Pattern of anterior cruciate ligament force in normal walking

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1 Journal of Biomechanics 37 (2004) Journal of Biomechanics Award Pattern of anterior cruciate ligament force in normal walking Kevin B. Shelburne a, *, Marcus G. Pandy b, Frank C. Anderson c, Michael R. Torry a a Steadman-Hawkins Sports Medicine Foundation, Vail, Colorado, USA b Department of Biomedical Engineering, University of Texas at Austin, USA c Department of Mechanical Engineering, Stanford University, USA Accepted 29 October 2003 Abstract The goal of this study was to calculate and explain the pattern of anterior cruciate ligament (ACL) loading during normal level walking. Knee-ligament forces were obtained by a two-step procedure. First, a three-dimensional (3D) model of the whole body was used together with dynamic optimization theory to calculate body-segmental motions, ground reaction forces, and leg-muscle forces for one cycle of gait. Joint angles, ground reaction forces, and muscle forces obtained from the gait simulation were then input into a musculoskeletal model of the lower limb that incorporated a 3D model of the knee. The relative positions of the femur, tibia, and patella and the forces induced in the knee ligaments were found by solving a static equilibrium problem at each instant during the simulated gait cycle. The model simulation predicted that the ACL bears load throughout stance. Peak force in the ACL (303 N) occurred at the beginning of single-leg stance (i.e., contralateral toe off). The pattern of ACL force was explained by the shear forces actingat the knee. The balance of muscle forces, ground reaction forces, and joint contact forces applied to the legdetermined the magnitude and direction of the total shear force acting at the knee. The ACL was loaded whenever the total shear force pointed anteriorly. In early stance, the anterior shear force from the patellar tendon dominated the total shear force applied to the leg, and so maximum force was transmitted to the ACL at this time. ACL force was small in late stance because the anterior shear forces supplied by the patellar tendon, gastrocnemius, and tibiofemoral contact were nearly balanced by the posterior component of the ground reaction. r 2003 Elsevier Ltd. All rights reserved. Keywords: Knee modeling; Human gait; Ligament; Muscle; Force; Joint 1. Introduction A number of studies have inferred anterior cruciate ligament (ACL) loading from in vivo measurements of bony motion at the knee (Kvist and Gillquist, 2001; Lafortune et al., 1992; Li et al., 1996; Marans et al., 1989; Zhanget al., 2003), but very few studies have calculated knee-ligament forces in gait. Of those that have, there appears to be some disparity in the results. Morrison (1970) used an inverse dynamics approach to estimate muscle, ligament, and joint contact forces at the knee duringnormal level walking. By assumingthat only some of the muscles and ligaments bear load at any instant, Morrison was able to reduce the number of unknown muscle, ligament, and contact forces to that of the available equations of mechanics. His calculations *Correspondingauthor. 945 Meadow Run, Golden, CO 80403, USA. Tel.: ; fax: address: kevin.shelburne@shsmf.org(k.b. Shelburne). showed that the ACL was loaded throughout the stance phase of walking, and that peak ACL force was 156 N (B 1 4 body weight (BW)). Using a similar approach, Harrington (1976) also found that the ACL was loaded throughout stance, but the peak force transmitted to the ligament was estimated to be much higher (B 1 2 BW). Collins and O Connor (1991) used a sagittal plane model of the knee to study the interactions between the muscles, ligaments, and bones during normal level walking. Their analysis took into account simultaneous agonist and antagonist muscle activity, and peak ACL forces were found to be much higher ( BW) than the values obtained by Morrison and Harrington. Using the same sagittal plane model of the knee, Collins (1995) subsequently modeled the entire lower limb and applied optimization techniques to estimate legmuscle forces duringnormal gait. His results predicted peak ACL forces that were lower ( BW) than those of Collins and O Connor (1991) /$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi: /j.jbiomech

