Vol. 92, Issue 1, 129-134, January 2002
In vivo determination of fascicle curvature in contracting
human skeletal muscles
Tadashi
Muramatsu1,
Tetsuro
Muraoka2,
Yasuo
Kawakami2,
Akira
Shibayama2, and
Tetsuo
Fukunaga2
1 Institute of Physical Education, Keio University, Kohoku,
Yokohama 223-8521; and 2 Laboratory of Sports Sciences,
Department of Life Sciences, Graduate School of Arts and Sciences,
University of Tokyo, Meguro, Tokyo 153-8902, Japan
 |
ABSTRACT |
Fascicle curvature of human medial gastrocnemius muscle (MG) was
determined in vivo by ultrasonography during isometric contractions at
three (distal, central, and proximal) locations (n = 7)
and at three ankle angles (n = 7). The curvature
significantly (P < 0.05) increased from rest to
maximum voluntary contraction (MVC) (0.4-5.2 m
1). In
addition, the curvature at MVC became larger in the order dorsiflexed,
neutral, plantar flexed (P < 0.05). Thus both
contraction levels and muscle length affected the curvature.
Intramuscular differences in neither the curvature nor the fascicle
length were found. The direction of curving was consistent along the
muscle: fascicles were concave in the proximal side. Fascicle length
estimated from the pennation angle and muscle thickness, under the
assumption that the fascicle was straight, was underestimated by
~6%. In addition, the curvature was significantly correlated to
pennation angle and muscle thickness. These findings are particularly
important for understanding the mechanical functions of human skeletal
muscle in vivo.
medial gastrocnemius muscle; intramuscular variability
| |
INTRODUCTION |
THE ARCHITECTURE OF THE
SKELETAL muscle has been defined as the arrangement of the
fascicle (bundle of the muscle fibers) within the muscle (10, 12,
18). Knowledge of the architecture of the skeletal muscle is
essential to understanding its function (force and excursion ability)
(10, 18). Therefore, many studies have dealt with the
architecture in animals (7, 23, 35), human cadavers
(6, 34), and human in vivo (3, 13, 21, 25,
28). In many cases, muscle architecture is characterized by the
length (fascicle length), the angle (pennation angle) with respect to
the tendinous tissues, and the thickness of the muscle (3, 13,
21, 25, 28).
Another parameter that is necessary for better understanding the
architecture is the curvature (reciprocal of the radius of a circle) of
the fascicle (26, 32). Curved fascicles could produce
pressure toward the concave side (2, 5, 30, 32). It has
been reported, from the prediction using a muscle model, that curving
of the fascicle increases the intramuscular pressure and therefore
affects blood flow (30, 32). In addition, Sejersted et al.
(30) reported that fascicle curving reduces the force transmitted to the bone and that fascicle stress could be calculated if
the curvature, intramuscular pressure, and recording depth were known.
Thus it is clear that information on the degree and distribution of
fascicle curving is essential to understanding skeletal muscle
function. As for the degree of fascicle curving, no study has reported
experimental data on humans, although several studies have
qualitatively observed curving in humans in vivo (3, 8, 13,
21). Several studies also observed that the degree of curving
increased when muscle length decreased (13, 35). As for
the distribution of fascicle curving, Otten (26) predicted
that fascicles in a unipennate muscle curved in a way that induced
bulging of the whole muscle, whereas Van Leeuwen and Spoor
(32) predicted, as for the human MG, that fascicles curved
in the same direction with the same curvature along the muscle.
However, no study has reported experimental data on humans in vivo.
Intersubject variability has been reported for the architectural
parameters. Between muscle thickness and pennation angle, consistent
relationships have been reported (11, 12, 14, 15).
Similarly, the degree of fascicle curving could be related to other
architectural parameters, considering that curving has been related to
pennation angle (13).
The purposes of the present study were 1) to estimate the
fascicle curvature in vivo at different contraction levels (at rest and
during contraction) and at different muscle lengths, 2) to examine the intramuscular distribution of the fascicle curvature, and
3) to examine the relationships between fascicle curvature and the other architectural parameters. Our hypotheses were that 1) the fascicle curvature is increased by contraction and by
decrease in muscle length, 2) the direction and the degree
of curving are uniform along the muscle, and 3) the fascicle
curvature is related to the other architectural parameters.
| |
METHODS |
Subjects
The subjects of this study were 11 healthy men [age 26.5 ± 3.8 yr, height 174.6 ± 7.2 cm, weight 71.9 ± 8.3 kg
(means ± SD)]. The purposes and procedures were explained to the
subjects before their consent to participate in the study was obtained.
