabilized knee. In Knee ligaments-structure, Junction, injury, and repair (Eds D. M. Daniel, W. H. Akeson and J. J. OConnor), 1990, pp. 239-278 (Raven Press, New York). 21 O‘Connor, J., SherclifT, T., FitzPatrick, D., Biden, E. and Goodfellow, J. Mechanics of the knee. In Knee ligaments-structure, function, injurv, and reDair (Eds D. M. Daniel. W. H. Akeson and J. “J.OConnor),-l990, pp. 201-238 (Raven Press, New York). 22 Winter, D. A. Biomechanics of human mooement, 1979 (Wiley and Sons, New York). 23 Woltring, H. J. A FORTRAN package for generalized, crossvalidatory spline smoothing and differentiation. Ado. Engng Software, 1986,8(2), 104113. 24 Alexander, R. McN. and Vernon, A. The dimensions of knee and ankle muscles and the forces they exert. J. Hum. Mooement Stud., 1975.1, 115-123. 25 Sutherland, D., Olshen, R., Biden, E. and Wyatt, M. Development ofmature walking, 1988 (MacKeith Press, London). 26 Collins, J. J. Joint mechanics-modelling of the lower limb. Doctoral thesis, 1990, University of Oxford. 27 Noyes, F. R. and Grood, E. S. The strength of the anterior cruciate ligament in humans and rhesus monkeys. J. Bone Jt Surg., 1976, %A, 107&1082. 28 FitzPatrick, D. P. and O’Connor, J. J. Theoretical modelling of the knee applied to the anterior drawer test. IMechE conference on The Changing Role of Engineering in Orthopaedics, London, 1989, paper C384/033, pp. 79-83 (Mechanical Engineering Publications, London).
Part H : Journal of Engineering in Medicine
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IMechE 1991
1 INTRODUCTION
Although there have been numerous studies of the forces transmitted by the human knee during gait, very few of them have attempted to analyse ligament forces (1). This is an important omission in view of the frequency of ligament injury and the increased interest in its surgical treatment. The problem of determining the forces transmitted by the muscles, ligaments and articular surfaces at the knee is dynamically indeterminate. Considering only the resultant tensile forces transmitted by the patellar tendon, by the medial and lateral heads of hamstrings and gastrocnemius, by the iliotibial tract, by the two cruciate and the two collateral ligaments, and the resultant compressive forces transmitted by the medial and lateral compartments, there are twelve separate forces to be determined. By considering the dynamics of the lower leg, from the cleft of the knee joint down to the ground, only six independent dynamics equations are available for the determination of twelve forces at the knee. In theory, an infinite number of solutions that could satisfy the dynamic equations are possible. However, it is clear from electromyography (EMG) that not all muscles are active all the time during gait. Moreover, since ligaments can transmit only tensile force and since relative movements of the bones tend to stretch some ligaments and slacken others, it is likely that not all of the ligaments are loaded at every instant. If, in the limit, it is assumed that six of the twelve possible forces are zero, the values of the other six forces sufficient to satisfy the equations of dynamics can be The M S was received on 3 M a y 1990 and was accepted for publication on 1 1 February 1991. * Present address: NeuroMuscular Research Center, Boston University, Boston, Massachusetts, U S A , 02215, t o whom correspondence should be addressed. H02090 0 IMechE 1991
obtained. At every instant, there are 924 sets of such limiting solutions, lZC, .* The problem has usually been simplified by ignoring ligament forces and obtaining additional equations on the muscle forces by proposing some criterion to govern their selection. MacConnaill postulated in the ‘principle of minimal total muscular force’ that ‘no more total muscular force is used than is both necessary and sufficient for the task to be performed’ (2, p. 77). Variations of the minimal muscle force principle have been applied to models of the lower limb (35).Hardt (6)optimized a linear objective function based on the energy requirements of muscles. Others used inequality constraints which limited muscle stress (7-11). All of these studies, with the exception of Mikosz et al. (ll),considered the flexion axis of the knee to be fixed in relation to the bones, an assumption that could lead to substantial errors in the magnitude and sign of the moment of the knee (12). Mikosz et al. (11) took some account of the relative rolling and sliding movements of the articular surfaces of the knee but they assumed the directions of the flexor muscles to be fixed in relation to the tibia and did not attempt to evaluate ligament forces explicitly. The only knee model known to the authors that has been used in gait analysis to evaluate ligament forces was that of Morrison (1). From EMG, he chose the principal muscle group active at each instant and reduced the number of unknown forces to the number of available equations. However, he did not attempt to account for antagonistic or synergistic muscle action. * A system with m unknown forces and n available equations of mechanics, where m > n, is statically or dynamically indeterminate. If. in the limit, it is assumed that m - n of the m possible forces are zero, the absolute values of the n other forces sufficient to satisfy the equations of mechanics can be obtained. In the present paper, each combination of n of the m possible unknown forces will be referred to as a limiting solution.