2 798 ARTICLE IN PRESS K.B. Shelburne et al. / Journal of Biomechanics 37 (2004) The purpose of the present study was to calculate and explain the pattern of ACL force in normal level walking. Knee-ligament forces were obtained by a two-step procedure. First, a three-dimensional (3D) model of the whole body was used together with dynamic optimization theory to calculate body-segmental motions, ground reaction forces, and leg-muscle forces for one cycle of gait. Joint angles, ground reaction forces, and muscle forces obtained from the gait simulation were then input into a musculoskeletal model of the lower limb that incorporated a 3D model of the knee. The relative positions of the femur, tibia, and patella and the forces induced in the knee ligaments were found by solvinga static equilibrium problem at each instant duringthe simulated gait cycle. Based on previous estimates of ACL loadingobtained for more vigorous exercises such as maximum isometric knee extension (Shelburne and Pandy, 1997), we hypothesized that peak ACL force in normal walkingwould be much lower than 1 BW and that the pattern of ACL loading would be determined mainly by quadriceps muscle activity. 2. Methods A 3D model of the body was used to calculate legmuscle forces in normal walking. Details of this model are given by Anderson and Pandy (1999, 2001a), so only a brief description is provided here. The skeleton was modeled as a 10-segment, 23 degree-of-freedom (d.o.f.) articulated chain. The pelvis was given six d.o.f. so that it could translate and rotate freely in space. A 3-d.o.f. ball-and-socket joint was placed at the level of the third lumbar vertebra to model relative movements of the upper body and pelvis. Each hip was modeled as a 3- d.o.f. ball-and-socket joint; each knee, as a 1-d.o.f. hinge; and each ankle, as a 2-d.o.f. universal joint. Two segments were used to model each foot: a hindfoot and a toes segment. The hindfoot and toes segments were separated by a 1-d.o.f. hinge joint. Five damped springs were placed under the sole of each foot to simulate interaction with the ground; each spring applied forces in all three coordinate directions simultaneously. The inertial properties of the segments were based on anthropometric measures obtained from five healthy adult males (age 2673 yr, height cm, and mass kg). The walkingmodel was actuated by 54 musculotendinous units. Six abdominal and back muscles controlled the relative movements of the pelvis and upper body, and 24 muscles actuated each leg. Each musculotendon actuator was represented as a 3-element muscle in series with tendon (Pandy et al., 1990; Zajac, 1989). Parameters definingthe force producingproperties of each actuator are given by Anderson and Pandy (1999). Muscle excitation contraction (activation) dynamics was approximated as a first-order process (Zajac, 1989). The dynamic optimization problem for normal walkingwas to find the muscle excitation histories, muscle forces, and body-segmental motions correspondingto minimum metabolic energy consumed per unit distance moved. Bilateral symmetry was assumed, and so only half the gait cycle was simulated. The initial states of the model were found by averaging kinematic and force plate data obtained from gait experiments performed on the five subjects. The final time was fixed to 0.56 s, which was the average time taken by the subjects to complete one-half the gait cycle. Terminal constraints were applied to the joint angles, joint angular velocities, and muscle forces to enforce symmetry of the gait cycle. Details of the dynamic optimization problem are given by Anderson and Pandy (2001a). Details of the model used to estimate muscle metabolic energy consumption are presented by Bhargava et al. (2004). A computational solution was found by convertingthe dynamic optimization problem to a parameter optimization problem (Pandy et al., 1992). Knee-ligament forces were calculated using another musculoskeletal model of the lower limb that incorporated a 3D model of the knee. Five segments were used to represent the lower limb in this model: thigh, shank, patella, hindfoot, and toes. These segments were connected together by five joints: hip, tibiofemoral joint, patellofemoral joint, ankle, and metatarsal joint (Fig. 1). The pelvis was treated as the base segment and remained fixed to the ground. The hip, ankle, and metatarsal joints were represented in exactly the same way as in the walkingmodel. Details of the models used to represent these joints are presented as Supplementary Material on the Webpage of the Journal of Biomechanics. Six generalized