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Tapering of the intrafascicular endings of muscle fibers and its implications to relay of force.

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THE ANATOMICAL RECORD 236:390-398 (1993)
Tapering of the lntrafascicular Endings of Muscle Fibers and Its
implications to Relay of Force
E. ELDRED, M. OUNJIAN, R.R. ROY, AND V.R. EDGERTON
Brain Research Institute and Departments of Anatomy and Cell Biology, and
Physiological Science, University of California at Los Angeles, California
of
ABSTRACT
The geometric shape of the filamentous, intrafascicular
type of muscle fiber ending was reconstructed as a basis for understanding
the pattern in relay of the fiber’s force to the muscle tendon. Single motor
units (MUs)identified physiologically as being fast and slow, respectively,
were isolated in cat tibialis muscles and glycogen-depletedfor recognition
in cross sections of the muscle frozen at its Lo. Serial measurements of
cross-sectional area (CSA) using an image processing system were made
along 14 intrafascicular endings of MU fibers and an additional seven, nondepleted fibers identified histochemically as slow. Comparison of coefficients of variation for the linear relation of the CSAs and of the equivalent
diameters with length along the taper indicated that in both fast and slow
fibers the areas bore a closer relationship, that is, the taper had the equivalent of a parabolic, rather than a conical outline. The implications of these
two conformations to relav of the fiber’s contractile force to surrounding
structures are displayed graphically. 0 1993 Wiley-Liss, Inc.
Commonly one or both ends of a skeletal muscle fiber
do not reach the muscle’s tendon or aponeurotic surface, and terminate within the surrounding fascicle a s
a filamentous strand (Huber, 1916; Adrian, 1925).
Such a n “intrafascicular ending” (Bardeen, 1903) can
taper over a distance of a centimeter or more (Barrett,
1962; Richmond et al., 1985; Loeb et al., 1987; Ounjian
et al., 1991), so that its precise shape is not appreciated
by observing fibers isolated using conventional techniques of maceration and teasing. If the geometric conformation were known, insight might be gained a s to
the rearrangement within the fiber underlying its progressive decline in caliber, and how this tapering
translates to relay of the fiber’s contractile force to its
surroundings and eventually to the muscle’s tendon.
The shape of such a n ending can be reconstructed
from measurements of fiber area taken on serial, frozen
cross-sections in which identification of the individual
fiber is aided through prior glycogen depletion of the
motor unit (MU) of which it is a part. When the areas
of fibers from fast MUs of the cat tibialis anterior (TA)
muscle were plotted against length from the tip of the
fiber, they appeared to decrease more or less along a
straight line (Ounjian et al., 1991), that is, the crosssectional area (CSA) varied linearly as length. Necessarily a correlation of the fiber diameter with position
along the taper must also exist, and this raises the
question as to whether the decrease in fiber caliber is
more closely described a s a linear relation of area or of
diameter to level along the taper. These geometric
models are diagrammed in Figure 1,where in A diameters decrease linearly along the taper length as in a
true cone, while in B the fiber diameters follow a parabolic course as they must when areas decline linearly
with length. The cross section outlines of muscle fibers
0 1993 WILEY-LISS, INC.
are seldom perfectly symmetrical due to ellipticity,
facetization, etc., but if such deviations are sustained
along the taper length the change in fiber caliber can
still be described as being more akin to a parabolic
solid or a cone, and the model will illustrate the pattern in relay of the fiber’s force. There are other conceivable relationships, e.g., a decrease in area as a constant percentage or following some power function, but
a s a first approximation to description of the geometry
our attention has been channeled to the two hypotheses: that the taper is better expressed as a linear relationship of CSA to length, or of diameter to length. The
question was approached by determining the closeness
of fit of measurements to a straight-line regression relationship. Perimeter measurements were included in
the comparison, as they reflect on the mean diameters
of irregular fiber outlines and when compared with the
circumference of a hypothetical circle of equal crosssectioned area give a n idea of the degree of irregularity. The results indicate that the shape more nearly
approached that of a parabolic solid.