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He also assumed that the joint flexes and extends about an axis fixed in the bones. Harrington (13) used a quasistatic version of Morrison's model to compare the gait of normal and pathological subjects during stance phase. In summary, there is not available in the literature an analysis of cruciate ligament forces in gait that takes account of the movements of the flexion axis relative to the bones and that, in that context, considers the possibility of antagonistic and synergistic muscle action. In the present paper, a two-dimensional mathematical model of the knee will be used to calculate the values of the muscle, cruciate ligament and tibio-femoral contact forces developed at the knee during normal level walking. Each of the limiting solutions, reduced in two dimensions to a maximum of twenty, is examined to see how many of them are consistent with the EMG evidence and how many of them imply the necessity of ligament forces. While a two-dimensional analysis is obviously a simplification, one can nonetheless gain some understanding of possible muscle-ligament interactions in walking without having to confront their full complexities. 2 METHODS
2.1 Experimental
Ten normal adult subjects, four female and six male (body weight: 49.C95.5 kg, mean 66.5 kg; height: 1.63-
(b)
1.88 m, mean 1.72 m), were studied in the gait laboratory of the Oxford Orthopaedic Engineering Centre. A description of the Vicon gait analysis system (Oxford Metrics, Oxford, UK OX2 OJB), Kistler force platforms (Kistler Instruments Limited, Hartley Wintney, UK RG27 8RN) and the methods of calculating limb position and joint angles from the Vicon data was given by Whittle (14). The present methods have also been used to study events around heelstrike (15). The kinematic and kinetic data were sampled at 50 Hz.
2.2 Model of the knee joint A mathematical model of the knee in the sagittal plane (16-18) (Fig. 1) was used to determine the lines of action of the tibio-femoral contact force and the cruciate and muscle forces at each position of the joint together with the lengths of the muscle lever-arms at the knee (Fig. 2). The model is based on a four-bar linkage comprising the femur, the tibia and the two cruciates. Knee and lower leg model parameters were estimated from anthropometrical studies (19). The model was validated by comparing its predictions with in uitro measurements of muscle and ligament forces (20).In order to assess the errors associated with using averaged anthropometrical measurements, a subject-specific set of parameters was obtained for one representative subject (one of the authors) from lateral X-rays of his right knee.
(C)
Fig. 1 Computer model of the knee. The positions of the bones on each other are controlled by the four-bar cruciate linkage ABCD. The perpendicular to the tibio-femoral surfaces at their point of contact passes through the instant centre I of the linkage. The patellar tendon and hamstrings tendons are attached to the tibia at R and H respectively. The gastrocnemius tendon is attached to the femur at G and to the calcaneal tendon (not shown). The quadriceps and patellar tendons intersect at P. In (a), at full extension, the gastrocnemius tendon intersects the tibia1 plateau at J. In (b), at 70" of flexion, the lever-arms of the muscle tendons are shown. In (c), at 140" of flexion, the quadriceps tendon wraps around the patellar facet of the femur @ IMechE 1991
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MUSCLE-LIGAMENT INTERACTIONS AT THE KNEE DURING WALKING
13
Table 2 Self-equilibrating muscle, ligament, contact force systems. The notation is the same as that used in Table 1 Two muscles, one ligament, one contact Three muscles, one contact
E
s
E E 1.2
g E
E
QHAC, QHPC, QGAC, QGPC QHGC
1
40_-1---: 30
20
10
01 0
20
40
60
80
100
120
140
s
amstrings
0.6
Posterior cruciate
Flexion angle deg
Fig. 2 The calculated values of subject-specific muscle leverarms plotted against flexion angle. The symbols b, = quadriceps, b, = patellar tendon, b, = gastrocnemius, b, = hamstrings
;'77 0
20
40
' Patellar tendon
60
80
100
\ 120
140
Flexion angle
2.3 Limiting solutions of knee joint forces
deg
In the sagittal plane, the lower leg, from the joint cleft of the knee down t o the sole of the foot, has three degrees of freedom (ignoring the joints of the ankle and foot). The equations of mechanics can thus be used to determine the absolute values of only three of the knee joint forces simultaneously. In the modelling, forces transmitted by six elements were considered : quadriceps Q, hamstrings H, gastrocnemius G, anterior cruciate ligament A, posterior cruciate ligament P and tibio-femoral contact C. There are twenty limiting solutions of three of these forces, 6C3 (Table 1).
