Pergamon

0021-9290(95)ooo55-0

J Biomechanics. Vol. 29. Nu 3 pp 331 341. 1996 Elsevier Saence Lrd

Printed in Grea! Britain 0021 -9290’96 $15 c&l + .Nl

HUMAN WRIST MOTORS: BIOMECHANICAL DESIGN AND APPLICATION TO TENDON TRANSFERS

G. J. Loren,* S. D. Shoemaker,* T. J. Burkholder,* M. D. Jacobson,* J. FridCnf and R. L. Lieber* *Departments of Orthopaedics and Bioengineering, Biomedical Sciences Graduate Group, University of

California and Veterans Administration Medical Centers, San Diego, U.S.A.; and tDepartment of Orthopaedi~,~~vision of Hand Surgery, Giiteborg University, Sweden.

Abstract-Moment am, muscle architecture, and tendon compliance in cadaver& human forearms were deter- mined and used to model the wrist torque-joint angle relation (i.e. wrist torque profile). Instantaneous moment arms were calculated by differentiating tendon excursion with respect to joint rotation. Maximum isometric tension of each wrist muscle-tendon unit was predicted based on muscle physiological cross-sectional area. Muscle forces were subsequently adjusted for sarcomere length changes resulting from joint rotation and tendon strain. Torque profiles were then calculated for each prime wrist motor (i.e. muscle-tendon unit operating through the corresponding moment arm). Influences of moment arm, muscle force, and tendon compliance on the torque profile of each motor were quantified. Wrist extensor motor torque varied considerably throughout the range of motion. The contours of the extensor torque profiles were determined primarily by the moment arm-joint angle relations. In contrast, wrist flexor motors produced near-maximal torque over the entire range of motion. Flexor torque profiles were less influenced by moment arm and more dependent on muscle force variations with wrist rotation and with tendon strain. These data indicate that interactions between the joint, muscle, and tendon yield a unique torque profile for each wrist motor. This information has significant implications for biomechanical modeling and surgical tendon transfer.

Keywords: Moment arm; Muscle architecture; Sarcomere length; Tendon; Wrist joint strength; Tendon transfer.

1NTROI)UCTION

Strength is the most common clinical parameter used to assess neuromuscular function. Because strength results from interactions between the joint, muscle, and tendon, it may be altered by variations in one or more of these factors. Accurate interpretation of strength, therefore, requires an understanding of joint kinematics, muscle tension, tendon compliance, and their interactions.

Muscle tension applied to a moment arm produces joint torque. Maximum muscle force is determined prim- arily by muscle architecture (Powell er at., 1984; Roy et al., 1991), though muscle force generation is considerably influenced by sarcomere length (Gordon et af., 1966) and tendon behavior (Zajac, 1989). A moment arm is deter- mined by the line of muscle-tendon unit force and the center of joint rotation. Alterations in either muscle force or moment arm affect torque output. For example, joint torque may be diminished by tendon subluxation that decreases the moment arm or as a consequence of neur- opathy or myopathy that compromises muscle force gen- eration.

Few studies have addressed the interactions between the joint, muscle, and tendon in producing torque (Hoy et al., 1990, Lieber and Shoemaker, 1992). Such informa- tion regarding the biomechanical determinants of

Received injinalform 21 March 1995. Address correspondence to: Richard L. Lieber, Ph.D., Depart-

ment of Orthopaedics, U.C. San Diego School of Medicine and V.A. Medical Center, 9500 Gilman Drive, La Jola, CA 92093- 9151, U.S.A.

strength is requisite for understanding normal wrist func- tion and for planning surgical procedures in which muscle-tendon units are transferred from one insertion to another. Literature describing normal operating ranges of the wrist motors is scarce and related data concerning tendon transfers is necessarily vague. Opera- tive techniques recommended to establish ‘normal ten- sion’ (Brand, 1985; Mayer, 1916) during tenorrhaphy (i.e. end-to-end tendon attachment) are not precise, since it is often assumed that muscles will simply remodel to the altered level of use and biomechnical requirements (Will- iams and Goldspink, 1973).

Thus, the purpose of this investigation was to measure the instantaneous moment arms of the prime wrist mo- tors and combine these data with muscle forces predicted from architecutral and biomechanical information to model the human wrist torque profile. The results permit discussion of torque motor design and provide a ration- ale for surgical restoration of wrist function. Portions of this work have been presented (Jacobson et al., 1993; Loren and Lieber, 1994).

METHODS

Fresh cadaveric specimens were intact from the mid- humeral level and free of arthritis and other apparent musculoskeletal defects. The same five human upper ex- tremities were used for determination of moment arms, muscle architecture, and tendon compliance. This deci- sion was made because joint, muscle, and tendon proper- ties may complement one another within a given speci- men to produce a torque different than that predicted by

331

332 G. J. Loren et al.

considering the average properties of each component (Griffiths, 1989: Hoffer et al., 1989; Hoy et al., 1990). The prime wrist muscle-tendon units (MTUs)-extensor carpi radialis brevis (ECRB), extensor carpi radialis lon- gus (ECRL), extensor carpi ulnaris (ECU), flexor carpi radialis (FCR) and flexor carpi ulnaris (FCU)-were identified by forearm dissection. Care was taken to main- tain the integrity of the skin and associated tissues of the wrist with specific attention to the extensor retinaculum.

Moment arm determination

The forearm was mounted onto a mechanical jig (Fig. 1); the distal humerus was secured by Steinman pins to vertical braces while an additional pin engaged the middle third of the radius allowing forearm pronation and supination. Thirty-gauge stainless steel sutures were secured to the distal tendon stumps of each muscle (n = 25, five MTUs from five specimens) and routed subcutaneously over the muscle belly to the media1 or lateral epicondyle recreating the line of force of each muscle. Steel sutures were then secured to toothed nylon cables and connected to nonbacklash gears mounted to potentiometers as described by An et al. (1983) and placed under 500 g tension. Tendon excursions were measured as the individual steel sutures rotated gears interfaced with potentiometers, providing voltage

changes which corresponded linearly to suture and thus tendon excursion. A nonlinkage electrogoniometer (Penny and Giles, M series twin-axis goniometer) placed over the radiocarpal articulation measured joint angle (4) in either the sagittal or corona1 plane. The elec- trogoniometer was secured to the distal radius and third metacarpal with Steinman pins and custom mounts. In pilot studies, explicit comparison between setting of elec- trogoniometer and radiographic joint angle was shown to agree within 2.4 f 2.1” (mean + S.D.). Neutral (0”) was defined by alignment of the third metacarpal and the distal radius for both the corona1 and sagittal planes. The sensors were interfaced to a Macintosh IIci computer (Apple Computer Inc., Cupertino, CA) using the Super- Scope application and MacADIOS II analog interface (Version 1.0, GW Instruments Inc., Somerville, MA).

