- Email: [email protected]
Biarticular hip extensor and knee flexor muscle moment arms of the feline hindlimb
ARTICLE IN PRESS
Journal of Biomechanics 40 (2007) 3448–3457 www.elsevier.com/locate/jbiomech www.JBiomech.com
Biarticular hip extensor and knee flexor muscle moment arms of the feline hindlimb Lisa N. MacFaddena,b, Nicholas A.T. Browna,b, a
Department of Orthopaedics, University of Utah, Salt Lake City, UT 84108, USA Department of Bioengineering, University of Utah, Salt Lake City, UT 84112, USA
b
Accepted 18 May 2007
Abstract Moment arms are important for understanding muscular behavior and for calculating internal muscle forces in musculoskeletal simulations. Biarticular muscles cross two joints and have moment arms that depend on the angle of both joints the muscles cross. The tendon excursion method was used to measure the joint angle-dependence of hamstring (biceps femoris, semimembranosus and semitendinosus) moment arm magnitudes of the feline hindlimb at the knee and hip joints. Knee angle influenced hamstring moment arm magnitudes at the hip joint; compared to a flexed knee joint, the moment arm for semimembranosus posterior at the hip was at most 7.4 mm (25%) larger when the knee was extended. On average, hamstring moment arms at the hip increased by 4.9 mm when the knee was more extended. In contrast, moment arm magnitudes at the knee varied by less than 2.8 mm ðmean ¼ 1:6 mmÞ for all hamstring muscles at the two hip joint angles tested. Thus, hamstring moment arms at the hip were dependent on knee position, while hamstring moment arms at the knee were not as strongly associated with relative hip position. Additionally, the feline hamstring muscle group had a larger mechanical advantage at the hip than at the knee joint. r 2007 Elsevier Ltd. All rights reserved. Keywords: Intrafascicular multielectrode stimulation; Musculoskeletal modeling; Tendon excursion
1. Introduction The contribution of a muscle’s force to a moment about a joint depends on that muscle’s level of activation, its contractile properties (Zajac, 1989) and its mechanical advantage or muscle moment arm at the joint (An et al., 1984; Pandy, 1999). Knowledge of each of these factors is required to estimate how a muscle contributes to a specific task (Crowninshield and Brand, 1981; Prilutsky et al., 1997; Pandy, 2001, 2003, 2005; Anderson and Pandy, 2001a, b, 2003; Buchanan et al., 2005), when evoking movements with functional electrical stimulation (FES), or to understand fundamentals of muscle function and coordination.
Corresponding author. Department of Orthopaedics, University of Utah, 590 Wakara Way, Room A100, Salt Lake City, UT 84108, USA. Tel.: +1 801 587 5200; fax: +1 801 587 5211. E-mail address: [email protected] (N.A.T. Brown).
0021-9290/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2007.05.021
Postural tasks such as sit-to-stand transitions or gait require ankle, knee and hip extension to generate a support moment (Winter, 1980). Electromyographic and anatomic studies of cat hindlimb muscles during locomotion provide evidence that hip extension is largely the result of activation of the biceps femoris (BF), semimembranosus (SM) and semitendinosus (ST) muscles (Goslow et al., 1973; English and Weeks, 1987; Buford and Smith, 1990; Aoyagi et al., 2004; Burkholder and Nichols, 2004). While these ‘hamstring’ muscles are the primary hip joint extension muscles in the cat, they also flex the knee joint. This biarticular arrangement likely produces joint stabilizing co-contractions between the hamstrings and quadriceps femoris muscle groups during movement incorporating hip extension, and may function to transfer power through the limb during running and jumping tasks (Gregoire et al., 1984; van Ingen Schenau et al., 1987; van Soest et al., 1993). Knowledge of hamstring muscle moment arms would allow quantification of the relative contribution of each component muscle to hip extension and knee flexion.
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
Traditional FES approaches to evoking movement invariably result in recruitment of all muscles the target nerve innervates because the cuff electrode placed around the nerve tends to excite all motor units (Triolo et al., 1996; Marsolais et al., 2000). Intrafascicular stimulation of nerves in the feline hindlimb using arrays of penetrating micro-electrodes has shown promise to enable submaximal and selective recruitment of individual muscles (Branner et al., 2001; McDonnall et al., 2004). With this approach, it becomes increasingly necessary to understand how individual muscles contribute to a moment because force in individual muscles can be produced selectively. Feline hindlimb moment arms have been reported across a range of joint angles for the muscles about the ankle (Young et al., 1993) and for the quadriceps femoris muscles about the knee (Boyd and Ronsky, 1998). Additionally, Burkholder and Nichols (2004) reported moment arms for biceps femoris posterior (BFP), SM, and ST at the knee with the limb in a stance configuration. Moment arms for BF, SM, and ST have not been reported at the knee for a range of joint angles, nor have they been reported for the hip joint. The tendon excursion method allows calculation of a muscle’s moment arm for a range of joint angles through measurement of muscle excursion with respect to joint angle (An et al., 1983, 1984). Even with straight line representations of a muscle’s line of action (e.g., Goslow et al., 1973) it can be seen that the analytical derivative of muscle length with respect to joint angle represents each muscle’s moment arm and that these quantities are not constant across joint angle (Pandy, 1999). Further, moment arms can be shown to depend on the angle of both joints spanned by biarticular muscles. This multiple joint angle dependency has implications for the function of biarticular muscles such that variation in knee angle could increase or decrease the mechanical advantage of a muscle at the hip. Conversely, hip joint angle could affect the function of hamstring muscles flexing the knee. Thus, the purpose of this study was to quantify the moment arms for BF, SM, and ST at the feline knee and hip for a range of joint angles encountered in locomotor
3449
and postural activities. A second purpose of this study was to investigate whether the moment arm magnitudes of these muscles changed at the hip when knee joint angle was varied and, similarly, whether knee joint moment arms changed when hip joint angle was varied. 2. Materials and methods 2.1. Specimens Six fresh frozen hindlimbs from six domestic felines (felis silvestris catus) were thawed 24 h prior to testing. All animals were obtained and frozen immediately after euthanasia was induced at the end of an experiment involving the contralateral limb to that used in this study. Weight, gender, and segment lengths were recorded (Table 1). The experimental procedures used in this study were approved by the University of Utah Institutional Animal Care and Use Committee.
2.2. Specimen preparation The hindlimb of interest was skinned from the ankle to the hip, exposing the pelvis. Tissue was regularly hydrated with a saline spray and covered with moist gauze where possible. Muscles were bluntly dissected to separate the loose connective tissue. Each muscle was removed from its origin and an eyelet screw was placed at the geometric centroid of the origin. BF has been shown to have three distinct neuromuscular compartments; posterior (BFP), middle (BFM), and anterior (BFA) (English and Weeks, 1987). These compartments were separated from each other according to the divisions described by English and Weeks (1987). SM also has separate neuromuscular compartments (Engberg and Lundberg, 1969; Botterman and Cope, 1988; Pratt et al., 1991), and was separated into anterior (SMA) and posterior (SMP) sections along the anatomic midline of the muscle. No alterations were made to ST. A Krackow stitch was used to attach a stiff ultra high molecular weight polyethylene fiber (Innovative Textiles, Grand Junction, CO, USA) to the muscle’s mid belly. Muscle proximal to the suture was removed. The excess suture was then passed through the eyelet placed at the origin of each muscle (Fig. 1). The pelvis was rigidly fixed to ground via two threaded bone pins inserted into the ilium. A threaded bone pin in the tibia and one in the femur provided points of fixation for these bones (Fig. 1). Rigid body clusters of four infrared emitting diodes (IREDS) rigidly attached to the tibia, femur, and pelvis tracked bone motion. Each IRED cluster was tracked using a 3-camera motion capture system (OptoTRAK, Northern Digital, Waterloo, Ontario, Canada).
Table 1 Anthropometric and gender characteristics of the specimens Specimen characteristics
Tip toes–MTP (mm) MTP–ankle (mm) Ankle–knee (mm) Knee–hip (mm) Hip–pelvis (mm) Mass (kg) Gender (M/F) Limb tested (R/L)
Cat 1
Cat 2
Cat 3
Cat 4
Cat 5
Cat 6
Mean
Standard deviation
35.0 77.5 133.5 108.0 59.0 4.8 M L
35.5 81.5 125.0 110.0 54.0 4.8 M R
37.0 76.5 119.0 111.0 60.0 4.7 F R
39.0 87.0 138.0 125.0 65.5 7.0 M R
36.5 80.0 118.0 107.5 61.5 4.5 M R
36.5 84.5 134.5 107.0 63.0 4.8 M R
36.6 81.2 128.0 111.4 60.5 5.1
1.4 4.0 8.5 6.8 3.9 0.9
Segment lengths were measured twice and the average reported. MTP refers to the metatarsal phalangeal joint.
ARTICLE IN PRESS 3450
L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457 The maximum difference in ST measures across all experiments was 4.6 mm with an average absolute difference of 1.6 mm. A series of five flexion–extension cycles was performed for each joint and each muscle. The limb was moved by hand by a single experimenter who maintained smooth motion in the limb’s parasaggital plane. Motion was slow to reduce viscoelastic effects. Reference positions were recorded prior to each trial to equate the motion capture device’s reference system to anatomical joint angles. Data were recorded at 20 Hz.
