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The sensitivity of muscle force predictions to changes in physiologic cross-sectional area
THE SENSITIVITY OF MUSCLE FORCE PREDICTIONS TO CHANGES IN PHYSIOLOGIC CROSS-SECTIONAL AREA RICHARD A. BRAND, DOUGLAS R. PEDERSEN and JAMES A. FRIEDERICH Biomechanics Laboratory, Department of Orthopaedic Surgery, The University of Iowa, Iowa City, IA 52212. U.S.A. Abstract-The mechanical effects of a muscle are related in part to the size of the muscle and to its location relative to the joint it crosses. For more than a century, researchers have expressed muscle size by its ‘physiological cross-sectional area’ (PCSA). Researchers mathematically calculating muscle and joint forces typically use some expression of a muscle’s PCSA to constrain the solution to one which is reasonable (i.e. a solution in which small muscles may not have large forces, and large muscles have large forces when expected or when there is significant electromyographic activity). It is obvious that muscle mass (and therefore any expression of PCSA) varies significantly from person to person, even in individuals of similar weight and height. Since it is not practical to predict the PCSA of each muscle in a living subjects limb or trunk, it is important to generally understand the sensitivity of muscle force solutions to possible variations in PCSA. We used nonlinear optimization techniques lo predict 47 muscle forces and hip contact forces in a living subject. The PCSA (volume/muscle fiber length) of each of 47 lower limb muscle elements from two cadaver specimens and the 47 PCSA’s reported by Pierrynowski were input into an optimization algorithm to create three solution sets. The three solutions were qualitatively similar but at times a predicted muscle force could vary as much as two to eight times. In contrast, the joint force solutions were within 1174 of each other and. therefore, much less variable. When using optimization techniques to predict muscle forces, it must be recognized that the solution is sensitive to many assumptions and variables such as PCSA. The muscle force solutions are therefore best used to determine relative values (i.e. trends) in parametric studies. On the other hand, the joint force solutions are less sensitive to such variations, and the absolute values are more reliable.
lNTRODUCTlON
The mechanical effects of a muscle are related in part to the size of the muscle and to its location relative to the joint it crosses. This fact was noted by Weber (1846, 1851) and has been corroborated by many subsequent investigators (Fick, 1911; Haxton, 1944; Morris, 1948; Ikai and Fukunaga, 1968; Ripperger er al., 1980; Spector et al., 1980; Brand et al., 1981; Wickiewin et ai., 1983). Through the works of these researchers, a concept has evolved that a muscle’s potential force generation is related in part to its ‘physiologic crosssectional area’. However, the physiologic crosssectionalarea (PCSA) has been defined in several ways. Ordinarily, the muscle volume has been divided by either gross muscle length or muscle fiber length with or without taking the angle of pennation into account. While exploring methods to mathematically calculate muscle and joint forces in the lower limbs (Crowninshield et al., 1978; Brand and Crowninshield, 1980; Crowninshield and Brand, 1981; Brand et al., 1982) it became obvious that any quantitative prediction of muscle force needed to include some measure of muscle size. Formulation of the problem without accounting for muscle size can result in unrealistic muscle force predictions. For example, a force of 2100 N in the tibialis anterior muscle and no force in the gluteus maximus muscle during the early part of Received 13 December 1984; in revised form 5 February 1986.
stance phase were predicted with one such technique (Seireg and Arvikar, 1975). The question then arises: What is a proper way to account for varying muscle sizes? The question is important if a muscle force solution is sensitive to varying muscle sizes and/or differing definitions of PCSA. Therefore, we conducted the following study to determine whether or not either muscle size (within an anatomically reasonable range) or definition of PCSA could significantly affect muscle force solutions. MATERIALS AND iMETHODS
We first measured the length and approximate angle of pennation of all muscles in two embalmed cadaver lower limbs (in an anatomic position) before excising the muscles and then measured their volume (by water displacement) after excision and removal of all tendinous tissue. The methods were similar to those reported by many other investigators (Haxton, 1944; Morris, 1948; Spector et al., 1980; Brand et al., 1981; Wickiewicz et al., 1983). Muscle length was measured with a ruler from the estimated center of its origin to the estimated center of its insertion. The angles of pennation were estimated with a goniometer. The two cadaver specimens were of quite different sizes (a 183 cm, 91 kg, 37 yr old male, and a 163 cm, 59 kg, 63 yr old female) to obtain a wide range of muscle size values. We then measured the length of 10 to 20 muscle fibers from each of the muscles using a method similar to that reported by Spector et al. (1980). The excised
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PEDERSEN and
muscles were soaked in saline for 1-5 days, then placed in nitric acid for 24-48 h to macerate the tissue. Each muscle was then dissected into several bundles, and 10 to 20 muscle fibers (each about the diameter of a human hair) were separated under a dissecting microscope. Finally. the lengths of each of the individual muscle fibers were measured and an average fiber length for each muscle calculated. (The method is a tedious one, practically precluding the collection of fiber length data from all limb muscles from a large number ofcadavers.) We then defined the PCSA as the muscle volume divided by the averaged muscle fiber length (V/MFL). The PCSAs are reported in cm’. To calculate muscle and joint forces during selected activities, we formulated a two-part problem (Crowninshield et al.. 1978; Crowninshield and Brand, 1981). The first part of the problem may be called the ‘inverse dynamics problem’. Body segment inertial properties for the subject were estimated by the regression equations of Clauser et al. (1969, 1970), and Chandler rf al. (1975). Body segment displacement histories were recorded with biplanar photography (with 20 to 30 locations typically recorded for each gait cycle), digitized, and segment accelerations calculated with analytical differentiation (Crowninshield et