2.5 Analysis of forces at the knee
2.4 Self-equilibrating solutions In addition to the twenty limiting solutions of Table 1, there is the possibility that antagonistic muscles can contract simultaneously while keeping the limb position and the external loads unchanged. The muscle, ligament and contact forces generated at the joint by such isometric contractions are in addition to those required to balance the external loads and may be said to form selfequilibrating systems of forces, having no resultant (21). The five possible systems of four self-equilibrating forces are listed in Table 2. The three equations of mechanics are sufficient only to evaluate the magnitudes of three of the forces relative to a fourth (Fig. 3). Any of the selfequilibrating solutions of Table 2 can exist in isolation. However, superposition of such solutions upon each other or on the limiting solutions of Table 1 has to be
From gait analysis, the configuration of the subject's lower leg in space was known at every instant of the walking cycle. Data from the force platform defined the magnitude and direction of the resultant force applied by the ground to the foot and its relation in space to the lower leg. The inertial parameters for the lower leg were determined for each subject from Winter's (22) data according to weight and lower leg length. Quintic spline approximations were used to smooth and differentiate the time-displacement data t o determine the linear and angular accelerations of the lower leg (23). The values of the resultant force through the instant centre of the knee and the associated couple needed for dynamic equilibrium of the lower leg were calculated. The computer model of the knee was used to deduce the lines of action of the quadriceps, hamstrings and gastrocnemius tendons, the two cruciates and the tibio-
Fig. 3 Self-equilibrating forces. Loading of the cruciate liga-
ments by isometric quadriceps and hamstrings contractions. Patellar tendon, hamstrings, ACL and PCL forces per unit contact force plotted against flexion angle, using averaged parameters. The diagram shows that the combination QHAC is possible near extension, QHPC above 23" of flexion done with care because of the one-sided constraint that ligament and muscle forces must be tensile.
Table 1 Twenty limiting solutions of three out of six possible forces at the knee, where Q = quadriceps, H = hamstrings, G = gastrocnemius, A = anterior cruciate ligament, P = posterior cruciate ligament, C = tibio-femoral contact One muscle, distracting load One muscle, compressing load Two muscles, distracting load Two muscles, compressing load Three muscles, distracting load No muscle 0 IMechE
QAP, HAP, GAP QAC, QPC, HAC, HPC, GAC, GPC QHA, QHP, QGA, QGP, GHA, GHP QHC, QGC, HGC QHG APC Proc Instn Mech Engrs Vol 205
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- 20
3 RESULTS
Figure 4 shows the pattern of knee flexion angle during the gait cycle of the representative subject. The temporal patterns of the predicted limiting solutions and the value of the resultant flexion/extension moment of the loads about the instant centre of the knee for that subject using subject-specific and averaged parameters are presented in Fig. 5. The number of limiting solutions available at each data point for the subject-specific case is shown in Fig. 6. Figure 7 shows the corresponding EMG data. The temporal solution patterns consistent with the various minimum principles are presented in Fig. 8. Sample ligament force calculations for the representative subject are shown in Fig. 9, where limiting solutions are selected that most closely accord with Stance phase
To
To Swing phase
Stance phase
femoral contact at each sampling point of the walking cycle. For each data point on the walking cycle, the values of the forces in all twenty limiting solutions of Table 1 were calculated. A solution was rejected if any of the three forces were negative-corresponding to compressive muscle or ligament forces or to a tensile contact force. Four possible minimum principles of muscle selection-minimal muscle force, muscle stress, ligament force and contact force-were examined. The limiting solutions consistent with each principle were selected at each sampling point on the walking cycle. The effective cross-sectional areas of quadriceps, hamstrings and gastrocnemius needed to calculate muscle stress were taken from Alexander and Vernon (24).