The arm was positioned with the elbow in 90” of flexion and the wrist passed manually from flexion (pal- marflexion) to extension (dorsiflexion) and from radial to ulnar deviation in either supination, pronation, or neu- tral forearm rotation. Individual tendon excursions of the five MTUs and joint angular displacements were meas- ured simultaneously through the range of flexion-exten- sion and radial-ulnar deviation in each of the three fore- arm rotations. Each of these six conditions was repeated three times for a total of 18 data sets for each MTU with

TO

TENWN EXTENSoR i i

/’

Fig. 1. Apparatus used to measure wrist tendon excursions and joint angle. Tendon insertions and MTU lines of force were preserved along with associated soft tissues. Differentiation of tendon excursion data

with respect to joint angle yielded moment arm-joint angle relations.

Biomechanical determinants of wrist joint strength 333

each condition generating approximately 180 data points and fiber, to change as a function of muscle length. The through the range of wrist motion.

Tendon excursion vs joint angle data were differenti- reference state for each muscle was defined by the meas- ured architectural properties. For any muscle length,

ated with respect to joint angte (An et a!., 1983) yielding fiber length can be derived from the area of the triangular moment arm as a function of joint angle. Moment ‘muscle’, arm-joint angie relations were fit by stepwise ~lynomial regression using an algorithm developed to minimize the influence of the fitting method on the resulting equation (Burkholder and Lieber, in press). This was done by

and the law of sines,

including only the polynomial terms which significantly improved the curve fit (F-to-enter = 4.00) and not requir- L Lf L =-

sin[l80-(cc+/?)] sin[fl=&’ (21 ing all lower order terms to be included beneath the highest order term. Peak moment arm and angle at peak moment arm were calculated from these functions. Mean

where /I represents the angle between the fibers and the

moment arm-joint angle relation for a given MTU in muscle line of action, and tt is the angle between the

each forearm rotation was subsequently obtained by aponeurosis and the muscle line. Equation (2) yields

fitting the set of moment arm curves of the five specimens /I and Lf from L,, L,, and ~1. The normahzed

by stepwise polynomial regression. This method of mo- force-length property was based on the relation present-

ment arm calculation was calibrated by measuring plexi- ed by Cutts (1988). The force-length relation plateau

glass templates with moment arms varying from 5 to extended from sarcomere lengths of 2.5-2.8 ,um. To ac-

15 mm. Our calculated values were within 1.4 + 1.8 mm count for tendon lengthening due to muscle force genera-

of measured values and showed no systematic deviation tion, tendon force was iteratively matched to muscle

as a function of moment arm magnitude. force while MTU length was held constant. The isometric joint torque profile was then calculated by simply multi-

Muscle architecture and tendon properties plying estimated muscle force by the corresponding mo-

Muscle architecture was subsequently determined ac- ment arm. Mean sarcomere operating ranges of the prime MTUs for the five specimens were determined

cording to the methods developed by Sacks and Roy indepedently of the ensemble average muscle force (1982) as previously implemented (Lieber ef nt, 1990, joint-angle and torque-joint angle relations. 1992b). MTUs were dissected free and fixed in 10% buffered Formalin. Muscle length (L,) was measured as Statisticul analysis the distance from the origin of the most proximal muscle fibers to the insertion of the most distal fibers. The

To estimate the influence of each torque dete~inant

tendons were removed and measured (L,), and the (i.e. moment arm, muscle force, or tendon compliance) on

muscles weighed (tw). Then, utilizing the L,:L, ratios the torque profile, simulations of the torque profiles of

previously reported (Lieber et al., 1990; since these values the five motors were created using either a constant mo-

demonstrated little interspecimen variability), muscle ment arm or invariant muscle force or inelastic tendon.

fiber length (Lf) was calculated. Surface pennation angle Each of these reduced modeis was compared to the

(0) was also incorporated from the previous study. corresponding torque profile determined from the meas-

Physiological cross-sectional area (PCSA) was calculated ured moment arm, predicted muscle force, and physiolo-

(Sacks and Roy, 1982) to predict maximum tetanic ten- gical tendon compliance. Coefficients of determination

sion ( PO) based on a muscle specific tension of 0.25 MPa (r’) were calculated to quantify the influence of each

(Close, 1972; Powell et al., 1984). Following determina- determinant on the torque profile (Lieber and Shoe-

tion of sarcomere lengths in each fixed muscle specimen maker, 1992). Thus, an r2 value of 0.99 for a constant

by laser diffraction (Lieber et al. 1990, 1984), L, and Lf moment arm simulation (i.e. incorporating a constant

were normalized to a sarcomere length (L,) of 2.5 pm to moment arm with normal muscle force variation and

compensate for length changes occurring during fixation. physiological tendon strain) indicated that the variability

Tendons from each MTU were loaded to their respect- in the torque profile was not greatly influenced by mo-

ive PO to determine individual tendon compliance under ment arm variability with joint rotation. If the coefficient

physiological tensions (Loren and Lieber, 1995). of determination were lower, we would infer a greater influence of moment arm on the torque profile shape.

Wrist torque model Comparisons among peak joint torque, moment arms,

and wrist angles in the various forearm rotations were The muscle model was based on an arbitrary muscle performed using two-way analysis of variance (ANOVA),

with variable architectural and fiber type properties, sim- using MTU and forearm rotation as grouping variables. ilar to that of Zuurbier and Huijing (Zuurbier and Huij- Multipie paired comparisons were performed ~usf-~oc ing, 1992). Briefly, each muscle was modeled as half using Fisher’s protected least-squares difference method. a parallelogram, with sides of fiber length (Lr), muscle Data were analyzed using StatView software (Version length (L,), and aponeurosis length (La). The area of this 4.0, Abacus Concepts Inc., Berkeley, CA). Significance triangle was held independent of muscle length, which level (a) was selected as 0.05. Data are expressed as requires pennation angle, the angle between aponeurosis mean &- S.E.

334

RESULTS

G. J. Loren er al.

Flexor Torques (Nm)

Moment Arms (mm) Forcei (N, Moment arms

Moment arms varied considerably throughout the range of joint motion. The extension moment arms of ECRB and ECRL were smallest in wrist flexion and increased nearly linearly with progressive wrist extension (Fig. 2). Radial moment arms were maximum with the wrist deviated toward the ulna (Fig. 4). ECU extension moment arm was greatest near neutral wrist extension (Fig. 2). With forearm rotation from supination to prona- tion, maximum ECU ulnar moment arm significantly increased (p < 0.01) while maximum ECU extension mo- ment arm significantly decreased (p < 0.01).

Flexor moment arms were greatest with the wrist flexed and decreased with extension (Fig. 3). Maximum FCR radial moment arm occurred in ulnar deviation (Fig. 4). Conversely, maximum FCU ulnar moment arm was noted in radial deviation (Fig. 5).

Predicted muscle forces

Wrist extensor MTUs were predicted to operate prim- arily on the plateau of the sarcomere length-tension curve (Fig. 6). Only the ECRB was predicted to operate at sarcomere lengths corresponding to less than 80% PO in the normal range of motion.