al., 1978; Johnston et al., 1979; Brand and Crowninshield, 1980; Crowninshield and Brand, 1981). Foot-floor reactions were measured with a piezoelectrical force plate. The body segment inertial properties, segment accelerations and foot-floor reactions were input into a Newtonian formulation of the equations of motion, yielding the intersegmental resultant forces and moments. These resultants do not represent the forces in any anatomic structure, but rather the vector sums of the forces inall thestructures (muscles, ligaments,joint surfaces) and the vector sums of the moments generated by those forces. In order to calculate forces in the individual anatomic structures, we must know the location of the muscle relative to any joint ‘center’ (i.e. its moment arm) at any given time. This information is specified by a straight-line muscle model based on anatomic dissections of six cadaver limbs in which all muscle origins and insertions were marked and located in bony reference frames (Brand et al., 1982). The model includes a scaling scheme which allows estimation of the location of the origins and insertions of 47 lower limb muscle elements relative to the joint centers in a living subject at any time during a gait cycle. Since there are more unknown muscle forces and joint contact forces than possible equations of motion, the problem of calculating individual forces is indeterminate. This requires formulation of the second part of the problem: the ‘distribution’ problem in which the intersegmental resultants are distributed or apportioned to the anatomic structures. This problem may be solved by the mathematical technique of optimization (Penrod et ol., 1974; Seireg and Arvikar, 1975; Crowninshield et al., 1978; Hardt, 1978; Pedotti er al.. 1978; Johnston et al., 1979; Crowninshield and
J. A. FRIEDERICH
Brand, 1981; Pierrynowski, 1982). Each of the reported methods differed in various respects although all were conceptually similar. Our method differs from each of the others in one or more of the following ways: the use of gait (i.e. kinematic and kinetic) data from the same subject whose inertial properties were estimated, the completeness of the muscle model. the ability to estimate muscle origin and insertion locations in the living subject being studied, the inclusion of two-joint muscles, the use of nonlinear optimization methods, and the optimization criteria. The details and rationale for our particular optimization criterion have been reported previously (Crowninshield and Brand, 1981). The criterion is based on the known nonlinear relationship between muscle force and muscle contraction time: that is, the greater the contraction force, the less time the muscle will be able to maintain the force. The degree to which this relationship is nonlinear is reported to be between 1.5 and 5 in the literature (see Crowninshield and Brand, 1981). We chose a value of three as an intermediate value in the range reported. Specifically, we attempt to maximize endurance by minimizing (optimizing) the sum of the muscle ‘stresses’ cubed. Muscle stresses are defined as the muscle force divided by the PCSA. Our optimization programs (i.e. computer algorithms which predict muscle forces) allow us to input any desired PCSA for each muscle. The details of this program have been reported elsewhere (Arora, 1984). For this study, we calculated all lower limb muscle forces throughout a single gait cycle of a normal subject using three data sets: the PCSA (V/MFL) for each of our two specimens and the PCSAs used by Pierrynowski (1982). (Pierrynowski used various information collected from the literature, or estimates when information on a particular muscle was not available.) From the calculated muscle forces, we calculated the hip contact forces by vectorially subtracting the muscle forces from the intersegmental resultant hip forces. This provided three sets of muscle forces and hip contact force solutions. For each solution set we determined the peak muscle force for each of 47 muscle elements in the lower limb and the peak hip contact force. A linear regression analysis was performed on the relationship between the muscle force vs PCSA for each of the three solution sets.
RESULTS
The PCSAs for each muscle from our two specimens and the PCSAs for the corresponding muscles from Pierrynowski’s report are shown in Table 1. Sometimes a muscle from one of our specimens had the largest PCSA while at other times those from Piertynowski’s report had the largest PCSA. Therefore, based only on the data from these three solution sets, we suspect that it would not be possible to accurately predict PCSA for a given living individ-
Sensitivity
Table
I Physiologic cross-sectional
areas the three data sets
PC% (Volume/muscle Xluscle Adductor brevis (S) Adductor brevis (I) Adductor longus Adduclor magnus 1 Xdductor magnus 2 .Adductor magnus 3 Gluteus maximus I Glutrus maximus 2 Gluteus maximus 3 Gluteus medius I Gluteus medius 2 Gluteus medius 3 Gluteus minimus 1 Gluteus minimus 2 Gluteus minimus 3 Iliacus Psoas Inferior gemellus Obturator externus Obturator internus Pectineus Piriiormis Quadriceps femoris Superior gemellus Biceps femoris Gracilis Rcctus fcmoris Sartorius Semimembranosus Semitcndinosus Tensor fascia latae Gastrocnemius (M) Gastrocnemius (L) Biceps femoris (S) Vastus intermedius Vastus lateralis Vastus medialis Tlbialis anteriosus Ext dig comm Ext hall long Flexor dig Flexor hall long Peronius brevis Pcronius longus Peronius tertius Tlbialis posterior solcus
I* 11.52 5.34 22.73 25.52 18.35 16.95 ‘0.20 19.59 20.00 25.00 16.21 21.21 6.76 8.20 11.98 23.33 25.70 4.33 2.7 1 9.07 9.03 20.54 21.00 2.13 27.34 3.74 42.96 2.90 46.33 13.05 8.00 50.60 14.30 8.14 82.00 64.4 I 66.87 16.88 1.46 4.49 6.40 18.52 19.61 24.65 4.14 26.27 186.69
(PC%) in cm’ for
fiber length-cm’) 3 2 1.92 1.97 9.75 10.91 6.97 5.93 14.59 9.60 8.10 13.09 10.43 9.64 7.45 8.76 1.97 8.82 3.70 1.51 4.88 9.30 1.24 9.16 0.10 1.45 9.12 0.79 9.20 2.68 13.99 3.12 2.46 17.04 8.60 4.69 17.20 16.48 15.60 8.48 4.04 2.12 5.86 8.96 5.29 7.61 1.23 18.89 57.72
of muscle force predictions
3.73 3.74 IS.30 4.24 11.35 29.36 10.25 35.33 15.19 16.48 11.55 12.70 7.09 5.50 5.35 18.36 14.27 2.14 18.96 16.12 7.11 5.12 8.79 1.24 47.99 4.30 54.07 4.80 93.38 9.52 7.13 63.36 34.50 19.35 87.91 147.75 45.65 39.53 23.55 11.93 12.91 26.53 20.08 39.92 5.31 50.85 187.15
Maxt 1 1 1 1 1 3 1 3 I I I I 2 2 1 1
1 1 3 3 1 1 1 1 3 3 3 3 3 1 I 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3
* l-male specimen; 2-female specimen; 3-PCSA given by Pierrynowski. : Max indicates which of the three data sets had the largest PC%.