Grounded Pelvis Muscle
Femoral Bone Pin
Suture
M
Muscle Excursion
Mass Joint Angle
Tibial Bone Pin IREDS
Fig. 1. The experimental setup with the hip rigidly fixed to ground. The knee joint angle was fixed for other trials by attaching an aluminum bar between the femoral and tibial bone pins. A stiff suture was attached to the proximal and distal edges of the distal muscle stump and passed through an eyelet placed at that muscle’s origin. Infrared emitting diode (IRED) clusters attached to a mass suspended from the suture allowed measurement of the muscle’s excursion during joint rotations. Clusters of IREDs were also rigidly attached to the pelvis, femur and tibia to track bone motion.
2.3. Experimental design Both the hip and the knee joints were immobilized at different times during the experiment. The order of the joint immobilization was randomized. Knee immobilization permitted examination of hamstring moment arms at the hip. The knee was immobilized with an aluminum fixture attaching the tibial bone pin to the femoral bone pin. Hip immobilization was achieved by fixing the femoral bone pin to ground. The hip was tested with the knee immobilized at 58:3 4:1 and 107:5 7:6 and the knee was tested with the hip held rigidly at 73:3 8:76 and 100:0 4:1 . These angles represent the mid range of joint angles observed during sit-to-stand and gait behaviors (Goslow et al., 1973). Immobilization of each joint was confirmed by examining motion of the IRED clusters. To allow tendon creep to stabilize, a mass of 500 g was suspended from each muscle for at least 2 min before each trial. An IRED cluster was placed on the weight to measure displacement (resolution: þ0:1 mm) of the weight which was used to calculate muscle excursion as each joint was moved through its range of motion.
2.4. Data acquisition The six muscle compartments (BFA, BFM, BFP, SMA, SMP, and ST) were tested in a randomized order. ST was repeated as the seventh trial for each experiment to quantify measurement reliability (Fig. 2).
2.5. Data analysis Of the trials collected from six cats, there were several cases where a muscle tore from its insertion, the eyelet failed at its origin, or the muscle was not tested as part of the protocol. Therefore, between three and six trials were considered and averaged over common joint angles for each muscle. A second order low pass Butterworth filter with a cutoff frequency of 5 Hz was applied to the tendon excursion and joint angle data. Data sets were split into flexion and extension cycles from approximately 5 higher than the maximum (negative) flexion angle to approximately 5 lower than the maximum (positive) extension angle (Fig. 3A, B). This approach reduced the effects that sudden changes in direction at the end of a motion had on the excursion magnitudes. Four flexion and four extension cycles were selected for analysis (Fig. 3). Flexion and extension cycles were considered separately (Fig. 3D). A fourth order polynomial was applied to the excursion versus joint angle data (Fig. 3C), and the analytic derivative of the polynomial was computed to obtain the moment arms for each muscle over the range of motion (Delp et al., 1999). Resultant moment arm magnitudes were therefore a third order polynomial with respect to joint angle. First through fifth order polynomial fits of the data were explored. A fourth order polynomial was chosen because the error between the data and the polynomial was the lowest and provided a high correlation between the data and the polynomial. Knee extension moment arms are negative, and hip flexion moment arms are positive (Burkholder and Nichols, 2004).
3. Results 3.1. Knee joint moment arms The two muscles that insert farthest from the knee, BFP and ST, had the largest knee flexion moment arms and varied most across the range of motion (Fig. 4). ST moment arm increased 10 mm (25%) from the most flexed and extended knee joint positions to the middle of the joint range, and BFP showed a 20 mm change (33%) in moment arm magnitude over the same knee joint angles. The maximum moment arm for ST and BFP occurred at approximately 100 of knee flexion (Fig. 4). The remaining muscle compartments had smaller knee flexion moment arms of less than 10 mm. SMA had a negligible moment arm about the knee and BFA had a small extension moment arm about the knee which switched to a small knee flexion moment arm as the knee was extended. The two different fixed hip angles had little effect on hamstring moment arm magnitudes at the knee joint (Table 2) with a mean difference 1.6 mm in flexed ð73 Þ versus extended ð100 Þ fixed hip joint positions. The moment arm magnitude for ST differed the most of the knee flexor muscles due to changes in hip joint angle with an maximum increase of 2.8 mm when the hip was moved
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
3451
Semitendinosus knee moment arms for hip fixed at 73.3 degrees
Knee Moment Arm (mm)
45
Cat 3 Trial 1 Cat 3 Trial 2 Cat 4 Trial 1 Cat 4 Trial 2 Cat 5 Trial 1 Cat 5 Trial 2 Cat 6 Trial 1 Cat 6 Trial 2
40 35 30 25 20 15 10 5 0
20
40
60
80
100
120
140
160
180
120
140
160
180
Knee Angle (degrees)
Error
Cat 3, max error = 3.1 mm Error in Moment Arm (mm)
6
Cat 4, max error = 3.1 mm Cat 5, max error = 3.7 mm
4
Cat 6, max error = 4.6 mm 2 0
0
20
40
60
80
100
Knee Angle (degrees)
Fig. 2. Inter- and intra-animal variability for semitendinosus moment arms at the knee joint with the hip fixed in a flexed position ð73:3 Þ. Semitendinosus tendon excursions were collected as the first and seventh trials in each series of trials. The peak difference between these repeated measures was taken to represent measurement repeatability of up to 4.6 mm. The average absolute error for the range of joint angles shown here was 1.6 mm.
from an extended to flexed position (Fig. 4, Table 2). The peak moment arm magnitudes for ST were also offset by approximately 10 of knee joint angle (Fig. 4). 3.2. Hip joint moment arms SMA and SMP had the largest hip extension moment arms (Fig. 5). ST, BFA, and BFM all had maximum hip extension moment arms of approximately 30 mm that varied up to 20 mm (33–66%) across the joint range studied. The BFA and BFM compartments had similar moment arm magnitudes and patterns across the range of hip flexion angles because they passed through the same origin eyelet at the hip. For ST, the maximum magnitude occurred near the mid range of motion, while for BFA and BFM the maximum magnitude occurred when the hip was extended (Fig. 5). The BFP had the smallest hip extension moment arm of 20 mm (contrast with BFP having the largest knee flexion magnitude, Fig. 4).
The two different static knee angles tested influenced hamstring moment arm magnitudes at the hip (Fig. 5). For all muscles, flexed knee angles ð58 Þ resulted in maximum moment arms that were on average 4.9 mm (range: 3.0–7.4 mm) larger than extended fixed knee angles ð107 Þ (Table 2). Hamstring moment arms at the hip were always larger when the knee was extended compared to flexed. 4. Discussion Motivated by an interest in the function of muscles of the cat hindlimb for functional electrical stimulation and a need to provide data for muscle paths of a musculoskeletal (MS) model, the purposes of this study were to quantify the hamstring moment arms about the hip and the knee and to assess whether these magnitudes were dependent on the angle of the adjacent joint. All muscles of the hamstring group had large extension moment arms (20–40 mm) about
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
3452
Cat 6, Biceps Femoris Anterior
160
Mass Position (mm)
Knee Angle (degrees)
180
140 120 100
Full Knee Motion Flexion (dashed) Extension (dotted)
80 0
2
4
6
8
10
12
Full Knee Motion Flexion Extension 0
2
4
Knee Extension Moment Arm (mm)
Muscle Excursion (mm)
Actual Extension Polyfit Extension Actual Flexion Polyfit Flexion 100
110
120 130 140 150 Knee Angle (degrees)
6
8
10
12
Time (seconds)
Time (seconds)
160
170
Flexion Extension Flexion mean Extension mean Overall mean
100
110
120
130
140
150
160
170
Knee Angle (degrees)
Fig. 3. Data analysis for biceps femoris anterior moment arms determined at the hip joint with the knee fixed in an extended position for a representative trial from Cat 6. The joint angle (A) and tendon excursion (B) were divided into four flexion cycles (dotted lines, A, B) and four extension cycles (dashed lines, A, B). Tendon excursion versus joint angle figures (C) were produced and fourth order polynomials were applied to all extension and flexion data. Note small differences in excursion versus joint angle but similarities among the slopes of these data. The analytical derivative of the polynomial yielded the (third order) muscle moment arm versus joint angle relationships (D). Moment arm magnitudes differ most at the ends of the range of motion.
