## SUMMARY

The use of sarcomere length to normalize fiber length in architectural
studies is commonly practiced but has not been explicitly validated. Using
mouse hindlimb muscles as a model system, ankle joints were intentionally set
to angles ranging from 30° to 150° and their muscles fixed. Tibialis
anterior (TA), extensor digitorum longus (EDL) and soleus muscles were removed
and their raw fiber length measured. Sarcomere length was then measured for
each fiber length sample and fiber length was normalized to a standard
sarcomere length. As expected, raw fiber length was dependent on tibiotarsal
angle (*P*<0.0005 for all muscles, *r*^{2} range
0.22–0.61), while sarcomere length normalization eliminated the
joint-angle dependent variation in fiber length (*P*>0.24,
*r*^{2} range 0.001–0.028). Similarly, one-way ANOVA
revealed no significant differences in normalized fiber length among ankle
angles for any of the three muscles (*P*>0.1), regardless of animal
size. To determine the resolution of the method, power calculations were
performed. For all muscles studied, there was >90% chance of detecting a
15% fiber length difference among muscles and >60% chance of detecting
fiber length differences as small as 10%. We thus conclude that the use of
sarcomere length normalization in architectural studies permits resolution of
fiber length variations of 15% and may even be effective at resolving 10%
fiber length variations.

- sarcomere compliance
- muscle architecture
- modeling
- surgery
- fiber length
- physiological cross-sectional area

## Introduction

An accurate knowledge of skeletal muscle architecture provides an
understanding of muscle design and performance
(Gans, 1982;
Sacks and Roy, 1982), the
ability to plan surgical procedures
(Fridén and Lieber,
2001; Fridén and Lieber,
2002), and elucidates motor control strategies utilized by the
neuromuscular system (Walmsley et al.,
1978; Walmsley and Proske,
1981). Undoubtedly, the two most important parameters obtained
from architectural studies are physiological cross-sectional area (PCSA in
cm^{2}) and fiber length (*L*_{f} in cm). This is
because PCSA is a good predictor of maximum tetanic tension
(Powell et al., 1984) while
*L*_{f} predicts maximum velocity and excursion
(Bodine et al., 1987), a
muscle's most important functional properties.

Physiological cross-sectional area is typically calculated from muscle
specimens (see, for example, Lieber et
al., 1992), using the equation:
(1)
where *M*_{m} is muscle mass (g), ρ is muscle density andρ
is fiber pennation angle. Alternatively, PCSA may be calculated from
imaging studies (see, for example,
Fukunaga et al., 1992), using
the equation:
(2)
where muscle volume (*V*_{m} in cm^{3}) is quantified
from reconstructed images. In either case, estimates of PCSA rely strongly and
nonlinearly on measurements of *L*_{f} (because it is in the
denominator of both Eq. 1 and 2) and thus it is clear that reliable
architectural information depends on obtaining accurate *L*_{f}
values. One difficulty in obtaining accurate *L*_{f} values, is
compensating for the natural fiber length variation that occurs simply because
muscles are fixed at different joint angles, or because images are obtained
with limbs at different joint angles. It is nearly impossible to set joint
angles, precisely due to the masking effects of soft tissues that overlie
bones (Fig. 1A). Another
difficulty is the natural variation in fiber length that occurs within
muscles, making it difficult to compare either between muscles or between
treatments.

The procedure typically used to standardize *L*_{f} values
and compensate for such variation is to measure sarcomere length within a
specimen, select a standard sarcomere length, and then normalize all raw fiber
lengths using the equation:
(3)
where *L*_{f} is the normalized fiber length (cm),
*L*_{f}′ is the experimentally measured (raw) fiber
length (cm), *L*_{s} is a standard sarcomere length (μm) and
*L*_{s}′ is the experimentally measured sarcomere length
at the experimentally measured fiber length (μm). This commonly used
equation assumes that sarcomere length and fiber length are linearly related.
However, other sources of compliance such as *Z*-disks, myofilaments,
tendons and the muscle–tendon junction itself, may compromise this
assumption, or the *L*_{f} variation may be natural and not
associated with sarcomere length. As a result, sarcomere length could change
nonlinearly with fiber length. In addition, fast and slow muscle fibers, with
varying *Z*-disk widths and muscle–tendon geometry
(Eisenberg, 1983), could
differentially affect the validity of this assumption. If any of these
assumptions are invalid, it calls into question the use of sarcomere length
for normalizing *L*_{f} values in architectural studies. These
assumptions have not been explicitly tested. Thus, the purpose of this study
was to intentionally set identical muscles to different lengths and use the
normalization method shown in Eq. 3 to test whether *L*_{f}
could be accurately normalized with high resolution.