coordinates described the relative movements of the thigh and shank in the lower-limb model (Fig. 1). The orientation of the shank relative to the thigh was defined by a sequence of three rotations about three mutually perpendicular joint axes: varus valgus, internal external rotation, and flexion extension. The displacement of the origin of the shank reference frame relative to the origin of the thigh reference frame was defined by translations alongthe three joint axes: anterior posterior translation, proximal distal translation, and medial lateral translation, respectively. The geometry of the distal femur, proximal tibia, and patella was based on parasagittal sections of the bones obtained from 23 cadaveric knees (Gargand Walker, 1990). The shapes of the medial and lateral tibial plateau, the medial and lateral femoral condyles, the medial and lateral patellar surfaces of the femur, and the medial and lateral patellar facets were reproduced by fittingpolynomials to the digitized cadaver data. The

3 K.B. Shelburne et al. / Journal of Biomechanics 37 (2004) Fig. 1. Kinematic structure of the lower-limb model used to calculate knee-ligament forces during gait. The position and orientation of the legrelative to a reference frame fixed on the pelvis were described by 18 generalized coordinates: flexion extension of the hip, q 1 ; adduction abduction of the hip, q 2 ; internal external rotation of the hip, q 3 ; anterior posterior translation of the shank relative to the thigh, q 4 ; proximal distal translation of the shank relative to the thigh, q 5 ; medial lateral translation of the shank relative to the thigh, q 6 ; varus valgus rotation of the knee, q 7 ; internal external rotation of the knee, q 8 ; flexion extension of the knee, q 9 ; anterior posterior translation of the patella relative to the thigh, q 10 ; proximal distal translation of the patella relative to the thigh, q 11 ; medial lateral translation of the patella relative to the thigh, q 12 ; patellar rotation, q 13 ; patellar tilt, q 14 ; patellar flexion extension, q 15 ; plantarflexion dorsiflexion of the ankle, q 16 ; inversion eversion of the subtalar joint, q 17 ; flexion extension of the metatarsal joint, q 18. The locations of the centers of the mass of the thigh, shank, hindfoot, and toes segments are as indicated; the patella was treated as a massless body in the model. contactingsurfaces of the femur and tibia were modeled as deformable (Pandy et al., 1998). Six generalized coordinates described movements of the patella relative to the thigh (Fig. 1). The orientation of the patella was defined by a sequence of three rotations about three mutually perpendicular joint axes: patellar rotation, patellar tilt, and flexion extension. The displacement of the origin of the patellar reference frame relative to the origin of the thigh reference frame was defined by three translations alongthe three joint axes: anterior posterior translation, proximal distal translation, and medial lateral translation, respectively. The model of patellofemoral mechanics was based on the assumptions that the patellar tendon was inextensible and that interpenetration between the facets of the patella and the patellar surfaces of the femur can be neglected. These two assumptions define three holonomic constraints for movement of the patella on the femur. These three constraints can be combined with the six force and moment equilibrium equations for the patella to yield a set of six non-linear algebraic equations for patellofemoral mechanics (Eq. (2) below). The geometry of the cruciate and collateral ligaments, the posterior capsule, and the anterolateral structures of the knee was modeled using13 elastic elements (Fig. 2). The ACL and PCL were each represented by an anterior and a posterior bundle. The MCL was represented by two layers: a superficial layer comprised of three bundles, and a deep layer comprised of two bundles. The LCL and the ALS were each represented by one bundle, while the posterior capsule was represented by two: a medial and a lateral bundle. Each ligament bundle was assumed to be elastic, and its properties were described by a non-linear force strain curve (Blankevoort et al., 1991). The origin and insertion sites of the ligament bundles were based on data reported by Garg and Walker (1990). Stiffness and reference strain values assumed for each ligament bundle were adjusted until the response of the model in anterior posterior draw