J
Walking cycle %
Fig. 4 Knee flexion angle plotted over the walking cycle for the representative subject
EMG findings, both published (25) and the authors' own (Fig. 7). 3.1 Temporal solution patterns The following description applies to the temporal patterns for the representative subject using subject-specific parameters (Fig. 5a). For a brief period following heelstrike, a flexing moment was required at the knee. Three limiting solutions were found-two single muscle solutions, HPC (hamstrings, posterior cruciate ligament and tibio-femoral contact force) and GAC To
Stance phase
Swing phase
-
Swing phase
-
-
-
+xion
20
60
40
80
QAC QPC QHC HQC QGC GQC HAC HPC GAC GPC HGC QAP HAP GAP QHA QHP QGA QGP HGA HGP QHG 100
Extension
Extension Walking cycle
Walking cycle
%
%
(a) Subject-specific parameters
(b) Averaged parameters
Fig. 5 Temporal patterns of admissible limiting solutions for the representative subject. The notation is the same as that used for Table 1. Knee flexion/extension moment of the external loads about the instant centre of the knee is plotted at the bottom @
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MUSCLE-LIGAMENT INTERACTIONS AT THE KNEE DURING WALKING
Stance phase
.n
EMG temporal patterns Stance phase To
Swing phase
To
15
Swing phase
:
-t
Quadriceps
Hamstrings
Gastrocnemius
0
20
60
40
100
80
Walking cycle %
Fig. 7 Periods of muscle activity for the representative subject, as indicated by EMG
.
0
.
20
.
. . .
. 60
40
80
, 100
Walking cycle %
The number of admissible limiting solutions found at each data point of the walking cycle for the representative subject, using subject-specific parameters Stance phase
(gastrocnemius, anterior cruciate ligament and tibiofemoral contact force), and one synergistic solution, HGC. During the remainder of early stance, an extending moment was required at the knee. Only one limiting solution, QAC (quadriceps, anterior cruciate ligament and tibio-femoral contact force), was found to be physically realizable. For the bulk of mid-stance, two single muscle solutions, HPC and GAC, and one synergistic Stance phase
Swing phase
To
-
To
Swing phase
QAC HPC
-
GAC
QAP HAP
QAP HAP
i
.
20
.
40
60
80
1
100
40
20
60 Walking cycle
Walking cycle %
%
(b) Principle of minimal muscle stress
(a) Principle of minimal muscle force Stance phase
Swing phase
To
To
Stance phase
Swing phase
.
.)
0
20
40
60
100
80
80
100
0
20
60
40
Walking cycle %
Walking cycle
(c) Principle of minimal ligament force
(d) Principle of minimal contact force
80
. 100
%
Fig. 8 Temporal patterns of admissible limiting solutions consistent with the tested minimum principles of muscle selection Proc Instn Mech Engrs Vol 205
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Table 3 Subject-to-subject
4
force patterns
o
Anterior cruciate (QAC) Anterior cruciate (GAC)
A
Posterior cruciate (HPC)
A
Anterior cruciate (HAC)
3
F? L
variability:
temporal
Period of gait cycle
Admissible limiting solution(s)
Number of subjects (9 total)
Early stance Mid-stance Late stance Early swing Mid-swing Late swing
QAC HAC, GAC QAC QAC, QHC Distracting solutions HAC, GAC, HGC
8 7 7 9
9 9
number of admissible limiting solutions for any subject was six, occurring only at one data point in the swing phase. 0
0
20
40
60
80
100
3.4 Minimum principles of muscle selection
Walking cycle %
Fig. 9 Sample cruciate ligament forces for the representative subject over the gait cycle, where limiting solutions have been selected that most closely accord with EMG findings. Adjusted subject-specific parameters (solid lines), averaged parameters (dashed lines) solution, HGC, were again obtained. During the early and late portions of mid-stance, the antagonistic solution, HQC, was also found. Throughout most of late stance, only one solution, QAC, was obtained. However, for two frames preceding toe-off, the antagonistic solution, QHC, was also found. Following toe-off, two extensor solutions, QAC and QHC, were found. Mid-swing was briefly characterized by distracting solutions. During most of late swing, three flexor solutions were obtained-HPC, GAC and HGC. The antagonistic solution, HQC, was found only for a few data frames. It was not found peri-heelstrike where indicated by EMG.