the hip joint. In contrast only ST and BFP had large flexion moment arms ð430 mmÞ about the knee joint. If all of the feline hamstring muscles were maximally activated at their optimal fiber length and these muscle forces were multiplied by each muscle’s maximal moment arm (Fig. 6, Table 3), the feline hamstrings have a greater mechanical advantage at the hip than at the knee joint. While many factors affect force generation by muscles, for the biarticular hamstring muscles, it appears the orientation of the distal knee joint is important for torque production at the proximal hip joint. This finding has implications for coordination and control of locomotor and jumping activities where knee and hip joint angles vary considerably during movement (Goslow et al., 1973). During walking for example, the feline knee extends during mid stance as the hip joint is maintained at a relatively constant angle. Similar relationships during trotting, galloping and jumping have also been noted (Goslow et al., 1973). Because hamstring moment arms at the hip
joint are larger for extended knee joint positions, the capacity of the hamstring muscle to generate hip joint torque for a given muscle force may increase as mid stance progresses even though hip joint angle does not change. Smaller hamstring moment arm magnitudes could also allow larger hip joint rotations and higher angular velocities for a given magnitude of muscle shortening when the knee is more flexed. Conversely, because hamstring moment arms at the knee appeared to vary little with hip joint position, the torque-generating capacity of the hamstring muscles at the knee joint may remain relatively constant for many limb (hip joint) configurations. Moment arms were not constant across joint range and varied as much as 66% as the knee and hip joints were rotated (Figs. 4, 5). These results are consistent with the predictions obtained from deriving moment arms from the muscle length functions provided by Goslow et al. (1973). The magnitudes of the knee joint moment arms are also consistent with Burkholder and Nichols (2004) except for
Knee Moment Arm (mm)
0
0
10
10 20
20 hip flexed, m = 5 hip extended, m = 4
30 60
70
80
90
100
110
hip flexed, m = 5 hip extended, m = 4
30
120
130
140
60
150
70
80
90
BFP
100
110
130
140
150
130
140
150
140
150
120
ST
hip flexed, m = 6 hip extended, m = 4
30 40 50
hip flexed, m = 5 hip extended, m = 4
60 60
70
80
90
100
110
120
130
140
150
60
70
80
90
110
120
SMP
SMA 0
0
10
10 20
20 hip flexed, m = 3 hip extended, m = 3
30 60
100
70
80
90
100
110
120
Knee Flexion Angle, deg
hip flexed, m = 5 hip extended, m = 4
30 130
140
150
60
70
80
90
100
110
120
130
Knee Flexion Angle, deg
Fig. 4. Average knee moment arm magnitudes for semitendinosus (ST) and each compartment of biceps femoris (BFA—anterior, BFM—middle, BFP—posterior) and semimembranosus (SMA— anterior, SMP—posterior). The ST and BFP have the largest knee flexion moment arm magnitudes, while SMA and BFA had negligible moment arm magnitudes about the knee. The BFA had an extension moment arm that switched to a flexion moment arm as the hip was extended. There was no appreciable difference between moment arm magnitudes at the two fixed hip angles tested (m is the number of muscles).
ARTICLE IN PRESS
Knee Moment Arm (mm)
BFM
L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
Knee Moment Arm (mm)
BFA
3453
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
3454
Table 2 Absolute differences in moment arms (MA) at the hip with the knee in flexed and extended positions, and at the knee with the hip in flexed and extended positions. Hamstring moment arm magnitudes at the hip were larger when the knee was extended compared to flexed. There was a small effect of hip angle on hamstring moment arms at the knee Muscle
Hip moment arms (MA)
Hip Moment Arm (mm)
SMA SMP ST BFA BFM BFP
Hip Moment Arm (mm)
Max. MA w/knee flexed (mm)
Max. MA w/knee extended (mm)
Absolute difference (mm)
Max. MA w/hip flexed (mm)
Max. MA w/hip extended (mm)
Absolute difference (mm)
26.6 29.5 26.3 25.0 24.8 11.0
32.1 36.8 29.9 28.0 28.2 17.7
5.5 7.3 3.6 3.0 3.4 6.7
0.2 7.7 38.5 0.4 5.0 45.6
0.1 6.0 35.7 1.0 7.1 43.3
0.1 1.8 2.8 0.6 2.1 2.3
BFA
BFM
knee flexed, m = 3 knee extended, m =3
40
30
20
20
10
10 70
80
90
100
110
knee flexed, m = 3 knee extended, m = 3
40
30
60
120
130
60
70
80
BFP
30
20
20
10
10 80
90
110
120
130
100
110
120
130
60
120
130
120
130
knee flexed, m = 3 knee extended, m = 4 70
80
90
100
110
SMP
SMA 40
40
30
30 20
20 knee flexed, m = 3 knee extended, m = 4
10 60
100
40
30
70
90
ST
knee flexed, m = 3 knee extended, m = 4
40
60
Hip Moment Arm (mm)
Knee moment arms (MA)
70
80
90
100
110
knee flexed, m = 3 knee extended, m = 3
10 120
130
Hip Flexion Angle (degrees)
60
70
80
90
100
110
Hip Flexion Angle (degrees)
Fig. 5. Hip moment arm magnitudes for semitendinosus (ST) and each compartment of biceps femoris (BFA—anterior, BFM—middle, BFP—posterior) and semimembranosus (SMA—anterior, SMP—posterior). Semimembranosus had the largest hip extension moment arms. Note that BFP and ST had an offset in hip extension moment arm magnitudes with respect to the two different knee joint angles tested and differences in hip joint moment arm magnitudes were noted for BFA and SMP with respect to changes in knee joint angle (m is the number of muscles).
the knee joint moment arm for BFP. Burkholder and Nichols estimated BFP to have a moment arm magnitude twice the values reported here and twice that calculated from Goslow et al.’s equations. Burkholder and Nichols studied smaller cats (mean 3.5 kg) than those used here and Goslow et al.’s animals perhaps accounting for some of the differences noted in knee joint moment arm magnitudes.
In MS models (Murray et al., 1995; Delp et al., 1999; Brown et al., 2003), the joint torque ðTÞ produced by a muscle equals the product of the muscle’s moment arm ðrÞ, and its force ðF Þ; T ¼ F r. During the stance phase of gait, cat knee extension torques reach approximately 2 Nm (McFadyen et al., 1999; Gregor et al., 2006). For sake of this example, if only the BFP was assumed to produce this
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
torque, the force required in this muscle could vary from 32 to 44 N if the moment arm is considered at its maximum (0.062 m) and minimum (0.045 m) values, respectively. This difference represents a considerable change in required muscle force and illustrates the importance of not assuming a constant moment arm with respect to joint angle. BF and SM muscles were separated and each compartment was tested independently. Despite the separation at the distal end of each compartment, the sutures attached proximally were passed through one eyelet for BFA, BFM and BFP, and one eyelet for SMA and SMP. Consequently, the hip moment arm magnitudes for the compartments of BF and SM were expected to be similar. Hip extension moment arms were similar for SMA compared to SMP and for BFA compared to BFM (Fig. 5). However, hip extension moment arms about the hip for BFP were 5–10 mm lower than BFA and BFM and did not follow the same pattern across hip joint angle. Compared to BFA and BFM, it appears that BFP’s more distal insertion on the tibia created a muscle line of action that led to differences in muscle excursion with hip flexion and extension. Despite the considerable variability of animal body weight and segment lengths there was not a great variation in moment arm magnitudes among animals. While specimen 4 had the largest mass (7.0 kg, Table 1), almost 1.4 times larger than the second largest specimen (Cat 6, 4.8 kg), it did not have proportionally larger segment lengths (femur length was 125 mm, just over 1.1 times larger than the second largest femur length of 111 mm). When moment arms were scaled to femur segment length or other anthropometric measures, they did not converge in a way that allowed prediction by a single equation. Therefore, the average values presented here are appropriate for animals with segmental lengths similar to the animals reported in this study. A polynomial was used to describe the excursion versus angle data and may have introduced error into the moment arm calculations. Of all of the polynomials applied to these data, only 7 (of 1112 total) demonstrated correlations with the experimental data between 0:80oR2 o0:95; all others were above 0.95. Polynomials lower than the fourth order approach yielded lower correlations. The difference between the experimental data and polynomials were always greatest when the limb was moved into end range flexion and extension and smallest in the mid range of the motion Fig. 6. Torque produced about the hip and knee joints calculated by multiplying the force produced in each muscle when maximally stimulated at its optimal length by the maximal moment arm noted in this study. Forces were estimated using each muscle’s physiological cross sectional area (Sacks and Roy, 1982) and a maximum muscle stress of 300 kPa (Wilson et al., 2001). The BFP produced a large knee flexion and small hip extension torque. ST produced large hip and knee joint torques while SMA was found to be a strong hip extensor. Even considering maximal muscle force and maximal moment arm magnitudes for BFA, BFM and SMP, these muscles had lower moment generating capacities at the hip and knee in comparison to their hamstring agonists.
3455
(Fig. 2). Thus, although maximal errors were up to 4.6 mm, this represents values at the end of range and errors in mid range were approximately half this value.
BFM
BFA 2.13 Nm
2.13 Nm
0.08 Nm
0.72 Nm
BFP
ST 3.84 Nm
1.76 Nm
5.03 Nm
4.76 Nm
SMA 4.18 Nm
SMP 2.05 Nm
0.53 Nm 0.03 Nm
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
3456
Table 3 Percentage of torque contributions about the hip and knee joints when all the hamstrings are maximally stimulated Muscle
Muscle torque contributions Knee
BFA BFM BFP ST SMA SMP All muscles
Hip
Maximum torque (Nm)
Percent contribution (%)
Maximum torque (Nm)
Percent contribution (%)
0.1 0.7 4.8 5.0 0.03 0.5 11.0
0.7 6.6 43.3 45.8 0.3 4.8
2.1 2.1 1.8 3.8 4.2 2.1 16.1
13.2 13.2 10.9 23.9 26.0 12.7
Average maximal moment arm values multiplied by maximal muscle force were used to calculated torques. ST contributed the most to the knee joint moment while SMA provided the largest hip joint moment contribution. BFA had a positive extension moment about the knee.