and axial rotation matched in vitro data reported in the literature. The parameters assumed for the model ligaments are given in Table 1 of the Supplementary Material presented on the Webpage of the Journal of Biomechanics. The behavior of the meniscus was approximated by applyinga shear force to the shank. Specifically, the action of the posterior horn of the medial meniscus was modeled by applyinga posterior shear force to constrain anterior movement of the shank relative to the thigh. The slope and intercept of the curve definingthis shear force were found by simulatingthe experiments of Shoemaker and Markolf (1986) and Levy et al. (1982), and then adjustingthe model parameters until the anterior posterior laxity of the model matched the in vitro data. Thirteen muscles were represented in the lower-limb model (Fig. 3). The paths of all muscles, except vasti, hamstrings, and gastrocnemius, were identical with those incorporated in the walkingmodel. Whereas vasti, hamstrings, and gastrocnemius were each represented as

4 800 ARTICLE IN PRESS K.B. Shelburne et al. / Journal of Biomechanics 37 (2004) one muscle in the walkingmodel, the separate portions of each of these muscles were included in the lower-limb model. Forces in the separate portions of these muscles were estimated based on the relative cross-sectional area of each muscle. For example, vastus medialis force was found by multiplyingvasti force obtained from the walkingsimulation by the cross-sectional area of vastus medialis and then dividingby the total cross-sectional area of vasti. The dynamical equations of motion for the lowerlimb model can be written in general form as AðqÞ.q þ Cðq; qþþm mus ðqþf mus þ M lig ðqþf lig þ M GRF ðqþf GRF þ GðqÞþTðq; qþ ¼0; p i ðq hip; q tf ; q pf ; F Q Þ¼0; i ¼ 1; 6; ð2þ ð1þ Fig. 2. Schematic of the knee ligaments represented in the lower-limb model. The diagram shows the paths of the anterior and posterior bundles of the anterior cruciate ligament (aacl and pacl, respectively); the anterior and posterior bundles of the posterior cruciate ligament (apcl and ppcl); the anterior, central, and posterior bundles of the superficial medial collateral ligament (amcl, cmcl, and pmcl); the anterior and posterior bundles of the deep medial collateral ligament (acm and pcm); the lateral collateral ligament (LCL); the anterolateral structures (ALS) of the knee; and the medial and lateral posterior capsule (mcap and lcap). where q ¼ðq hip ; q tf ; q pf ; q ankle ; q mt Þ is a 18 1 vector containingthe generalized coordinates used to describe the configurations of the hip, tibiofemoral joint, patellofemoral joint, ankle, and metatarsal joint; AðqÞ Zisa1218 matrix representingthe mass matrix of the system; Cðq; qþ is a 12 1 vector containingthe Coriolis and centrifugal forces and torques arising from the motion of the thigh, shank, hindfoot, and toes segments (the patella was treated as massless); F mus is a 13 1 vector containingthe forces applied by the 13 muscles included in the model; M mus ðqþ is a matrix describingthe moment arms of the applied muscle forces; F lig is a 13 1 vector containingthe forces applied by the knee ligaments in the model; M lig ðqþ is a matrix describingthe moment arms of the kneeligament forces; F GRF is a 15 1 vector containingthe forces applied by the ground springs to the hindfoot and toes segments; M GRF ðqþ is a matrix of moment arms for the ground reaction forces (GRF); F Q is the magnitude of the force applied by the quadriceps tendon Table 1 Comparison of peak ACL force for normal level walkingas calculated in the current study and reported previously in the literature Task Peak ACL force Reference Walking1000 N at early stance Collins and O Connor (1991) Walking900 N at early stance Collins (1995) Max. isokinetic extension at 30 /s 590 N at 10 Serpas et al. (2002) Max. isometric extension 520 N at 15 Shelburne and Pandy (1997) Max. isokinetic extension at 180 /s 450 N at 5 Serpas et al. (2002) Walking411 N at early stance Harrington (1976) Max. isokinetic extension at 300 /s 340 N at 5 Serpas et al. (2002) Walking303 N at early stance Shelburne et al. Static squat 200 N at 0 Shelburne and Pandy (1998) Active knee extension 160 N at 0 Shelburne and Pandy (1997) Walking156 N at early stance Morrison (1970) Squatting N Toutoungi et al. (2000) Dynamic squat-to-stand 20 N at 25 Shelburne and Pandy (2002) Also shown for comparison are peak ACL forces estimated for other tasks.