3.2 Subject-specific versus averaged parameters The resultant flexion/extension moment at the knee (Figure 5) for the representative subject using subjectspecific and averaged parameters was nearly identical. The magnitudes of the subject-specific muscle tendon lever-arms (Fig. 1) were generally larger than those calculated using averaged parameters. There was similarity in the temporal patterns (Fig. 5) during early stance, late stance and early swing. Significant differences in temporal patterns occurred during mid-stance and late swing.
3.3 Subject-to-subject variability The patterns of limiting solutions at different phases of the walking cycle for the remaining nine subjects, using averaged parameters, are summarized in Table 3. There was remarkable unanimity in the predicted patterns of seven of the subjects during the stance phase and for all of them during swing. The ninth subject walked so as to eliminate the need for flexor activity during stance. In nearly all of the subjects, the single muscle solution, QAC, proved to be the only theoretical possibility over 35-60 per cent of the walking cycle, mainly in early and late stance. Of the twenty possible, the maximum
The validity of each minimum principle (Fig. 8) can be evaluated by comparing the obtained limiting solutions with temporal EMG patterns (Fig. 7). Three of the minimum theories-minimal muscle force, ligament force and contact force-selected gastrocnemius solutions during late swing and the period immediately following heelstrike, whereas EMG did not indicate such activity. The principles of minimal muscle stress, minimal ligament force and minimal contact force preferred hamstrings solutions during mid-stance, while EMG demonstrated only gastrocnemius action. 4 DISCUSSION
4.1 Number of admissible limiting solutions Although twenty limiting solutions could arise at each point of the walking cycle, the simple constraint that ligament and muscle forces should be tensile and contact forces compressive ruled out most of them (Figs 5 and 6). The locomotor system is less redundant than might appear at first sight. Possible solutions were found to be inappropriate for a number of reasons. When the resultant force transmitted across the joint had a compressive component, the distractive limiting solutions of Table 1 were inappropriate. This was the case over most of the gait cycle. Flexing loads ruled out the single muscle flexor solutions and vice versa. Since the ACL pulls the tibia backwards and the PCL pulls it forwards, one or other can always provide a shear force in the direction required for dynamic equilibrium so that a single muscle limiting solution involving only one cruciate ligament force was found at every instant. The same is not true of the muscle forces; at times in the walking cycle, some antagonistic and synergistic solutions were ruled out because the antagonistic or synergistic pull had a forwards component parallel to the tibia1 plateau when a backwards pull was required, or vice versa. Thus, many of the non-distractive solutions were ruled out because the directions of the soft tissue elements, muscles or ligaments, were inappropriate.