ual based on data from a group of cadaver specimens using height or weight alone as scaling factors. The three sets of muscle force solutions were generally similar throughout the gait cycle; i.e. the various muscle groups had similar peaks and valleys (Fig. I). Large forces were generally predicted in large muscles and small forces in small muscles (Table 2). However, for a given muscle we could not predict which of the three PCSAs would result in the largest muscle force. The correlation coefficients, r, for the linear correlations of muscle force to PCSA for
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the male specimen. the female specimen and Pierrynowski’s data were 0.40, 0.40 and 0.27 respectively. The absence of strong correlations are emphasized by the most extreme cases. When a nontrivial force (i.e. over 25 N) was predicted with all three data sets. the peak muscle forces could be two to eight (e.g. soleus and gluteus maximus 2) times greater for one of the PCSAs than for another (Fig. 2). Regardless of the significant sensitivity ofthe muscle force solutions to the choice of PCSA. the peak hip contact solutions were much less sensitive. The peak joint forces for our male specimen, our female specimen, and Pierrynowski‘s data were 5.2 times body weight (BW), 5.0 BW and 4.7 BW respectively (i.e. the largest was 111~~of the smallest). In addition to the relatively small differences in peak hip contact forces, the peak hip contact forces occurred shortly after heel strike in all three cases. DISCUSSION It is obvious that a muscle’s ability to generate force is in some way related to its size. However, the nature of this relationship is not well understood. The concept of physiologic cross-sectional area (PCSA) represents a good first approximation of the relationship, but the fact that there are differing definitions of PCSA in and of itself implies that there are differing (or at least evolving) ideas as to the physiologic implications of muscle size. The presence of differing fiber types also implies that current concepts of PCSA alone do not strictly define a muscle’s ability to generate force. However, at the present time, there are no other reasonably straightforward concepts which allow an estimation of a muscle’s force generating capability. Muscle forces cannot be readily measured in uioo owing to obvious technical, biological and, in the case of humans, ethical and legal problems. However, it is desirable to be able to know muscle forces during a variety ofactivities and under many conditions, and we do have methods to mathematically estimate them. Most of these methods require some estimate of the muscle’s ability to generate force, i.e. a knowledge of its PCSA (Penrod ef al., 1974; Chao and An, 1978; Pedotti et al., 1978; Takashima et al., 1979). However, even assuming that we knew the best definition of PCSA and its precise relationship with force generating ability, we still would have no way of accurately determining the PCSA of all muscles in a living subject whose muscle forces we wished to know. Therefore, it is desirable to know a range of muscle and joint force predictions which might reasonably be expected when making differing assumptions about a subject’s PCSA. Not unexpectedly, our data suggests that a muscle force solution can be very sensitive to PCSA. As noted in the introduction, it is possible to utilize optimization techniques to predict muscle forces without accounting for muscle size. Those techniques, while useful for demonstrating particular aspects of the formulation of
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optimization procedures, have not to date allowed reasonable estimates of muscle forces (e.g. Seireg and Arvikar. 1975; Hardt, 1978). Muscle force solutions will be dependent not only on PCSA but also on the particular way in which the problem is formulated (e.g. the choice of optimization criterion). Additionally, muscle force solutions can depend not only on PCSA values, but also other modeled neurophysiological variables and on moment arm values. The effects of
one cannot be simply and clearly isolated from the other. Methods using muscle stress (as contrasted to muscle force) in the optimization criterion demonstrate sensitivity to PCSA, particularly with a linear formulation of muscle stress or with an exponent of two or less. However, as the exponent increases beyond three, the solution becomes less sensitive to the value of the exponent (Crowninshield and Brand, 1981). Since it is presently impossible to determine the
(4
Fig. 1. (a) and (b)
Sensitivity of muscle force predictions
/
(4 Fig. 1. Muscle force solutions throughout the gait cycle using each of the three PCSA data bases. (a) Male specimen, (b) female specimen, (c) Pierrynowski’s data. The gait cycle is divided into multiple time intervals of the stance and swing phase of gait on one axis. The second axis represents the 47 individual muscles in the muscle model and are shown in the same order as theTables. The third (vertical) axis is force (N)and the peak forces correspond to the forces presented in Table 2. While the illustration connects the force predictions for subsequent muscles at identical times, it should be noted that these force predictions are discrete numbers, rather than continuous functions; the connections are made merely to provide time interval marks throughout the illustration.
Gale
cycle
Fig. 2. (A-D)
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R. A. BRAND. D. R. PEDERSENand J. A. FRIEDERICH
F
ObcIn,
Fig. 2. Three muscle force solutions throughout the gait cycle for threesets ofsimilar solutions (A-C) and the three most dissimilar solutions (D-F). PC1 indicates solutions for our male specimen; PC2 indicates solutions for our female specimen; SKY indicates solutions for Pierrynowski’s data. The dotted vertical line marks toe off (TO) at about 58 “/! of the gait cycle.
of these muscle force solutions,
Heel Strike
Toe Off
Hv3l Strike
Fig. 3. Joint force solutions for the hip from the studies of Paul (1965) and Seireg and Arvikar (1975) and this study. These three solutions were obtained using very different techniquesand data: Paul used gait data from a living subject, but instead of using an optimization technique to calculate muscle forces. he grouped various musclesto create a series of determinate problems; Seireg and Arvikat, on the other band, used a linear optimization technique to calculate muscle forces, but used hypothetical subject data; our data were derived from a living subject and used a nonlinear optimization technique.