Wrist flexor MTUs were predicted to operate pre- dominantly on the ascending and steep ascending limbs of the length-tension curve. Peak muscle forces for both

Extensor Torques (Nm)

Moment Arms (mm) Forces (N) lea

CRB

CRL

cu

-40 -20 0 20 40

Flexion Joint Angle Extension

Fig. 2. Determinants of extension torque. Torque profiles oi extensor MTUs are shown enlarged with moment arm-joint angle relations and muscle force-joint angle relations provided as insets. Note the considerable influence of ECRB moment arm variability with joint rotation on the torque profile. (Data pre- sented for neutral forearm rotation. Shaded area represents

SEM of five cadaveric specimens.)

-40 -20 0 20 40

Flexion Joint Angie Extension

Fig, 3. Determinants of flexion torque. Torque profiles of exten- sor MTUs are shown enlarged with moment arm-joint angle relations and muscle force-joint angle relations provide as in- sets. Note the considerable influence of muscle force variability on the torque profiles. (Data presented for neutral forearm

rotation.)

the FCR and the FCU occurred in full wrist extension (Fig. 6). In radial-ulnar deviation as in flexion-extension, the wrist flexors achieved maximal muscle forces at joint angles corresponding to long muscle lengths, i.e. the FCR in ulnar deviation and the FCU in radial deviation (Figs 4 and 5).

Determinants of wrist joint torque

Wrist extensor torque varied markedly as a function of joint angle (Fig. 2). Extensor torque profiles were domin- ated by the moment arm-joint angle relations (Table 1). For example, in a simulated ECRB torque motor with invariant muscle force, variability in the moment arm with joint rotation correlated highly with the torque profile (r’ = 0.990). Similar relationships were noted for the ECRL (r* = 0.996) and ECU (r’ = 0.981). Muscle force-joint angle curves generally contributed less to extensor torque profiles (Table 1). An ECRL or ECU torque motor with constant moment arm produced a torque-joint angle relation that correlated poorly with the predicted torque profile (r2 = 0.482 or r* = 0.322, respectively), indicating a small influence of muscle force variability on joint torque. Given the large ECRB muscle force variability with wrist rotation, an expectedly larger influence of muscle force on torque output was apparent (r* = 0.960). The influence of tendon compliance on ex- tensor torque profiles was negligible (r2 range, 0.954- 0.991; Table 1).

Biomechanical determinants of wrist joint strength 335

Radial Deviation Torque (Nm)

Momem Am (mm) Force(N) 3 *I

ECRL

FCA

-10

Radial

-5 0 Joint Angle

5 10 15

Ulnar

Fig. 4. Determinants of radial torque. Torque profiles of exten- sor MTUs are shown enlarged with moment arm-joint angle relations and muscle force-joint angle relations provided as

insets. (Data for neutral forearm rotation.)

Ulnar Deviation Torque (Nm)

Force(N)

ECU Moment Am (mm) 1.0

FCU

4 y”.5t I

&/ -1 -5 0 10 15 5, -10 -5 0 5 10 15

Joint Angle

’ Sarcoiere Lew$h 4 5 (pn)

Fig. 6. Operating ranges of the wrist motors on the isometric sarcomere length-tension relation, Extensors operated prim- arily on the plateau region while the flexors operated predomi- nantly along the ascending and steep ascending limbs. Lines at end of shaded areas represent the standard error for the data set. (Data presented for flexion-extension in neutral forearm rota- tion.) Mean sarcomere operating ranges were determined inde- pendently of the ensemble average muscle force- and torque-joint angle relations. Operating ranges when plotted against the isometric force-length relation may deviate slightly

from the ensemble average force-joint angle relations.

extension were attenuated by decreasing moment arms (Fig. 3). Consequently, torque profiles were dominated by neither muscle force nor moment arm relations (Table 1). Tendon compliance had a greater effect on the flexor torque profiles compared to those of the extensors. This resulted from the relatively greater influence of muscle force on the flexor torque profile with tendon compliance permitting muscle force. changes.

Moment arm variability with radial-ulnar deviation compared to flexion-extension was less influential on torque profiles of the radial extensors (Table 1). The muscle force-joint angle relation, though, with radial deviation in the constant moment arm simulation con- siderably influenced the ECRB and ECRL torque pro- files (r’ = 0.875 and r2 = 0.808, respectively). Further- more, muscle force variability substantially influenced FCR radial torque profile (r* = 0.839) although the mo- ment arm curve was the primary determinant (r2 = 0.999). Given the relatively constant ECU moment arm with ulnar deviation, the ECU ulnar torque profile was dominated by the muscle force-joint angle relation (r2 = 0.999). The FCU ulnar torque profile, like the flexion profile, was influenced by both moment arm and muscle force variability (r2 = 0.994 and r2 = 0.934, re- spectively).

Fig. 5. Determinants of ulnar torque. Torque profiles of exten- sor MTUs are shown enlarged with moment arm-joint angle relations and muscle force-joint angle relations provided as

insets. (Data for neutral forearm rotation.)

Flexor torque demonstrated comparatively less varia- bility with wrist rotation. The virtually constant flexor torque was not due to invariant muscle force and con- stant moment arm. Rather, increasing muscle forces with

DISCUSSION

The purpose of this investigation was to define the biomechanical basis of the isometric joint torque profiles of the human wrist motors and to use this information to provide guidelines for tendon transfers. We incorporated the angular dependence of moment arm and muscle force and accounted for tendon compliance to illustrate the interactions between these factors. Torque output for the

33h G. J. Loren et al.

Table 1. Wrist motor torque profile coefficients of determination (r’)* __--

Flexion-extension Radial-ulnar deviation

Constant Invariant Constant Invariant moment muscle Inelastic moment muscle Inelastic

arm force tendon arm force tendon

ECRB 0.960 0.990 ECRI.‘ 0.482 0.996 ECU 0.322 0.981 FCR 0.307 0.123 FCU 0.943 0.790

*Data presented for neutral forearm rotation.

0.954 0.875 0.684 0.998 0.991 0.808 0.288 0.985 0.976 0.999 0.000 0.922 0.415 0.839 0.999 0.999 0.444 0.933 0.994 0.990

wrist motors was not constant throughout the physiolo- resulted in large sarcomere excursions and greater gical range of motion, reflecting varying influences of muscle force changes with joint rotation (Lieber and moment arm, muscle force, and tendon strain. Brown, 1993).