METHODS
Analysis was made of measurements on 21 fibers (22
tapers) from three cat TA muscles (Table 1).Eleven
fibers from a 4.5 kg cat were in a MU identified physiologically and histochemically as fast-fatigable (FF)
Received June 8, 1992; accepted December 16, 1992.
Address reprint requests to Dr. Roland R. Roy, Brain Research Institute, Center for the Health Sciences, University of California, Los
Angeles, Los Angeles, CA 90024-1761.
391
TAPERING OF MUSCLE FIBERS
CONICAL MODEL
DIAMETER = k . LENGTH
A
PARABOLIC MODEL
B
CSA= k.LENGTH
3
4
L
I
I
I
1
I
J
I
I
I
I
I
DIAMETER
Fig. I.Alternative models for the tapering, intrafascicular type of
muscle fiber ending. In A, the diameter of successive cross-sections
varies linearly with “length” from the tip of the taper, while in B, the
area varies with length. In both drawings diameter is plotted against
length, but a t a 100 x reduced scale for the latter, so that the images
appear greatly shortened lengthwise. Blackened areas on the upper
surfaces of successive segments (1 to 4) symbolize patterns of loss
in area that could result in the configurations. The cone configuration
in A could result either from losses restricted to zones of equal depth
in the segments, from losses distributed a t equal concentration across
each segment (dots),or a combination of these patterns. The parabolic
solid in B could result from loss of a peripheral zone of increasing
depth in successive segments, or through increasingly concentrated
attrition across them. Vertical arrows alongside each segment represent the upward directed vector of relayed force produced during contraction of that segment. The small hinged-out projections relate loss
in CSA to the corresponding strip of surface area (SfA) of that segment, and illustrate 1) for the conical model, a constant ratio between
loss in CSA (ie., relayed force) and surface area (concentration of
force per surface area), and 2) for the parabolic model, a curvilinearly
increasing ratio.
by the criteria of Burke et al. (1973). In a second animal of 2.9 kg three fibers in a similarly tested MU of
slow type and three fibers identified histochemically as
being slow, but not in the glycogen-depleted MU, were
traced. From a third muscle (2.7 kg cat) four more nondepleted fibers of histochemical slow type were measured. Procedures for the isolation of a MU, determining its optimal contractile length (Lo), depletion of its
glycogen, histological preparation as 20+m frozen sections, and measurement of fiber CSA using a n image
processing system have been given elsewhere (Bodine
et al., 1987). The muscles were held firmly to the Lo of
the particular MU a t the time of freezing in isopentane,
and fibers selected for measurement were contained in
a single fascicle or in one muscle three adjacent fascicles. Outlines on selected sections along the entire fiber
length were measured, but only those showing a progressive decline in area toward the fiber end, i.e., ta-
pering, were subjected to analysis. Values from 11to 23
sections at irregular intervals (0.5 to 2.0 mm) along
each taper were obtained. The image processing system
also measured the perimeter and calculated a value for
the diameter of a circle corresponding to the fiber CSA.
The angle of pinnation in the cat TA is only 7” (Sacks
and Roy, 1982) and no correction was made for it. Geometric formulae used to construct the graphs in Figure
3 include the following: for the volume of a cone, 1/3 TT
r2L, and of a parabolic solid, 112 T r2L; for the curved
surface area of the frustum (“segment” in Fig. 1) of a
cone, d r l + r2) v(L,-L2)2+(rI-r2)2, and for the
frustum of a parabolic solid, 413 TT k (L,3’2 - LZ3”).L,
and L, designate lengths from the fiber tip to the large
and smaller surfaces (top and bottom in Fig. 1)of the
frustum, and rl and r2 are the corresponding radii. The
following abbreviations are used: CSA, the cross-sec-
392
E. ELDRED ET AL.
TABLE 1. Correlation of length along the tapering ends of fast and slow fibers to three measures of their
cross-sectional outline
Fast fibers’
Fiber No.