PCSAs of the muscles in living subjects, we must recognize that any muscle force solution is subject to a range of error at least as great as that reported here. These errors limit the usefulness of the absolute values
although
certain
information regarding trends and relative muscle forces (Johnston et al., 1979; Brand and Crowninshield. 1980). It appears that the peak hip contact force solution is much less sensitive to changes in PCSA than is the muscle force solution, although there are seine differences in when the peak occurs during the gait cycle and slight differences in direction of loading for those peaks.This is not unexpected when one imagines that if one muscle is smaller in one person compared to another, all else being equal, another muscle must simply ‘take up the slack’ to maintain equilibrium (or in a strict sense, for a given circumstance the solution is always constrained by the intersegmental resultant joint force and moment regardless of the PCSA). Additionally, the fact that reasonably similar hip joint forces have been predicted with very different techniques and subject input data, suggests a relative insensitivity of joint force predictions to the muscle force solution (Fig. 3). Thus. muscle forces could be quite different without joint forces being radically different. This relative insensitivity of the joint force solution has implications for studies using such information: joint force problems, being less sensitive to such errors, may generally be more reasonable to study using current techniques than problems in which only muscle force solutions are considered. parametric
studies may still provide reasonable
Acknowledgement-The
authors wish to thank
Mrs. Rose
Britton for preparation of the manuscript. This research was supported in part by NlH AM 14486.
Grant
Sensitivity
of muscle force predictions
Table 2. The peak muscle forces predicted for each muscle for the three data sets shown in Table 1
Muscle Adductor brevis (S) Adductor brevis (I) Adductor longus hdductor magnus 1 Adductor magnus 2 Adductor mngnus 3 Gluteus maximus I Gluteus maximus 2 Gluteus maximus 3 Glutcus medius I Gluteus medius 2 Gluteus medius 3 Gluteus minimus I Gluteus minimus 2 Gluteus minimus 3 lliacus Psoas Inferior gemellus Obturator externus 0b:urator internus Pectineus Piriformis Quadriceps femoris Superior gemellus Biceps femoris Gracilis Rectus femoris Sartorius Semimembranosus Semitendinosus Tensor fascia latae Gastrocnemius (M) Gastrocnemius (L) Biceps femoris (S) Vastus intermedius Vastus lateralis Vastus medialis Tibialis anteriosus Ext dip. comm Ext ha?1 long Flexor die Flexor hall long Peronius brevis Peronius longus Peronius tertius Tibialis posterior Soleus
1* 1 2 217 0 10 96 329 147 82 601 264 240 95 87 74 487 505 II 2 28 70 203 0 5 429 5 290 45 697 139 650 208 480 661 62 58 527 535 372 424 16 923 66 1206 17 187 1008
Muscle forces (N) 2 3 9 13 320 13 20 16 383 56 42 494 284 221 286 295 186 495 146 19 51 140 23 215 0 17 363 3 302 88 612 22 379 284 539 915 47 160 591 416 640 325 44 1041 56 1164 13 208 868
0 2 52 0 0 2 254 451 11 520 332 336 133 81 74 451 145 3 31 al 34 39 0 5 182 a 507 56 625 12 270 502 710 1017 174 37 674 212 735 423 30 979 69 934 13 181 473
* l-male specimen; 2-female specimen; I-PCSA by Pierrynowski. t Max indicates which of the three data sets resulted largest peak muscle force.
Maxt 2 2 2 2 2 1 2 3
I 1 3 3 2 2 2 2 1 2 2 2 1 2 2
I 3 3 2 1 1 1 3 3 3 3 2 3 1 3 1 2 2 3 1 1 2 1 given in the
REFERENCES Arora, J. S. (1984) An algorithm for optimum structural design without line search. New Directions in Optimum Srru~rure Design (Edited by Atrek, E.. Gallagher; E. H., Ragsdell, K. M. and Zienkiewicr 0. C.). Ch. 20. DD. 429-441. John Wiley, New York. ” *’ Brand, P. W.. Beach, R. B. and Thompson, D. E. (1981) Relative tension and potential excursion of muscles in the forearm and hand. J. Hand Surg. 6, 209-219. Brand. R. A. and Crowninshield, R. D. (1980) The effect of
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cane use on hip contact force. Clin. Orrhop. Rd. Rer. 147. 181-184. Brand, R. A., Crowninshield. R. D.. Wittstock. C. E., Pedersen, D. R., Clark, C. R. and van Krieken. F. M. (1982) A model of lower extremity muscular anatomy. J. btomech. Engng 104, 304-310. Chandler, R. F.. Clauser. C. E.. McConville. J. T.. Reynolds, H. M. and Young. J. M. (1975) Investigation of inertial properties of the human body. Report DOT-HS-801: 430. Chao, E. Y. and An, K. N. (1978) Graphical interpretation of the solution to the redundant problem in biomechanics. J. biomech. Engng 100, 159-167. Clauser, C. E.. McConville. J. T. and Young. J. W. (1969) Weight, volume. and center of mass of segments of the human body. Air Force Systems Command. WriehtPatterson Air Force Base, AD-7 10: 622. Clauser, C. E.. McConville, J. T. and Young. J. M. (1970) Weight, volume, and center of mass of segments of the human body. National Technical Information Service, Report AMRL-TR-69-70. Crowninshield. R. D. and Brand, R. A. (1981) A physiologically based criterion of muscle force prediction in locomotion. J. Biomechanics 14, 793-801. Crowninshield. R. D., Johnston, R. C., hndrews. J. G. and Brand, R. A. (1978) A biomechanical investigation of the human hip. 1. Biomechonics 11, 75-85. Fick. R. (1911) Hondbuch der Anatomic und Mechunik der Gelenke. Vol. 3. Jena Verlag von Gustav Fischer. Hardt, D. E. (1978) Determining muscle forces in the leg during normal human walking-an application and evaluation of optimization methods. J. biomrch. Enyng 100. 72-78. Haxton, H. A. (1944) Absolute muscle force in the ankle flexors of man. J. Physiol. 103. 267-273. Ikai, M. and Fukunaga, T. R. (1968) Calculation of muscle strength per unit cross-sectional area of human muscle by means of ultrasonic measurements. Inc. Z. onyew*. Physiol. 26,
