Functional morphology of proximal hindlimb muscles in the frog Rana pipiens
1 The Neurosciences Institute, 10640 John Jay Hopkins Drive, San Diego, CA
92121, USA
2 Department of Biology, University of Pennsylvania, Philadelphia, PA 19129,
USA
* e-mail: kargo{at}nsi.edu
Accepted 3 May 2002
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Summary |
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Key words: muscle, hindlimb, musculoskeletal model, moment arm, force field, frog, Rana pipiens
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Introduction |
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Before one can understand how the neural system controls limb behaviors or
how molecular properties of muscle might affect performance, one must first
have a clear picture of the mechanics of the limb. In particular, a
substantial part of the control of any behavior is embedded in the anatomical
and geometric design of the limb (Lombard
and Abbot, 1907; Kubow and Full, 1999;
Mussa-Ivaldi et al., 1985
).
Anatomical design features that affect the transformation of neural commands
into force and movement may be classified as either macroscopic or microscopic
features of the limb mechanical system
(Lieber and Friden, 2000
).
Macroscopic features include those of the skeleton, e.g. bone lengths, joint
degrees of freedom, moments of inertia and limb configuration, and those of
the musculotendon complexes (MTCs), e.g. attachment sites, moment arms, muscle
fiber lengths, in-series connective tissue lengths, cross-sectional areas and
pennation angles. An important microscopic feature of the limb mechanical
system is the internal sarcomere length of MTCs with respect to limb
configuration (Burkholder and Lieber,
2001
). The integration of these design features determines the
movement ranges over which MTCs operate
(Lieber and Friden, 2000
), the
moment-generating capabilities at particular limb positions
(Murray et al., 2000
) and the
potential contributions of MTCs to endpoint force or limb stiffness
(Buneo et al., 1997
).
Anatomically realistic models, which integrate experimentally measured
properties of real animals, can be used to predict the operating ranges,
moment-generating capabilities and endpoint force capabilities of MTCs and to
estimate MTC trajectories during behaviors in which joint kinematics have been
measured (Arnold et al., 2000
;
Delp et al., 1998
;
Hoy et al., 1990
;
Pandy, 2001
).
In this study, we determined the anatomical properties of 13 proximal
muscles in the frog hindlimb. We incorporated these properties into an
accurate anatomical model of the frog. A previous study developed and
described the skeleton and joint subsystems of this model
(Kargo et al., 2002). We
validated the interaction between the hindlimb musculotendon and joint
subsystems by comparing moment arms measured across the configuration-space of
the hindlimb and sarcomere lengths measured at the starting and take-off
positions of a jump with moment arms and sarcomere lengths predicted by the
model at these same limb positions. We then used the model to describe the
static, whole-limb effects of each of the hindlimb muscles as a
three-dimensional force field. The force-field measurements summarize how a
muscle contraction will act to accelerate the limb from a large range of limb
configurations (Giszter et al.,
1993
; Loeb et al.,
2000
). We also use the model to predict MTC trajectories during a
number of hindlimb behaviors (wiping, kicking, swimming and jumping) and to
estimate the contractile function of specific MTCs during these behaviors. The
results of this study provide a useful summary of the static mechanics of the
pelvic/hindlimb system of the frog. More importantly, the model forms a
foundation upon which additional subsystems (e.g. neural systems) and more
sophisticated muscle models can be appended to examine the dynamic control of
limb behaviors.
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Materials and methods |
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The image file was imported into SIMM (Software for Interactive Musculoskeletal Modeling, Musculographics Inc., Santa Rosa, CA, USA), which is a graphics-based, biomechanical modeling package. A second laser-scanned image of the bone, one in which the muscles had been completely removed, was also imported into SIMM and overlaid on the first image. The attachments of virtual muscles in SIMM were manually positioned on this second bone segment. The hindlimb muscles whose attachment sites were determined were the semimembranosus (SM), gracilus major (GR), adductor magnus dorsal and ventral heads (ADd and ADv), cruralis (CR), gluteus magnus (GL), semitendinosus ventral and dorsal heads (STv and STd), iliofibularis (ILf), iliacus externus (ILe), iliacus internus (ILi), sartorius (SA) and tensor fascia latae (TFL).
In the model, the paths for 10 of the hindlimb muscles were represented as
a simple straight line from an origin point to an insertion point (all muscles
but STd, STv and ILe). The paths for STd and STv between the origin and
insertion points were constrained by an intermediate via-point added 2.0 mm
posterior to the knee joint. This via-point approximates the effect of a
connective-tissue loop, which constrains ST paths in real frogs
(Lombard and Abbot, 1907). The
path for ILe between its origin and insertion points was constrained by an
intermediate via-point positioned just ventral to the GL attachment on the
pelvis. The path for the triceps muscle group (CR, GL and TFL) was constrained
to wrap over the anterior knee joint. The shape of the wrap object that
deflected the triceps muscles approximated the distal surface of the femur. A
second wrap object, which approximated the geometry of the femoral head,
prevented muscles from penetrating the femoral head in the extreme ranges of
hip rotation. A third wrap object approximated the posterior surface of the
distal femur and deflected knee flexor muscles (ST, GR, ILf and SA) in the
extreme ranges of knee extension.
Moment arm measurements
The tendency of a muscle to rotate a bone segment is described by its
moment arm, which is the perpendicular distance from the muscle's line of
action to the instantaneous center of rotation. The instantaneous centers of
rotation at the hip and knee joints in Rana pipiens were measured in
a previous study, and this information was used to model the behavior of these
joints (Kargo et al., 2002).
In that study, hip kinematics was well approximated by a ball-and-socket joint
in which the instantaneous center of rotation was fixed. The behavior of the
knee joint was more complex. However, the primary range of knee motion
(flexionextension) was well approximated by a rolling joint in which
the instantaneous center of rotation was translated along the distal surface
of the femur. In this study, we measured moment arms of hindlimb muscles about
the three axes of the hip joint and about the primary axis of knee rotation.