There was a difference between the magnitudes of the moment arms during joint flexion and extension (Fig. 3D). This difference occurred primarily for the muscles with broader insertions (BFA, BFM) where the two ends of the suture attached to the outer edges of the muscle were separated by a larger distance (Fig. 1). While these differences may be attributed to tissue compliance of muscular insertions, observations during the experiment suggest that tendon excursions and therefore muscle moment arms may represent measurements from two different regions of the same muscle. During flexion, for example, the proximal fibers of BFM were loaded more than the distal fibers and because the proximal fibers are closer to the knee joint, the tendon excursion magnitudes were lower. Conversely, during extension the distal region of the muscle was loaded and the tendon excursions were larger. These findings parallel those of Blemker and Delp (2006), who modeled muscle fibers to have different moment arm magnitudes depending on their location within the muscles. For muscles with broad insertions, the average of the flexion and extension moment arms presented here likely represent the moment arm at the centroid of muscle insertion. Intrafascicular multielectrode stimulation of the muscular branch of the sciatic nerve in the feline hindlimb appears able to stimulate individual muscles of the hamstring group (unpublished observation). To generate a sit-to-stand behavior in the cat, muscles such as SMA and BFA could be targeted because they have large hip extension moment generating capabilities (Fig. 6) and contribute a large percentage of the hip extension moment while providing a small percentage of the knee flexion torque (Table 3). Muscles such as SMP and BFM would also be appropriate to target to provide a small degree of co-contraction against evoked quadriceps femoris muscle forces which produce a knee extension torque. Conversely, ST and BFP could be avoided for a sit-tostand behavior as these muscles have large knee flexion moment-generating capacities but would be very appropriate when greater knee joint stability is required from
muscle co-contractions and when knee flexion is required during gait. Conflict of interest The authors do not have any financial and personal relationships with other people or organisations that could inappropriately influence (bias) this work. Acknowledgment This work was supported by the NIH (NINDS/ NICHHD-2RO1-NS039677-05A1). References An, K.N., Ueba, Y., Chao, E.Y., Cooney, W.P., Linscheid, R.L., 1983. Tendon excursion and moment arm of index finger muscles. Journal of Biomechanics 16 (6), 419–425. An, K.N., Takahashi, K., Harrigan, T.P., Chao, E.Y., 1984. Determination of muscle orientations and moment arms. Journal of Biomechanical Engineering 106 (3), 280–282. Anderson, F.C., Pandy, M.G., 2001a. Dynamic optimization of human walking. Journal of Biomechanical Engineering 123 (5), 381–390. Anderson, F.C., Pandy, M.G., 2001b. Static and dynamic optimization solutions for gait are practically equivalent. Journal of Biomechanics 34 (2), 153–161. Anderson, F.C., Pandy, M.G., 2003. Individual muscle contributions to support in normal walking. Gait & Posture 17 (2), 159–169. Aoyagi, Y., Mushahwar, V.K., Stein, R.B., Prochazka, A., 2004. Movements elicited by electrical stimulation of muscles, nerves, intermediate spinal cord, and spinal roots in anesthetized and decerebrate cats. IEEE Transactions on Neural Systems and Rehabilitation Engineering 12 (1), 1–11. Blemker, S.S., Delp, S.L., 2006. Rectus femoris and vastus intermedius fiber excursions predicted by three-dimensional muscle models. Journal of Biomechanics 39 (8), 1383–1391. Botterman, B.R., Cope, T.C., 1988. Motor-unit stimulation patterns during fatiguing contractions of constant tension. Journal of Neurophysiology 60 (4), 1198–1214. Boyd, S.K., Ronsky, J.L., 1998. Instantaneous moment arm determination of the cat knee. Journal of Biomechanics 31 (3), 279–283.
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457 Branner, A., Stein, R.B., Normann, R.A., 2001. Selective stimulation of cat sciatic nerve using an array of varying-length microelectrodes. Journal of Neurophysiology 85 (4), 1585–1594. Brown, N.A., Pandy, M.G., Kawcak, C.E., McIlwraith, C.W., 2003. Force- and moment-generating capacities of muscles in the distal forelimb of the horse. Journal of Anatomy 203 (1), 101–113. Buchanan, T.S., Lloyd, D.G., Manal, K., Besier, T.F., 2005. Estimation of muscle forces and joint moments using a forward–inverse dynamics model. Medicine and Science in Sports Exercise 37 (11), 1911–1916. Buford, J.A., Smith, J.L., 1990. Adaptive control for backward quadrupedal walking. II. Hindlimb muscle synergies. Journal of Neurophysiology 64 (3), 756–766. Burkholder, T.J., Nichols, T.R., 2004. Three-dimensional model of the feline hindlimb. Journal of Morphology 261 (1), 118–129. Crowninshield, R.D., Brand, R.A., 1981. A physiologically based criterion of muscle force prediction in locomotion. Journal of Biomechanics 14 (11), 793–801. Delp, S.L., Hess, W.E., Hungerford, D.S., Jones, L.C., 1999. Variation of rotation moment arms with hip flexion. Journal of Biomechanics 32 (5), 493–501. Engberg, I., Lundberg, A., 1969. An electromyographic analysis of muscular activity in the hindlimb of the cat during unrestrained locomotion. Acta Physiologica Scandinavica 75 (4), 614–630. English, A.W., Weeks, O.I., 1987. An anatomical and functional analysis of cat biceps femoris and semitendinosus muscles. Journal of Morphology 191 (2), 161–175. Goslow Jr., G.E., Reinking, R.M., Stuart, D.G., 1973. The cat step cycle: hind limb joint angles and muscle lengths during unrestrained locomotion. Journal of Morphology 141 (1), 1–41. Gregoire, L., Veeger, H.E., Huijing, P.A., van Ingen Schenau, G.J., 1984. Role of mono- and biarticular muscles in explosive movements. International Journal of Sports Medicine 5 (6), 301–305. Gregor, R.J., Smith, D.W., Prilutsky, B.I., 2006. Mechanics of slope walking in the cat: quantification of muscle load, length change, and ankle extensor EMG patterns. Journal of Neurophysiology 95 (3), 1397–1409. Marsolais, E.B., Kobetic, R., Polando, G., Ferguson, K., Tashman, S., Gaudio, R., Nandurkar, S., Lehneis, H.R., 2000. The Case Western Reserve University hybrid gait orthosis. Journal of Spinal Cord Medicine 23 (2), 100–108. McDonnall, D., Clark, G.A., Normann, R.A., 2004. Selective motor unit recruitment via intrafascicular multielectrode stimulation. Canadian Journal of Physiology and Pharmacology 82 (8–9), 599–609. McFadyen, B.J., Lavoie, S., Drew, T., 1999. Kinetic and energetic patterns for hindlimb obstacle avoidance during cat locomotion. Experimental Brain Research 125 (4), 502–510.
3457
Murray, W.M., Delp, S.L., Buchanan, T.S., 1995. Variation of muscle moment arms with elbow and forearm position. Journal of Biomechanics 28 (5), 513–525. Pandy, M.G., 1999. Moment arm of a muscle force. Exercise and Sport Sciences Reviews 27, 79–118. Pandy, M.G., 2001. Computer modeling and simulation of human movement. Annual Review of Biomedical Engineering 3, 245–273. Pandy, M.G., 2003. Simple and complex models for studying muscle function in walking. Philosophical Transactions of the Royal Society of London, Series B, Biological Sciences 358 (1437), 1501–1509. Pandy, M.G., 2005. Mixed forward and inverse solutions in movement biomechanics. Theoretical Issues in Ergonomics Science 6 (3/4), 325–330. Pratt, C.A., Chanaud, C.M., Loeb, G.E., 1991. Functionally complex muscles of the cat hindlimb. IV. Intramuscular distribution of movement command signals and cutaneous reflexes in broad, bifunctional thigh muscles. Experimental Brain Research 85 (2), 281–299. Prilutsky, B.I., Herzog, W., Allinger, T.L., 1997. Forces of individual cat ankle extensor muscles during locomotion predicted using static optimization. Journal of Biomechanics 30 (10), 1025–1033. Sacks, R.D., Roy, R.R., 1982. Architecture of the hind limb muscles of cats: functional significance. Journal of Morphology 173 (2), 185–195. Triolo, R.J., Bieri, C., Uhlir, J., Kobetic, R., Scheiner, A., Marsolais, E.B., 1996. Implanted functional neuromuscular stimulation systems for individuals with cervical spinal cord injuries: clinical case reports. Archives of Physical and Medical Rehabilitation 77 (11), 1119–1128. van Ingen Schenau, G.J., Bobbert, M.F., Rozendal, R.H., 1987. The unique action of bi-articular muscles in complex movements. Journal of Anatomy 155, 1–5. van Soest, A.J., Schwab, A.L., Bobbert, M.F., van Ingen Schenau, G.J., 1993. The influence of the biarticularity of the gastrocnemius muscle on vertical-jumping achievement. Journal of Biomechanics 26 (1), 1–8. Wilson, A.M., McGuigan, M.P., Su, A., van Den Bogert, A.J., 2001. Horses damp the spring in their step. Nature 414 (6866), 859–895. Winter, D.A., 1980. Overall principle of lower limb support during stance phase of gait. Journal of Biomechanics 13 (11), 923–927. Young, R.P., Scott, S.H., Loeb, G.E., 1993. The distal hindlimb musculature of the cat: multiaxis moment arms at the ankle joint. Experimental Brain Research 96 (1), 141–151. Zajac, F.E., 1989. Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Critical Reviews in Biomedical Engineering 17 (4), 359–411.