1
2
3
4
5
6
7
8
9
10
11’
11’
Taper directed,
P
P
P
P
D
D
P
P
P
P
P
D
proximally; distally
No. measurements
15
12
14
14
12
14
14
7
14
13
13
13
Correlation coef.,
,969
,997
,988
,994
,983
,971
,991
,993
,975
.982
,982
,980
length: CSA
,960
,971
,952
,938
.927
Eq. diameter3
,937
,952
,991
,945
,964
,959
,968
,935
,976
,950
,935
,925
,941
,948
,924
,989
Perimeter
,968
,955
,974
3.54
3.11
3.14
3.58
3.65
Slope, CSA, pm’imm
2.60
3.83
3.88
3.99
4.29
3.81
1.86
56.9
38.1
38.3
54.5
45.2
56.3
Length, fiber, mm
45.3
47.0
51.3
58.0
42.6
11.8
10.3
12.0
11.2
11.5
6.7
13.1
10.4
ta~er*
12.1
10.4
10.5
13.4
Slow fibers1
Fiber No.
1
2
3
4
5
6
7
8
9
10
Mean 2 S.D.
12
,984 2 ,009
,955 2 .018*
,952 t .030*
358 ? 49.5
48.5 2 7.3
11.1 1.7
All tapers:
Mean? S.D. Mean 5 S.D.
Taper directed,
P
P
P
P
D
D
D
D
D
D
proximally; distally
No. measurements
15
9
17
13
22
12
17
26
19
22
10
Correlation coef.,
,983
,986
,972
.988
,963
length: CSA
,958
,980
,986
,984
,968 ,977 2 ,010
,974
,943
,960
,925
,965
,896
,916
,982
,949
Eq. diameter
,983 ,954 2 .029*
,950
,961
,885
.946
.963
.914
,950
Perimeter
,979
.941
,980 ,947 2 .029*
Slope, CSA,
1.21
1.50
1.81
1.45
1.68
1.50
2.21
p,m2/mm
2.35
2.50
2.62
188 t 38.7
29.5
21.5
19.8
18.9
35.6
25.2
26.4
36.0
Length, fiber, m m
26.7
21.0
26.2 t 5.9
11.8
12.9
8.2
11.4
11.2
11.0
14.1
11.5
11.8
11.6 2 1.6
10.0
taper
22
,980 2 ,010
,954 ? .023*
,949 2 .024*
%low fibers 1-3 were in one depleted unit, while 4-6 were non-depleted, slow fibers in the same muscle. Fibers 7-10 were non-depleted, slow
fibers in another muscle. Fast fibers were from a single, depleted MU.
‘This fiber tapered at both ends. Its values were not included in calculation of the mean for the slopes.
3The “equivalent diameter” or diameter of a circle equal in area to the measured cross-sectional area.
4Length along which data were taken as measured from the tip of the taper.
*Mean significantly different ( P < . O O l ) compared to mean for area. None of differences between diameter and perimeter was significant at
P<O.l.
tional area measured from the fiber outline; eqD, the
equivalent diameter of a circle having the same CSA;
eqC, the equivalent circumference of the hypothetical
circle; P, the perimeter as actually measured on the
fiber cross-section.
RESULTS
Configuration of Tapering
Graphs (Fig. 2) showing fiber cross-sectional dimensions plotted relative to “length” (L) from the tip of a
tapering intrafascicular ending gave the impression
that the relationship approached linearity more closely
when length was matched with CSA than with the calculated eqD of a circle of equal area (Fig. 2A,B,C) or the
measured perimeter (P) of the fiber (Fig. 2D). To substantiate this, the linear regression relationships of
CSA-to-L, and eqD-to-L were determined for each taper
(Table 1j. Also, because the cross-sectional outlines seldom took the form of a perfect circle and on such irregular outlines the perimeter bears relation to the mean
of the diameters about the center of the area, correlations of P-to-L were included in the comparisons. Two
approaches were taken to see which series yielded the
best fit. In one, the simple arithmetic mean (“cost”) of
those percentage amounts by which data points for a
taper strayed from prediction according to its linear
regression relationship were calculated. For 19 of the
22 tapers, the mean of such deviations was least for the
CSA-to-L relationship. The second approach was to
compare the means of the linear correlation coefficients
to see if the mean for the relation of CSA-to-L was
significantly higher than for the relationship of eqDto-L or P-to-L. Figure 3 illustrates expectations with
the conical (D = kL) and parabolic (CSA = kL) models. If
the fiber diameter (and circumference j increases precisely with length as in A, the correlation coefficient
(r), of course, is 1.0. The corresponding coefficient for
areas a s calculated for a n arbitrary sample consisting
of 60 points at 0.2 mm intervals would be 0.969. If
conversely areas are linearly related to length as in B,
a n r value of 0.982 is obtained for the D-to-L correlation. In the two models the curves for the corresponding
non-linear measures are seen to be concave in opposite
directions (CSA in Fig. 3A, diameter in 3B: cf. eqDs in
Fig. 2A-C).