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Johnston, R. C., Brand, R. A. and Crowninshield, R. D. (1979) Reconstruction of the hip-a mathematical approach to determine optimum geometric relationships. J. Bone Jr Surg. 61-A, 639-652. Morris, C. B. (1948) The measurement of the streneth of muscle relative to the cross section. Res. Q. Am. AssrHlrh, phys. Educ. 19, 295-303. Paul, J. P. (1965) Bio-engineering studtes of the forces transmitted by joints: Il. Engineering analysis. Biomechanics and Related Bio-Enuinrerinu Tonics (Edited by Kenedi, R. M.), pp. 369-380. Pkrgamon Press, Oxford. Pedotti, A., Krishnan, V. V. and Stark, L. (1978) Optimization of muscle-force sequencing in human locomotion. Math/. Biosci. 38, 57-76. Penrod, D. D.. Davy. D. T. and Singh. D. P. (1974) An optimization approach to tendon force analysis. J. Biomechanics 7, 123- 129. Pierrynowski. M. R. (1982) A physiologtcal model for the solution of individual muscle forces durmg normal human walking. Ph.D. Thesis, Simon Fraser University, Vancouver, British Columbia. Ripperger, R. R., Chao, E. Y. and Staulfer. R. N. (1980) A quantitative analysis of leg musculature. Trans. Orfhop. Res. Sot. 5, 52. Seireg, A. and Arvikar, R. J. (1975) The prediction of muscular load sharing and joint forces in the lower extremities during walking. J. Biomrchanics 8, 89-102. Spector, S. A., Gardiner, P. F., Zernicke. R. F., Roy, R. R. and Edgerton, V. R. (1980) Muscle architecture and forcevelocity characteristics of cat soleus and medial gastrocnemius: implications for motor control. J. Seurophysiol. 44, 951-960. Takashima. S. T., Singh. S. P., Haderspeck. K. A. and Schultz, A. B. (1979) A model for semi-quantitative studies of muscle actions. .I. Biomechanics 12, 929-939.
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Wrber. E. (1846) Wang& Horuiworterbuch der Physiologic. Braunschweiger, Vieweg. Webcr. E. F. (1851) Ueber die Langenverhaltnisse der Fleischfasern der Muskeln im Allgemeinen. Akad.
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Wickiewicz, T. L., Roy. R. R., Powell, P. L. and Edgerton. V. R. (1983) Muscle architecture of the human lower limb. Clin. Orthop. Rel. Res. 179, 275-283.
lNTRODUCTlON
The mechanical effects of a muscle are related in part to the size of the muscle and to its location relative to the joint it crosses. This fact was noted by Weber (1846, 1851) and has been corroborated by many subsequent investigators (Fick, 1911; Haxton, 1944; Morris, 1948; Ikai and Fukunaga, 1968; Ripperger er al., 1980; Spector et al., 1980; Brand et al., 1981; Wickiewin et ai., 1983). Through the works of these researchers, a concept has evolved that a muscle’s potential force generation is related in part to its ‘physiologic crosssectional area’. However, the physiologic crosssectionalarea (PCSA) has been defined in several ways. Ordinarily, the muscle volume has been divided by either gross muscle length or muscle fiber length with or without taking the angle of pennation into account. While exploring methods to mathematically calculate muscle and joint forces in the lower limbs (Crowninshield et al., 1978; Brand and Crowninshield, 1980; Crowninshield and Brand, 1981; Brand et al., 1982) it became obvious that any quantitative prediction of muscle force needed to include some measure of muscle size. Formulation of the problem without accounting for muscle size can result in unrealistic muscle force predictions. For example, a force of 2100 N in the tibialis anterior muscle and no force in the gluteus maximus muscle during the early part of Received 13 December 1984; in revised form 5 February 1986.
stance phase were predicted with one such technique (Seireg and Arvikar, 1975). The question then arises: What is a proper way to account for varying muscle sizes? The question is important if a muscle force solution is sensitive to varying muscle sizes and/or differing definitions of PCSA. Therefore, we conducted the following study to determine whether or not either muscle size (within an anatomically reasonable range) or definition of PCSA could significantly affect muscle force solutions. MATERIALS AND iMETHODS
We first measured the length and approximate angle of pennation of all muscles in two embalmed cadaver lower limbs (in an anatomic position) before excising the muscles and then measured their volume (by water displacement) after excision and removal of all tendinous tissue. The methods were similar to those reported by many other investigators (Haxton, 1944; Morris, 1948; Spector et al., 1980; Brand et al., 1981; Wickiewicz et al., 1983). Muscle length was measured with a ruler from the estimated center of its origin to the estimated center of its insertion. The angles of pennation were estimated with a goniometer. The two cadaver specimens were of quite different sizes (a 183 cm, 91 kg, 37 yr old male, and a 163 cm, 59 kg, 63 yr old female) to obtain a wide range of muscle size values. We then measured the length of 10 to 20 muscle fibers from each of the muscles using a method similar to that reported by Spector et al. (1980). The excised
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R. A. BRAND, D. R.