We then tested whether the model moment arms matched the moment arm
measurements made in experimental frogs.
The method used to measure moment arms experimentally was the `tendon
excursion method'. This method has been used previously in our laboratory and
described in detail (see Lutz and Rome,
1996b). Briefly, all muscles were removed from the hindlimb except
the muscle under study and small muscles surrounding the joints. One bone
segment (e.g. the pelvis) was secured into the fixed arm of a custom-built jig
apparatus, and its distal joint member (e.g. the femur/tibiofibula complex)
was secured into the movable arm of the jig. The movable arm permitted
180° of rotation and unopposed translation of the distal segment within
two orthogonal planes of motion. The muscle attachment on the fixed segment
was detached. A thread was tied to the detached tendon of the muscle and run
over a length scale and pulley. A 20 g weight was suspended from the end of
the thread to maintain a constant tension. The change in the length of the
muscle was measured as the moving arm of the jig was rotated. The moment arm
(r) about an axis of rotation was calculated using the following
equation:
![]() | (1) |
We used a modified technique, similar to that used by Delp et al.
(1999), to measure the moment
arms of smaller muscles and muscles with little tendon in which to tie the
thread around (ADd, ADv, ILe, ILi, ILf, STv, STd, SA). A miniature bone screw
was placed at the insertion site of the muscle in the moving segment. A suture
thread was tied around the screw. A minutien pin (i.e. an insect pin) with a
loop at one end was placed at the muscle origin on the fixed segment. The
suture was threaded through the loop and run over the length scale, and a 5 g
weight was suspended from the end of the thread. The change in the length of
the suture thread was measured as the moving arm of the jig was rotated. The
moment arm was calculated using equation 1.
The moment arms of muscles crossing the hip joint were measured with respect to an xyz coordinate system embedded in the femur (see Fig. 3). When all the bones rested in the horizontal plane, the z-axis of the femur pointed dorsally. For the right hip, clockwise rotation of the femur about the z-axis was extension and counterclockwise rotation was flexion. The x-axis of the femur pointed down its long axis. When looking up the x-axis (proximal to distal), clockwise rotation of the femur was external rotation and counterclockwise rotation was internal rotation. The y-axis of the femur pointed rostrally when the femur was positioned to the frog's side and in the horizontal plane. When looking up the y-axis (rostral to caudal), clockwise rotation of the femur was abduction and counterclockwise rotation was adduction.
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The moment arms of muscles crossing the knee were measured only with
respect to the z-axis of the knee joint (see
Fig. 3). The z-axis
pointed dorsally when the hindlimb was positioned in the horizontal plane and
was located along the distal surface of the femur. The z-axis was
translated along the distal surface of the femur with tibiofibula rotation
(see Kargo et al., 2002). For
the right knee, clockwise rotation of the tibiofibula about the
z-axis was flexion and counterclockwise rotation was extension.
In total, 27 frogs were used to measure moment arms at the hip and knee joints. Moment arm measurements performed in individual frogs were normalized to combine data among frogs. To normalize the data, we assumed that all frogs were geometrically similar. In our study, an averaged-sized Rana pipiens weighed 28±4 g (mean ± S.E.M.) and had a tibiofibula length of 30±3 mm (mean ± S.E.M.). Also, all frogs (three frogs) whose bones were laser-scanned to construct the hindlimb model weighed 28 g and had a tibiofibula length of 30 mm. Thus, moment arm measurements were normalized to a tibiofibula length of 30 mm. For example, a moment arm measurement of 3.0 mm made in a frog with a tibiofibula length of 32 mm was normalized to 2.8 mm, i.e. 3.0x30.0/32.0.
The moment arm about a single axis of hip rotation can vary as the angle
about the other two axes of the hip is changed
(Arnold and Delp, 2001). Since
the jig allowed simultaneous and independent rotations about two joint axes,
we examined the nature of such interactions for hindlimb muscles in the frog.
1 (e.g. hip abduction angle) was fixed at a specific value,
and
2 (e.g. hip extension angle) was changed in 10°
increments. The moment arm with respect to
2 was determined.
1 was then rotated to a new angle, and the same series of
2 rotations was imposed. The data for such an experiment
were evaluated using three-dimensional plots (Matlab, Mathworks Inc., Natick,
MA, USA). The horizontal axes in the plots represented the angles
1 and
2, and the vertical axis represented
the moment arm with respect to
2. Joint angle interactions
were tested for in four representative muscles that cross the hip joint: SM
(five frogs), GR (four frogs), SA (five frogs) and GL (three frogs).
Musculotendon architecture
We measured physiological cross-sectional area (PCSA), sarcomere
length/joint angle relationships, muscle fiber lengths and in-series
connective tissue lengths for each of the proximal hindlimb muscles. These
parameters have previously been measured for some muscles in Rana
pipiens. Calow and Alexander
(1973) and Lieber and Brown
(1992
) published values for
CR, Plantarus ankle extensor (PL), GL, SM, GR, STv and ILi. We determined
these parameters for six additional muscles in Rana pipiens and for
the same seven muscles for comparison purposes.
PCSA was determined using the following relationship:
![]() | (2) |
Sarcomere lengths were measured in both fixed and frozen muscle tissue at a single test position. The test position was a planar configuration in which the femur was extended by 90° relative to the long axis of the pelvis and the tibiofibula was extended by 90° relative to the femur (see Fig. 2). The pelvis/limb complex was secured in the test position using bone pins, fine steel wire and hardening epoxy resin. For fixed tissue measurements, the complex was sequentially immersed in 0.05% formalin solution for 8 h, 10% formalin solution for 24 h and 30% nitric acid for 4 h, and then washed in distilled water. Small fascicles were dissected from each hindlimb muscle, and their lengths were measured with a stage graticule. In-series connective tissue length was found by subtracting fascicle length from whole-muscle length. The fascicle was placed on a slide and mounted in glycerine. Sarcomere lengths were measured at three regions along the length of the fascicle by counting 30 sarcomeres in series, measuring the length from the first to the last sarcomere under a calibrated eyepiece graticule and dividing by 30. Care was taken to dissect fascicles from similar anatomical regions of each muscle in all the frogs. For example, in thinner strap-like muscles such as SA, fascicles were dissected from a middle region and from regions bordering adjacent muscles. For thicker, architecturally more complex, muscles such as CR, fascicles were dissected from superficial, middle and deep regions of the muscle belly.