Joint Position Settings and Torque Measurements
We used the same techniques as described in our laboratory's
recent study (24). Briefly, we used an electric myometer
(model Myoret RZ-450, Asics, Tokyo, Japan) to fix the ankle joint and to measure plantar flexion torque. Each subject lay prone on a bed,
with the left foot fixed to the myometer (Fig.
1). After a warm-up session, the subjects
were instructed to exert isometric torque from relaxation to MVC with a
visual aid of the developed torque on an oscilloscope with a ramp
increase in torque at 10% MVC/s. Average values over three
measurements were adopted.
View larger version (21K):
[in this window]
[in a new window]
|
Fig. 1.
Schematic illustration of the human medial gastrocnemius
muscle (MG). Measurement sites (distal, central, proximal) are also
shown.
|
|
Test 1 (three regions).
The ankle was set at neutral anatomic position, with the sole of the
foot at 90° to the tibia. The measurement was repeated three times
for each of the three parts (see Ultrasonography). We
executed this test for seven subjects.
Test 2 (three angles).
The ankle was set at 120 (plantar flexed), 90, and 75° (dorsiflexed).
The measurement was repeated three times for each of the three ankle
angles at the central part of the medial gastrocnemius (MG). We also
executed this test for seven subjects. That is, three subjects
participated in both tests, four subjects participated only in
test 1, and the other four subjects participated only in
test 2.
For each of test 1 and 2, at least 3 min of rest
were taken between trials. We adopted average values over three measurements.
Ultrasonography
The muscle tested in this study was human MG. The technique used
was also described in our laboratory's previous study
(24). Briefly, we used the ultrasonic apparatus (model
SSD-2000, Aloka, Tokyo, Japan) with an electronic linear array probe of
7.5-MHz wave frequency. The precision and linearity of the image using ultrasonography have been confirmed by Kawakami et al.
(11), who compared the distance between pins struck on an
acoustic standoff and the distance between the pins in the
reconstructed image. The probe was longitudinally attached to the
dermal surface by an adhesive tape, which restrained the probe from
sliding (4, 16, 22), over the mediolateral center of MG.
To obtain architectural parameters from different portions in MG, we
scanned distal, central, and proximal parts of MG by ultrasonography
(Fig. 1). In each part, we could find a fascicle whose echo was clear
from the superficial to the deep aponeuroses throughout the contraction
(Fig. 2). Therefore, it should be
reasonable to suppose that the plane of the ultrasonogram is parallel
to the fascicle (13). We randomized the order of experiments among each part. In each trial, the targeted fascicle moved
slowly enough for the investigator to scan successfully. The ultrasound
images were transferred to a personal computer (model Powerbook G3,
Apple, Tokyo, Japan) at 30 Hz for obtaining architectural parameters
(see below).
View larger version (91K):
[in this window]
[in a new window]
|
Fig. 2.
Ultrasonographic images of longitudinal sections of MG muscle at
distal, central, and proximal regions. Top: at rest;
bottom: at maximal voluntary contraction (MVC).
|
|
Estimation of Curvature and Other Architectural Parameters
On each of the ultrasonographic images, four architectural
parameters (
d, s, H, T) defined in Fig.
3 were measured three times, and averaged
values were adopted for further analyses. From d, s, and
H, the curvature of the fascicle was determined (see below).
The fascicle was assumed to be the arc (29, 32, 33) with
radius r (the curvature to be 1/r). Then the
following should be formed (for the meaning of each symbol, see Fig. 3)
We could solve these equations for the curvature
(1/r)
Then, to ascertain the validity of the value of the curvature,
we compared the fascicle length (Lfarc), calculated from
the curvature, with the actual fascicle length (Lftrc)
measured on the ultrasonographic image. Lfarc corresponds
to the arc P1P2 (Fig. 3) and is calculated by the following.
(Note that the unit of d and s is radian in this
equation.)
View larger version (63K):
[in this window]
[in a new window]
|
Fig. 3.