5 K.B. Shelburne et al. / Journal of Biomechanics 37 (2004) Fig. 3. Schematic diagram showing the 13 muscles incorporated in the lower-limb model. The path of each muscle was modeled as a straight line, except when it came into contact with another muscle or the surface of a bone. Via points and via cylinders were used to model the way a muscle wrapped around another muscle or bone (Garner and Pandy, 2000). The diagram shows the paths of the medial, intermediate, and lateral portions of vasti (VasMed, VasInt, VasLat); rectus femoris (RF); biceps femoris longhead (BFLH) and short head (BFSH); semimembranosus (MEM) and semitendinosus (TEN); the medial and lateral portions of gastrocnemius (GasMed, GasLat); and tensor fascia latae (TFL). Sartorius and gracilis were also included in the model, but are not shown here. to the patella; GðqÞ is a 12 1 vector of gravitational forces and torques actingon the thigh, shank, hindfoot, and toes segments; and Tðq; qþ is an 8 1 vector of external torques applied at the joints (see below). Eq. (1) was obtained by usinga commercial software package called SD/Fast (Parametric Technology Inc.). The anthropometry of the lower-limb model (i.e., mass and position of center of mass of each segment) was based on that of the walkingmodel (Anderson and Pandy, 1999, 2001a). Knee-ligament forces were calculated by assuming that the lower limb was in static equilibrium at each instant duringthe simulated gait cycle; specifically, the inertial contributions of the thigh, shank, patella, hindfoot, and toes segments were neglected in these calculations. Eqs. (1), (2) define a system of 18 nonlinear differential and algebraic equations in 18 unknowns. The muscle forces and ground reaction forces obtained from the walkingsimulation served as inputs to these equations. The joint angles of the hip, ankle, and metatarsal joints and the flexion extension and internal external rotation angles of the knee were also assumed to be known from the walkingsimulation. The 18 unknowns were the varus valgus angle of the knee, the three translations of the shank relative to the thigh, the six generalized coordinates for the position and orientation of the patella relative to the thigh, and the eight joint torques needed to hold the lower limb static at each instant. Thus, the static equilibrium problem can be stated as follows: Given the muscle forces, ground reaction forces, flexion extension and internal external rotation angles of the knee, and the joint angles of the hip, ankle, and metatarsal joints, find the positions of the shank and patella relative to the thigh as well as the magnitudes of the joint torques needed to equilibrate the lower limb at each instant duringthe gait cycle. A computational solution to this problem was found by integrating Eqs. (1) (2) forward at each time step of the walkingsimulation until the accelerations and velocities of all the joints approached zero. A standard proportional-integral-derivative control scheme was used to drive the joint velocities and accelerations to zero (see Webpage of the Journal of Biomechanics for details). Once the values of the generalized coordinates correspondingto static equilibrium of the lower limb had been determined, the forces transmitted to the knee ligaments were calculated using the force strain relationships assumed for the various ligament bundles incorporated in the model. 3. Results The pattern of muscle forces predicted by the dynamic optimization solution was consistent with EMG activity recorded from the subjects (Fig. 4). The joint angles and GRF obtained from the walkingsimulation were also similar to the same measures obtained from the subjects. For example, knee flexion angle predicted by the model was stereotypic, showingtwo peaks one in the stance phase and the other in swing(fig. 5A). The muscles that applied the largest forces at the knee were the vasti and gastrocnemius (Fig. 5B). Peak force in vasti was 1188 N, which occurred at CTO (Fig. 5B,

6 802 ARTICLE IN PRESS K.B. Shelburne et al. / Journal of Biomechanics 37 (2004) Fig. 4. Forces developed by hamstrings, vasti, and gastrocnemius as obtained from the walkingsimulation reported by Anderson and Pandy (2001b) (thick black lines). The wavy lines are EMG data from one subject who walked at his self-selected speed; these data were normalized by dividingby the maximum electrode voltage recorded duringa maximum voluntary contraction for each muscle. The gait cycle begins and ends with ipsilateral heel strike (HS). Other kinematic events identified at the top of the diagram are contralateral toe-off (CTO), contralateral heel strike (CHS), and ipsilateral toe-off (TO). Adapted from Anderson and Pandy (2001b). Fig. 5. (A) Knee flexion angle as obtained from the walking simulation reported