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For sarcomere length measurements in frozen tissue, the limb was secured in
the test position and glycerinated in cold rigor solution (15 ml potassium
phosphate buffer, 100 mmol l-1 potassium acetate, 5 mmol
l-1 K2EGTA, 1 mmol l-1 iodoacetic acid, 0.1
mmol l-1 leupeptin, 0.25 mmol l-1 phenylmethylsulfonyl
fluoride and 0.01 mmol l-1 pepstatin, pH 7.2) for approximately 2
days. Sosnicki et al. (1991)
determined that this method allowed fibers to go into complete rigor. The limb
complex was then quickly and entirely immersed in liquid-nitrogen-cooled
isopentane. Frozen blocks were cryo-sectioned along the long axis of the
muscle in sections 25 µm thick and examined under the light microscope.
Both techniques (fixation and freezing) were used because of trade-offs
between the two. Frozen tissue measurements have been shown under certain
circumstances to be more accurate for determining in vivo sarcomere
lengths (Sosnicki et al.,
1991
). However, the fixed tissue procedure allowed sarcomere
lengths to be measured simultaneously in more muscles, i.e. in frozen blocks,
it is difficult to distinguish muscles so only one or two muscles were left
intact. Thus, the freezing technique was used mainly to validate measurements
made in fixed tissue. We found that sarcomere lengths were, on average, 5-7%
shorter in fixed tissue than in frozen tissue. Thus, a correction factor
(0.05) was applied to all fixed tissue measurements, e.g. a sarcomere length
of 2.00 µm in fixed tissue was multiplied by 0.05 (+2.00 µm) to produce
a corrected sarcomere length of 2.10 µm.
Validating model predictions of sarcomere length
We measured sarcomere, fascicle and whole-muscle lengths of each muscle at
the test position in six frogs. We then positioned the model hindlimb at the
same test position. Because we measured the lengths of sarcomeres and muscle
fibers undergoing fixed-end contractions (i.e. when in the rigor state), we
could not simply assign each `non-contracting' muscle in the model the
experimental measurements. Sarcomeres are arranged in series with connective
tissue that is stretched during muscle contraction and may therefore shorten
by up to 20% during fixed-end contractions
(Lieber et al., 1991;
James et al., 1995
). To assign
the model muscles the correct, non-contracting values for in-series connective
tissue, muscle fiber and sarcomere length, we had to estimate the
non-contracting lengths. This was performed as detailed below.
First, we assumed that in-series connective tissue (for each muscle)
exhibited an ideal stress/strain relationship, which is similar to that
described for the frog plantarus tendon
(Trestik and Lieber, 1993),
and a strain at maximal tetanic tension equal to 3.5%. We chose 3.5% as a
general measure for each muscle because the in-series connective tissue of
frog muscles exhibits strains that range, on average, from 2 to 5%
(Lieber et al., 1991
;
Trestik and Lieber, 1993
;
Kawakami and Lieber, 2000
).
Second, we determined the ratio of connective tissue length to muscle fiber
length for each muscle at the test position (see
Table 1). Third, we assumed
that frog sarcomeres exhibit an ideal sarcomere length/tension relationship,
which has been described by Gordon et al.
(1966
). On the basis of these
three relationships and the measured sarcomere length at the test position, we
estimated the non-contracting sarcomere length. For example, muscle A
had a measured sarcomere length of 2.2 µm. Frog sarcomeres produce their
maximal tetanic force at this length
(Gordon et al., 1966
). This
level of force stretches the in-series connective tissue by 3.5%. Thus, if the
measured lengths of in-series connective tissue, muscle fiber and sarcomere
were 10.35 mm, 10.00 mm and 2.20 µm, respectively, the non-contracting
lengths would be 10.00 mm, 10.35 mm and 2.28 µm, respectively. However, the
13 proximal muscles of the frog hindlimb have a mean connective tissue/muscle
fiber ratio of only 1.04. Thus, the sarcomere shortening effect was not
substantial (i.e. these muscles are `stiff' actuators). This effect only
becomes substantial when ratios approach 5.0-10.0
(Zajac, 1989
;
Lieber et al., 1991
;
James et al., 1995
).
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We assigned the virtual muscles comprising the model the mean
(non-contracting) values for in-series connective tissue, muscle fiber and
sarcomere lengths. Since the model accurately reproduced moment arms at the
hip and knee (see Results), we could then use the model to predict the
(non-contracting) fascicle and sarcomere lengths at different limb
configurations. SIMM uses the following relationships to predict fascicle and
sarcomere lengths on the basis of moment arm variations across the
configuration-space of the limb:
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![]() | (4) |
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To test the model predictions, we compared sarcomere lengths measured in
experimental frogs at the starting and take-off positions of a jump with
sarcomere lengths calculated at these same positions in the model. The
three-dimensional kinematics of jumping was previously determined and used to
position both experimental frogs and the model
(Kargo et al., 2002). To
measure sarcomere lengths experimentally, the right limb was fixed at the
starting configuration of a jump by wrapping fine steel wire around bone
screws placed in the hindlimb segments. A hardening epoxy compound secured the
wires in place. This start position was 30° hip flexion, 15° internal
rotation, 18° hip adduction and 65° knee flexion. Angles were
determined in the jig apparatus. The left limb was then fixed at the
approximate take-off position. This position was -75° hip extension,
0° internal rotation, 0° hip adduction and -75° knee extension.