4.2 EMG/theory agreement and discrepancy Single muscle limiting solutions consistent with EMG phasic patterns were found over much of the gait cycle: 0 IMechE 1991
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MUSCLE-LIGAMENT INTERACTIONS AT THE KNEE DURING WALKING
for quadriceps during early and late stance, hamstrings during late swing and gastrocnemius throughout midstance. However, during the interval immediately preceding and following heelstrike, EMG indicated antagonistic quadriceps-hamstrings activity not found by the calculations. 4.3 Bi-articular muscles Quadriceps and hamstrings have elements that span the hip as well as the knee. Gastrocnemius spans the ankle as well as the knee. It is possible that mechanical considerations at the hip or ankle could influence the activity of these muscles at the knee and could, for instance, explain the quadriceps-hamstrings activity periheelstrike detected by EMG but not predicted by the present model. The fact that hip or ankle mechanics might demand flexor activity at the knee whereas knee mechanics demands extensor activity need not present a dificulty since the self-equilibrating solutions of Table 2 can always be superimposed on the load-based solutions of Table 1. If account were taken of the contact forces at the hip and ankle, both in magnitude and direction, as well as tensile forces in soleus, tibialis anterior, iliopsoas and gluteus maximus together with the six forces of the present model, there would be 498 possible limiting solutions for a two-dimensional study (26). Taking account of the bi-articular nature of muscles obviously complicates the problem. 4.4 Parameter sensitivity
The main sensitivity to parameter choice proved to be for those elements, ligaments or muscles, that can lie nearly perpendicular to the tibial plateau : the anterior cruciate ligament near extension, the posterior cruciate ligament at high flexion angles (not encountered in level walking), the patellar tendon near 90" of flexion, the hamstrings tendon near extension, the gastrocnemius tendon over most of the flexion range. Differences of 10 per cent in parameter values, well within the error bands of these measurements, could make a difference of 4-5" in the values of the muscle or ligament angles relative to the tibial plateau. When the muscle tendon angles are nearly 90°, such differences could make uncertain whether the muscle applies a backwards or a forwards pull to the tibia, the factor which mainly determines whether a given limiting solution is likely to be admissible or not. For instance, for the representative subject in early stance, when the hamstrings tendon is nearly perpendicular to the plateau, the averaged parameters gave the HAC solution only whereas the subject-specific parameters gave H P C and HGC. The main area of uncertainty in the present study lies in the calculated values of the anterior cruciate ligament force near extension, that is during the stance phase. The value of the ligament force is determined by the need to provide a shear force parallel to the tibial plateau, the shear force being equal to the horizontal component, L cos(B,,,), of the ligament force L. When the inclination of the ligament, Olig, is near 90", small changes in O,,, can result in large differences in the calculated value of L. The similarity between the resultant flexion/extension moment at the knee (Fig. 5) for the subject-specific and
17
averaged parameters reflected the fact that the calculated position of the instant centre relative to the line of action of the external loads was nearly identical in the two cases. The differences in the values of the muscle tendon lever-arms had consequences for the calculated forces-the larger subject-specific lever-arms resulted in smaller muscle forces at any instant during the gait cycle. The knee-model parameters were estimated from anthropometrical studies of cadaveric specimens, using X-ray and dissection techniques (19). These formed the basis of the set of averaged parameters used in the calculations. For all but the representative subject, individual measurements on the subjects in this study were limited to weight, lower leg length and marker position. Magnetic resonance imaging methods are now being developed that can detect more readily the points of soft tissue insertion into the bones and may make parameter estimation easier and more accurate for individual subjects.