by Anderson and Pandy (2001b). The thin gray lines indicate subject data 71 SD. The thick gray line represents the mean of the experimental data. (B) Resultant forces developed by quadriceps, hamstrings, and gastrocnemius as predicted by the dynamic optimization solution for normal gait. Data are from Anderson and Pandy (2001b). Kinematic events indicated at the top of the graph are as defined in the caption for Fig. 4. VAS). Peak force in gastrocnemius was lower at 849 N, and it occurred at CHS (Fig. 5B, GAS). The hamstrings developed much lower forces duringstance; peak force Fig. 6. Forces transmitted to the cruciate ligaments, the collateral ligaments, and the posterior capsule of the knee over one cycle of normal level walking. The knee ligaments were more heavily loaded in stance than in swing. The ACL bore the largest force: peak force transmitted to the ACL was around 1/2 BW at CTO. Peak force in the ACL coincided with the peak force developed by vasti (compare with Fig. 5). predicted for hamstrings was 495 N, which occurred at heel strike (Fig. 5B, HAMS). The ACL was loaded throughout stance. Peak ACL force was 303 N and occurred at CTO (Fig. 6). Thereafter, ACL force decreased and reached zero shortly after ipsilateral toe-off. The ACL was only lightly loaded duringswing. The PCL was unloaded in stance and only lightly loaded in swing. The LCL was loaded for only brief periods during stance and swing; peak force transmitted to this ligament was much less than that borne by the ACL (Fig. 6). Relatively small forces (o40 N) were also transmitted to the MCL and the posterior capsule. No shear force was borne by the posterior horn of the medial meniscus in the model. The patellar tendon, gastrocnemius, and the tibiofemoral contact force all applied anterior shear forces to the leg(shank+foot), while hamstrings and the resultant GRF applied posterior shear forces (Fig. 7). The total shear force contributed by all sources other than the knee ligaments and inertial forces was directed anteriorly throughout stance (Fig. 7, shaded region). The total shear force peaked at CTO and was around 260 N; thereafter, it decreased and remained much smaller. In early stance, the total shear force was dominated by the anterior pull of the patellar tendon (Fig. 7, PT). In late stance, the posterior component of the GRF was the major factor limitingthe magnitude of the total shear force. 4. Discussion The purpose of this study was to calculate and explain the pattern of force transmitted to the ACL in normal

7 K.B. Shelburne et al. / Journal of Biomechanics 37 (2004) Fig. 7. Shear forces acting on the leg (shank+foot) during the simulated gait cycle. The shaded region shows the total shear force borne by the knee ligaments in the model. Total shear force is the shear force due to the muscle forces, ground reaction forces, and joint contact forces, and does not include the inertial forces of the lowerlimb segments as the analysis assumed static equilibrium of the leg at each instant. Anterior shear forces tended to translate the leg anteriorly; posterior shear forces tended to translate the legposteriorly. Hamstrings always applied a posterior shear force to the leg because these muscles pass behind the knee and insert on the back of the tibia. The ground reaction force applied a posterior shear force to the legbecause the line of action of the resultant ground force passed behind the knee as indicated in the diagram above the graph. Symbols appearingin the diagram are PT (patellar tendon), HAMS (hamstrings); GAS (gastrocnemius), TF (tibiofemoral contact force), and GRF (ground reaction force). gait. The most significant aspects of the analysis are that (1) body motions, ground reaction forces, and legmuscle forces were obtained by solvinga large-scale dynamic optimization problem for normal level walking; and (2) knee-ligament forces were calculated using a musculoskeletal model of the lower limb that incorporated a 3D model of the knee. Perhaps the major limitation of the analysis was the assumption of static equilibrium in the calculation of knee-ligament loading, wherein the effects of centrifugal (velocity-dependent) and inertial forces were neglected. Anderson and Pandy (2003) showed recently that, except near heel strike, muscle and gravitational forces dominate the forces transmitted by the lower-limb joints throughout the stance phase of normal gait. This is in accord with results obtained much earlier by Bresler and Frankel (1950) and Harrington (1976), who found that the inertial forces contributed very little to the net muscle moments exerted about the joints in the sagittal plane when humans walked at their natural speeds. We expect, therefore, that the pattern of ACL force would look much the same as that shown in Fig. 