The muscle/limb complex was then fixed, the fascicles were dissected and the
sarcomere lengths were measured using the procedure described above. The
correction factor (0.05) was applied to account for the additional shortening
due to the fixative.
We used the following procedure to predict the length of `contracting'
sarcomeres in the model at the start and take-off positions of a jump. We
simulated fixed-end contractions for each musculotendon actuator at the two
limb positions. Each actuator produced a contractile force that was derived
from scaling generic musculotendon properties with five muscle-specific
parameters. The muscle-specific parameters were: PO, peak
tetanic force; lOM, optimal muscle fiber
length, , pennation angle; lOT, length
of in-series connective tissue; and
OT, strain of
in-series connective tissue when force in the tendon
PT=PO. PO was
estimated for each muscle by multiplying PCSA by muscle stress, which Lutz and
Rome (1996b
) measured in the
SM muscle to be 260 kN m-2. The SM muscle is composed of 85-90%
fast muscle fibers, and the other hindlimb muscles have similar high
percentages of fast muscle fibers (Lutz et
al., 1998
). Thus, assuming that all hindlimb muscles had a muscle
stress equal to 260 kN m-2 is reasonable. Although this assumption
will affect the contractile force that each model actuator is capable of
producing, it will not affect the calculation of sarcomere or muscle fiber
lengths in the model. The reason for this is that tendon properties were
assumed to be matched to muscle properties, i.e. tendon strain (at
PO) was 3.5% irrespective of how much force each actuator
produced.
In contrast to PO, we measured directly at the
test position, and lOM and
lOT were the muscle fiber and in-series
connective tissue lengths at the limb position in which sarcomere length was
2.2 µm. The generic musculotendon properties that were necessary for
calculating muscle fiber lengths during the fixed-end contraction were: the
ideal muscle sarcomere length/tension relationship described by Gordon et al.
(1966
), the ideal muscle fiber
velocity/force relationship (
CE/PCE)
described for the frog sartorius muscle by Edman et al. (1979) and the
exponential stress/strain (PT/
T)
relationship of the tendon described by Trestik and Lieber
(1993
) where
PT is the force in the tendon in-series connective tissue.
Thus, the contractile force in response to maximal activation
[a(t)=1.0] of a model actuator could be described by the
following:
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Determination of static muscle functions
We used the hindlimb model to describe the static mechanical effects of
each muscle. The state space of muscle effects was described as an isometric
force field (see Giszter et al.,
1993; Loeb et al.,
2000
). To construct a force field, the ankle of the model limb was
placed at 80 different positions throughout the hindlimb's reachable
workspace. The reachable workspace refers to the three-dimensional area over
which the ankle can be positioned. The workspace was divided into five levels.
The top level was 15 mm above the horizontal plane of the pelvis
(z=+15 mm), the bottom level was 15 mm below the plane of the pelvis
(z=-15 mm) and the middle level was at the plane of the pelvis
(z=0 mm). The other two planes were +7.5 mm above and -7.5 mm below
the plane of the pelvis. The ankle was placed at 16 different positions
(x1-16,y1-16) within each horizontal
level. These x,y positions were the same for each level. The 80
positions spanned the reachable workspace of the limb and formed a
three-dimensional box.
To construct muscle force fields, we simulated fixed-end contractions of
each musculotendon actuator at each position. The actuators were maximally
activated, and the contractile force was calculated 500 ms into the simulation
run. At each position, the contractile force of the muscle produced a set of
joint moments about the hip and knee. Joint moments were calculated
automatically in SIMM by multiplying muscle force by the respective moment
arm. The joint moments were then transmitted through the hindlimb to produce a
force at the ankle. This force (F) was calculated using the following
relationship (Tsai, 1999):
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Results |
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The knee-joint complex from three separate frogs was laser-scanned. The ST, ILf and CR, GL and TFL (triceps group) tendons were left intact on the tibiofibula in one knee complex. The GR, SA and SM tendons were left intact on a second complex. GR and SA attached to the tibiofibula and SM attached to the posterior surface of the distal femur and knee capsule. All the tendons were left intact on a third knee complex. This third complex is shown in Fig. 1B. The attachment sites of additional distal muscles (actions at the ankle and tarso-metatarsal joint) are also shown in Fig. 1B. These muscles are the plantarus (PL), tibialis anterior (TA) and peroneus (PE) muscles.
The modeled paths of the proximal hindlimb muscles are shown in Fig. 2. The top four panels show the paths of hipflexor muscles (CR, GL, ILe, ILf, ILi, Pec, SA and TFL). The bottom four panels show the paths of hip-extensor muscles (ADd, ADv, GR, OI, OE, QF, SM, STd, STv). Some muscle paths were constrained to wrap around certain skeletal features. The distal path of the triceps group (CR, GL and TFL) wrapped over the knee joint. The distal path of ILe wrapped over the femoral head. In the extreme ranges of hip flexion and hip extension, both extensor and flexor paths were constrained to wrap around the femur. In addition, in the extreme range of knee extension, the ST, ILf, GR and SA tendons were constrained to wrap around the posterior surface of the distal femur.