The coefficients (Table 1)calculated for the tapered
ends of 11 fibers in one FF MU, all in a single fascicle,
emphatically favor the model in which area increases
linearly with length. With one exception the correlation coefficient for area was higher than for either the
eqD calculated from the area, or the directly measured
P. Similarly, the CSA-to-L correlation was higher for 9
of the 10 slow fibers studied. The mean of the CSA-to-L
coefficients from all fibers was significantly higher
(P<O.OOl) than those for eqD-to-L and P-to-L. A further argument against a linear eqD-to-L expression of
the taper is that this would result in 19 of the 22 tapers
having a coefficient for CSA-to-L greater or equal to
393
TAPERING OF MUSCLE FIBERS
r
A
r B
I
FIBER 5
FIBER 4
2-
x
(u,
t
i
aw
fx
a
P
Q
0
I-
z
W
22
3
9
'0
J
a
Z
r
-
0
D
l0
w
cn
I
cn
cn
0
a
0
c
i
W
14
0
LENGTH, mm
Fig. 2. Changes in CSA and related measures as seen in selected
sections through the tapering endings of two fibers (A and B) of slow
(S)type and two (C and D) fast (FF)fibers from another cat. The scale
to the left of each graph refers to the CSA, and the horizontal scale to
length measured from the tip of the taper. In A-C the scale along the
right margin refers to the equivalent diameter of a circle equal in area
to that of the fiber CSA, while in D the scale represents both the
equivalent circumference of this hypothetical circle and the measured
perimeter. The right-hand scales were normalized to the maximum
CSA (and accordingly a t other levels are proportionate to the square
root of the CSA values). Light, straight lines indicate the linear regression relationship of CSA to length along the taper, and the correlation coefficient (r) is indicated. While the slopes of the regression
lines for the slow and fast fibers appear similar, the two-fold difference in scaling of CSA implies that loss in CSA occurred about twice
as rapidly in the fast fibers. A dotted line extending beyond the maximum CSA data point indicates the beginning of the non-tapering
portion of the fiber.
the 0.969 limit for CSA-to-L obtained in the example of
Figure 3A wherein diameter was assumed to vary linearly a s length.
The failure of the CSA-to-L correlation coefficients to
reach 1.0 could reflect a weak, secondary tendency for
diameters to vary linearly as fiber length. However,
the deviations of data points (N = 160) from the linear
regression lines for the CSA-to-L relationship for individual fibers did not show a preponderance of negative
values over a middle section of the taper length a s
would be expected (cf. upward directed concavity along
the CSA curve in Fig. 3A). Consistent with this, the
CSA-to-L relationship calculated as a second polyno-
mial had a positive constant for the x2 term for only 10
of the 22 tapers.1 Thus, failure for the CSA-to-L correlation coefficients to quite reach 1.0 was probably due
to variance (e.g., Fig. 2A,B) introduced in the histological preparation, and in tracing of the outlines.
'The near balance of positive and negative constants precluded
finding a consistent constant for the x2 term in a polynomial of the
second order applicable to all the slopes, although polynomials derived for individual fibers consistently gave higher correlation coeffcients than did first-order equations.