Journal of Biomechanics 40 (2007) 3448–3457 www.elsevier.com/locate/jbiomech www.JBiomech.com
Biarticular hip extensor and knee flexor muscle moment arms of the feline hindlimb Lisa N. MacFaddena,b, Nicholas A.T. Browna,b, a
Department of Orthopaedics, University of Utah, Salt Lake City, UT 84108, USA Department of Bioengineering, University of Utah, Salt Lake City, UT 84112, USA
b
Accepted 18 May 2007
Abstract Moment arms are important for understanding muscular behavior and for calculating internal muscle forces in musculoskeletal simulations. Biarticular muscles cross two joints and have moment arms that depend on the angle of both joints the muscles cross. The tendon excursion method was used to measure the joint angle-dependence of hamstring (biceps femoris, semimembranosus and semitendinosus) moment arm magnitudes of the feline hindlimb at the knee and hip joints. Knee angle influenced hamstring moment arm magnitudes at the hip joint; compared to a flexed knee joint, the moment arm for semimembranosus posterior at the hip was at most 7.4 mm (25%) larger when the knee was extended. On average, hamstring moment arms at the hip increased by 4.9 mm when the knee was more extended. In contrast, moment arm magnitudes at the knee varied by less than 2.8 mm ðmean ¼ 1:6 mmÞ for all hamstring muscles at the two hip joint angles tested. Thus, hamstring moment arms at the hip were dependent on knee position, while hamstring moment arms at the knee were not as strongly associated with relative hip position. Additionally, the feline hamstring muscle group had a larger mechanical advantage at the hip than at the knee joint. r 2007 Elsevier Ltd. All rights reserved. Keywords: Intrafascicular multielectrode stimulation; Musculoskeletal modeling; Tendon excursion
1. Introduction The contribution of a muscle’s force to a moment about a joint depends on that muscle’s level of activation, its contractile properties (Zajac, 1989) and its mechanical advantage or muscle moment arm at the joint (An et al., 1984; Pandy, 1999). Knowledge of each of these factors is required to estimate how a muscle contributes to a specific task (Crowninshield and Brand, 1981; Prilutsky et al., 1997; Pandy, 2001, 2003, 2005; Anderson and Pandy, 2001a, b, 2003; Buchanan et al., 2005), when evoking movements with functional electrical stimulation (FES), or to understand fundamentals of muscle function and coordination.
Corresponding author. Department of Orthopaedics, University of Utah, 590 Wakara Way, Room A100, Salt Lake City, UT 84108, USA. Tel.: +1 801 587 5200; fax: +1 801 587 5211. E-mail address: [email protected] (N.A.T. Brown).
0021-9290/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2007.05.021
Postural tasks such as sit-to-stand transitions or gait require ankle, knee and hip extension to generate a support moment (Winter, 1980). Electromyographic and anatomic studies of cat hindlimb muscles during locomotion provide evidence that hip extension is largely the result of activation of the biceps femoris (BF), semimembranosus (SM) and semitendinosus (ST) muscles (Goslow et al., 1973; English and Weeks, 1987; Buford and Smith, 1990; Aoyagi et al., 2004; Burkholder and Nichols, 2004). While these ‘hamstring’ muscles are the primary hip joint extension muscles in the cat, they also flex the knee joint. This biarticular arrangement likely produces joint stabilizing co-contractions between the hamstrings and quadriceps femoris muscle groups during movement incorporating hip extension, and may function to transfer power through the limb during running and jumping tasks (Gregoire et al., 1984; van Ingen Schenau et al., 1987; van Soest et al., 1993). Knowledge of hamstring muscle moment arms would allow quantification of the relative contribution of each component muscle to hip extension and knee flexion.
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
Traditional FES approaches to evoking movement invariably result in recruitment of all muscles the target nerve innervates because the cuff electrode placed around the nerve tends to excite all motor units (Triolo et al., 1996; Marsolais et al., 2000). Intrafascicular stimulation of nerves in the feline hindlimb using arrays of penetrating micro-electrodes has shown promise to enable submaximal and selective recruitment of individual muscles (Branner et al., 2001; McDonnall et al., 2004). With this approach, it becomes increasingly necessary to understand how individual muscles contribute to a moment because force in individual muscles can be produced selectively. Feline hindlimb moment arms have been reported across a range of joint angles for the muscles about the ankle (Young et al., 1993) and for the quadriceps femoris muscles about the knee (Boyd and Ronsky, 1998). Additionally, Burkholder and Nichols (2004) reported moment arms for biceps femoris posterior (BFP), SM, and ST at the knee with the limb in a stance configuration. Moment arms for BF, SM, and ST have not been reported at the knee for a range of joint angles, nor have they been reported for the hip joint. The tendon excursion method allows calculation of a muscle’s moment arm for a range of joint angles through measurement of muscle excursion with respect to joint angle (An et al., 1983, 1984). Even with straight line representations of a muscle’s line of action (e.g., Goslow et al., 1973) it can be seen that the analytical derivative of muscle length with respect to joint angle represents each muscle’s moment arm and that these quantities are not constant across joint angle (Pandy, 1999). Further, moment arms can be shown to depend on the angle of both joints spanned by biarticular muscles. This multiple joint angle dependency has implications for the function of biarticular muscles such that variation in knee angle could increase or decrease the mechanical advantage of a muscle at the hip. Conversely, hip joint angle could affect the function of hamstring muscles flexing the knee. Thus, the purpose of this study was to quantify the moment arms for BF, SM, and ST at the feline knee and hip for a range of joint angles encountered in locomotor
3449
and postural activities. A second purpose of this study was to investigate whether the moment arm magnitudes of these muscles changed at the hip when knee joint angle was varied and, similarly, whether knee joint moment arms changed when hip joint angle was varied. 2. Materials and methods 2.1. Specimens Six fresh frozen hindlimbs from six domestic felines (felis silvestris catus) were thawed 24 h prior to testing. All animals were obtained and frozen immediately after euthanasia was induced at the end of an experiment involving the contralateral limb to that used in this study. Weight, gender, and segment lengths were recorded (Table 1). The experimental procedures used in this study were approved by the University of Utah Institutional Animal Care and Use Committee.
2.2. Specimen preparation The hindlimb of interest was skinned from the ankle to the hip, exposing the pelvis. Tissue was regularly hydrated with a saline spray and covered with moist gauze where possible. Muscles were bluntly dissected to separate the loose connective tissue. Each muscle was removed from its origin and an eyelet screw was placed at the geometric centroid of the origin. BF has been shown to have three distinct neuromuscular compartments; posterior (BFP), middle (BFM), and anterior (BFA) (English and Weeks, 1987). These compartments were separated from each other according to the divisions described by English and Weeks (1987). SM also has separate neuromuscular compartments (Engberg and Lundberg, 1969; Botterman and Cope, 1988; Pratt et al., 1991), and was separated into anterior (SMA) and posterior (SMP) sections along the anatomic midline of the muscle. No alterations were made to ST. A Krackow stitch was used to attach a stiff ultra high molecular weight polyethylene fiber (Innovative Textiles, Grand Junction, CO, USA) to the muscle’s mid belly. Muscle proximal to the suture was removed. The excess suture was then passed through the eyelet placed at the origin of each muscle (Fig. 1). The pelvis was rigidly fixed to ground via two threaded bone pins inserted into the ilium. A threaded bone pin in the tibia and one in the femur provided points of fixation for these bones (Fig. 1). Rigid body clusters of four infrared emitting diodes (IREDS) rigidly attached to the tibia, femur, and pelvis tracked bone motion. Each IRED cluster was tracked using a 3-camera motion capture system (OptoTRAK, Northern Digital, Waterloo, Ontario, Canada).
Table 1 Anthropometric and gender characteristics of the specimens Specimen characteristics
Tip toes–MTP (mm) MTP–ankle (mm) Ankle–knee (mm) Knee–hip (mm) Hip–pelvis (mm) Mass (kg) Gender (M/F) Limb tested (R/L)
Cat 1
Cat 2
Cat 3
Cat 4
Cat 5
Cat 6
Mean
Standard deviation
35.0 77.5 133.5 108.0 59.0 4.8 M L
35.5 81.5 125.0 110.0 54.0 4.8 M R
37.0 76.5 119.0 111.0 60.0 4.7 F R
39.0 87.0 138.0 125.0 65.5 7.0 M R
36.5 80.0 118.0 107.5 61.5 4.5 M R
36.5 84.5 134.5 107.0 63.0 4.8 M R
36.6 81.2 128.0 111.4 60.5 5.1
1.4 4.0 8.5 6.8 3.9 0.9
Segment lengths were measured twice and the average reported. MTP refers to the metatarsal phalangeal joint.
ARTICLE IN PRESS 3450
L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457 The maximum difference in ST measures across all experiments was 4.6 mm with an average absolute difference of 1.6 mm. A series of five flexion–extension cycles was performed for each joint and each muscle. The limb was moved by hand by a single experimenter who maintained smooth motion in the limb’s parasaggital plane. Motion was slow to reduce viscoelastic effects. Reference positions were recorded prior to each trial to equate the motion capture device’s reference system to anatomical joint angles. Data were recorded at 20 Hz.