Moment arm magnitudes compared favorably to those previously reported for the wrist (Horii et al., 1991; Tolber et al., 1985; Youm et al., 1976). As suggested by Brand (1985), we found considerable moment arm varia- bility throughout the range of joint motion. Previous studies which have reported constant or linearly varying moment arms (Horii et at., 1991; Ketchum et ai., 1978; Youm et al., 1976) may not, therefore, accurately repres- ent normal wrist kinematics. Forearm rotation only sig- nificantly aftered ECU moment arm; an anatomical basis has been previously described (Brand, 1985; Youm et al.,

1976). Predicted muscle force changes with wrist rotation

variably influenced flexor and extensor torque profiles. Muscle force generation reflected the operating range of each muscle on the sarcomere length-tension relation. Sarcomere excursion with joint motion was determined by muscle fiber length and moment arm magnitude. Fiber lengths and moment arm magnitudes, however, varied among the wrist motors so muscle and sarcomere length changes with joint rotation were unique (Table 2). For example, the muscle with the shortest fibers was FCW. Of the prime wrist motors. FCU had a peak flexion-extension moment arm of intermediate magni- tude. Consequently, the FCU fiber length to maximum moment arm ratio (L,: r) was the lowest (Table 2). As the wrist was flexed, sarcomere length change per degree of joint rotation (d&/d& was greatest for FCU, resulting in substantial sarcomere length variation and, therefore, muscle force change over the physiological range of motion. It follows that maximal muscle tension was gen- erated over a limited angular range. Conversely, ECRL had the longest fiber length of the prime wrist motors, an intermediate extension moment arm, and the highest L,: r (Table 2). As the wrist extended, dLs/dc# was minim- al. Muscle force generation was virtually constant as sarcomeres remained on the plateau of their length-ten- sion relation. High &:r was therefore indicative of small sarcomere length changes per degree of joint rotation and consequently minimal muscle force variability throughout the range of motion. Conversely, low Lf:r

The most common action of the human wrist is rota- tion from extension-radial deviation to flexion-ulnar de- viation. Moment arm magnitudes, though, for this com- bined motion of the wrist motors were not defined in this study. Utilizing the vector sum of the ffexion-extension and radial-ulnar deviation moment arms at 0” of wrist rotation, combined moment arm magnitudes were esti- mated. Of the radial extensors, ECRB had a greater combined moment arm. Lf:r and dLs/d$ for ECRB and ECRL were comparable to values obtained for wrist extension, The combined FCU moment arm, Lf:r, and dLs/dQ, were intermediate to values measured for pure flexion or ulnar deviation (Table 1).

Tendon strain under physiological loads varied sub- stantially among wrist MTUs and was significantly greater for wrist flexors than extensors (Loren and Lieber, 1994, 1995). Muscle fiber shortening at the ex- pense of tendon len~hening (maths, 1989; Hoffer et nl., 1989) skews the sarcomere-length tension curve and al- ters the MTU operating range (Lieber et al., 1991, 1992a; Xajac, 1989). In FCR and FCU, with short fiber lengths and limited angular ranges of maximal muscle force genera- tion, skew of the sarcomere length-tension relations due to tendon compliance increased muscle tensions at greater degrees of wrist extension by allowing sarcomere shortening from the descending limb of the isometric force curve. If the flexors were in series with a noncom- pliant tendon, maximal mu&e forces for FCR and FCU would occur at approximately 19” and 27” less extension, respectively; resulting joint torque would be maximum for FCR at 12” and for FCU at 26” wrist extension (compared to 35” and 39” extension for the compliant tendon). It is not clear how or if such a shift in peak torque angle would alter strength and coordination given that peak flexor torque remained in wrist extension where manual function is optimal (Kraft and JXels, 1972; Pryce, 1980).

Previous investigations relating in uit;o sarcomere lengths to joint motion have reported operating ranges on the ascending limb (Herzog et al., 1991; Rack and Westbury, 1969), the plateau (Lieber et al., 1992~; Rome

Biomechanical determinants of wrist joint strength

Table 2. Wrist MTU properties and torque determinants*

137

ECRB ECRL ECU FCR FGU

PtAW 58.8 i 5.0 Mmm)t 70.8 * 1.7 Tendon E (%)t 1.99 4_ 0.20 -L: Lft 2.89 + 0.11

Flex&-Extension: rmal (mm) 19.3 + 1.2 Angle at rmar 1”) 29 + 2 (E) F,,, (N) 58.8 + 5.0 Angle at F,,, (“) 11 f 1 (E) rmar. (Nm) 1.07 + 0.11 Angle at z,,, (“) 17 + 3 (E) L,:r 3.21 + 0.24 dLdd, (nmi”) 11.4 rt 1.1

Radial-Ulnar Deivation rmar (mm) 16.9 + 0.8 :““‘“(ii rmar (“1 1 rt 3(U)

nax 58.8 If: 5.0 ;.fe$t ;m.%x (“)

rffe r: a?T,., I”)

1.01 14 + + 0.10 1 (U)

0+3(u) 3.62 + 0.16

d&&P (nmi”) 11.5 + 0.6

31.9 f 2.7 127.3 + 5.6

1.78 f 0.14 2.10 f 0.18

15.5 & 0.8 31 +2(E)

31.9 f 2.7 20&4(E)

0.50 + 0.07 30 5 2 (E)

7.48 f 1.39 3.93 + 0.52

23.9 k 0.9 3 &4(U)

31.9 + 2.7 16+0(U)

0.77 k 0.06 2+4(U)

4.74 + 0.71 8.27 + 0.74

51.5 * 3.4 58.8 f 1.7 2.35 + 0.30 3.67 IO.13

8.51 $- 1.00 7+8(E)

51.4 & 3.5 15 f 7 (F)

0.43 + 0.06 5 + 7 (E1

6.07 + 1.13 4.87 + 0.72

29.3 + 1.8 1 k 3 CR)

51.4 * 3.5 6+0(R)

1.51 rf: 0.11 1 + 3(R)

1.65 + 0.14 24.6 f 1.3

Combined Extension-Radial Deviation and Fkxion-Ulnar Deviation: r, (mm) 22.6 f 2.0 14.6 + 1.6 10.2 5 1.3 Lr:r, 2.78 & 0.26 8.19 & 1.6 5.01 * .75

51.9 -t- 3.7 59.8 * 1.5 2.48 & 0.45 3.86 + 0.12

17.3 f 0.55 36 + 5(F)

51.2 * 3.7 45+

0.68 + 0.05 35 rt 7 (E)

3.32 + 0.17 11.5 & 0.5

8.21 * 1.70 1212(U)

40.0 & 2.8 16 +0(U)

0.38 rt 0.09 14 + 2 (U)

12.2 f 6.4 4.48 + 1.27

21.7 f 0.9 2.65 & .13

89.1 * 8.4 41.9 2 I.6

3.68 & 0.31 4.96 + 0.18

f6.8 & 1.4 20 f 17 (F)

87.5 f 8.5 45'

1.19 If 0.18 39 + 4 (E)

2.54 It 0.23 14.2 + 0.6

27.5 + 2.1 9?;2@)

68.6 i 5.6 12 &0.0(R)

2.40 & 0.17 11 ?; 1 (R)

1.57 t- 0.16 22.4 rt 1.0

19.5 4 1.2 2.17 F .14

*Data presented for neutral forearm rotation. iData from Loren and Lieber, 1995. + Peak force occurred at extreme range of motion Abbreviations: P,, maximum tetanic tension; .&, fiber length; tendon s, ph~siolo~cal tendon strain; L,: L,,

tendon length to fiber length ratio; I,,,, rn~rn~ moment arm; F,,,, maximum muscle force; t,,,, maximum joint torque;(F), flexion; (E), extension; (R) radial deviation; (U), ulnar deviation; &: r, fiber length to maximum moment arm ratio; d&/d@, change in sarcomere length per degree ofjoint rotation; rs, combined moment arm at neutral flexion-extension and radial-ulnar deviation; L,:r,, fiber length to combined moment arm ratio. Data expressed as mean + SE.

