Pergamon

J Biomechanics.

Vol. 29. Nu

0021-9290(95)ooo55-0

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 andBioengineering, Biomedical Sciences GraduateGroup,Universityof CaliforniaandVeteransAdministrationMedicalCenters,SanDiego,U.S.A.;andtDepartmentof Orthopaedi~,~~vision of HandSurgery,GiiteborgUniversity,Sweden. Abstract-Moment am, musclearchitecture, andtendoncompliance in cadaver& humanforearms weredeterminedandusedto modelthewristtorque-jointanglerelation(i.e.wristtorqueprofile).Instantaneous moment armswerecalculatedby differentiatingtendonexcursionwith respectto joint rotation.Maximumisometric tensionofeachwristmuscle-tendon unitwaspredicted based on muscle physiological cross-sectional area. Muscle forces were subsequently adjusted for sarcomere length changes resulting from joint rotation and tendon strain.

Torqueprofileswerethencalculated for eachprimewristmotor(i.e.muscle-tendon unit operatingthroughthe corresponding moment arm). Influences of moment arm, muscle force, and tendon compliance on the torque profile of eachmotorwerequantified. Wrist extensormotortorquevariedconsiderably throughouttherangeof 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:

Momentarm;Musclearchitecture; Sarcomere length;Tendon;Wristjoint strength;Tendontransfer.

1NTROI)UCTION

strengthis requisitefor understandingnormal wrist function and for planning surgical proceduresin which muscle-tendonunits are transferredfrom one insertion to another. Literature describing normal operating rangesof the wrist motors is scarceand related data concerningtendon transfersis necessarilyvague.Operative techniquesrecommendedto establish‘normal tension’(Brand, 1985;Mayer, 1916)during tenorrhaphy (i.e. end-to-endtendonattachment)are not precise,sinceit is often assumedthat muscleswill simply remodelto the alteredlevel of useand biomechnicalrequirements(Williamsand Goldspink, 1973). Thus,the purposeof this investigationwasto measure the instantaneousmoment armsof the prime wrist motors andcombinethesedata with muscleforcespredicted from architecutral and biomechanicalinformation to modelthe humanwrist torque profile. The resultspermit discussionof torque motor designand provide a rationale for surgicalrestoration of wrist function. Portions of this work have been presented(Jacobsonet al., 1993; Loren and Lieber, 1994).

Strength is the most commonclinical parameterusedto assess neuromuscularfunction. Becausestrength results from interactionsbetweenthe joint, muscle,and tendon, it may be altered by variations in one or more of these factors. Accurate interpretation of strength, therefore, requires an understandingof joint kinematics,muscle tension,tendon compliance,and their interactions. Muscle tension applied to a moment arm produces joint torque. Maximum muscleforce is determinedprimarily by musclearchitecture (Powell er at., 1984;Roy et al., 1991),though muscleforce generationis considerably influencedby sarcomerelength (Gordon et af., 1966)and tendon behavior (Zajac, 1989).A moment arm is determined by the line of muscle-tendonunit force and the center of joint rotation. Alterations in either muscleforce or momentarm affect torque output. For example,joint torque may be diminishedby tendon subluxation that decreases the momentarm or asa consequence of neuropathy or myopathy that compromises muscleforcegeneration. Few studieshave addressedthe interactionsbetween METHODS the joint, muscle,and tendon in producing torque (Hoy et al., 1990,Lieber and Shoemaker,1992).SuchinformaFresh cadavericspecimens were intact from the midtion regarding the biomechanical determinants of humeral level and free of arthritis and other apparent musculoskeletaldefects.The samefive humanupper extremities were usedfor determinationof moment arms, Received injinalform 21 March 1995. musclearchitecture,and tendon compliance.This deciAddress correspondence to: Richard L. Lieber, Ph.D., Departjoint, muscle,and tendon propermentof Orthopaedics, U.C.SanDiegoSchoolof Medicineand sionwasmadebecause ties may complement one another within a given speciV.A. Medical Center, 9500 Gilman Drive, La Jola, CA 92093mento producea torque different than that predictedby 9151, U.S.A. 331

G. J. Loren

332

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), extensorcarpi radialislongus (ECRL), extensor carpi ulnaris (ECU), flexor carpi radialis (FCR) and flexor carpi ulnaris (FCU)-were identifiedby forearmdissection.Carewastaken to maintain the integrity of the skin and associatedtissuesof the wrist with specificattention to the extensorretinaculum. Moment

arm determination

The forearm was mounted onto a mechanicaljig (Fig. 1);the distal humeruswassecuredby Steinmanpins to vertical braceswhile an additional pin engagedthe middle third of the radius allowing forearm pronation and supination.Thirty-gauge stainlesssteelsutureswere securedto the distal tendon stumps of each muscle (n = 25, five MTUs from five specimens)and routed subcutaneouslyover the musclebelly to the media1or lateral epicondyle recreating the line of force of each muscle.Steelsutureswerethen securedto toothed nylon cablesand connectedto nonbacklashgearsmounted to potentiometers as describedby An et al. (1983) and placed under 500g tension. Tendon excursionswere measuredas the individual steelsutures rotated gears interfaced with potentiometers, providing voltage

et al.