## Materials and methods

### Hindlimb fixation

Nine adult male mice (strain, 129Sv; mass range, 25–30 g) were
weighed and killed by cervical dislocation. Both lower extremities were
disarticulated at the hip, skinned and the feet secured to cork with Vetrap
bandaging tape (3M Corporation, St Paul, MN, USA) to maintain alignment of the
tarsal and digital bones. Limbs were divided into five groups
(*N*=3–4/group) based on the nominal tibiotarsal angles at which
their ankles were to be fixed: 30°, 60°, 90°, 120° or
150°. The limbs, along with the cork splints, were pinned onto cork to
maintain the knee at a nominal angle of 90° of flexion and the ankle at
the specified tibiotarsal angle (Fig.
1A). Limb muscles were then fixed in 10% buffered formalin for 48
h and rinsed in 1× phosphate buffered saline (PBS) for 72 h (3 rinses×
24 h/rinse). To determine the actual tibiotarsal and tibiofemoral
angles of fixation, lateral radiographs were obtained of the limbs (X-ray
System 43805 N; Faxitron X-ray, Palo Alto, CA, USA) at settings of 30 kV for a
45 s exposure, and joint angles digitized (ImageJ, version 1.33, NIH).
Tibiotarsal angle was defined as the included angle between the axis of the
metatarsals and the line from the center of the distal end of the tibia
through the center of the tibial plateau, while tibiofemoral angle was defined
as the included angle between the long axis of the femur and the line through
the tibia as just described (Fig.
1B). Repeatability of digitization was ±0.41°
(±s.d., *N*=3 repeat measures of 16
tibiotarsal angles and 16 tibiofemoral angles).

### Fiber bundle dissection

Tibialis anterior (TA), extensor digitorum longus (EDL), and soleus muscles
were removed from the limbs, digested in 15% H_{2}SO_{4} for
30 min to facilitate fiber bundle isolation and stored in 1× PBS at room
temperature until fiber bundle dissection. Small fiber bundles (5–50
fibers) were dissected from the whole muscle in 1× PBS under a
dissecting microscope using 8–20× magnification
(Sacks and Roy, 1982). Special
care was taken to remove the entire bundle, from tendon to tendon. At least
three bundles were isolated from different regions of each soleus and TA
muscle while four bundles were isolated from the EDL, one from each muscle
belly (Chleboun et al., 1997).
Fiber bundle length was measured under the microscope with a digital caliper
to the nearest 0.01 mm. Only fibers that remained straight after fixation were
measured. This criterion excluded soleus muscle bundles fixed at ankle joint
angles of 150° since, at this angle, fibers had a distinctly wavy
appearance and length could not be reliably measured with calipers. Sarcomere
length was measured at three different points along each mounted bundle using
laser diffraction as previously described
(Lieber et al., 1990). Fibers
were used only if at least two useable sarcomere lengths were obtained. Such a
criterion was necessary to preclude the possibility of normalizing sarcomere
length in damaged muscles, where severe nonhomogeneities can exist
(Talbot and Morgan, 1996).
This criterion excluded 17 fibers from soleus muscles. In general, diffraction
pattern quality from the soleus was poor compared to TA or EDL, as has been
previously observed (Burkholder et al.,
1994).

Raw fiber lengths were normalized to a standard sarcomere length of 2.5μ
m (assumed to be mouse muscle optimal sarcomere length; see
Walker and Schrodt, 1973)
using Eq. 3. Two separate statistical approaches were used to determine the
quantitative effect of sarcomere length normalization. First, raw fiber length
was regressed against ankle and knee angle using simple and multiple linear
regression to determine whether fiber length varied with either ankle or knee
joint angle. Then, the same procedure was performed on normalized fiber length
to determine whether this variation was eliminated by the sarcomere length
normalization procedure. Next, to quantify the resolution of the normalization
procedure, raw and normalized fiber lengths were compared across groups by
one-way analysis of variance (ANOVA). Attempted and actual joint angles were
compared by one-sampled *t*-test. Resolution of the method was defined
based on the statistical power (1-β) achieved using standard statistical
power equations (Sokal and Braumann,
1980). This is because the normalization procedure must still have
adequate resolution in experimental studies, even after sarcomere length is
accounted for. For all tests, critical *P*-value (α) was set to
0.05 and values are reported in the text as mean ±
s.e.m. unless otherwise noted.

## Results and discussion

As anticipated, raw fiber bundle length (referred to as raw fiber length)
was strongly dependent on tibiotarsal angle
(Fig. 2A–C). As the
tibiotarsal angle increased (i.e, as the foot was plantarflexed), TA and EDL
fiber length significantly increased (Fig.
2A,B) while soleus fiber length decreased
(Fig. 2C; *P*<0.0005
for all muscles, *r*^{2} range from 0.22–0.61), although
the magnitude of the change was muscle-dependent, based on the different fiber
length–moment arm relationships of each muscle–joint system
(Lieber, 1997).

Variation between intended and actual joint angles was significant for six
of the 14 groups of muscles within each intended joint angle
(*P*<0.05). The average joint angle variability around the intended
angle was 7.9±1.4° for the knee joint and 4.4±0.8° for
the ankle joint. The fact that the variability for the knee was greater
probably reflected the greater volume of proximal muscle mass on the femur
(Fig. 1A).