Moment arms about the hip joint
We measured the moment arms about the flexionextension axis of the
femur (z-axis) in experimental frogs (see z-axis in
Fig. 3). The limb configuration
in Fig. 3 was the test position
from which moment arms were measured. Counterclockwise rotation of the femur
about the z-axis was hip flexion, and clockwise rotation was hip
extension. Fig. 4A shows
averaged moment arms (± 1 S.D.) about the z-axis of the femur
for 12 of the muscles tested. All moment arms varied with the hip
flexionextension angle. SM, GR, ADd, ADv, STd and STv extended the
femur at all positions. For each extensor, the largest moment arm was found
between -5° and -35° of hip extension. GR had the largest extensor
moment arm (-3.9 mm). ILi, ILe, CR, TFL and SA flexed the femur at all
positions. The hip position at which the largest flexor moment arm was
measured varied between muscles: TFL and SA had peak moment arms at the most
flexed hip positions, whereas CR, ILe and ILi had peak moment arms at more
neutral hip positions near the test position. TFL had the largest flexor
moment arm (+3.8 mm). ILf and GL were bifunctional with respect to rotation
about the z-axis: their moment arms acted to flex the femur at flexed
hip positions and to extend the femur at extended positions. The magnitude of
these moment arms was relatively minor (at most 1-1.5 mm) compared with the
peak moment arms of the other muscles.
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We next measured moment arms about the abductionadduction axis of the femur (y-axis; see Fig. 3). The y-axis points rostrally at the test position. Clockwise rotation of the femur about the y-axis (looking up the y-axis) was hip abduction, and counterclockwise rotation was hip adduction. Fig. 4B shows averaged moment arms measured about the y-axis of the femur. Like flexionextension moment arms, abductionadduction moment arms were configuration-dependent. SM, STd, GL, TFL, ILe, ILf and ILi abducted the femur from all positions. TFL had the largest abduction moment arm (-3.1 mm). ADv, SA and STv adducted the femur from all positions. ADv had the largest adduction moment arm (+2.8 mm). CR, GR and ADd were bifunctional with respect to rotation about the y-axis: they had moment arms that acted to abduct the femur at abducted hip positions and to adduct the femur at adducted positions.
We then measured moment arms about the internalexternal rotation axis of the femur (x-axis; see Fig. 3). The x-axis points down the long axis of the femur. Counterclockwise rotation about the x-axis from the test position was termed hip internal rotation, and clockwise rotation was termed hip external rotation. Fig. 4C shows averaged moment arms measured about the x-axis of the femur. SM, GR, STd, ILf and ILi rotated the femur internally at all positions. ILi had the largest peak moment arm (+1.5 mm). GL, SA and TFL rotated the femur externally at all positions. SA had the largest peak moment arm (-1.0 mm). The rest of the muscles were bifunctional with respect to rotation about the x-axis: they rotated the femur externally or internally depending on the current rotation angle.
We tested whether the hindlimb model correctly predicted the moment arms measured experimentally. Model moment arms about the z-axis (hip flexionextension) and y-axis (hip abductionadduction) lay within one standard deviation of the mean moment arms measured experimentally. To obtain such a good fit for each muscle, we had to move certain muscle attachment sites slightly (by less than 1 mm in the x, y and z directions) and adjust the geometry of the wrap objects. Model moment arms about the x-axis (hip internalexternal rotation) lay within one standard error of the mean of the averaged values measured experimentally. The reason for the reduced fit of moment arms about the x-axis was that these moment arms were 2-4 times smaller than the moment arms about the z-axis and y-axis and, thus, the signal-to-noise ratio was more substantial.
We tested for configuration-dependent interactions about the axes of the hip joint in four representative muscles (ADv, GL, SA and SM) and examined whether the model reproduced these interaction effects. The hindlimb model reproduced the interaction effects measured experimentally at the hip joint. The top row of Fig. 5 shows data for SM and the bottom row shows data for SA. The left column of each panel (Fig. 5A-C) represents model data and the right column represents data from experimental frogs. The first observed effect was a reduction in both hip flexor and extensor moment arms when the femur was adducted or abducted away from the test position. These effects ranged in magnitude from 5 to 25 % decreases in the flexor or extensor moment arm. For example, the SM moment arm was 4.0 mm when the hip was extended by 30° from the test position but was only 3.0 mm at this same position when the hip was abducted by 40°. This effect is shown in Fig. 5A, in which the vertical axis represents the moment arm measured about the z-axis of the femur for SA (flexor) and SM (extensor; this axis is inverted and is therefore positive to compare SA and SM interaction effects). The left axis represents the flexionextension angle at the hip, and the right axis represents the abductionadduction angle. Qualitatively similar effects were observed for GL and ADv, i.e. hip extensor and flexor moment arms were largest when the femur rested in the horizontal plane and were 5-25 % smaller when the femur was lowered or raised above this plane.
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The second observed interaction was the effect on abductionadduction moment arms when the femur was flexed and extended away from the test position. This effect is shown in Fig. 5B (left column, model data; right column, real frog). At extended hip positions, abduction moment arms for SM (and GL; not shown) varied by as little as 5 % across the entire range of abductionadduction (abduction moment arms inverted to positive values to compare with SA measurements shown below). Thus, SM had nearly equal capacities to abduct the femur at all positions in which the hip was extended. In contrast, at flexed hip positions, abduction moment arms varied by as much as 30-40 % across the range of abductionadduction, thereby greatly affecting the capacity of SM (and GL; not shown) to abduct or raise the femur. The opposite effect was observed for adduction moment arms for SA (and ADv; not shown). That is, adduction moment arms varied to a greater extent at flexed hip positions (25-35 %) than at extended hip positions (5-10 % variation).
The final observed interaction effect was the effect of hip flexionextension on externalinternal rotation moment arms. This effect is shown in Fig. 5C (left column, model data; right column, real frog). Internal rotation moment arms for SM (and ADv; not shown) were largest at extended hip positions (approximately 1.0 mm) and negligible at flexed hip positions (approximately 0 mm). The opposite was the case for the external rotation moment arm of SA (and GL; not shown). External rotation moment arms were largest at flexed positions (approximately 1.0 mm) and negligible at extended positions (approximately 0 mm). In summary, the model captured the main interaction effects observed at the hip joint in experimental frogs.
Moment arms about the knee joint
Most muscles that cross the hip also cross the knee joint. These include
STd, STv, ILf, SA, GR and the triceps group (CR, GL and TFL). SM has a
negligible flexor moment arm about the knee (<0.1 mm;
Lutz and Rome, 1996b), so we
did not measure SM moment arms experimentally. However, we did place the
distal attachment site of SM on the tibiofibula of the model, i.e. SM had a
small moment arm (see Fig. 4D).