4.5 The role of ligaments in force transmission during gait Despite uncertainties about force magnitudes arising from parameter sensitivity, these studies show that the cruciate ligaments play an essential mechanical role in normal gait. Whenever EMG shows that a single muscle group, extensor or flexor, is active, it is a certainty that some of the ligaments are under load. Single muscle group activity is found over most of the cycle for normal level walking so that it can be concluded that the muscles do not normally act to protect the ligaments. Figure 9 shows selected cruciate ligament force calculations for the representative subject. These results are consistent with the EMG data for that subject, except in late swing and early stance. The results for the ten subjects were similar: the ACL was shown to have a possible role in 65-100 per cent of the cycle (Table 3). It is interesting to compare these calculations with those of Morrison (1, Fig. 8). Both sets of solutions show three waves of ACL load transmission during stance, but the present calculations yielded much higher force values. Morrison found PCL action during stance in two of his three subjects whereas it was not found in nine subjects in the present study. The calculations for the representative subject, using subject-specific parameters, gave possible PCL action with hamstrings during mid-stance, but this limiting solution was not confirmed with EMG. Both models found PCL action during swing, persisting in Morrison's results and in the case of our representative subject into early stance. Using subject-specific parameters for the representative subject, a maximum ACL force of 3.5 x BW (2.35 kN) was calculated for the single muscle solution QAC during early stance. This value is high compared to the experimental results of Noyes and Grood (27), who found that human anterior cruciate ligaments failed in vitro at maximum tensile loads of 1.7 kN & 0.7 kN, decreasing rapidly with advancing age. Apart from parameter sensitivity discussed above, there are several factors neglected in the theoretical analysis which could further reduce the predicted ligament loading. For example, ligament elasticity would allow some anteroposterior translation of the tibia rela-
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tive to the femur and would decrease the ligament force required to resist an arbitrary shear force acting along the plateau. Moreover, allowing for participation by the collateral ligaments would further reduce the estimates of cruciate ligament forces (28). The assumption of inextensible ligaments therefore provides an upper bound solution to the value of the cruciate ligament forces. Finally, the maximum value of the ACL force was found for the single muscle solution QAC in early stance where EMG indicated that hamstrings and quadriceps were acting simultaneously at the knee, although the antagonistic limiting solution QHC was not obtained by the model. It is theoretically possible, however, to superimpose the self-equilibrating solution QHPC onto the limiting solution QAC to give an antagonistic solution with ligament loading, that is QHAC. The addition of hamstrings action at the knee could reduce the loading of the ACL. 4.6 Muscle selection criteria
These studies, taking account of possible ligamentous contributions to load bearing, appear to rule out many of the global criteria previously considered as possible bases for the selection of muscle activity. Any contribution from the self-equilibrating solutions of Table 2 to the load-related single muscle limiting solutions would increase the predicted muscle and contact forces but may increase or decrease ligament forces (Fig. 3). The single muscle limiting solutions therefore represent the minimal muscle force response of the knee joint to load. In the presence of flexing loads acting at the knee, single extensor muscle limiting solutions, that is QAC and QPC, also represent the minimal contact force response of the knee. In the presence of extending loads acting at the knee, the minimal contact force response of the joint is characterized by synergistic flexor muscle limiting solutions, that is HGC (Fig. 8d). The present geometric and mechanical analysis suggests that it may be unrealistic to seek a single optimality criterion for muscle selection that applies throughout the walking cycle. ACKNOWLEDGEMENTS
The authors would like to thank Professor M. W. Whittle and Dr R. J. Jefferson for their support and encouragement during this investigation. The authors also thank Professor H. J. Woltring for providing them with a FORTRAN package for spline smoothing and differentiation. Dr E. Biden and Mrs T. L. Shercliff contributed to the development of the knee model and associated programming and were supported by grants from the Arthritis and Rheumatism Council. JJC held a Rhodes Scholarship for the duration of the study. REFERENCES 1 Morrison, J. B. The mechanics of the knee joint in relation to normal walking. J . Biomech., 1970, 3, 51-61.