PEDERSEN and
muscles were soaked in saline for 1-5 days, then placed in nitric acid for 24-48 h to macerate the tissue. Each muscle was then dissected into several bundles, and 10 to 20 muscle fibers (each about the diameter of a human hair) were separated under a dissecting microscope. Finally. the lengths of each of the individual muscle fibers were measured and an average fiber length for each muscle calculated. (The method is a tedious one, practically precluding the collection of fiber length data from all limb muscles from a large number ofcadavers.) We then defined the PCSA as the muscle volume divided by the averaged muscle fiber length (V/MFL). The PCSAs are reported in cm’. To calculate muscle and joint forces during selected activities, we formulated a two-part problem (Crowninshield et al.. 1978; Crowninshield and Brand, 1981). The first part of the problem may be called the ‘inverse dynamics problem’. Body segment inertial properties for the subject were estimated by the regression equations of Clauser et al. (1969, 1970), and Chandler rf al. (1975). Body segment displacement histories were recorded with biplanar photography (with 20 to 30 locations typically recorded for each gait cycle), digitized, and segment accelerations calculated with analytical differentiation (Crowninshield et al., 1978; Johnston et al., 1979; Brand and Crowninshield, 1980; Crowninshield and Brand, 1981). Foot-floor reactions were measured with a piezoelectrical force plate. The body segment inertial properties, segment accelerations and foot-floor reactions were input into a Newtonian formulation of the equations of motion, yielding the intersegmental resultant forces and moments. These resultants do not represent the forces in any anatomic structure, but rather the vector sums of the forces inall thestructures (muscles, ligaments,joint surfaces) and the vector sums of the moments generated by those forces. In order to calculate forces in the individual anatomic structures, we must know the location of the muscle relative to any joint ‘center’ (i.e. its moment arm) at any given time. This information is specified by a straight-line muscle model based on anatomic dissections of six cadaver limbs in which all muscle origins and insertions were marked and located in bony reference frames (Brand et al., 1982). The model includes a scaling scheme which allows estimation of the location of the origins and insertions of 47 lower limb muscle elements relative to the joint centers in a living subject at any time during a gait cycle. Since there are more unknown muscle forces and joint contact forces than possible equations of motion, the problem of calculating individual forces is indeterminate. This requires formulation of the second part of the problem: the ‘distribution’ problem in which the intersegmental resultants are distributed or apportioned to the anatomic structures. This problem may be solved by the mathematical technique of optimization (Penrod et ol., 1974; Seireg and Arvikar, 1975; Crowninshield et al., 1978; Hardt, 1978; Pedotti er al.. 1978; Johnston et al., 1979; Crowninshield and
J. A. FRIEDERICH
Brand, 1981; Pierrynowski, 1982). Each of the reported methods differed in various respects although all were conceptually similar. Our method differs from each of the others in one or more of the following ways: the use of gait (i.e. kinematic and kinetic) data from the same subject whose inertial properties were estimated, the completeness of the muscle model. the ability to estimate muscle origin and insertion locations in the living subject being studied, the inclusion of two-joint muscles, the use of nonlinear optimization methods, and the optimization criteria. The details and rationale for our particular optimization criterion have been reported previously (Crowninshield and Brand, 1981). The criterion is based on the known nonlinear relationship between muscle force and muscle contraction time: that is, the greater the contraction force, the less time the muscle will be able to maintain the force. The degree to which this relationship is nonlinear is reported to be between 1.5 and 5 in the literature (see Crowninshield and Brand, 1981). We chose a value of three as an intermediate value in the range reported. Specifically, we attempt to maximize endurance by minimizing (optimizing) the sum of the muscle ‘stresses’ cubed. Muscle stresses are defined as the muscle force divided by the PCSA. Our optimization programs (i.e. computer algorithms which predict muscle forces) allow us to input any desired PCSA for each muscle. The details of this program have been reported elsewhere (Arora, 1984). For this study, we calculated all lower limb muscle forces throughout a single gait cycle of a normal subject using three data sets: the PCSA (V/MFL) for each of our two specimens and the PCSAs used by Pierrynowski (1982). (Pierrynowski used various information collected from the literature, or estimates when information on a particular muscle was not available.) From the calculated muscle forces, we calculated the hip contact forces by vectorially subtracting the muscle forces from the intersegmental resultant hip forces. This provided three sets of muscle forces and hip contact force solutions. For each solution set we determined the peak muscle force for each of 47 muscle elements in the lower limb and the peak hip contact force. A linear regression analysis was performed on the relationship between the muscle force vs PCSA for each of the three solution sets.