Arc P1P2 corresponds to the fascicle. L1 corresponds to
the tangent of the deep aponeurosis at P1, which is the intersection
made by the fascicle and the deep aponeurosis. L2 corresponds to the
tangent of the fascicle at P1. L3 corresponds to the tangent of the
fascicle at P2, which is the intersection made by the fascicle and the
superficial aponeurosis. r, Radius of circle; H,
length of perpendicular dropped from P2 to L1; T, muscle
thickness (minimum distance between P2 and deep aponeurosis); d,
angle made by L1 and L2; s, angle made by L1 and L3.
|
|
We measured Lftrc by tracing along its path with the
curvature taken into consideration on the ultrasonographic images
(13, 20). In addition, fascicle length estimated from the
following equation (Lfhyp), which had often been used in
previous studies (12, 15, 17, 19), was calculated
This method of estimating the fascicle length was based on
the assumption that the fascicle was not curved. We also compared Lfhyp with Lftrc to understand, although
indirectly, the effect of the fascicle curvature on the estimation of
the fascicle length. The comparison among Lfarc,
Lfhyp, and Lftrc was made only at the central
part of MG. Note that H and T in Fig. 3 are
different. In the present study, we did not use the assumption that the
aponeuroses were straight, which could induce errors in estimating
architectural parameters.
Reproducibility
We evaluated the reproducibility of calculating the fascicle
curvature, as for test 1, through three procedures
(24) on the basis of a coefficient of variation (SD/mean)
(9, 25): 1) interday reproducibility, which was
tested for three subjects on two separate occasions, was on average
9.8%; 2) reproducibility of three trials for all subjects
was on average 9.7%; and 3) reproducibility of measuring
from the same ultrasonic image (images for all trials were digitized
three times) was on average 6.1%.
Statistics
Values are presented as means ± SD. A two-way ANOVA with
repeated measures was used to analyze the effects of 1)
contraction levels (rest vs. MVC) and locations (distal, central, and
proximal) on curvature (1/r) and fascicle length
(Lftrc); 2) contraction levels (rest vs. MVC)
and methods (Lfarc, Lfhyp, and
Lftrc) on the estimated fascicle length; and 3)
contraction levels and ankle angles (120, 90, and 75°) on curvature.
Significant differences among means were detected by using
Tukey-Kramer's post hoc tests. To test the significance of the
relationship of 1) the fascicle curvature and pennation
angle, 2) the fascicle curvature and muscle thickness, and
3) the fascicle length between Lftrc and the
other methods (Lfarc and Lfhyp), Pearson's
correlation coefficient was calculated. A P < 0.05 level of confidence was set for all analyses.
| |
RESULTS |
Test 1
The calculated curvature and measured Lftrc at
three parts at rest and MVC are shown in Table
1. For each of the curvature and
Lftrc, the effect of the contraction level was significant, whereas the effect of the location was not significant. The direction of curving was consistent along the muscle. That is, fascicles were
concave on the proximal side. We found significant correlation between
1) the curvature and pennation angle at distal and central regions during MVC (Fig. 4),
2) the curvature and muscle thickness at distal and central
regions during MVC (Fig. 5), and
3) Lftrc and each of Lfarc and
Lfhyp [Fig. 6, A
(at rest) and B (at MVC)]. Lfhyp was
significantly smaller than Lftrc, whereas significant difference was not found between Lfarc and
Lftrc. At MVC, Lfhyp was ~6% smaller than
Lftrc.
View larger version (15K):
[in this window]
[in a new window]
|
Fig. 4.
Relationship between pennation angle (d) and fascicle
curvature at 3 regions ( and  , distal;
 and  , central;  and
 , proximal). Open symbols indicate plots at MVC, and
solid symbols indicate plots at rest.
|
|
View larger version (13K):
[in this window]
[in a new window]
|
Fig. 5.
Relationship between muscle thickness and fascicle
curvature at 3 regions ( and , distal;
and , central; and
, proximal). Open symbols indicate plots at MVC, and
solid symbols indicate plots at rest.
|
|
View larger version (16K):
[in this window]
[in a new window]
|
Fig. 6.
Calculated fascicle length plotted against measured length
(Lftrc) at rest (A) and at MVC (B).
 , Length calculated under the assumption that the
fascicles were curved (Lfarc); , length
calculated under the assumption that the fascicles were straight
(Lfhyp).
|
|
Test 2
The calculated curvature at three ankle angles at rest and
MVC is shown in Table 2. The effect of
the difference of the ankle angle was significant. At MVC, significant
difference among three angles was found: the value was in the order
120° > 90° > 75°.
| |
DISCUSSION |
This is, to the best of our knowledge, the first study that
quantitatively showed the fascicle curvature of the human skeletal muscle in vivo. We showed that 1) the curvature was
increased by contractions and by decrease in muscle length,
2) intramuscular variability was observed neither in the
degree nor the direction of curving, and 3) the curvature
was correlated to muscle thickness and pennation angle.