2 MacConnaill, M. A. The ergonomic aspects of articular mechanics. In Studies ofthe anatomy andfunction ofbones and joints (Ed. F. G. Evans), 1967, pp. 69-80 (Springer, Berlin). 3 Seireg, A. and Arvikar, R. J. The prediction of muscular load sharing and joint forces in the lower extremities during walking. J . Biomech., 1975,8, 89-102. 4 Pedotti, A., Krishnan, V. V. and Stark, L. Optimization of muscle force sequencing in human locomotion. Math. Biosci., 1978, 38, 57-76. 5 Rohrle, H., Scholten, R., Sigolotto, C. and Sollbach, W. Join! forces
in the human pelvis-leg skeleton during walking. J. Biomech., 1984,17,409424. 6 Hardt, D. E. Determining muscle forces in the leg during normal human walking-an application and evaluation of optimization methods. Trans. ASME, J. Biomech. Engng, 1978,100,72-78. 7 Crowninshield, R. D. and Brand, R. A. A physiologically based criterion of muscle force prediction in locomotion. J. Biomech., 1981, 14,793-801. 8 Patriarco, A. G., Mann, R. W., Simon, S. R. and Mansour, J. M. An evaluation of the approaches of optimization models in the prediction of muscle forces during human gait. J. Biomech., 1981, 14,513-525. 9 An, K. N., Kwak, B. M., Chao, E. Y. and Morrey, B. F. Determination of muscle and joint forces: a new technique to solve the indeterminate problem. Trans. ASME, J. Biomech. Engng, 1984, 106,364367. 10 Calderdale, P. M. and Scelfo, G. A mathematical model of the locomotor apparatus. Engng Med., 1987,16, 147-161. 11 Mikosz, R. P., Andriacchi, T. P. and Andersson, G. B. J. Model analysis of factors influencing the prediction of muscle forces at the knee. J . Orthop. Res., 1988,6,205-214. I2 Nissan, M. Review of some basic assumptions in knee biomechanics. J . Biomech., 1980, 13, 37S381. 13 Harrington, I. J. A bioengineering analysis of force actions at the knee in normal and pathological gait. Biomed. Engng, 1976, 11(5), 167-172. 14 Whittle, M. W. Calibration and performance of a threedimensional television system for kinematic analysis. J . Biomech., 1982,15, 185-196. 15 Jefferson, R. J., Collins, J. J., Whittle, M. W., Radin, E. L. and O’Connor, J. J. The role of quadriceps in controlling impulsive forces around heelstrike. Proc: Instn h e c h . Engrs, Part H; 1990, 204(H1),21-28. 16 O’Connor, J., Shercliff, T. and Goodfellow, J. The mechanics of the knee in the sagittal plane. In Surgery and arthroscopy of the knee (Eds W. Muller and W. Hackenbruck), Proceedings of the Second Congress of the European Society for Surgery and Arthroscopy of the Knee, 1988, pp. 12-30 (Springer-Verlag, Berlin). 17 O’Connor, J. J., Shercliff, T. L., Biden, E. and Goodfellow, J. W. The geometry of the knee in the sagittal plane. Proc-. Instn Mech. Engrs, Part H, 1989,203(H4), 223-233. 18 O’Connor, J., Shercliff, T., FitzPatrick, D., Bradley, J., Daniel, D., Biden, E. and Goodfellow, H. Geometry of the knee. In Knee ligaments-structure, Junction, injury, and repair (Eds D. M. Daniel, W. H. Akeson and J. J. O’Connor), 1990, pp. 163-200 (Raven Press, New York). 19 Bradley, J., FitzPatrick, D., Daniel, D., Shercliff, T. and O’Connor, J. Orientation of the cruciate ligament in the sagittal plane: a method of predicting length change vs. knee flexion. J . Bone Jt Surg., 1988,70B, 9499. 20 O’Connor, J., Biden, E., Bradley, J., FitzPatrick, D., Young, S., Kershaw, C., Daniel, D. and Goodfellow, J. The muscle-stabilized knee. In Knee ligaments-structure, Junction, injury, and repair (Eds D. M. Daniel, W. H. Akeson and J. J. OConnor), 1990, pp. 239-278 (Raven Press, New York). 21 O‘Connor, J., SherclifT, T., FitzPatrick, D., Biden, E. and Goodfellow, J. Mechanics of the knee. In Knee ligaments-structure, function, injurv, and reDair (Eds D. M. Daniel. W. H. Akeson and J. “J.OConnor),-l990, pp. 201-238 (Raven Press, New York). 22 Winter, D. A. Biomechanics of human mooement, 1979 (Wiley and Sons, New York). 23 Woltring, H. J. A FORTRAN package for generalized, crossvalidatory spline smoothing and differentiation. Ado. Engng Software, 1986,8(2), 104113. 24 Alexander, R. McN. and Vernon, A. The dimensions of knee and ankle muscles and the forces they exert. J. Hum. Mooement Stud., 1975.1, 115-123. 25 Sutherland, D., Olshen, R., Biden, E. and Wyatt, M. Development ofmature walking, 1988 (MacKeith Press, London). 26 Collins, J. J. Joint mechanics-modelling of the lower limb. Doctoral thesis, 1990, University of Oxford. 27 Noyes, F. R. and Grood, E. S. The strength of the anterior cruciate ligament in humans and rhesus monkeys. J. Bone Jt Surg., 1976, %A, 107&1082. 28 FitzPatrick, D. P. and O’Connor, J. J. Theoretical modelling of the knee applied to the anterior drawer test. IMechE conference on The Changing Role of Engineering in Orthopaedics, London, 1989, paper C384/033, pp. 79-83 (Mechanical Engineering Publications, London).
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