Grounded Pelvis Muscle
Femoral Bone Pin
Suture
M
Muscle Excursion
Mass Joint Angle
Tibial Bone Pin IREDS
Fig. 1. The experimental setup with the hip rigidly fixed to ground. The knee joint angle was fixed for other trials by attaching an aluminum bar between the femoral and tibial bone pins. A stiff suture was attached to the proximal and distal edges of the distal muscle stump and passed through an eyelet placed at that muscle’s origin. Infrared emitting diode (IRED) clusters attached to a mass suspended from the suture allowed measurement of the muscle’s excursion during joint rotations. Clusters of IREDs were also rigidly attached to the pelvis, femur and tibia to track bone motion.
2.3. Experimental design Both the hip and the knee joints were immobilized at different times during the experiment. The order of the joint immobilization was randomized. Knee immobilization permitted examination of hamstring moment arms at the hip. The knee was immobilized with an aluminum fixture attaching the tibial bone pin to the femoral bone pin. Hip immobilization was achieved by fixing the femoral bone pin to ground. The hip was tested with the knee immobilized at 58:3 4:1 and 107:5 7:6 and the knee was tested with the hip held rigidly at 73:3 8:76 and 100:0 4:1 . These angles represent the mid range of joint angles observed during sit-to-stand and gait behaviors (Goslow et al., 1973). Immobilization of each joint was confirmed by examining motion of the IRED clusters. To allow tendon creep to stabilize, a mass of 500 g was suspended from each muscle for at least 2 min before each trial. An IRED cluster was placed on the weight to measure displacement (resolution: þ0:1 mm) of the weight which was used to calculate muscle excursion as each joint was moved through its range of motion.
2.4. Data acquisition The six muscle compartments (BFA, BFM, BFP, SMA, SMP, and ST) were tested in a randomized order. ST was repeated as the seventh trial for each experiment to quantify measurement reliability (Fig. 2).
2.5. Data analysis Of the trials collected from six cats, there were several cases where a muscle tore from its insertion, the eyelet failed at its origin, or the muscle was not tested as part of the protocol. Therefore, between three and six trials were considered and averaged over common joint angles for each muscle. A second order low pass Butterworth filter with a cutoff frequency of 5 Hz was applied to the tendon excursion and joint angle data. Data sets were split into flexion and extension cycles from approximately 5 higher than the maximum (negative) flexion angle to approximately 5 lower than the maximum (positive) extension angle (Fig. 3A, B). This approach reduced the effects that sudden changes in direction at the end of a motion had on the excursion magnitudes. Four flexion and four extension cycles were selected for analysis (Fig. 3). Flexion and extension cycles were considered separately (Fig. 3D). A fourth order polynomial was applied to the excursion versus joint angle data (Fig. 3C), and the analytic derivative of the polynomial was computed to obtain the moment arms for each muscle over the range of motion (Delp et al., 1999). Resultant moment arm magnitudes were therefore a third order polynomial with respect to joint angle. First through fifth order polynomial fits of the data were explored. A fourth order polynomial was chosen because the error between the data and the polynomial was the lowest and provided a high correlation between the data and the polynomial. Knee extension moment arms are negative, and hip flexion moment arms are positive (Burkholder and Nichols, 2004).
3. Results 3.1. Knee joint moment arms The two muscles that insert farthest from the knee, BFP and ST, had the largest knee flexion moment arms and varied most across the range of motion (Fig. 4). ST moment arm increased 10 mm (25%) from the most flexed and extended knee joint positions to the middle of the joint range, and BFP showed a 20 mm change (33%) in moment arm magnitude over the same knee joint angles. The maximum moment arm for ST and BFP occurred at approximately 100 of knee flexion (Fig. 4). The remaining muscle compartments had smaller knee flexion moment arms of less than 10 mm. SMA had a negligible moment arm about the knee and BFA had a small extension moment arm about the knee which switched to a small knee flexion moment arm as the knee was extended. The two different fixed hip angles had little effect on hamstring moment arm magnitudes at the knee joint (Table 2) with a mean difference 1.6 mm in flexed ð73 Þ versus extended ð100 Þ fixed hip joint positions. The moment arm magnitude for ST differed the most of the knee flexor muscles due to changes in hip joint angle with an maximum increase of 2.8 mm when the hip was moved
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
3451
Semitendinosus knee moment arms for hip fixed at 73.3 degrees
Knee Moment Arm (mm)
45
Cat 3 Trial 1 Cat 3 Trial 2 Cat 4 Trial 1 Cat 4 Trial 2 Cat 5 Trial 1 Cat 5 Trial 2 Cat 6 Trial 1 Cat 6 Trial 2
40 35 30 25 20 15 10 5 0
20
40
60
80
100
120
140
160
180
120
140
160
180
Knee Angle (degrees)
Error
Cat 3, max error = 3.1 mm Error in Moment Arm (mm)
6
Cat 4, max error = 3.1 mm Cat 5, max error = 3.7 mm
4
Cat 6, max error = 4.6 mm 2 0
0
20
40
60
80
100
Knee Angle (degrees)
Fig. 2. Inter- and intra-animal variability for semitendinosus moment arms at the knee joint with the hip fixed in a flexed position ð73:3 Þ. Semitendinosus tendon excursions were collected as the first and seventh trials in each series of trials. The peak difference between these repeated measures was taken to represent measurement repeatability of up to 4.6 mm. The average absolute error for the range of joint angles shown here was 1.6 mm.
from an extended to flexed position (Fig. 4, Table 2). The peak moment arm magnitudes for ST were also offset by approximately 10 of knee joint angle (Fig. 4). 3.2. Hip joint moment arms SMA and SMP had the largest hip extension moment arms (Fig. 5). ST, BFA, and BFM all had maximum hip extension moment arms of approximately 30 mm that varied up to 20 mm (33–66%) across the joint range studied. The BFA and BFM compartments had similar moment arm magnitudes and patterns across the range of hip flexion angles because they passed through the same origin eyelet at the hip. For ST, the maximum magnitude occurred near the mid range of motion, while for BFA and BFM the maximum magnitude occurred when the hip was extended (Fig. 5). The BFP had the smallest hip extension moment arm of 20 mm (contrast with BFP having the largest knee flexion magnitude, Fig. 4).
The two different static knee angles tested influenced hamstring moment arm magnitudes at the hip (Fig. 5). For all muscles, flexed knee angles ð58 Þ resulted in maximum moment arms that were on average 4.9 mm (range: 3.0–7.4 mm) larger than extended fixed knee angles ð107 Þ (Table 2). Hamstring moment arms at the hip were always larger when the knee was extended compared to flexed. 4. Discussion Motivated by an interest in the function of muscles of the cat hindlimb for functional electrical stimulation and a need to provide data for muscle paths of a musculoskeletal (MS) model, the purposes of this study were to quantify the hamstring moment arms about the hip and the knee and to assess whether these magnitudes were dependent on the angle of the adjacent joint. All muscles of the hamstring group had large extension moment arms (20–40 mm) about
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
3452
Cat 6, Biceps Femoris Anterior
160
Mass Position (mm)
Knee Angle (degrees)
180
140 120 100
Full Knee Motion Flexion (dashed) Extension (dotted)
80 0
2
4
6
8
10
12
Full Knee Motion Flexion Extension 0
2
4
Knee Extension Moment Arm (mm)
Muscle Excursion (mm)
Actual Extension Polyfit Extension Actual Flexion Polyfit Flexion 100
110
120 130 140 150 Knee Angle (degrees)
6
8
10
12
Time (seconds)
Time (seconds)
160
170
Flexion Extension Flexion mean Extension mean Overall mean
100
110
120
130
140
150
160
170
Knee Angle (degrees)
Fig. 3. Data analysis for biceps femoris anterior moment arms determined at the hip joint with the knee fixed in an extended position for a representative trial from Cat 6. The joint angle (A) and tendon excursion (B) were divided into four flexion cycles (dotted lines, A, B) and four extension cycles (dashed lines, A, B). Tendon excursion versus joint angle figures (C) were produced and fourth order polynomials were applied to all extension and flexion data. Note small differences in excursion versus joint angle but similarities among the slopes of these data. The analytical derivative of the polynomial yielded the (third order) muscle moment arm versus joint angle relationships (D). Moment arm magnitudes differ most at the ends of the range of motion.