394
E. ELDRED ET AL.
D =kL
A=k L
BASE OF TAPER
L d - - L
3
BASE OF TAPER
3
A
2
C
-----h---h--.,---hrt
DECREASE IN CSA PER SEGMENT
ACCUMULATIVE
DECREASE IN CSA
t
DECREASE IN CSA
0
4
DECREASE IN CSA TO
SURFACE AREA OF SEGMENT
8
12 0
TAPER LENGTH, m m
Fig. 3. Correlations between morphological features (in A-D) and
functional consequences (in parentheses in C and D) predicted under
the two hypotheses that the diameter (A and C) or the CSA (B and D)
of a fiber varies linearly with distance along the taper. Maxima for
diameter and area have been scaled to equal height. In the parabolic
model at D the horizontal line labeled “decrease in C S A represents
the constant loss in area per millimeter-thick segment with progression toward the taper tip, while in the conical model (in C) the linear
4
8
I
decline in size of the segmental losses is indicated. The sum of successive losses (accumulative decrease in CSA), hence total excess force
relayed to the fiber’s surroundings, necessarily rises linearly toward
the taper tip in D and curvilinearly in C. The ratio of a segment’s loss
in CSA to loss in external surface area, i.e., concentration of force at
the surface, remains constant in the conical model (C), but trends
asymptotically upward in the parabolic solid (D). Surface areas of the
segments are not graphed.
fiber’s perimeter, as measured by the image-processing
system, to the eqC of a circle equal in area to that of the
The CSA-to-L slopes expressed a s the decrease in fiber
cross section. The perimeters on sections along a
fiber area per millimeter of taper length averaged 358 given fiber clearly exceeded the eqC by a n increasing
bm2/mm for the fast fibers, and for slow fibers only 188
with progress toward the base of the taper (Fig.
wm2/mm (Table 1).The ranges were almost exclusive margin
2D).
The
ratio of the perimeter to the eqC, however,
(Fig. 4).However, since only one fast MU was reprewas
more
stable, the majority of values falling between
sented and the slow fibers, depleted and non-depleted,
1.20
and
1.45
with only a suggestion of a gradual incould have come from as few a s three MUs, it cannot be
crease
(Fig.
5).
The mean for the ratios along a zone
safely inferred that the difference was related to the
extending
from
8 to 12 mm beyond the taper’s tip did
basic distinction in metabolic and contractile characsignificantly
exceed
(P<0.001) that for sections beteristics of these fiber types. As expected for the TA
(Bodine et al., 19871, the fast fibers had larger CSAs as tween 1 and 5 mm (i.e., excluding the very small cross
measured over a 5-mm, non-tapering segment of the sections within the first mm beyond the tip). The ratios
fiber, and tests for correlation of this measure of fiber over 10 mm of taper increased from about 1.30 to 1.40,
caliber with the taper slopes in the samples of fast and which is more impressive if considered as being a n inslow fibers yielded positive coefficients (Fig. 4)with a crease in irregularity by a third. The ratios of the 14
mean of 0.921 for all monopolarly tapering fibers. This fibers that had been subjected to the rigorous procedure
suggests that fiber caliber was one determinant of the of glycogen depletion and those for 7 nondepleted fibers
degree of slope. Any explanation of the development of did not differ significantly.
tapering should take cognizance of the one bipolarly
tapering fast fiber in which the slope a t one end was
twice that a t the other.
DISCUSSION
SLOPES
Perimeter-to-Equivatnt Circumference (P/eqC) Ratio
Geometry of the Taper and Its Functional implications
The degree to which a fiber cross section deviated
from a circular outline was gauged by the ratio of the
Tapering along the intrafascicular endings of the
muscle fibers conformed more closely to the shape of a
395
TAPERING OF MUSCLE FIBERS
5-
E
E 4I+
Q
X
N
E
=4
d
z2 3 z
sa
(3
$
2V
z
#a
W
8
W
1-
0
1
0
I
1
I
I
1
I
2
4
6
CROSS-SECTIONAL AREA I N NON-TAPERING ZONE, Am2 x lo3mrn
I
8
Fig. 4. Relation of rate of decrease in CSA (slope) in individual fibers
to the mean of areas measured along an approximately 5-mm, more
centrally located, non-tapering segment of the fiber. Included are 10
fibers from a single fast motor unit, and 10 depleted and non-depleted
slow fibers from two other muscles (bracketed at a and b). The one,
bipolarly tapering fiber encountered in the study was excluded. The
dashed lines represent the linear regression relationships for the fast
and slow fibers, and the solid line that for all fibers.
parabolic solid than that of a cone.2 This observation, of
course, could be conditional upon the experimental circumstances: use of the cat TA muscle, the freezing and
subsequent thawing, and the histochemical staining
procedure. Also, the muscle was under stretch. However, a fiber should retain a parabolic conformation
when under less tension providing that sarcomere intervals along the taper length shortened equally.