We directly measured the moment arms of the other muscles about the
flexionextension axis of the knee. This axis points dorsally when the
frog is in the test position (see Fig.
3) and rolls along the distal surface of the femur, i.e. knee
flexionextension is represented as a rolling joint.
Averaged moment arm measurements are shown in
Fig. 4D (solid lines represent
mean ± 1 S.D.). All muscles in the triceps group had the same moment
arm since these muscles inserted into a common tendon. The triceps moment arm
varied little over the range of knee flexionextension (mean of
approximately 1.9 mm). The other muscles all primarily flexed the tibiofibula.
The muscle with the largest flexor moment arm was ST (peak of 3.0 mm; both STd
and STv insert into a common tendon at the knee). GR, ILf and SA had moderate
flexor moment arms. In some frogs, GR and SA were bifunctional with respect to
rotation about the z-axis of the tibiofibula: at extended knee
positions (50° and beyond), they had extensor moment arms and at other
positions, they had flexor moment arms. The bifunctional effects of GR and SA
have been reported previously (Lombard and
Abbot, 1907).
We found that the hindlimb model accurately predicted measured moment arms about the knee joint in experimental frogs (Fig. 4H shows model data). All model moment arms lay within one standard deviation of the experimental means.
Sarcomere lengthjoint angle relationships
Musculotendon complex lengths, muscle fascicle lengths and sarcomere
lengths were measured in experimental frogs at the test position. The mean
values from six frogs are shown in Table
1. The hindlimb model was then placed in the test position, and
the virtual muscles composing the model were assigned the mean values in
Table 1 (for a thorough
description, see Materials and methods).
Because the hindlimb model reproduced the MTC moment arms from the test position, it could be used to predict sarcomere and fascicle lengths at different limb configurations. To test whether the model accurately predicted sarcomere lengths in experimental frogs and accounted for simultaneous changes in hip and knee angles, we measured sarcomere lengths at the starting and take-off positions of a jump in six frogs. We then placed the hindlimb model at these same two positions and determined what the predicted sarcomere lengths would be for each muscle. Data for experimental frogs (± 1 S.D.) and data predicted by the hindlimb model (solid horizontal arrows) are shown in Fig. 6. The arrow tail marks the predicted starting sarcomere length and the arrow head marks the predicted final sarcomere length. For most muscles (11/13), the model predictions lay within ± 1 S.D. of the mean values measured in the group of six frogs (standard deviations ranged from 0.10 to 0.25 µm).
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The sarcomere length predictions for CR, TFL and ILF lay outside ± 1 S.D. of the experimental means. The CR predictions may be affected by the fact that CR is highly pinnate (20-25°) and the CR muscle model did not account for pennation angle changes with MTC length change or rigor contraction. Thus, our predictions of CR sarcomere length at the take-off position were longer (by approximately 8-14 %) than sarcomere lengths measured experimentally. In contrast to pennation angle effects, TFL and ILF predictions may instead be affected by the fact that both muscles have a high in-series connective tissue length/muscle fiber length ratio (2.0-3.0) and these muscle models may not have adequately captured the in-series connective tissue properties (e.g. either the exponential stress/strain relationship or strain at maximum tetanic tension). Thus, model predictions were longer (by approximately 5-12 %) than sarcomere lengths measured experimentally. It will be necessary to perform sensitivity analyses to see how inaccuracies in modeling CR, TFL and ILF sarcomere lengths affect the dynamic behavior of the model and whether better models should be used, e.g. that account for pennation angle changes and muscle-specific connective tissue properties.
In Fig. 6, the classic
isometric force/length curve (for SA;
Gordon et al., 1966) is
overlaid on the sarcomere length measurements to provide a general indication
of where on the curve each of these muscles might operate during jumping. In
general, most of the muscles appeared to operate over a range of sarcomere
lengths where at least 80 % of the maximum contractile force could be
produced. Nonetheless, it is important to stress that, because sarcomere
measurements were performed under static conditions, in the absence of any
tendon recoiling effects and velocity-dependent reductions in contractile
force, the operating ranges reflect static ranges only and might be
substantially different from ranges during jumping.
Architectural properties
We measured the muscle mass, pennation angle and PCSA for each of the 13
proximal muscles of the frog hindlimb in a total of six frogs. The data are
shown in Table 2.
Static muscle functions
We constructed three-dimensional force fields to describe the multi-joint
effects of muscle contraction. Force fields were constructed by placing the
ankle of the model at a range of positions and maximally activating each
musculotendon actuator at each position. The maximum contractile force of the
actuator was calculated on the basis of a simulation of a fixed-end
contraction (see Materials and methods). The static joint moments and the peak
force produced at the ankle were calculated. The peak forces at each limb
position were then plotted in the form of a three-dimensional force field.
Each force field was normalized to the maximum force within the field to
compare the fields produced by muscles with different tension-generating
capabilities (e.g. CR generated four times the force of ADd).
Fig. 7A shows muscle force fields for the three primary hip extensors (SM, top row; GR, middle row; ADd, bottom row). The left column shows a top view. The right column shows a side view. Each vector represents the peak force exerted by the ankle (against a virtual force sensor) at that particular limb position. If the limb were suddenly freed to move, the force vector would represent the initial direction in which the ankle would be accelerated. In three-dimensional space, there will be six forcing functions along which the limb could be accelerated: elevation and depression, caudal and rostral, and medial and lateral. The top view (left column) captures the caudalrostral and mediallateral forcing functions, and the side view (right column) captures the caudalrostral and elevationdepression forcing functions.
|
Examination of the top and side views for the hip extensor force fields in Fig. 7A shows that each muscle was multifunctional in terms of the six forcing functions. SM functions to elevate, caudally direct and medially direct the limb, with the balance of forcing functions changing across limb positions. ADd functions mainly to depress, caudally direct and medially direct the limb, with the balance of functions changing across positions. GR functions mainly to direct the limb caudally and medially and to bring the limb to the horizontal plane. Of these muscles, GR will have the largest effect on accelerating the ankle. GR produced a maximum ankle force of 0.74 N that was 1.37 times greater than that produced by SM (0.54 N) even though GR only produced a maximum contractile force that was 1.07 times greater than that produced by SM. This enhanced effect was because GR produced substantial hip and knee moments while SM produced only a very small knee moment.