the hip joint. In contrast only ST and BFP had large flexion moment arms ð430 mmÞ about the knee joint. If all of the feline hamstring muscles were maximally activated at their optimal fiber length and these muscle forces were multiplied by each muscle’s maximal moment arm (Fig. 6, Table 3), the feline hamstrings have a greater mechanical advantage at the hip than at the knee joint. While many factors affect force generation by muscles, for the biarticular hamstring muscles, it appears the orientation of the distal knee joint is important for torque production at the proximal hip joint. This finding has implications for coordination and control of locomotor and jumping activities where knee and hip joint angles vary considerably during movement (Goslow et al., 1973). During walking for example, the feline knee extends during mid stance as the hip joint is maintained at a relatively constant angle. Similar relationships during trotting, galloping and jumping have also been noted (Goslow et al., 1973). Because hamstring moment arms at the hip
joint are larger for extended knee joint positions, the capacity of the hamstring muscle to generate hip joint torque for a given muscle force may increase as mid stance progresses even though hip joint angle does not change. Smaller hamstring moment arm magnitudes could also allow larger hip joint rotations and higher angular velocities for a given magnitude of muscle shortening when the knee is more flexed. Conversely, because hamstring moment arms at the knee appeared to vary little with hip joint position, the torque-generating capacity of the hamstring muscles at the knee joint may remain relatively constant for many limb (hip joint) configurations. Moment arms were not constant across joint range and varied as much as 66% as the knee and hip joints were rotated (Figs. 4, 5). These results are consistent with the predictions obtained from deriving moment arms from the muscle length functions provided by Goslow et al. (1973). The magnitudes of the knee joint moment arms are also consistent with Burkholder and Nichols (2004) except for
Knee Moment Arm (mm)
0
0
10
10 20
20 hip flexed, m = 5 hip extended, m = 4
30 60
70
80
90
100
110
hip flexed, m = 5 hip extended, m = 4
30
120
130
140
60
150
70
80
90
BFP
100
110
130
140
150
130
140
150
140
150
120
ST
hip flexed, m = 6 hip extended, m = 4
30 40 50
hip flexed, m = 5 hip extended, m = 4
60 60
70
80
90
100
110
120
130
140
150
60
70
80
90
110
120
SMP
SMA 0
0
10
10 20
20 hip flexed, m = 3 hip extended, m = 3
30 60
100
70
80
90
100
110
120
Knee Flexion Angle, deg
hip flexed, m = 5 hip extended, m = 4
30 130
140
150
60
70
80
90
100
110
120
130
Knee Flexion Angle, deg
Fig. 4. Average knee moment arm magnitudes for semitendinosus (ST) and each compartment of biceps femoris (BFA—anterior, BFM—middle, BFP—posterior) and semimembranosus (SMA— anterior, SMP—posterior). The ST and BFP have the largest knee flexion moment arm magnitudes, while SMA and BFA had negligible moment arm magnitudes about the knee. The BFA had an extension moment arm that switched to a flexion moment arm as the hip was extended. There was no appreciable difference between moment arm magnitudes at the two fixed hip angles tested (m is the number of muscles).
ARTICLE IN PRESS
Knee Moment Arm (mm)
BFM
L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
Knee Moment Arm (mm)
BFA
3453
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
3454
Table 2 Absolute differences in moment arms (MA) at the hip with the knee in flexed and extended positions, and at the knee with the hip in flexed and extended positions. Hamstring moment arm magnitudes at the hip were larger when the knee was extended compared to flexed. There was a small effect of hip angle on hamstring moment arms at the knee Muscle
Hip moment arms (MA)
Hip Moment Arm (mm)
SMA SMP ST BFA BFM BFP
Hip Moment Arm (mm)
Max. MA w/knee flexed (mm)
Max. MA w/knee extended (mm)
Absolute difference (mm)
Max. MA w/hip flexed (mm)
Max. MA w/hip extended (mm)
Absolute difference (mm)
26.6 29.5 26.3 25.0 24.8 11.0
32.1 36.8 29.9 28.0 28.2 17.7
5.5 7.3 3.6 3.0 3.4 6.7
0.2 7.7 38.5 0.4 5.0 45.6
0.1 6.0 35.7 1.0 7.1 43.3
0.1 1.8 2.8 0.6 2.1 2.3
BFA
BFM
knee flexed, m = 3 knee extended, m =3
40
30
20
20
10
10 70
80
90
100
110
knee flexed, m = 3 knee extended, m = 3
40
30
60
120
130
60
70
80
BFP
30
20
20
10
10 80
90
110
120
130
100
110
120
130
60
120
130
120
130
knee flexed, m = 3 knee extended, m = 4 70
80
90
100
110
SMP
SMA 40
40
30
30 20
20 knee flexed, m = 3 knee extended, m = 4
10 60
100
40
30
70
90
ST
knee flexed, m = 3 knee extended, m = 4
40
60
Hip Moment Arm (mm)
Knee moment arms (MA)
70
80
90
100
110
knee flexed, m = 3 knee extended, m = 3
10 120
130
Hip Flexion Angle (degrees)
60
70
80
90
100
110
Hip Flexion Angle (degrees)
Fig. 5. Hip moment arm magnitudes for semitendinosus (ST) and each compartment of biceps femoris (BFA—anterior, BFM—middle, BFP—posterior) and semimembranosus (SMA—anterior, SMP—posterior). Semimembranosus had the largest hip extension moment arms. Note that BFP and ST had an offset in hip extension moment arm magnitudes with respect to the two different knee joint angles tested and differences in hip joint moment arm magnitudes were noted for BFA and SMP with respect to changes in knee joint angle (m is the number of muscles).
the knee joint moment arm for BFP. Burkholder and Nichols estimated BFP to have a moment arm magnitude twice the values reported here and twice that calculated from Goslow et al.’s equations. Burkholder and Nichols studied smaller cats (mean 3.5 kg) than those used here and Goslow et al.’s animals perhaps accounting for some of the differences noted in knee joint moment arm magnitudes.
In MS models (Murray et al., 1995; Delp et al., 1999; Brown et al., 2003), the joint torque ðTÞ produced by a muscle equals the product of the muscle’s moment arm ðrÞ, and its force ðF Þ; T ¼ F r. During the stance phase of gait, cat knee extension torques reach approximately 2 Nm (McFadyen et al., 1999; Gregor et al., 2006). For sake of this example, if only the BFP was assumed to produce this
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
torque, the force required in this muscle could vary from 32 to 44 N if the moment arm is considered at its maximum (0.062 m) and minimum (0.045 m) values, respectively. This difference represents a considerable change in required muscle force and illustrates the importance of not assuming a constant moment arm with respect to joint angle. BF and SM muscles were separated and each compartment was tested independently. Despite the separation at the distal end of each compartment, the sutures attached proximally were passed through one eyelet for BFA, BFM and BFP, and one eyelet for SMA and SMP. Consequently, the hip moment arm magnitudes for the compartments of BF and SM were expected to be similar. Hip extension moment arms were similar for SMA compared to SMP and for BFA compared to BFM (Fig. 5). However, hip extension moment arms about the hip for BFP were 5–10 mm lower than BFA and BFM and did not follow the same pattern across hip joint angle. Compared to BFA and BFM, it appears that BFP’s more distal insertion on the tibia created a muscle line of action that led to differences in muscle excursion with hip flexion and extension. Despite the considerable variability of animal body weight and segment lengths there was not a great variation in moment arm magnitudes among animals. While specimen 4 had the largest mass (7.0 kg, Table 1), almost 1.4 times larger than the second largest specimen (Cat 6, 4.8 kg), it did not have proportionally larger segment lengths (femur length was 125 mm, just over 1.1 times larger than the second largest femur length of 111 mm). When moment arms were scaled to femur segment length or other anthropometric measures, they did not converge in a way that allowed prediction by a single equation. Therefore, the average values presented here are appropriate for animals with segmental lengths similar to the animals reported in this study. A polynomial was used to describe the excursion versus angle data and may have introduced error into the moment arm calculations. Of all of the polynomials applied to these data, only 7 (of 1112 total) demonstrated correlations with the experimental data between 0:80oR2 o0:95; all others were above 0.95. Polynomials lower than the fourth order approach yielded lower correlations. The difference between the experimental data and polynomials were always greatest when the limb was moved into end range flexion and extension and smallest in the mid range of the motion Fig. 6. Torque produced about the hip and knee joints calculated by multiplying the force produced in each muscle when maximally stimulated at its optimal length by the maximal moment arm noted in this study. Forces were estimated using each muscle’s physiological cross sectional area (Sacks and Roy, 1982) and a maximum muscle stress of 300 kPa (Wilson et al., 2001). The BFP produced a large knee flexion and small hip extension torque. ST produced large hip and knee joint torques while SMA was found to be a strong hip extensor. Even considering maximal muscle force and maximal moment arm magnitudes for BFA, BFM and SMP, these muscles had lower moment generating capacities at the hip and knee in comparison to their hamstring agonists.
3455
(Fig. 2). Thus, although maximal errors were up to 4.6 mm, this represents values at the end of range and errors in mid range were approximately half this value.
BFM
BFA 2.13 Nm
2.13 Nm
0.08 Nm
0.72 Nm
BFP
ST 3.84 Nm
1.76 Nm
5.03 Nm
4.76 Nm
SMA 4.18 Nm
SMP 2.05 Nm
0.53 Nm 0.03 Nm
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457
3456
Table 3 Percentage of torque contributions about the hip and knee joints when all the hamstrings are maximally stimulated Muscle
Muscle torque contributions Knee
BFA BFM BFP ST SMA SMP All muscles
Hip
Maximum torque (Nm)
Percent contribution (%)
Maximum torque (Nm)
Percent contribution (%)
0.1 0.7 4.8 5.0 0.03 0.5 11.0
0.7 6.6 43.3 45.8 0.3 4.8
2.1 2.1 1.8 3.8 4.2 2.1 16.1
13.2 13.2 10.9 23.9 26.0 12.7
Average maximal moment arm values multiplied by maximal muscle force were used to calculated torques. ST contributed the most to the knee joint moment while SMA provided the largest hip joint moment contribution. BFA had a positive extension moment about the knee.