A linear decrease in CSA (and volume) with distance
down the taper implies loss of a constant fraction of the
fiber's myofibrillar field if the subsarcolemmal zone occupied by non-fibrillar elements is overlooked. Given
uniform packing density of the myofibrils and constant
sarcomere intervals, there must be disappearance of
the same number of myofibrils per millimeter in the
robust base of the taper as in the filamentous terminal.
The loss cannot take the form of a dropout within each
segment of a set number of ranks of the most peripherally located myofibrils (as depicted by the black zones
in Fig. lA), €or that process would yield a cone. If fibrils
in the periphery are more susceptible, as for example
they are to atrophy (Engel and Stonnington, 1974)or in
spread of activation across a sarcomere (Constantin
and Taylor, 1973), then as the muscle fiber narrows,
deeper ranks of fibrils must drop out (black zones to the
left side in Fig. 1B) as modeled by Haggqvist (1925).
Alternative arrangements would be for attrition to be
spread across the entire myofibrillar field with its concentration increasing as the segments became smaller
(represented by black dots, Fig. lB), or for tapering to
result from losses concentrated centrally as depicted by
Heidenhain (in Haggqvist, 1925). Observations on the
interrelation between fibrils and sarcolemma in the
blunt type of ending (e.g., Schwarzacher, 1960; Muir,
1961; Trotter et al., 1983; Tidball, 1983; Tidball and
Daniels, 1986) and elsewhere along the fiber (e.g.,
Street, 1983; Burridge et al., 1988) strongly indicate
that the major loss is peripheral. The rate of loss in
area, 358 (*m2/mmfor the fast fibers, translates to a
reduction of 0.90 pm2 for a sarcomere of 2.5 pm length,
and since the cross-section of a myofibril is of this order
of magnitude (Eisenberg et al., 1974), the equivalent of
one myofibril drops out per sarcomere interval. Undoubtedly this occurs a t the myofilament level, rather
than through loss of some single, uniquely situated
fibril.
'It is of historical interest that diameter measurements a t several
levels along the 25 mm taper of a fiber of the rabbit abdominal external oblique given by Bardeen in 1903yield a correlation coefficient
of 0.997 for the relation of equivalent areas to I. and only 0.957 for
diameter to L.
Relay of Force
Prediction of the consequences of tapering to relay of
the fiber's force is diagrammed in Figures 1 and 3, and
summarized in Table 2. Two assumptions were made:
1)that the force produced is proportional to the CSA of
the fiber; and 2) that in each segment force produced at
its base in excess of that at the small end of the segment is relayed outside the fiber. In the model found to
E. ELDRED ET AL
396
FAST (FF) FIBERS
N=ll
0
.
1.00
0
I
I
2
I
I
I
4
I
I
6
I
0
I
I
10
I
I
12
LENGTH, mm
Fig. 5. Relationship of 1)the ratio between the measured perimeter
of a fiber cross-section to the circumference of a circle of equal CSA
(ordinate) to 2) length from the tip of the taper (abscissa). Eleven
fibers from a MU of fast (FF) type contained within a single fascicle
are represented. The straight line represents the linear regression
relationship which has a slope of only 0.0075 and a correspondingly
low correlation coefficient of 0.397. However, the means for the bracketed ratios differed significantly (P < 0.001).