et al., 1988; Rome and Sosnicki, 1991) and on the de- scending limb (Herzog and ter Keurs, 1988; Lieber and Boakes, 1988; Lieber and Brown, 1993) of the sarcomere length-tension relation. Our data support such special- ization within the muscle-joint system of the human wrist. Wrist flexors operated primarily on the ascending limb with higher sarcomere length changes per degree of wrist rotation while extensors contracted predominately along the plateau region with smaller sarcomere excur- sions. The average slope (d&/dqb) of the sarcomere length-joint angle relations of ECRB (11.4 nmf) and the predicted ECRB sarcomere operating range (2.18-3.33 pm) were almost identical with recent in- traoperative human muscle sarcomere length measure- ments (Lieber et al., 1994) [Fig. 7(A)]. Data obtained on the FCU during wrist rotation also demonstrated an identical d&/d+ as that predicted by the model (Fridkn and Lieber, unpublished data) [Fig. 7(B)] but are offset to longer sarcomere lengths by about 1 pm. This probably reflected that sarcomere lengths were predicted with the elbow flexed to 90” but intraoperative measurements were made with the elbow extended. As these two studies

were performed using independent methodologies and different specimens, they provide strong support for the validity of our model. Wrist motor torque profiles illustrated a complex in-

teraction between skeletal muscle and articular motion. Several authors have demonstrated that maximum muscle force and peak joint torque occur at different joint angles (Herzog et al., 1991; Hay et al., 1990; Lieber and Boakes, 1988; Lieber and Shoemaker, 1992) and that the angle of maximum joint torque is not necessarily the joint angle where the muscle generates maximum force nor the angle where the moment arm is maximized. Wrist exten- sor torque profiles paralleled the moment arm curves with both peak torque and rn~irn~ moment arm oc- curring in wrist extension. The moment arm was poten- tiated by maximal muscle tension in wrist extension, functionally maximizing muscle strength at the expense of restricting the range of wrist motion over which peak torque was produced. In contrast, the opposing angular dependence of the flexor moment arms and muscle forces produced a torque profile with a broad range of near- maximal torque and limited the torque magnitude.

6. J. toren et af.

l

a.01 , z x -90 -60 -30 0 30 60 90

Wrist Angie (degrees) l l

/, ,__, -&I -30 0 30 60

EXt3”SsoOn

Wrist Angle (degrees)

Fig. 7. (A) Comparison between predicted ECRB sarcomere length-joint angle relation (solid line) and that measured in- traoperatively as reported by Lieber et al. 1994 (filled cimles) Sarcomere lengths were corrected for tendon compliance during muscle activation. Predicted slope and sarcomere length range compare favorably to experimental data. (B) Comparison be- tween predicted FCU sarcomere length-joint angle relation (solid line) and that measured intraoperatively (filled circles; Friden and Lieber, unpublished data). Note that the predicted slope compares favorably with experimental data but sareomere length range is offset by about 1 pm. This is probably due to differences in elbow flexion-extension angle for predicted com-

pared to experimental data (see discussion.)

The summated wrist extensor torque was dominated by ECRB, given its largest moment arm and greatest force generating capacity. Although the isometric muscle force of ECU was comparable, the poor mechanical advantage limited its functional contribution as a wrist extensor. Peak extensor torque was 1.8 Nm at 18” of wrist extension. Although FCR generally maintained a larger flexion moment arm, the summed flexor torque profile reflected a greater FCU contribution given the higher muscle forces generated. Summated flexor torque was 1.7 Nm near the limit of wrist extension.

These data provide isolation relevant to the selec- tion of a donor muscle for tendon transfer used to restore lost function. A comparable r * PCSA product is desirable to restore maximum torque capacity and a similar Lf:r ratio should be sought to replicate joint excursion (Zajac, 1992) and the torque profile. Physiolo~ca~ tendon com- pliance and tendon length to fiber length ratio may also be considered. The relative import of such determinants, however, is variable among motors (Table 1) and does not take into account the possibility that the MTUs may adapt (see below).

Because the extensor torque profiles are dominated by the moment arm-joint angle relation, surgical reconstruc-

-40 -20 0 2b 40 sb

Wrist Angle (degrees)

ISO -40 -20 0 20 40 so FlSWil Wrist Angle (degrees) Extension

Fig. 8. Simulated FCU to ECRL tendon transfer torque profile. (A) Muscle force as a function of wrist joint angle for sarcomere lengths 2.1 m(O), 3.Ojun (A), and 3.9 pm(m) at neutral wrist rotation.(B) Wrist joint torque resulting from tendon transfer at different resting sacrcomere lengths: 2.1 pm (*I, 3.0 pm (A), and 3.9pm (a). Note the greater disparity in muscle force-joint angle relations compared to torque profiles. This is because the ECRL moment arm dominates the torque profile. Combined extensor moment for ECU f ECRB $ ECRL is provided {O)

for comparison.

tion of wrist extension is anticipated to have a good clinical result if the MTU line of force is replicated. A common tendon transfer to improve functional wrist extension and restore motor balance in spasticity is transfer of the FCU MTU to the ECRL tendon. The FCU muscle and tendon then act through the ECRL moment arm. As a wrist extensor, FCU operates through a moment arm which is greatest in extension. FCU muscle force generation, though, is substantially in- fluenced by its operating range on the sarcomere length-tension relation.

The tension imposed during reconstruction establishes the resting sarcomere length and thus the active sar- comere operating range. Adjusting the tension of the tendon transfer may profoundly affect the resulting torque profile. A slack tenorrhaphy (e.g. sarcomere length = 2.1 pm at neutral wrist rotation) is predicted to yield an FCU-ECRL motor operating primarily on the amending limb of the sarcomere length tension curve with peak muscle force in wrist flexion [Fig. g(A)] and consequently a markedly compromised joint torque [Fig. 8(B)]. However, tenorrhaphy of the FCU-ECRL tendon under tension (e.g. sarcomere length = 3.0 pm at neutral) may more accurately replicate the pre&ted ECRL sarcomere operating range on the plateau and approximate the extensor torque profile [Fig. 8(A)]. Fur-

Biomechanical determinants of wrist joint strength 339

ther manipulation of the resting sarcomere length during tendon transfer by intraoperative laser diffraction may optimize the magnitude of the torque product. A sar- comere length of 3.9 pm at 0” flexion-extension of the FCU-ECRL motor is predicted to produce a maximum torque in wrist extension comparable to the summated torque of ECRB, ECRL, and ECU, yet the transfer torque profile differs markedly from the normal relation [Fig. 8(B)]. With transfer then, the lost elegance of ECRL motor design is overcome by the greater FCU muscle force. Manipulation of the donor operating range by setting tenorrhaphy length at surgery, then, may estab- lish the desired torque profile and favorably influence the ultimate clinical result.