RESULTS
The PCSAs for each muscle from our two specimens and the PCSAs for the corresponding muscles from Pierrynowski’s report are shown in Table 1. Sometimes a muscle from one of our specimens had the largest PCSA while at other times those from Piertynowski’s report had the largest PCSA. Therefore, based only on the data from these three solution sets, we suspect that it would not be possible to accurately predict PCSA for a given living individ-
Sensitivity
Table
I Physiologic cross-sectional
areas the three data sets
PC% (Volume/muscle Xluscle Adductor brevis (S) Adductor brevis (I) Adductor longus Adduclor magnus 1 Xdductor magnus 2 .Adductor magnus 3 Gluteus maximus I Glutrus maximus 2 Gluteus maximus 3 Gluteus medius I Gluteus medius 2 Gluteus medius 3 Gluteus minimus 1 Gluteus minimus 2 Gluteus minimus 3 Iliacus Psoas Inferior gemellus Obturator externus Obturator internus Pectineus Piriiormis Quadriceps femoris Superior gemellus Biceps femoris Gracilis Rcctus fcmoris Sartorius Semimembranosus Semitcndinosus Tensor fascia latae Gastrocnemius (M) Gastrocnemius (L) Biceps femoris (S) Vastus intermedius Vastus lateralis Vastus medialis Tlbialis anteriosus Ext dig comm Ext hall long Flexor dig Flexor hall long Peronius brevis Pcronius longus Peronius tertius Tlbialis posterior solcus
I* 11.52 5.34 22.73 25.52 18.35 16.95 ‘0.20 19.59 20.00 25.00 16.21 21.21 6.76 8.20 11.98 23.33 25.70 4.33 2.7 1 9.07 9.03 20.54 21.00 2.13 27.34 3.74 42.96 2.90 46.33 13.05 8.00 50.60 14.30 8.14 82.00 64.4 I 66.87 16.88 1.46 4.49 6.40 18.52 19.61 24.65 4.14 26.27 186.69
(PC%) in cm’ for
fiber length-cm’) 3 2 1.92 1.97 9.75 10.91 6.97 5.93 14.59 9.60 8.10 13.09 10.43 9.64 7.45 8.76 1.97 8.82 3.70 1.51 4.88 9.30 1.24 9.16 0.10 1.45 9.12 0.79 9.20 2.68 13.99 3.12 2.46 17.04 8.60 4.69 17.20 16.48 15.60 8.48 4.04 2.12 5.86 8.96 5.29 7.61 1.23 18.89 57.72
of muscle force predictions
3.73 3.74 IS.30 4.24 11.35 29.36 10.25 35.33 15.19 16.48 11.55 12.70 7.09 5.50 5.35 18.36 14.27 2.14 18.96 16.12 7.11 5.12 8.79 1.24 47.99 4.30 54.07 4.80 93.38 9.52 7.13 63.36 34.50 19.35 87.91 147.75 45.65 39.53 23.55 11.93 12.91 26.53 20.08 39.92 5.31 50.85 187.15
Maxt 1 1 1 1 1 3 1 3 I I I I 2 2 1 1
1 1 3 3 1 1 1 1 3 3 3 3 3 1 I 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3
* l-male specimen; 2-female specimen; 3-PCSA given by Pierrynowski. : Max indicates which of the three data sets had the largest PC%.
ual based on data from a group of cadaver specimens using height or weight alone as scaling factors. The three sets of muscle force solutions were generally similar throughout the gait cycle; i.e. the various muscle groups had similar peaks and valleys (Fig. I). Large forces were generally predicted in large muscles and small forces in small muscles (Table 2). However, for a given muscle we could not predict which of the three PCSAs would result in the largest muscle force. The correlation coefficients, r, for the linear correlations of muscle force to PCSA for
591
the male specimen. the female specimen and Pierrynowski’s data were 0.40, 0.40 and 0.27 respectively. The absence of strong correlations are emphasized by the most extreme cases. When a nontrivial force (i.e. over 25 N) was predicted with all three data sets. the peak muscle forces could be two to eight (e.g. soleus and gluteus maximus 2) times greater for one of the PCSAs than for another (Fig. 2). Regardless of the significant sensitivity ofthe muscle force solutions to the choice of PCSA. the peak hip contact solutions were much less sensitive. The peak joint forces for our male specimen, our female specimen, and Pierrynowski‘s data were 5.2 times body weight (BW), 5.0 BW and 4.7 BW respectively (i.e. the largest was 111~~of the smallest). In addition to the relatively small differences in peak hip contact forces, the peak hip contact forces occurred shortly after heel strike in all three cases. DISCUSSION It is obvious that a muscle’s ability to generate force is in some way related to its size. However, the nature of this relationship is not well understood. The concept of physiologic cross-sectional area (PCSA) represents a good first approximation of the relationship, but the fact that there are differing definitions of PCSA in and of itself implies that there are differing (or at least evolving) ideas as to the physiologic implications of muscle size. The presence of differing fiber types also implies that current concepts of PCSA alone do not strictly define a muscle’s ability to generate force. However, at the present time, there are no other reasonably straightforward concepts which allow an estimation of a muscle’s force generating capability. Muscle forces cannot be readily measured in uioo owing to obvious technical, biological and, in the case of humans, ethical and legal problems. However, it is desirable to be able to know muscle forces during a variety ofactivities and under many conditions, and we do have methods to mathematically estimate them. Most of these methods require some estimate of the muscle’s ability to generate force, i.e. a knowledge of its PCSA (Penrod ef al., 1974; Chao and An, 1978; Pedotti et al., 1978; Takashima et al., 1979). However, even assuming that we knew the best definition of PCSA and its precise relationship with force generating ability, we still would have no way of accurately determining the PCSA of all muscles in a living subject whose muscle forces we wished to know. Therefore, it is desirable to know a range of muscle and joint force predictions which might reasonably be expected when making differing assumptions about a subject’s PCSA. Not unexpectedly, our data suggests that a muscle force solution can be very sensitive to PCSA. As noted in the introduction, it is possible to utilize optimization techniques to predict muscle forces without accounting for muscle size. Those techniques, while useful for demonstrating particular aspects of the formulation of
592
R. A. BRAND.D. R. PEDERSEN and J. A. FRIEDERICH
optimization procedures, have not to date allowed reasonable estimates of muscle forces (e.g. Seireg and Arvikar. 1975; Hardt, 1978). Muscle force solutions will be dependent not only on PCSA but also on the particular way in which the problem is formulated (e.g. the choice of optimization criterion). Additionally, muscle force solutions can depend not only on PCSA values, but also other modeled neurophysiological variables and on moment arm values. The effects of
one cannot be simply and clearly isolated from the other. Methods using muscle stress (as contrasted to muscle force) in the optimization criterion demonstrate sensitivity to PCSA, particularly with a linear formulation of muscle stress or with an exponent of two or less. However, as the exponent increases beyond three, the solution becomes less sensitive to the value of the exponent (Crowninshield and Brand, 1981). Since it is presently impossible to determine the
(4
Fig. 1. (a) and (b)
Sensitivity of muscle force predictions
/
(4 Fig. 1. Muscle force solutions throughout the gait cycle using each of the three PCSA data bases. (a) Male specimen, (b) female specimen, (c) Pierrynowski’s data. The gait cycle is divided into multiple time intervals of the stance and swing phase of gait on one axis. The second axis represents the 47 individual muscles in the muscle model and are shown in the same order as theTables. The third (vertical) axis is force (N)and the peak forces correspond to the forces presented in Table 2. While the illustration connects the force predictions for subsequent muscles at identical times, it should be noted that these force predictions are discrete numbers, rather than continuous functions; the connections are made merely to provide time interval marks throughout the illustration.