Fig. 7B shows muscle force fields for the triceps group of muscles (CR, top row; GL, middle row; TFL, bottom row). These muscles were also multifunctional, and the balance of forcing functions was configuration-dependent. CR functions mainly to direct the limb laterally and rostrally. At elevated positions, CR elevated the limb and at depressed positions CR depressed the limb. GL functions mainly to elevate the limb. At rostral workspace positions, GL functions to direct the limb laterally when the ankle is held at low levels (due to hip adduction) and to direct the limb rostrally when the ankle is held at high levels (due to hip abduction). TFL functions mainly to direct the limb rostrally and laterally, and to elevate it. Because of the sarcomere/limb configuration relationship of TFL, this muscle produced little force at the ankle in the most rostral positions. Of these muscles, CR will have the largest effect on accelerating the limb. CR produced a maximum force of 0.90 N at the ankle compared with 0.39 N for GL and 0.15 N for TFL.
Fig. 8A shows muscle force fields for the two monoarticular hip flexors (ILi, top row; ILe, bottom row). ILi functions mainly to direct and elevate the limb rostrally, with a stronger elevator effect at caudal workspace positions. ILe functions to elevate the limb at mid to caudal positions, to direct the limb rostrally at rostral workspace positions and to depress the limb at elevated positions in the rostral workspace. The depressor function of ILe was due to a shift from producing an abduction moment at the hip to producing an adduction moment in combination with a small internal rotation moment at these rostral positions.
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Fig. 8B shows muscle force fields for two hip adductor muscles (Adv, top row; SA, bottom row). ADv functions mainly to depress the limb and to direct it caudally and medially. SA functions mainly to depress the limb, but as opposed to ADv, to direct it rostrally. Thus, both muscles were multifunctional, and the balance of forcing functions was configuration-dependent. SA was particulary effective at directing the ankle rostrally at rostral (i.e. flexed) limb positions.
Fig. 8C shows muscle force fields for ST (combined activation of STv and STd) and ILf. ST (top row) functions mainly to direct the limb medially. ILf functions mainly to elevate the limb. ILf exhibited an interesting bifunctionality. At the lowest level in the limb's workspace, ILf directed the limb caudally (i.e. acted to extend the ankle away from the body), while at the highest levels ILf directed the limb rostrally (i.e. acted to flex the ankle towards the body).
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Discussion |
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In the present study, we described the multi-joint mechanical effects
resulting from isometric muscle contraction as a force field. We simulated
fixed-end muscle contractions in which each musculotendon actuator making up
the hindlimb model was maximally activated at a number of limb positions (80
in total). The contractile forces at each limb position produced joint moments
that were transmitted through the hindlimb and resulted in a force at the
ankle. This force represents the force that the ankle would exert against an
immovable object, e.g. a torque-force sensor, and points in the initial
direction of ankle acceleration were the object to have been suddenly removed.
Previous studies used direct muscle stimulation in frogs to measure
two-dimensional muscle force fields
(Giszter et al., 1993;
Loeb et al., 2000
). Frog
muscles fatigue quickly because of the high percentage of fast muscle fibers
(Lutz et al., 1998
;
Peters, 1994
), so only a
limited number of positions were tested in those studies (i.e. 15-30). In
addition, the results of using direct electrical stimulation were complicated
by the effects of stimulus spread, by electrode movement that occurs with
repeated contractions and by the selection of the stimulus parameters used to
evoke contraction. By using a model that captured the essential anatomical
properties of real frogs, we avoided these complications and were able to
describe muscle function over a complete state space. The set of force fields
described here provides a useful summary of how each proximal muscle acts to
accelerate the hindlimb from a large range of configurations.
The main finding of using the force field approach was that each hindlimb
muscle was multifunctional with respect to its static, whole-limb effects. We
described muscle function with respect to six forcing functions (see also
Loeb et al., 2000). The six
forcing functions were related to the six (extrinsic) directions in which the
ankle could be accelerated (or forces applied to an object) in
three-dimensional space. In the present study, we described the extrinsic
directions as elevation and depression of the ankle within the gravitational
plane, caudal and rostral movement of the ankle along the long axis of the
frog, and medial and lateral movement of the ankle within the horizontal
plane. At a single limb position, all muscles produced forces that had two
primary vector components (i.e. forcing functions), but most often all three
vector components were substantial. Interestingly, the balance of forcing
functions changed dramatically across the workspace of the hindlimb for nearly
every muscle, e.g. a muscle that primarily directed the limb rostrally at one
position might primarily elevate the limb at a different position. These
configuration-dependent changes in muscle effects are likely to have a great
impact on motor pattern selection and on the utilization of feedback to adjust
motor patterns initiated from different starting configurations (see, for
example, Kargo and Giszter,
2000b
).