In the present study, to confirm the validity of the estimation
of the curvature, the fascicle length calculated from the curvature (Lfarc) was compared with the measured fascicle
length (Lftrc). There was remarkable conformity (Fig. 6).
In addition, we superimposed the arc, drawn from the calculated
curvature, onto the fascicle echo of the ultrasonographic image and
found no visible deviation from each other. From these observations, we
believe that the procedure to estimate the curvature was valid. Fascicle length (Lfhyp) has sometimes been estimated from
the pennation angle and muscle thickness (12, 15, 17, 19). The result of the present study suggests that this method significantly produces errors in estimating the fascicle length if the fascicle curves. The average underestimation was ~6% (Fig. 6B),
which might not seem to be critically large. However, the error would
be substantially large in muscles with long fascicles such as vastus
lateralis (8) and triceps brachii (12).
Fascicles were almost straight at rest, which agrees with the report of
Narici et al. (25), who described the MG of human cadavers. On the other hand, at MVC, we observed substantial curving of
the fascicles as reported by several previous studies (13, 21). Although quantitative data of the curvature has not been reported as for the human skeletal muscles, assumptions have been made:
Van Leeuwen and Spoor (32) assumed the curvature for the human MG to be 5.1 m1, and Sejersted et al.
(30) assumed it for the human vastus medialis to be
5-6.7 m1. These assumed values of the curvature were
on the same order with those of the present study.
The effects of muscle length (ankle angle) on the curvature was
substantial (Table 2), which supports the observation of Kawakami et
al. (13) and Zuurbier and Huijing (35), who
reported that the fiber curvature of rat gastrocnemius resulted in an
underestimation of fiber length of 2% when the muscle length was above
optimum and 5% when the muscle length was below optimum length. In
other words, fiber curvature was larger when the muscle was shorter, which agrees with the present results. The measured fascicle length at
three ankle angles at MVC was 28.7 ± 4.3, 33.1 ± 6.0, and
43.7 ± 5.6 mm, for 120, 90, and 75°, respectively. That is,
fascicles were longer when muscle is longer. Therefore, we could say
that the curvature was larger at shorter fascicle length at MVC. At rest, muscle length did not significantly affect the curvature (Table
2). These results suggest that the curvature is affected by both
contraction levels and fascicle length.
In the present study, fascicles curved in the same direction with
similar curvature along the muscle. This result coincided with that of
Van Leeuwen and Spoor (32) on the human MG but was
contrary to the findings of Otten (26), who predicted that fascicles in a unipennate muscle were curved in a way that induced bulging of the muscle; the direction of curving at distal part was
different from that at proximal part. The former study reported that
the compressive force produced by soleus and tibia onto the MG
tendinous tissue should play an important role in arranging the
tendinous tissue and therefore fascicles within the muscle, whereas the
latter study did not assume such a compressive force. Muscles in
vivo are necessarily compressed by surrounding tissues that
should affect the architecture of the muscle, which would be one reason
that the architecture in vitro is not always applicable to that in vivo
(27).
Curving of fascicles has been related to the intramuscular
pressure (1, 31). From the law of Laplace, the difference in pressure between the concave and the convex side of the fascicle has
been supposed to be proportional to the curvature and stress of the
fascicle (30). That is, when the tensed fascicle is
curving, the pressure would be generated in the concave side (2,
5, 32). For example, Ameredes and Provenzano (1)
assumed that the muscle bulging [onionlike configuration
(30)] during contractions contributed to the high
pressure in the central region in the canine gastrocnemius muscle. In
the present study, the fascicles were concave in the proximal side.
Therefore, the pressure could be higher in the proximal region in human
MG during contractions, although we could not confirm it because there
are no published data available concerning the distribution of human
gastrocnemius intramuscular pressure.
There were tendencies that curvature was proportional to muscle
thickness and pennation angle (Figs. 4 and 5), which supports the
prediction of Kawakami et al. (13). Styf et al.
(31) reported that the intramuscular pressure would
increase if the pennation angle increased. In the present study,
subjects with larger pennation angle generally had larger curvature,
which also convinces us to assume that subjects with larger pennation
angle had larger intramuscular pressure.