Gale
cycle
Fig. 2. (A-D)
594
R. A. BRAND. D. R. PEDERSENand J. A. FRIEDERICH
F
ObcIn,
Fig. 2. Three muscle force solutions throughout the gait cycle for threesets ofsimilar solutions (A-C) and the three most dissimilar solutions (D-F). PC1 indicates solutions for our male specimen; PC2 indicates solutions for our female specimen; SKY indicates solutions for Pierrynowski’s data. The dotted vertical line marks toe off (TO) at about 58 “/! of the gait cycle.
of these muscle force solutions,
Heel Strike
Toe Off
Hv3l Strike
Fig. 3. Joint force solutions for the hip from the studies of Paul (1965) and Seireg and Arvikar (1975) and this study. These three solutions were obtained using very different techniquesand data: Paul used gait data from a living subject, but instead of using an optimization technique to calculate muscle forces. he grouped various musclesto create a series of determinate problems; Seireg and Arvikat, on the other band, used a linear optimization technique to calculate muscle forces, but used hypothetical subject data; our data were derived from a living subject and used a nonlinear optimization technique.
PCSAs of the muscles in living subjects, we must recognize that any muscle force solution is subject to a range of error at least as great as that reported here. These errors limit the usefulness of the absolute values
although
certain
information regarding trends and relative muscle forces (Johnston et al., 1979; Brand and Crowninshield. 1980). It appears that the peak hip contact force solution is much less sensitive to changes in PCSA than is the muscle force solution, although there are seine differences in when the peak occurs during the gait cycle and slight differences in direction of loading for those peaks.This is not unexpected when one imagines that if one muscle is smaller in one person compared to another, all else being equal, another muscle must simply ‘take up the slack’ to maintain equilibrium (or in a strict sense, for a given circumstance the solution is always constrained by the intersegmental resultant joint force and moment regardless of the PCSA). Additionally, the fact that reasonably similar hip joint forces have been predicted with very different techniques and subject input data, suggests a relative insensitivity of joint force predictions to the muscle force solution (Fig. 3). Thus. muscle forces could be quite different without joint forces being radically different. This relative insensitivity of the joint force solution has implications for studies using such information: joint force problems, being less sensitive to such errors, may generally be more reasonable to study using current techniques than problems in which only muscle force solutions are considered. parametric
studies may still provide reasonable
Acknowledgement-The
authors wish to thank
Mrs. Rose
Britton for preparation of the manuscript. This research was supported in part by NlH AM 14486.
Grant
Sensitivity
of muscle force predictions
Table 2. The peak muscle forces predicted for each muscle for the three data sets shown in Table 1
Muscle Adductor brevis (S) Adductor brevis (I) Adductor longus hdductor magnus 1 Adductor magnus 2 Adductor mngnus 3 Gluteus maximus I Gluteus maximus 2 Gluteus maximus 3 Glutcus medius I Gluteus medius 2 Gluteus medius 3 Gluteus minimus I Gluteus minimus 2 Gluteus minimus 3 lliacus Psoas Inferior gemellus Obturator externus 0b:urator internus Pectineus Piriformis Quadriceps femoris Superior gemellus Biceps femoris Gracilis Rectus femoris Sartorius Semimembranosus Semitendinosus Tensor fascia latae Gastrocnemius (M) Gastrocnemius (L) Biceps femoris (S) Vastus intermedius Vastus lateralis Vastus medialis Tibialis anteriosus Ext dip. comm Ext ha?1 long Flexor die Flexor hall long Peronius brevis Peronius longus Peronius tertius Tibialis posterior Soleus
1* 1 2 217 0 10 96 329 147 82 601 264 240 95 87 74 487 505 II 2 28 70 203 0 5 429 5 290 45 697 139 650 208 480 661 62 58 527 535 372 424 16 923 66 1206 17 187 1008
Muscle forces (N) 2 3 9 13 320 13 20 16 383 56 42 494 284 221 286 295 186 495 146 19 51 140 23 215 0 17 363 3 302 88 612 22 379 284 539 915 47 160 591 416 640 325 44 1041 56 1164 13 208 868
0 2 52 0 0 2 254 451 11 520 332 336 133 81 74 451 145 3 31 al 34 39 0 5 182 a 507 56 625 12 270 502 710 1017 174 37 674 212 735 423 30 979 69 934 13 181 473
* l-male specimen; 2-female specimen; I-PCSA by Pierrynowski. t Max indicates which of the three data sets resulted largest peak muscle force.
Maxt 2 2 2 2 2 1 2 3
I 1 3 3 2 2 2 2 1 2 2 2 1 2 2
I 3 3 2 1 1 1 3 3 3 3 2 3 1 3 1 2 2 3 1 1 2 1 given in the
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