The multifunctional effects described above resulted from three fundamental
properties of the hindlimb musculoskeletal system. First, each proximal limb
muscle exhibited at least three moment arms about the hip
(flexionextension, internalexternal rotation,
abductionadduction) and most muscles exhibited a fourth moment arm
about the flexionextension axis of the knee. Importantly, the moment
arms about a single joint axis changed with rotation about that axis and with
rotations about adjacent joint axes (see Figs
3,
4). Thus, the ratio of moment
arms exhibited by a muscle was configuration-dependent. In addition to moment
arm variations, sarcomere lengths and therefore tension-producing capabilities
changed with limb configuration. Thus, the balance and absolute magnitude of
joint moments produced by a muscle were configurationdependent, which
has previously been noted in human studies (Friden and Lieber, 2000;
Pandy, 1999). Finally, the
Jacobian matrix, which determines how joint moments are transmitted through a
multi-jointed limb to a point of contact with the environment or an object, is
configuration-dependent (Tsai,
1999
). Because of this, the forcing functions produced by a
constant set of joint moments will depend on limb configuration
(Mussa-Ivaldi et al.,
1985
).
It is important to stress that we did not quantify the dynamic effects of
muscle contraction in the present study. In theory, if the ankle were a point
mass in a frictionless, gravity-less environment, the vectors comprising each
muscle force field would represent the trajectory along which the ankle would
be accelerated by muscle contraction. However, free limb trajectories are more
complicated because of limb inertia, dynamic mechanical effects arising from
multi-segmental motion, passive forces arising from stretched and shortened
connective tissue structures and sensory feedback effects
(Zajac, 1993;
Crago, 2000
). In addition,
passive mechanical effects arising from motion of the large
astragalus/calcaneus and foot segments in the frog are likely to have a large
effect on the ankle trajectory. Nonetheless, force-field descriptions might
provide some insight into muscle function that is complementary to functions
observed with other experimental methods. For example, measurements of in
vivo muscle length and force trajectories during specific behaviors
showed that muscles function as motors, brakes, springs or struts in the
context of the types of contraction performed, i.e. shortening, lengthening,
lengthening/shortening and isometric contractions respectively (for a review,
see Dickinson et al., 2000
).
In the following, we show how force-field descriptions might relate muscle
function in terms of contraction type during specific behaviors to muscle
function in terms of multijoint limb effects.
Anatomically realistic models can be used to predict the length changes and
contraction types of MTCs during specific behaviors when the kinematics and
motor patterns for these behaviors are known
(Arnold et al., 2000;
Delp et al., 1998
;
Hoy et al., 1990
). For
example, when we moved our model through the swimming kinematic cycle
described by Peters et al.
(1996
) and activated SM at
experimentally determined times (Kamel et
al., 1996
; Gillis and
Biewener, 2000
), the SM musculotendon complex shortened during its
period of activation and therefore functioned as a motor.
SM function can also be described in a more global sense. Specifically, the
ankle force vector produced by SM contraction (small black arrow in
Fig. 9A, lower panel) pointed
in the same direction as the velocity of the ankle during extension (gray
arrow). The dot product between these two vectors at every time point during
the swim cycle indicates that SM is activated mainly when it supports ankle
acceleration (top panel of Fig.
9A; dot product more than 0.75). In contrast to muscle motors,
muscle springs or brakes appear to produce forces at the ankle that are at
180° to the ongoing ankle velocity (dot product less than -0.75). For
example, when we moved the model through a hindlimb wiping cycle and activated
CR at experimental times (Kargo and Giszter,
2000a,
b
), CR produced forces that
initially opposed and then supported ankle acceleration, which is consistent
with a spring-like function (see Fig.
9B). Also, when we moved the model through a defensive kicking
cycle and activated SA at experimental times (D'Avella et al., 2000), SA
produced forces that opposed the entire extension phase, which is consistent
with a braking function (see Fig.
9C).
Finally, some muscles might not be easily classified as motor, spring or
brake. For example, when we moved the model through the jump extension phase
(Kargo et al., 2002) and
activated ST throughout, ST produced forces that acted neither to accelerate
nor to decelerate the body. This was determined by calculating force
directions generated by ST at the tip of the astragalus segment since this is
the center of pressure for much of jumping
(Calow and Alexander, 1973
). We
calculated the dot product of the ST force vectors with the (inverted) ground
reaction force vectors published by Calow and Alexander
(1973
). ST forces were oriented
at 90° to the forces applied to the ground. Therefore, ST was not helping
to drive the astragalus into the ground but instead was acting in less obvious
ways, e.g. redistributing moments or finely tuning the ground reaction
force.
In the above analyses, we were concerned only with the direction of the
ankle force, which depends solely on the geometrical properties of the muscle
or its configuration-dependent set of moment arms. However, the magnitude of
the ankle force and therefore a muscle's relative contribution to ankle
acceleration/deceleration is difficult to predict under dynamic conditions and
during behaviors in which the muscle is submaximally activated. First, the
instantaneous velocity of the activated muscle fibers will limit the ankle
force produced by a muscle contraction. Second, submaximal activation of a
muscle may shift the force/length and force/velocity relationships of
activated fibers (Sandercock and Heckman,
2001; Winters,
2000
; Huijing,
2000
). Third, tendon elasticity will affect the instantaneous
velocity of contracting fibers, the instantaneous sarcomere lengths and the
dynamic force profile (e.g. with recoiling effects;
Marsh, 1999
). Hence,
isometrically measured force fields might help to categorize muscle actions in
the context of multi-joint movements, but such an approach does not capture
exactly how individual muscles will participate under dynamic conditions. For
example, Giszter and Kargo
(2001
) found that, when
hindlimb models were driven with isometrically measured force fields, model
trajectories deviated substantially from experimental trajectories especially
during periods of limb deceleration. Thus, eccentric contractions, secondary
and tertiary muscle properties and sensory feedback will have significant
effects on the dynamic control of limb behaviors.
In summary, in the present study, we measured the anatomical properties of the proximal hindlimb muscles in Rana pipiens and incorporated these properties into a realistic biomechanical model. We used the model to describe the diversity of hindlimb muscle functions in terms of isometric force fields. The model forms a structural foundation for adding other subsystems (e.g. neural), enhancing subsystem complexity (e.g. more sophisticated muscle models) and testing motor control ideas through forward dynamic simulation.
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Acknowledgments |
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