In conclusion, we showed that the curvature of the fascicle could be
calculated from ultrasonographic images in vivo and that the curvature
is affected by both contraction levels and fascicle length. In
addition, the degree and the direction of curving were uniformly
distributed. These findings are particularly important for
understanding the mechanical functions of human skeletal muscle in vivo
and for more accurate muscle modeling in future studies.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: T. Muramatsu, 4-14-4 Setagaya, Setagaya-ku, Tokyo 154-0017, Japan (E-mail: mura{at}hc.cc.keio.ac.jp).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 15 June 2001; accepted in final form 27 August 2001.
| |
REFERENCES |
1.
Ameredes, BT,
and
Provenzano MA.
Regional intramuscular pressure development and fatigue in the canine gastrocnemius muscle in situ.
J Appl Physiol
83:
1867-1876,
1997[Abstract/Free Full Text].
2.
Angelillo, M,
Boriek AM,
Rodarte JR,
and
Wilson TA.
Shape of the canine diaphragm.
J Appl Physiol
89:
15-20,
2000[Abstract/Free Full Text].
3.
Fukunaga, T,
Ichinose Y,
Ito M,
Kawakami Y,
and
Fukashiro S.
Determination of fascicle length and pennation in a contracting human muscle in vivo.
J Appl Physiol
82:
354-358,
1997[Abstract/Free Full Text].
4.
Fukunaga, T,
Ito M,
Ichinose Y,
Kuno S,
Kawakami Y,
and
Fukashiro S.
Tendinous movement of a human muscle during voluntary contractions determined by real-time ultrasonography.
J Appl Physiol
81:
1430-1433,
1996[Abstract/Free Full Text].
5.
Hill, AV.
The pressure developed in muscle during contraction.
J Physiol (Lond)
107:
518-526,
1948.
6.
Huijing, PA.
Architecture of the human gastrocnemius muscle and some functional consequences.
Acta Anat (Basel)
123:
101-107,
1985[ISI][Medline].
7.
Huijing, PA,
van Lookeren Campagne AA,
and
Koper JF.
Muscle architecture and fibre characteristics of rat gastrocnemius and semimembranosus muscles during isometric contractions.
Acta Anat (Basel)
135:
46-52,
1989[ISI][Medline].
8.
Ichinose, Y,
Kawakami Y,
Ito M,
and
Fukunaga T.
Estimation of active force-length characteristics of human vastus lateralis muscle.
Acta Anat (Basel)
159:
78-83,
1997[ISI][Medline].
9.
Ito, M,
Kawakami Y,
Ichinose Y,
Fukashiro S,
and
Fukunaga T.
Nonisometric behavior of fascicles during isometric contractions of a human muscle.
J Appl Physiol
85:
1230-1235,
1998[Abstract/Free Full Text].
10.
Jacobson, MD,
Raab R,
Fazeli BM,
Abrams RA,
Botte MJ,
and
Lieber RL.
Architectural design of the human intrinsic hand muscles.
J Hand Surg
17A:
804-809,
1992[Medline].
11.
Kawakami, Y,
Abe T,
and
Fukunaga T.
Muscle-fiber pennation angles are greater in hypertrophied than in normal muscles.
J Appl Physiol
74:
2740-2744,
1993[Abstract/Free Full Text].
12.
Kawakami, Y,
Abe T,
Kuno S,
and
Fukunaga T.
Training-induced changes in muscle architecture and specific tension.
Eur J Appl Physiol
72:
37-43,
1995[ISI].
13.
Kawakami, Y,
Ichinose Y,
and
Fukunaga T.
Architectural and functional features of human triceps surae muscles during contraction.
J Appl Physiol
85:
398-404,
1998[Abstract/Free Full Text].
14.
Kawakami, Y,
Ichinose Y,
Kubo K,
Ito M,
Imai M,
and
Fukunaga T.
Architecture of contracting human muscles and its functional significance.
J Appl Biomech
16:
88-98,
2000.
15.
Kearns, CF,
Abe T,
and
Brechue WF.
Muscle enlargement in sumo wrestlers includes increased muscle fascicle length.
Eur J Appl Physiol
83:
289-296,
2000[Medline].
16.
Kubo, K,
Kawakami Y,
and
Fukunaga T.
Influence of elastic properties of tendon structures on jump performance in humans.
J Appl Physiol
87:
2090-2096,
1999[Abstract/Free Full Text].
17.
Kumagai, K,
Abe T,
Brechue WF,
Ryushi T,
Takano S,
and
Mizuno M.
Sprint performance is related to muscle fascicle length in male 100-m sprinters.
J Appl Physiol
88:
811-816,
2000[Abstract/Free Full Text].
18.
Lieber, RL,
and
Friden J.