There was a difference between the magnitudes of the moment arms during joint flexion and extension (Fig. 3D). This difference occurred primarily for the muscles with broader insertions (BFA, BFM) where the two ends of the suture attached to the outer edges of the muscle were separated by a larger distance (Fig. 1). While these differences may be attributed to tissue compliance of muscular insertions, observations during the experiment suggest that tendon excursions and therefore muscle moment arms may represent measurements from two different regions of the same muscle. During flexion, for example, the proximal fibers of BFM were loaded more than the distal fibers and because the proximal fibers are closer to the knee joint, the tendon excursion magnitudes were lower. Conversely, during extension the distal region of the muscle was loaded and the tendon excursions were larger. These findings parallel those of Blemker and Delp (2006), who modeled muscle fibers to have different moment arm magnitudes depending on their location within the muscles. For muscles with broad insertions, the average of the flexion and extension moment arms presented here likely represent the moment arm at the centroid of muscle insertion. Intrafascicular multielectrode stimulation of the muscular branch of the sciatic nerve in the feline hindlimb appears able to stimulate individual muscles of the hamstring group (unpublished observation). To generate a sit-to-stand behavior in the cat, muscles such as SMA and BFA could be targeted because they have large hip extension moment generating capabilities (Fig. 6) and contribute a large percentage of the hip extension moment while providing a small percentage of the knee flexion torque (Table 3). Muscles such as SMP and BFM would also be appropriate to target to provide a small degree of co-contraction against evoked quadriceps femoris muscle forces which produce a knee extension torque. Conversely, ST and BFP could be avoided for a sit-tostand behavior as these muscles have large knee flexion moment-generating capacities but would be very appropriate when greater knee joint stability is required from
muscle co-contractions and when knee flexion is required during gait. Conflict of interest The authors do not have any financial and personal relationships with other people or organisations that could inappropriately influence (bias) this work. Acknowledgment This work was supported by the NIH (NINDS/ NICHHD-2RO1-NS039677-05A1). References An, K.N., Ueba, Y., Chao, E.Y., Cooney, W.P., Linscheid, R.L., 1983. Tendon excursion and moment arm of index finger muscles. Journal of Biomechanics 16 (6), 419–425. An, K.N., Takahashi, K., Harrigan, T.P., Chao, E.Y., 1984. Determination of muscle orientations and moment arms. Journal of Biomechanical Engineering 106 (3), 280–282. Anderson, F.C., Pandy, M.G., 2001a. Dynamic optimization of human walking. Journal of Biomechanical Engineering 123 (5), 381–390. Anderson, F.C., Pandy, M.G., 2001b. Static and dynamic optimization solutions for gait are practically equivalent. Journal of Biomechanics 34 (2), 153–161. Anderson, F.C., Pandy, M.G., 2003. Individual muscle contributions to support in normal walking. Gait & Posture 17 (2), 159–169. Aoyagi, Y., Mushahwar, V.K., Stein, R.B., Prochazka, A., 2004. Movements elicited by electrical stimulation of muscles, nerves, intermediate spinal cord, and spinal roots in anesthetized and decerebrate cats. IEEE Transactions on Neural Systems and Rehabilitation Engineering 12 (1), 1–11. Blemker, S.S., Delp, S.L., 2006. Rectus femoris and vastus intermedius fiber excursions predicted by three-dimensional muscle models. Journal of Biomechanics 39 (8), 1383–1391. Botterman, B.R., Cope, T.C., 1988. Motor-unit stimulation patterns during fatiguing contractions of constant tension. Journal of Neurophysiology 60 (4), 1198–1214. Boyd, S.K., Ronsky, J.L., 1998. Instantaneous moment arm determination of the cat knee. Journal of Biomechanics 31 (3), 279–283.
ARTICLE IN PRESS L.N. MacFadden, N.A.T. Brown / Journal of Biomechanics 40 (2007) 3448–3457 Branner, A., Stein, R.B., Normann, R.A., 2001. Selective stimulation of cat sciatic nerve using an array of varying-length microelectrodes. Journal of Neurophysiology 85 (4), 1585–1594. Brown, N.A., Pandy, M.G., Kawcak, C.E., McIlwraith, C.W., 2003. Force- and moment-generating capacities of muscles in the distal forelimb of the horse. Journal of Anatomy 203 (1), 101–113. Buchanan, T.S., Lloyd, D.G., Manal, K., Besier, T.F., 2005. Estimation of muscle forces and joint moments using a forward–inverse dynamics model. Medicine and Science in Sports Exercise 37 (11), 1911–1916. Buford, J.A., Smith, J.L., 1990. Adaptive control for backward quadrupedal walking. II. Hindlimb muscle synergies. Journal of Neurophysiology 64 (3), 756–766. Burkholder, T.J., Nichols, T.R., 2004. Three-dimensional model of the feline hindlimb. Journal of Morphology 261 (1), 118–129. Crowninshield, R.D., Brand, R.A., 1981. A physiologically based criterion of muscle force prediction in locomotion. Journal of Biomechanics 14 (11), 793–801. Delp, S.L., Hess, W.E., Hungerford, D.S., Jones, L.C., 1999. Variation of rotation moment arms with hip flexion. Journal of Biomechanics 32 (5), 493–501. Engberg, I., Lundberg, A., 1969. An electromyographic analysis of muscular activity in the hindlimb of the cat during unrestrained locomotion. Acta Physiologica Scandinavica 75 (4), 614–630. English, A.W., Weeks, O.I., 1987. An anatomical and functional analysis of cat biceps femoris and semitendinosus muscles. Journal of Morphology 191 (2), 161–175. Goslow Jr., G.E., Reinking, R.M., Stuart, D.G., 1973. The cat step cycle: hind limb joint angles and muscle lengths during unrestrained locomotion. Journal of Morphology 141 (1), 1–41. Gregoire, L., Veeger, H.E., Huijing, P.A., van Ingen Schenau, G.J., 1984. Role of mono- and biarticular muscles in explosive movements. International Journal of Sports Medicine 5 (6), 301–305. Gregor, R.J., Smith, D.W., Prilutsky, B.I., 2006. Mechanics of slope walking in the cat: quantification of muscle load, length change, and ankle extensor EMG patterns. Journal of Neurophysiology 95 (3), 1397–1409. Marsolais, E.B., Kobetic, R., Polando, G., Ferguson, K., Tashman, S., Gaudio, R., Nandurkar, S., Lehneis, H.R., 2000. The Case Western Reserve University hybrid gait orthosis. Journal of Spinal Cord Medicine 23 (2), 100–108. McDonnall, D., Clark, G.A., Normann, R.A., 2004. Selective motor unit recruitment via intrafascicular multielectrode stimulation. Canadian Journal of Physiology and Pharmacology 82 (8–9), 599–609. McFadyen, B.J., Lavoie, S., Drew, T., 1999. Kinetic and energetic patterns for hindlimb obstacle avoidance during cat locomotion. Experimental Brain Research 125 (4), 502–510.
3457
Murray, W.M., Delp, S.L., Buchanan, T.S., 1995. Variation of muscle moment arms with elbow and forearm position. Journal of Biomechanics 28 (5), 513–525. Pandy, M.G., 1999. Moment arm of a muscle force. Exercise and Sport Sciences Reviews 27, 79–118. Pandy, M.G., 2001. Computer modeling and simulation of human movement. Annual Review of Biomedical Engineering 3, 245–273. Pandy, M.G., 2003. Simple and complex models for studying muscle function in walking. Philosophical Transactions of the Royal Society of London, Series B, Biological Sciences 358 (1437), 1501–1509. Pandy, M.G., 2005. Mixed forward and inverse solutions in movement biomechanics. Theoretical Issues in Ergonomics Science 6 (3/4), 325–330. Pratt, C.A., Chanaud, C.M., Loeb, G.E., 1991. Functionally complex muscles of the cat hindlimb. IV. Intramuscular distribution of movement command signals and cutaneous reflexes in broad, bifunctional thigh muscles. Experimental Brain Research 85 (2), 281–299. Prilutsky, B.I., Herzog, W., Allinger, T.L., 1997. Forces of individual cat ankle extensor muscles during locomotion predicted using static optimization. Journal of Biomechanics 30 (10), 1025–1033. Sacks, R.D., Roy, R.R., 1982. Architecture of the hind limb muscles of cats: functional significance. Journal of Morphology 173 (2), 185–195. Triolo, R.J., Bieri, C., Uhlir, J., Kobetic, R., Scheiner, A., Marsolais, E.B., 1996. Implanted functional neuromuscular stimulation systems for individuals with cervical spinal cord injuries: clinical case reports. Archives of Physical and Medical Rehabilitation 77 (11), 1119–1128. van Ingen Schenau, G.J., Bobbert, M.F., Rozendal, R.H., 1987. The unique action of bi-articular muscles in complex movements. Journal of Anatomy 155, 1–5. van Soest, A.J., Schwab, A.L., Bobbert, M.F., van Ingen Schenau, G.J., 1993. The influence of the biarticularity of the gastrocnemius muscle on vertical-jumping achievement. Journal of Biomechanics 26 (1), 1–8. Wilson, A.M., McGuigan, M.P., Su, A., van Den Bogert, A.J., 2001. Horses damp the spring in their step. Nature 414 (6866), 859–895. Winter, D.A., 1980. Overall principle of lower limb support during stance phase of gait. Journal of Biomechanics 13 (11), 923–927. Young, R.P., Scott, S.H., Loeb, G.E., 1993. The distal hindlimb musculature of the cat: multiaxis moment arms at the ankle joint. Experimental Brain Research 96 (1), 141–151. Zajac, F.E., 1989. Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Critical Reviews in Biomedical Engineering 17 (4), 359–411.