TABLE 2. Changes in morphology and related functions with progression from the base to
the tip of a tapering ending
Morphology: Loss in CSAisegment
Function: Relayed forceisegment
Morphology: Accumulative loss in CSA
Function: Total relayed force
Morphology: Area of curved surface of segment
Loss in CSAisurface area of segment
Function: Relayed forceiunit of surface area
Morphology: Scarf angle
Function: Breaking threshold to sheer stress
be favored, wherein area is linearly related to length,
the longitudinal vector of force (arrows in Fig. 1B) relayed through the sarcolemma to the endomysium and
beyond would be constant in each taper segment (Fig.
3D,cf. “decrease in CSA per segment”), and the accumulation of such force would rise linearly (“accumulative decrease in area”). Since surface areas of the successive segments of the parabolic solid undergo an
accelerating decrease (not shown), the concentration of
force per sarcolemmal area must rise at increasingly
steeper rates toward the fiber tip (in Fig. 3D,“ratio of
decrease in CSA to surface area of segment”; also schematically illustrated by the horizontal, projected sectors in Fig. 1B). In response to this rise there should be
either progressive reinforcement of the structural ar-
Conical model
(D = kL)
Linearly .1
Linearly .1
t at decreasing rate
t at decreasing rate
5 linearly
Constant
Constant
Constant
Constant
Parabolic model
(CSA = kL)
Constant
Constant
t a t constant rate
t a t constant rate
J a t decreasing rate
f a t increasing rate
t at increasing rate
t at increasing rate
.1 at increasing rate
rangement for transfer of stress or a gradual change in
design for accomplishing this. In contrast, with the conical model the linear decrease in force relayed per segment along the taper (Fig. 3C)would be exactly paralleled by the decrease in surface area (not shown), so
that the concentration of relayed force per surface area
wouid remain constant (Figs. l A , 3C)and no structural
modification would be needed.
The changing slope along the surface of a parabolic
solid implies that the relative roles played by shear and
tensile stresses between the sarcolemma and elements
of the endomysium continuously shift, as pointed out
by Tidball (1983)for the paraboloid projections on the
blunt type of ending of frog muscle fibers. He found
that the angle at which actin filaments approached the
TAPERING OF MUSCLE FIBERS
convoluted sarcolemmal surface averaged 4.3". Trotter
(1991) from diameter measurements made on scanning
electronmicrographs of intrafascicular endings of biceps femoris (BF) fibers, and assuming the tapers were
conical, obtained a n angle of 0.72'. Along the taper of
cat TA fibers, the average angle as derived from the
mean for the slope in areas was 0.61" for fast and 0.45"
for slow fibers, but over the basal two thirds of the
taper length the angle would be smaller yet, since most
of the longitudinal curvature appears toward the tip.
With a 12-mm taper of parabolic configuration having
a maximal radius of 50 pm, the scarf angle (angle between the fiber axis or line of force, and the longitudinal tangent to the curved surface at a given level along
the taper) would be smaller and the shear strength
higher than if the ending were conical except in the
last millimeter preceding the tip, where the scarf angle
would finally exceed the steady level found in the conical model. During contraction of the fiber, shear stress
would not remain constant, a s predicted for a conical
outline (Trotter and Purslow, 1992). Moreover, the
more prominent increase in scarf angle would occur in
the presence of increasing surface concentration of relayed force (Fig. 3D). In response to the increasing shift
to emphasis on tensile strength adaptations in the endomysial contacts would be expected to become increasingly rugged toward the taper tip. Such appearances
have been described (Huber, 1916; Barrett, 1962; Swatland, 1975; Trotter, 1991).
397
fibers of larger size. Through comparison of perimeter
measurements made a t two levels of resolution, he estimated that 30% of the excess in the ratio above 1.00
was due to "roughness" of the fiber surface. Presumably contributing to this was the fact that only 18% of
the fibers studied were from fascicles stretched to their
full physiological length, for under stretch, longitudinal angularity (Blinks, 1965) and fine transverse folds
of the sarcolemma (Dulhunty and Franzini-Armstrong,
1975) are reduced, and in the biceps fibers the form
factor did become smaller. Also significant to explanation of the difference in the light and electronmicroscopic findings is that in the latter the angle of 0.72"
Trotter found for the average biceps femoris taper
would a t the 10.2 pm diameter correspond to a taper
length of 0.4 mm assuming the taper were conical and
even less if parabolic; the measurements on TA fibers
extended to 11 mm of taper length. Thus, the findings
in the two studies may represent the geometry a s seen
from different viewpoints, one of the gross taper, and
the other viewing the fiber close to its tip with a degree
of resolution capable of revealing incipient roughness
like that of the blunt, musculotendinous type of fiber
ending. The small increase in irregularity toward the
base of the taper indicated by the light microscope findings would reflect increased ellipticity or facetization
of the fibers.