A general recommendation in tendon transfer surgery is to perform tenorrhaphy under slight traction (Beasley, 1970; Hovius, 1993; Omer, 1968), although the physiolo- gical rationale for this is not clear. It is believed and even tacitly stated that muscles will adapt their sarcomere number (and fiber type distribution?) to the new bi- omechanical requirements imposed upon the transferred unit. However, this may be an overstatement of muscle’s ability to adapt in general. First, the magnitude of muscle adaptation to altered use varies largely between muscles [reviewed in Chap 4 and 5 of Lieber (1992); raw data in Lieber et al. (1988, 1986a,b 1989); Simard et al. (1982); Spector et al. (1982)] and it would not be clear which of the wrist MTUs would adapt to which extent. Second, since transfer alters muscle use and length in a way which is not completely understood, it is not clear exactly how strong a ‘remodeling stimulus’ will been provided to the muscle. Since the subtleties of such adaptations among human wrist MTUs have not been experimentally determined, we cannot incorporate them into the present study.

Application of these results to actual experimental measurements of human wrist torque must be made with caution based on the assumptions built into the study design and biomechanical model. First, we have modeled the muscle as an amplified sarcomere with ideal sar- comere length-tension properties (Gordon et al., 1966). Evidence exists which demonstrates that the descending limb of the length-tension curve may be modified in isolated single fibers due to intersarcomere dynamics (Altringham and Bottinelli, 1985; Lieber and Baskin, 1983; ter Keurs et al., 1978) but it is not clear whether this extends to the whole muscle, where endomysial connec- tions between fibers are present. This uncertainty would only affect our model at sarcomere lengths above 2.8 pm where these phenomena occur. Second, the prime wrist motors are biarticular with definable moment arms at the elbow. Our study excluded this parameter by fixing the elbow 90“ of flexion where most manipulative activity occurs. Given the available data on elbow moment arms (Brand, 1985), we anticipate that the effects of elbow motion on the torque profile are greatest for ECRL with diminishing effects for FCR, FCU, and ECRB and ECU. If elbow moment arm were considered, the ECRL would operate at sarcomere lengths closer to the plateau of the length-tension curve with elbow flexion and at even longer sarcomere lengths with elbow extension (ECRB

and ECU with negligible elbow moment arms would be less effected). The opposite situation would be true for the wrist flexors-FCR and FCU would operate at sar- comere lengths closer to the plateau with elbow flexion and at even shorter lengths with elbow extension. Obvi- ously, further experimentation is required to quantify the magnitude of this effect. Third, since the length-tension relationship is only strictly valid for maximally activated muscle fibers, the model may not predict the shape of the torque profile during submaximal activation. Since force-length properties of submaximally activated units generally show optimal length at longer muscle lengths (Heckman et al., 1992; Rack and Westbury, 1969) consid- eration of this factor would skew operating ranges to longer sarcomere lengths. Finally, torque magnitudes predicted in this study appear low compared to experi- mentally measured values (Grierson et al., in press). This probably reflects the use of cadaveric specimens from elderly subjects and a relatively low specific muscle ten- sion (0.25 MPa) compared to that measured for intact human muscle (Fukunaga et al., 1992; Schantz et al., 1983). It should also be noted that experimental measure- ment of wrist flexion and extension torque inherently include a substantial torque contribution of the digital motors.

In summary, this investigation integrated moment arm determinations with previous reports detailing muscle architecture (Lieber et nl., 1990) and tendon bi- omechanics (Loren and Lieber, 1995) to understand the determinants of wrist strength. Extensor strength was primarily dependent on the moment arm-joint angle relation while flexor torque was influenced both by muscle architecture and tendon compliance. The contri- butions of a MTU’s biomechanical design to the joint strength profile therefore deserves emphasis when con- sidering restoration of extremity function. Furthermore, surgical manipulation of sarcomere length during tendon transfer may assist in replicating the desired torque pro- file of the recipient (nonfunctional) motor and favorably influence the functional result in the reconstructed limb.

Acknowledgements-This work was supported by the Veterans Administration and NIH grant AR35192. The authors acknow- ledge Drs Paul Brand, Michael Botte, Reid Abrams, David Pierotti and Richard Braun for helpful suggestions and dis- cussions, Paul Yeatman and Brett Sokoloff for data analysis, and Christian Giangreco and John Butler for technical assist- ance. We thank Drs Scott Delp and Thomas Buchanan (North- western University) for access to their experimental wrist torque data.

REFERENCES

Altringham, J. D. and Bottinelli, R. (1985) The descending limb of the sarcomere length-force relation in single muscle fibers of the frog. J. Muscle Res. Cell Mod. 6, 585400.

An, K. N., Ueba, Y., Chao, E. Y., Cooney, W. P. and Linscheid, R. L. (1983) Tendon excursion and moment arm of index finger muscles. J. Biomechanics 16, 419-425.

Beasley, R. W. (1970) Tendon transfers for radial nerve palsy. Orthop. Clin. North Am. 1, 439-445.

Brand, P. W. (1985) Clinical Mechanics of the Hand. Mosby, St. Louis.

340 G. J. Loren et al.

Bukholder, T. J. and Lieber, R. L. Stepwise regression as an alternative to spines for fitting noisy data. J. Biomechanics (in press).

Close, R. I. (1972) Dynamic properties of mammalian skeletal muscles. Physiol. Rev. 52, 129-197.

Cutts, A. (1988) The range of sarcomere lengths in the muscles of the human lower limb. J. Anat. 160, 79-88.

Fukunaga, T., Roy. R. R., Shellock, F. G., Hodgson, J. A., Day, M. K., Lee, P. L., Kwong, F. H. and Edgerton, V. R. (1992) Physiological cross-sectional area of human leg muscles based onmagnetic resonance imaging. J. Orthop. Res. 10, 928-934.

Gordon. A. M.. Huxlev. A. F. and Julian. F. J. f1966) The _I \ ,

variation in isometric tension with sarcomere length in verte- brate muscle fibres. J. Physiol. (Lond.) 184, 170-192.

Grierson, A. E., Delp, S. L. and Buchanan, T. S. Maximum isometric moments generated by the wrist muscles in flexion-extension and radial-ulnar deviation. J. Biomechan- its (in press).

Lieber, R. L., Brown, C. G. and Trestik. C. L. (1992a) Model of muscle-tendon interaction during frog semitendinosis hxed- end contractions. J Biomechanics25, 421-428.

Lieber. R. L.. Fazeli. B. M. and Botte. M. J. (1990) Architecture ofselectedwrist flexor and extensor muscles. J. Hand Stag. 15, 244-250.

Lieber, R. L., Friden, J. O., Hargens, A. R.. Danzig. L. A. and Gershuni, D. H. (1988) Differential response of the dog quad- riceps muscle to external skeletal fixation of the knee. Muscle Nerve 11, 193-201.

Lieber, R. L., Friden, J. O., Hargens, A. R. and Feringa. E. R. (1986a) Long-term effects of spinal cord transection of fast and slow rat skeletal muscle--II. Morphometric properties. Exp. Neural. 91, 435-448.

Griffiths, R. I. (1989) The mechanics of the medial gastrocnemius muscle in the freely hopping wallaby (Thylogale billardierii). J. Exp. Biol. 147, 439-456.

Heckman, C. J., Weytjens, J. L. and Loeb, G. E. (1992) Effect of velocity and mechanical history on the forces of motor units in the cat medial gastrocnemius muscle. J. Neurophys. 68, 1503-1515.