Multiple regression analysis revealed that, for the EDL only, raw fiber
length was significantly correlated with knee joint angle
(*P*<0.05). This is reasonable based on the fact that the EDL is the
only one of the three muscles studied that crosses the knee joint. In spite of
this significant correlation, inclusion of knee joint angle in the regression
relationship only increased the coefficient of determination marginally (from
0.26 to 0.31). One-way ANOVA results supported regression results in that raw
fiber length was significantly different across ankle joint angles for all
three muscles (*P*<0.02).

Sarcomere length normalization effectively eliminated the joint-angle
dependent variation in fiber length, evidenced by both linear regression and
one-way ANOVA analysis. Linear regression of normalized fiber length on ankle
joint angle yielded no significant relationship between the two variables for
any of the three muscles (Fig.
2D-F; *P*>0.2, *r*^{2} range
0.001–0.028), while one-way ANOVA revealed no significant differences in
normalized fiber length across all angles for any of the three muscles
(*P*>0.1). This result validates statistically, the use of sarcomere
length for fiber length normalization.

A concern in this analysis was that systematic size differences between
animals could affect the experimental results. For example, some of the raw
fiber length differences between groups could simply reflect animal size.
Thus, to determine whether animal size varied significantly among groups, both
animal mass and tibial length were compared across groups. Neither animal mass
nor tibial length varied significantly among groups, as revealed by one-way
ANOVA (*P*>0.5) and neither were significantly correlated with ankle
angle as determined by linear regression (animal mass:
*r*^{2}=0.008, *P*>0.7; tibial length:
*r*^{2}=0.008, *P*>0.7). Thus, animal size did not
affect the fiber length analysis presented above.

Having determined that sarcomere length effectively normalizes fiber length, a practical consideration becomes defining the resolution of the normalization method. It could be that, due to the large degree of natural variability that occurs in fiber length within muscles fixed at the same joint angle, the lack of significant differences among angles after normalization renders the method imprecise. Note that, in Fig. 2, a great deal of natural fiber length variation occurs within angles that is not eliminated, even after sarcomere length normalization (average within-group coefficient of variation for all muscles at all angles was 11.1±0.9%). This large natural variability is a real phenomenon that has been observed in animal muscles and reported for comparatively large human muscles (Fridén et al., 2001, 2004). Since architectural studies often attempt to define fiber length differences among muscles (see, for example, Lieber et al., 1992) or fiber length changes after surgical intervention (see, for example, Burkholder and Lieber, 1998), it is of interest to determine the ability of such architectural analysis to resolve various relative fiber length differences. This relationship is shown for the EDL, soleus and TA muscles in Fig. 3. The abscissa represents the percentage change in fiber length that could be resolved for a given experiment. As expected, because experimental variability was lowest for the TA muscle, statistical power was highest for this muscle. Both soleus and EDL demonstrate a slight decrease in power at the same relative fiber length difference due to slightly higher intramuscular fiber length variability. Overall, it can be seen that, for all muscles studied, there is >90% chance of detecting a 15% fiber length difference among muscles and >60% chance of detecting fiber length differences as small as 10%. We thus conclude that the use of sarcomere length normalization in architectural studies permits resolution of fiber length variations of 15% and may even be effective at resolving 10% fiber length variations. Of course, this conclusion assumes that fiber length variability in the experimental system chosen is similar to that reported here for the mouse hindlimb. If the variability is much greater than that reported here, it may be necessary to restrict fiber sampling to analogous anatomical regions of muscles or to increase sample size to obtain the same resolving power as presented here.

It should be noted that, while the method of measuring sarcomere length in
the current study was laser diffraction, any sarcomere-length measuring
approach can be used. This could include phase microscopy of fixed fibers or
even paraffin embedding of muscle with longitudinal sectioning of stained
tissue. The main point is that sarcomere length must be measured in some way
that avoids unnecessary intermuscular variation. The data from
Fig. 2 demonstrate that, in the
mouse hindlimb system, there is no joint angle at which fiber lengths become
`less variable' such that normalization would become unnecessary. This is
supported by the fact that, for all muscles, one-way ANOVA reveals no
significant difference among joint angles for coefficient of variation
(*P*>0.6). Alternatively, one might simply try to fix joints at the
angle that corresponds to the reference sarcomere length. However, as can be
appreciated from Fig. 2, there
is no single angle that fulfills this criterion for all muscles, which
emphasizes the need for direct sarcomere length normalization.

## ACKNOWLEDGEMENTS

The authors gratefully the support of the Department of Veterans Affairs and the National Institutes of Health, grants AR40050, AR40539 and HD44822. We appreciate the helpful comments of Drs Jan Fridén and Ilona Barash. We also appreciate the technical assistance of Ms Debbie Trudell.

- © The Company of Biologists Limited 2005