Compliance in Series With Tapering
Symmetry of Fiber Cross-Sectional Outlines
One way of reducing the concentration of force a t the
sarcolemma would be to contour its surface, a s occurs
to a lavish extent in the blunt type of ending (Trotter et
al., 1983; Tidball, 1983; Tidball and Daniel, 1986). Perimeters of the fiber cross-section outlines did exceed
those of equivalent circles by 30 to 40%. This P/eqC
ratio would be influenced very little by deviation in
course of the fibers (Ounjian et al., 1991), by pinnation,
or by technical failure to section perpendicular to the
muscle axis, since the eqC being derived from the increased CSA also increases a s the perimeter enlarges,
so that the ratio of the circumference of a n ellipse to
that of a circle of equal area does not exceed 1.01 until
the obliquity rises above 30".
The increased ratio, then, arose from true ellipticity
or other irregularity of the fiber cross-sectional outline.
This qualification a s to the fiber outlines should not
alter the pattern of relay of force per mm of taper,
which is dictated by the decline in cross-sectional area
whatever form that takes. Also, along the taper the
ratios of relayed force to surface area, while reduced,
would probably follow the same pattern a s in the hypothetical model.
The P/eqC ratio was seen to increase toward the
base, rather than tip of the taper. Aquin et al. (1980) in
comparison of fiber CSA with the calculated area of a
circle having the longest diameter measured on the
fiber cross section (guinea pig soleus), also found increasing asymmetry with greater CSA. On the other
hand, Trotter (1991) in measurements made on electronmicrographic images of rabbit biceps femoris fascicles found the PieqC ratio (his "form factor") to be
higher in fibers limited to eqD's below 10.2 pm than in
A fiber taper of parabolic shape contains approximately 50% greater volume of myofibrillar substance
than one of conical outline (V = 112 A.L vs. 1/3 A.L, A
being the cross-sectional area at the base), so that
given two fibers with a taper of 12 mm length, one of
parabolic shape would act against the equivalent of 3
mm less passive tissue arranged in series. Overall, the
12-mm-long taper characteristic of a TA fiber should
add the equivalent of 6 mm to the gap filled by connective tissue and passive muscle through which this end
of the fiber must act. For comparison, the average distance from the end of the taper to the tendon for the 11
fast fibers was 19.9 mm. Further contribution to compliance is predicted because of slowing in the spread of
excitation along the taper, since conduction velocity
decreases as a function of fiber size (HAkansson, 1956;
Gydikov e t al., 1986). Along the 12 mm taper of a fast
TA fiber which conducts in its non-tapering segment a t
6 mm/ms (Eccles and Connor, 1939; Shimada et al.,
19631, additional milliseconds would be needed for arrival of excitation at the tip than if the fiber were cylindrical, and those sarcomeres awaiting excitation
would further prolong the contraction time. The additional compliance facing a tapering fiber should distinguish to a degree the force pattern registered a t the
tendon from that of a fiber which inserts a t both ends
onto firm tendon, and MUs that differ in ratio of fibers
with intrafascicular endings should differ in amplitude
and timing of force patterns a t the tendon. In view of
the contributions that tapering makes to compliance, it
is paradoxical that available data suggest that in the
cat TA it is the fibers with short contraction times, the
fast type, in which intrafascicular terminations seem
to be more common (Ounjian et al., 1991).
398
E. ELDRED ET AL.
ACKNOWLEDGMENTS
The authors are grateful to Daniel B. Eldred and
Drs. Alan Garfinkel, James Tidball, and Donald 0.
Walter for their critical review of the concepts presented. The work was supported by NIH grant NS
16333.
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