6 had the effects of centrifugal and inertial forces been taken into account. The predictions of knee-ligament loading were also compromised by the fact that the dynamic optimization solution for walkingwas not fully converged. Symmetry of the gait cycle was enforced in the walking simulation by constrainingthe values of the joint angles, joint angular velocities, and muscle forces to be equal at the beginning and end of the simulation (i.e., at contralateral and ipsilateral toe-off, respectively). However, the computed solution did not meet the terminal constraints exactly due to the large size and high non-linearity of the optimization problem solved (see Anderson and Pandy (2001a) for details). As a result, small discontinuities were evident in the joint angles, the ground reaction forces, and the muscle forces predicted by the model (e.g., VAS in Fig. 4 at CTO and TO). The discontinuity in vasti muscle force at contralateral toe-off obtained from the walkingsimulation led to a discontinuity in ACL force at contralateral toe-off (see Fig. 6, ACL at CTO). This is because the calculated value of ACL force in the model depended heavily on the value of vasti force. Even if the dynamic optimization solution had converged fully so that the terminal constraints were met with arbitrary tolerance, the magnitudes of the muscle forces predicted by the simulation would not be very different from those used in this study (Anderson and Pandy, 2001a, b). The reason is that the muscle forces were determined mainly by the minimum metabolic energy cost function specified in the gait simulation. Had the dynamic optimization solution converged fully, the muscle forces would be equal at the beginning and end of the gait cycle, as required by the terminal constraints. Since peak ACL force depends mainly on the value of vasti force at the beginning of the simulation, and because the initial value of vasti force was determined mainly by the cost function, peak ACL force would be practically the same as that shown in Fig. 6 had the gait simulation converged fully. The only difference would be that the discontinuity in ACL force at contralateral toe-off would vanish. Collins (1995), Collins and O Connor (1991), Harrington (1976), Morrison (1970) all found two peaks of ACL loadingin stance, with the maximum occurringin early stance, right around the time of contralateral toeoff. Although the pattern of ACL loading obtained in these studies is similar to that shown in Fig. 6, there are significant differences in the predicted values of peak ACL force. Morrison (1970) calculated a maximum ACL force of 156 N (B0.2 BW), whereas Harrington (1976) predicted forces of about 411 N (B0.7 BW), and Collins and O Connor (1991) and Collins (1995) obtained even higher forces in the range BW and BW, respectively. Peak ACL force for

8 804 ARTICLE IN PRESS K.B. Shelburne et al. / Journal of Biomechanics 37 (2004) normal walkingobtained in the current study was less than 1/2 BW, and is similar to the levels of ACL loading predicted for other tasks, includingisokinetic kneeextension exercise which is commonly performed in rehabilitation (see Table 1). The pattern of ACL force duringwalkingis explained by the shear forces actingat the knee. The balance of muscle forces, ground reaction forces, and joint contact forces applied to the legdetermined the magnitude and direction of the total shear force actingat the knee. The model ACL was loaded whenever the total shear force pointed anteriorly. In early stance, the shear force from the patellar tendon dominated the total shear force applied to the leg, and so maximum force was transmitted to the ACL at this time. Patellar tendon shear force was large in early stance because quadriceps force was large and also because the line of action of the patellar tendon was inclined anteriorly relative to the longaxis of the tibia. Just prior to contralateral heel strike, the posterior component of the ground reaction force was nearly equal to the sum of the anterior shear forces supplied by the patellar tendon, gastrocnemius, and the tibiofemoral contact force (Fig. 7). Gastrocnemius applied an anterior shear force to the shank because the knee was nearly fully extended just before contralateral heel strike, and at small flexion angles gastrocnemius wraps around the back of tibia. Tibiofemoral contact force applied an anterior shear force to the legdue to the posterior slope of the tibial plateau. The ground reaction force applied a posterior shear force to the leg because the line of action of the resultant ground force passed behind the knee. The posterior shear force caused by the ground reaction increased prior to contralateral heel strike because the angle between the shank and the ground increased at this time. The sum of