Clinical significance of skeletal muscle architecture.
Clin Orthop
383:
140-151,
2001.
19.
Maganaris, CN,
and
Baltzopoulos V.
Predictability of in vivo changes in pennation angle of human tibialis anterior muscle from rest to maximum isometric dorsiflexion.
Eur J Appl Physiol
79:
294-297,
1999.
20.
Maganaris, CN,
Baltzopoulos V,
Ball D,
and
Sargeant AJ.
In vivo specific tension of human skeletal muscle.
J Appl Physiol
90:
865-872,
2001[Abstract/Free Full Text].
21.
Maganaris, CN,
Baltzopoulos V,
and
Sargeant AJ.
In vivo measurements of the triceps surae complex architecture in man: implications for muscle function.
J Physiol (Lond)
512:
603-614,
1998[Abstract/Free Full Text].
22.
Maganaris, CN,
and
Paul JP.
Load-elongation characteristics of in vivo human tendon and aponeurosis.
J Exp Biol
203:
751-756,
2000[Abstract].
23.
Muhl, Z.
Active length-tension relation and the effect of muscle pinnation on fiber lengthening.
J Morphol
173:
285-292,
1982[ISI][Medline].
24.
Muramatsu, T,
Muraoka T,
Takeshita D,
Kawakami Y,
Hirano Y,
and
Fukunaga T.
Mechanical properties of tendon and aponeurosis of human gastrocnemius muscle in vivo.
J Appl Physiol
90:
1671-1678,
2001[Abstract/Free Full Text].
25.
Narici, MV,
Binzoni T,
Hiltbrand E,
Fasel J,
Terrier F,
and
Cerretelli P.
In vivo human gastrocnemius architecture with changing joint angle at rest and during graded isometric contraction.
J Physiol (Lond)
496:
287-297,
1996[ISI][Medline].
26.
Otten, E.
Concepts and models of functional architecture in skeletal muscle.
Exerc Sport Sci Rev
16:
89-137,
1988[Medline].
27.
Otten, E,
and
Hulliger M.
A finite-elements approach to the study of functional architecture in skeletal muscle.
Zoology
98:
233-242,
1994/1995.
28.
Rutherford, OM,
and
Jones DA.
Measurement of fibre pennation using ultrasound in the human quadriceps in vivo.
Eur J Appl Physiol
65:
433-437,
1992.
29.
Sejersted, OM,
and
Hargens AR.
Intramuscular pressures for monitoring different tasks and muscle conditions.
Adv Exp Med Biol
384:
339-350,
1995[Medline].
30.
Sejersted, OM,
Hargens AR,
Kardel KR,
Blom P,
Jensen O,
and
Hermansen L.
Intramuscular fluid pressure during isometric contraction of human skeletal muscle.
J Appl Physiol
56:
287-295,
1984[Abstract/Free Full Text].
31.
Styf, J,
Ballard R,
Aratow M,
Crenshaw A,
Watenpaugh D,
and
Hargens AR.
Intramuscular pressure and torque during isometric, concentric and eccentric muscular activity.
Scand J Med Sci Sports
5:
291-296,
1995[Medline].
32.
Van Leeuwen, JL,
and
Spoor CW.
Modelling mechanically stable muscle architectures.
Philos Trans R Soc Lond B Biol Sci
336:
275-292,
1992[ISI][Medline].
33.
Van Leeuwen, JL,
and
Spoor CW.
A two dimensional model for the prediction of muscle shape and intramuscular pressure.
Eur J Morphol
34:
25-30,
1996[Medline].
34.
Wickiwicz, TL,
Roy RR,
Powell PL,
and
Edgerton VR.
Muscle architecture of the human lower limb.
Clin Orthop
179:
275-283,
1983.
35.
Zuurbier, CJ,
and
Huijing PA.
Changes in geometry of actively shortening unipennate rat gastrocnemius muscle.
J Morphol
218:
167-180,
1993[ISI][Medline].
J APPL PHYSIOL 92(1):129-134
8750-7587/02 $5.00
Copyright © 2002 the American Physiological Society
TOP
ABSTRACT
INTRODUCTION
![1/r=‖cos 2<IT>&thgr;</IT>d<IT>−</IT>cos 2<IT>&thgr;</IT>s<IT>‖/</IT>[2<IT>H·</IT>(cos <IT>&thgr;</IT>d<IT>+</IT>cos <IT>&thgr;</IT>s)]](/content/vol92/issue1/fulltext/129/img006.gif)