Herzog, W. and ter Keurs, H. E. D. J. (1988) A method for the determination of the force-length relation of selectd in viuo human skeletal muscles. Pfliigers Archives Eur. J. Physiol. 411, 637-641.

Lieber, R. L., Jacobson, M. D., Fazeli, B. M., Abrams, R. A. and Botte, M. J. (1992b) Architecture of selected muscles of the arm and forearm: anatomy and implications for tendon trans- fer. .I. Hand. Surg. 17, 787-798.

Lieber, R. L., Johansson, C. B., Vahlsing, H. L., Hargens, A. R. and Feringa, E. R. (1986b) Long-term effects of spinal cord transection on fast and slow rat skeletal muscle-~-I. Contrac- tile properties. Exp. Neurol. 91, 423-434.

Lieber, R. L., Leonard, M. E., Brown, C. G. and Trestik, C. L. (1991) Frog semitendinosus tendon load-strain and stress-strain properties during passive loading. Am. J. Physiol. 261, C86-C92.

Herzog, W., Read, L. J. and ter Keurs, H. E. D. J. (1991) Experimental determination of force-length relations of in- tact human gasrocnemius muscles. Clin. Eiomech. 6, 230 -238.

Lieber, R. L., Loren, G. J. and Friden, J. (1994) In viuo measure- ment of human wrist extensor muscle sarcomere length cha- nges. J. Neurophys. 71, 874-881.

Lieber, R. L., McKee-Woodburn, T., Fridin, J. and Gershuni, D. H. (1989) Recovery of the dog quadriceps after ten weeks of immobilization followed by four weeks of remobilization. J. Orthop. Res. 7, 408-412.

Hoffer, J. A., Caputi, A. A., Pose, I. E. and Griffiths, R. I. (1989) Roles of muscle activity and load on the relationship between muscle spindle length and whole muscle length in the freely walking cat. Prog.~Brain Res. 80, 75-85. -

Horii. K. N.. An. W. P., Coonev. W. P. and Linscheid. R. L. (1991) Kinematics and tendon excursion of wrist movers. J. Hand Surg. 18A, 83-90.

Lieber, R. L., Raab, R., Kashin, S. and Edgerton, V. R. (1992~) Sarcomere length changes during swimming. J. Exp. Biol. 169, 251-254.

Hovius, S. E. R. (1993) Musculo-tendinous transfers of the hand and forearm. Clin. Neural. Neurosurg. 95, S92-94.

Hoy, M. G., Zajac, F. E. and Gordon, M. E. (1990) A musculos- keletal model of the human lower extremity: the effect of muscle, tendon, and moment arm on the moment-angle rela- tionship of musculotendon actuators at the hip, knee, and ankle. J. Biomechanics 23, 157-169.

Jacobson, M. D.. Loren, G. J., Yeatman, P., Sokoloff, B. and Liebcr, R. L. (1993) Relation between moment arms and muscle properties during wrist joint rotation. Trans. Orthop. Res. Sot. 18, 318.

Lieber, R. L. and Shoemaker, S. D. (1992) Muscle, joint, and tendon contribution to the torque profile of frog hip joint. Am. J. Physiol. 263: R586-590.

Lieber, R. L., Yeh, Y. and Baskin, R. J. (1984) Sarcomere length determination using laser diffraction. Effect of beam and fiber diameter. Biophys. J. 45, 1007-1016.

Loren, G. J. and Lieber, R. L. (1994) Tendon biomechanical properties enhance human wrist muscle specialization. Trans. Orthop. Res. Sot. 19, 691.

Loren, G. J. and Lieber, R. L. (1995) Tendon biomechanical properties enhance human wrist muscle specialization, J. Bio- mechanics 28: 791-799.

Mayer, L. (1916) The physiological method of tendon transplan- tation-11. Operative technique. Surg. Gynecol. Obstet. 22, 298-306.

Ketchum, L. D., Brand, P. W., Thompson, D. and Pocock, G. S. (1978) The determination of moments for extension of the wrist generated by muscles of the forearm. J. Hand Stag. 3A, 205-210.

Omer, G. E. (1968) Evaluation and reconstruction of the forearm and hand after acute traumatic peripheral nerve injuries. J. Bone Jt Surg. SO-A, 1454-1478. . _

Powell. P. L.. Rov. R. R.. Kanim. P.. Bello. M. and Edaerton V. ter Keurs, H. E. D. J., Iwazumi, T. and Pollack, G. H. (1978) The

Length-Tension Relation in Skeletal Muscle: Reoisited in Cross-Bridge Mechanism in Muscle Contraction (Edited by Pollack, G. H. and Sugi, H.). University Park Press, Baltimore.

R. (1984) Predictability of skeletal muscle tension from archi- tectural determinations in guinea pig hindlimbs. J. Appl. Physiol. 57, 1715-1721.

Kraft, G. H. and Detels, P. E. (1972) Position of function of the wrist. Arch. Phvs. Med. Rehab. 53. 272-275.

Lieber. R. L. (1992) Skeletal Muscle Structure and Function: Implications for Physical Therapy and Sports Medicine. Will- iams and Wilkins, Baltimore, MD.

Lieber, R. L. and Baskin, R. J. (1983) Intersarcomere dynamics of single muscle fibers during fixed-end tetani. J. Gen. Physiol. 82, 347-364.

Pryce, J. C. (1980) The wrist position between neutral and ulnar deviation that facilitates the maximum power grip strength. J. Biomechanics 13, 505-511.

Rack, P. M. H. and Westbury, D. R. (1969) The effects of length and stimulus rate on tension in the isometric cat soleus

Lieber. R. L. and Boakes, J. L. (1988) Sarcomere length and joint kinematics during torque production in the frog hindlimb. Am. J. Phvsiol. 254. C759-C768.

muscle. J. Physiol. (Lond.) 2@4, 443-460. Rome, L. C., Funke, R. P., Alexander, R. M., Lutz, G., Aldridge,

H., Scott, F. and Freadman, M. (1988) Why animals have different muscle fiber types. Nature 335, 824-827.

Rome L. C. and Sosnicki, A. A. (1991) Myofilament overlap in swimming carp II. Sarcomere length changes during swim-

Lieber, R. L: and Brown, C. G. (1993) Sarcomere length-joint angle relationships of seven frog hindlimb muscles. Acta Anat. 145, 289-295.

ming. Am J. Physiol. 163, 281-29s. - - Roy, R. R., Baldwin, K. M. and Edgerton, V. R. (1991) 7’he

Plasticity of Skeletal Muscle: Effects of Neuromuscular Activ- ity. In Exercise and Sport Sciences Reviews, pp. 269-312. Williams and Wilkins, Baltimore.

Biomechanical determinants of wrist joint strength 341

Sacks, R. D. and Roy, R. R. (1982) Architecture of the hindl~b muscles of cats: functional significance. J. Morptiot. 173, 185-195.

Schantz, P., Randall, F. E., Hutchinson, W., Tyden, A. and Astrand, P. 0. (1983) Muscle fibre type distribution, muscle cross-sectional area and maximal voluntary strength in hu- mans. Acta Physiol. Stand. 117, 219-226.