all these shear forces resulted in a total shear force that was almost zero, which explains why ACL force was relatively small in late stance. Acknowledgements Supported in part by the National Science Foundation, Engineering Research Centers Grant EEC and by Sulzer Orthopedics Inc., Austin, Texas. References Anderson, F.C., Pandy, M.G., A dynamic optimization solution for vertical jumpingin three dimensions. Computer Methods in Biomechanics and Biomedical Engineering 2, Anderson, F.C., Pandy, M.G., 2001a. Dynamic optimization of human walking. Journal of Biomechanical Engineering 123, Anderson, F.C., Pandy, M.G., 2001b. Static and dynamic optimization solutions for gait are practically equivalent. Journal of Biomechanics 34, Anderson, F.C., Pandy, M.G., Individual muscle contributions to support in normal walking. Gait Posture 17, Bhargava, L., Pandy, M.G., Anderson, F.C., A phenomenological model for estimating metabolic energy consumption in muscle contraction. Journal of Biomechanics 37, Blankevoort, L., Kuiper, J.H., Huiskes, R., Grootenboer, H.J., Articular contact in a three-dimensional model of the knee. Journal of Biomechanics 24, Bresler, B., Frankel, J.P., The forces and moments in the leg duringlevel walking. Transactions of the American Society of Mechanical Engineers 72, 27. Collins, J.J., The redundant nature of locomotor optimization laws. Journal of Biomechanics 28, Collins, J.J., O Connor, J.J., Muscle-ligament interactions at the knee during walking. Proceedings of the Institute of Mechanical Engineers [H] 205, Garg, A., Walker, P.S., Prediction of total knee motion using a three-dimensional computer-graphics model. Journal of Biomechanics 23, Garner, B.A., Pandy, M.G., The obstacle set method for representingmuscle paths in musculoskeletal models. Computer Methods in Biomechanics and Biomedical Engineering 3, Harrington, I.J., A bioengineering analysis of force actions at the knee in normal and pathological gait. Biomedical Engineering 11, Kvist, J., Gillquist, J., Anterior positioningof tibia during motion after anterior cruciate ligament injury. Medicine and Science in Sports and Exercise 33, Lafortune, M.A., Cavanagh, P.R., Sommer 3rd, H.J., Kalenak, A., Three-dimensional kinematics of the human knee during walking. Journal of Biomechanics 25, Levy, I.M., Torzilli, P.A., Warren, R.F., The effect of medial meniscectomy on anterior-posterior motion of the knee. Journal of Bone and Joint Surgery, American Volume 64, Li, X.M., Liu, B., Deng, B., Zhang, S.M., Normal six-degree-offreedom motions of knee joint duringlevel walking. Journal of Biomechanical Engineering 118, Marans, H.J., Jackson, R.W., Glossop, N.D., Young, C., Anterior cruciate ligament insufficiency: a dynamic three-dimensional motion analysis. American Journal of Sports and Medicine 17, Morrison, J.B., The mechanics of the knee joint in relation to normal walking. Journal of Biomechanics 3, Pandy, M.G., Zajac, F.E., Sim, E., Levine, W.S., An optimal control model for maximum-height human jumping. Journal of Biomechanics 23, Pandy, M.G., Anderson, F.C., Hull, D.G., A parameter optimization approach for the optimal control of large-scale musculoskeletal systems. Journal of Biomechanical Engineering 114, Pandy, M.G., Sasaki, K., Kim, S., A three-dimensional musculoskeletal model of the human knee joint. Part 1: theoretical construct. Computer Methods in Biomechanics and Biomedical Engineering 1, Serpas, F., Yanagawa, T., Pandy, M.G., Forward-dynamics simulation of anterior cruciate ligament forces developed during isokinetic dynamometry. Computer Methods in Biomechanics and Biomedical Engineering 5, Shelburne, K.B., Pandy, M.G., A musculoskeletal model of the knee for evaluatingligament forces duringisometric contractions. Journal of Biomechanics 30,

9 K.B. Shelburne et al. / Journal of Biomechanics 37 (2004) Shelburne, K.B., Pandy, M.G., Determinants of cruciateligament loading during rehabilitation exercise. Clinical Biomechanics 13, Shelburne, K.B., Pandy, M.G., A dynamic model of the knee and lower limb for simulatingrisingmovements. Computer Methods in Biomechanics and Biomedical Engineering 5, Shoemaker, S.C., Markolf, K.L., The role of the meniscus in the anterior-posterior stability of the loaded anterior cruciate-deficient knee. Effects of partial versus total excision. J Bone Joint SurgAm 68, Toutoungi, D.E., Lu, T.W., Leardini, A., Catani, F., O Connor, J.J., Cruciate ligament forces in the human knee during rehabilitation exercises. Clinical Biomechanics 15, Zajac, F.E., Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Critical Review Biomedical Engineering 17, Zhang, L.Q., Shiavi, R.G., Limbird, T.J., Minorik, J.M., Six degrees-of-freedom kinematics of ACL deficient knees duringlocomotion-compensatory mechanism. Gait Posture 17,