Simard, C. P., Spector, S. A. and Edgerton, V. R. (1982) Contrac- tile properties of rat hindlimb muscles immobilized at differ- ent lengths. Exp. Neuroi. 77, 467-482.

Spector, S. A., Simard, C. P., Fourier, M., Sternlicht, E. and Edgerton, V. R. (1982) Architectural alterations of rat hin- dlimbs skeletal muscles immobilized at different lengths. Exp. New-d. 76,94- 110.

Tolber, J. R., Blair, W. F., Andrews, J. G. and Cro~inshield, R. D. (1985) The kinetics of normal and prosthetic wrists. J. Biomechunics 18, 887-897.

Williams, P. and Goldspink, G. (1973) The effect of immobiliz-

ation on the lon~tudi~l growth of striated muscle fibers. J. Anat. 116,4S-55.

Youm, Y., Ireland, D. R., Flatt, A. E. and Sprague, B. L. (1976) A Moment Arm Analysis of the Prime Wrisrist Motors. Fifth International Congress of Biomechanics. University Park Press, Baltimore.

Zajac, F. E. (1989) Muscle and Tendon: Properties, Models, Scaling und Application to Biomechanics and Motor Control, in C.R.C. Critical Reviews in Biomedical Engineering, pp. 359-411. C.R.C. Press, Inc., Boca Raton.

Zajac, F. E. (1992) How musculotendon architecture and joint geometry affect the capacity of muscle to move and exert force on objects: A review with application to arm and forearm tendon transfer design. J. Hand Surg. 178, 799-804.

Zuurbier, C. J. and Huijing, P. A. (1992) Influence of muscle geometry on shortening speed of fibre, aponeurosis and muscle. J. Biomechunics 25, 1017-1026.

APPENDIX

Muscle

ECU

FCU

ECRB

FCU

ECRL

Joint Moment at Different Forearm Rotations (Nm)*

~exion-Extension Motion Pronated Neutraf Supinated

3.3 E-7 x2+ 254 E-7 x3 0.36 - 1.2E-4x’ - 2.9E-10x5 0.40 i- 2.9E-3 x - f.7E-4 x2 - 1.6E-6 x3 + LOE-12 x6

- 1.0 -0.012x + l.lE-4x2 - 0.90 - 8.4E-3x + 8.2E-5 x2 - 0.80 - 4SE-3 x + 9.1E-5 x2 + 3.78E-10x5 - 3.OE-7 x3

1.05 + 0.010 x - 2.0E-4 x2 - 4.OE-6 x’ 0.94 + 8.1E-3 x - 8.6E-5 x2 0.87 + 7.68-3 x - 8.7E-5 x2 - 3.2E-6 x3 - 4SE-8 x4 - 2.8E-6 x3 - 9.4E-9 x4

- OS2 - 9.9E-7 x3 + 3.2E-12 x6 -0.61 - l.lE-3x + 7.lE-5x2 - 0.63 - 2.7E-3 x + 8.2E-5 x2 - 7.6E-7 x3 - 8.6E-7 x3 + 8.OE - 14 x7

0.43 -I- 6.OE-3 x - 2.4E-6 x2 0.35 + 6.OE-3 x - 2.38-6 x3 0.32 + 2.8E-3 x - l.OE-6x” - 2.OE-6 x3 - 2.4E-8 x4 - 4.6E-12 x6

Radix-~nar Deviation Motion

ECU 1.4 - 4.lE-3 x -8&E-4 x2 1.4 - 6.2B3 x - 7.4E-4 x2 - 1.1 E-6 x3 0.98 - l.OE-4 x3 FCU 1.16 -0.020x 1.5 - 0.037 x - 8.OE-10x’ 1.4 -0.044x ECRB -0.78 -0.014x + l.lE-3x2 -0.93 - 3.lE-3x + 4.1E-6x4 -0.95 - 1.9E-3x + 7.lE-4x2 FCU -0.32 -0.013x - 0.25 - 0.011 x + 4.58-4 x’ ECRL - 0.65 - 3.1E-3 x + 4.OE-4 x2

-0.19-0.013x+2.0~7x5 - 0.736 - 2.3E-3 x + S.OE-4 x2 - 0.78 + l.OE-3 x2 - 2.lE-5 x3

Muscle

ECU FCU ECRB FCU ECRL

Muscle Force at Different Forearm Rotations (N)*

Flexion-Extension Motion Pronated Neutral Supinated

50 49.0 - 0.068 x 50.0 - 0.096 x - 1.4E-3 x2 62.0 + 0.76 x - 3.3E-3 x2 62.0 + 0.69 x - 3.8E-3 x2 62.0 + 0.65 x - 4.3E-3 x2 58.0 - 6.2E-3 x2 59.0 - 5.5E-3 x2 59.0 - 5.OE-3 x2 40.0 f 0.28 x - 7.68-7 x4 40.0 + 0.33 x - 2.2E-3 x2 40.0 + 0.32 x - 1.5E-3 x2 31 32 32

Radial-Ulnar Deviation Motion ECU 50.0 - 0.34 x - 0.018 x2 50.0 - 0.33 x - 0.018 x2 FCU 62.0 - 0.80 x 62.0 - 0.98 x ECRB 59 59 FCU 40.0 +0.17x 40.0 + 0.11 x ECRL 32 32

*Values given yield absolute muscle force (N) as a function of joint angle (degrees)

49.0 - 0.33 x 62.0 - 0.95 x 59 40.0 +0.13x 32

342 G. J. Loren et al.

__-----_ _.-. .-. -.

MLiSCk Pronated

Moment Arm at Different Forearm Rotations (mm)

Fiexion-Extension Motion Neutral Supj~ated

ECU FCU ECRB

FCU ECRL

0.94 - 160 . + 92E-4*x2 . 18 0 + 0.18*x - 1 . 8E-3*x2 --6.IE-5%3

- 13.0 + 0.045*x 13.0 + 0.17*x

7.2 - 2.2E-3*x2 - 4S*E-9*x5 - 14.0 + 0.028*x

16.0 + 0.16*x + 8.7&4*x2 - 5.8E-5*x3 - 1.2*&6*x4 - 15.0 + 0.082*x 10.0 + 0.17*x

Radial-Ulnar Deviation Motion

7.8 - 2.8E-3*x’ - 13.0 + 0.055*x 15.0 + 0.14”~ - 4.7E-5*x3 - 2.5*E-7*x4 - 16.0 + 0.057*x

9.8 + 0.087*x

ECU 28.0 + 0.082*x 28 21 FCU 19 0 - 9 4E-4*x3

-’ 13.0 10.24*x f 0.018*x2 23.0 - 0.24*x - 4.0*&6*x5 22.0 - 0.45*x

ECRB - 16.0 + 5.3&5*x4 - 16.0 + 0.010*x2 FCU - 8.7 - 0.21*x - 5.7 - 0.27*x -t 8.OE-5*x4 - 7.3 - 0.26*x + 0.017*x2 ECRL - 20 * 0 - 0 . 095*x -I- 0.014*x2 - 23.0 - 0.072*x f 0.017*x2 - 24.0 + 0.033*x2 - 6.3E-4*x3