changeswhich correspondedlinearly to sutureand thus tendon excursion. A nonlinkage electrogoniometer (Penny and Giles,M seriestwin-axis goniometer)placed over the radiocarpal articulation measuredjoint angle (4) in either the sagittal or corona1 plane. The electrogoniometerwassecuredto the distal radiusand third metacarpalwith Steinmanpins and custom mounts.In pilot studies,explicit comparisonbetweensettingof electrogoniometer and radiographicjoint angle wasshown to agreewithin 2.4 f 2.1”(mean+ S.D.).Neutral (0”)was defined by alignment of the third metacarpaland the distalradiusfor both the corona1and sagittalplanes.The sensorswere interfaced to a Macintosh IIci computer (Apple Computer Inc., Cupertino, CA) usingthe SuperScopeapplication and MacADIOS II analog interface (Version 1.0,GW InstrumentsInc., Somerville,MA). The arm was positioned with the elbow in 90” of flexion and the wrist passedmanually from flexion (palmarflexion)to extension(dorsiflexion)and from radial to ulnar deviation in either supination,pronation, or neutral forearmrotation. Individual tendonexcursionsof the five MTUs and joint angular displacements were measured simultaneouslythrough the rangeof flexion-extensionand radial-ulnar deviation in eachof the three forearm rotations. Each of thesesix conditionswasrepeated three timesfor a total of 18data setsfor eachMTU with

TO

EXTENSoR i i

TENWN

/’

Fig. 1. Apparatususedto measure wristtendonexcursions andjoint angle.Tendoninsertions andMTU linesof forcewerepreserved alongwith associated soft tissues. Differentiationof tendonexcursiondata with respectto joint angleyieldedmomentarm-jointanglerelations.

Biomechanicaldeterminants of wrist joint strength each condition generating approximately 180 data points through the range of wrist motion. Tendon excursion vs joint angle data were differentiated with respect to joint angte (An et a!., 1983) yielding moment arm as a function of joint angle. Moment 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 including only the polynomial terms which significantly improved the curve fit (F-to-enter = 4.00) and not requiring all lower order terms to be included beneath the highestorder term. Peakmomentarm and angleat peak momentarm werecalculatedfrom thesefunctions.Mean moment arm-joint angle relation for a given MTU in each forearm rotation was subsequentlyobtained by fitting the setof momentarm curvesof the five specimens by stepwisepolynomial regression.This method of moment arm calculationwascalibratedby measuringplexiglasstemplateswith moment arms varying from 5 to 15mm. Our calculatedvalueswerewithin 1.4 + 1.8mm of measuredvaluesand showedno systematicdeviation asa function of moment arm magnitude.

333

and fiber, to change as a function of muscle length. The reference state for each muscle was defined by the measured architectural properties. For any muscle length, fiber length can be derived from the area of the triangular ‘muscle’,

and the law of sines,

L sin[l80-(cc+/?)]

=-

Lf

sin[fl=&’

L

(21

where/I representsthe anglebetweenthe fibersand the muscleline of action, and tt is the angle betweenthe aponeurosisand the muscleline. Equation (2) yields /I and Lf from L,, L,, and ~1.The normahzed force-length property wasbasedon the relation presented by Cutts (1988).The force-length relation plateau extendedfrom sarcomerelengthsof 2.5-2.8 ,um.To account for tendonlengtheningdueto muscleforce generation, tendon force was iteratively matched to muscle force while MTU lengthwasheld constant.The isometric joint torque profile wasthen calculatedby simplymultiplying estimatedmuscleforce by the correspondingmoMuscle architecture and tendon properties ment arm. Mean sarcomereoperating ranges of the Muscle architecture wassubsequentlydeterminedac- prime MTUs for the five specimenswere determined cording to the methodsdeveloped by Sacksand Roy indepedently of the ensembleaverage muscle force (1982)as previously implemented(Lieber ef nt, 1990, joint-angle and torque-joint anglerelations. 1992b).MTUs were dissectedfree and fixed in 10% bufferedFormalin. Muscle length (L,) wasmeasuredas Statisticul analysis the distancefrom the origin of the mostproximal muscle To estimatethe influenceof eachtorque dete~inant fibers to the insertion of the most distal fibers. The (i.e.momentarm,muscleforce,or tendoncompliance)on tendons were removed and measured(L,), and the the torque profile, simulationsof the torque profiles of musclesweighed(tw). Then, utilizing the L,:L, ratios the five motors werecreatedusingeither a constantmopreviouslyreported (Lieberet al., 1990;sincethesevalues ment arm or invariant muscleforce or inelastictendon. demonstrated little interspecimenvariability), muscle Each of these reduced modeis was compared to the fiber length (Lf) wascalculated.Surfacepennationangle correspondingtorque profile determinedfrom the meas(0) was also incorporated from the previous study. ured momentarm, predictedmuscleforce, and physioloPhysiologicalcross-sectional area(PCSA)wascalculated gical tendon compliance.Coefficientsof determination (Sacksand Roy, 1982)to predict maximumtetanic ten(r’) were calculated to quantify the influence of each sion(PO)basedon a musclespecifictensionof 0.25MPa determinant on the torque profile (Lieber and Shoe(Close,1972;Powell et al., 1984).Following determina- maker, 1992).Thus, an r2 value of 0.99 for a constant tion of sarcomerelengthsin eachfixed musclespecimen moment arm simulation (i.e. incorporating a constant by laserdiffraction (Lieber et al. 1990,1984),L, and Lf moment arm with normal muscleforce variation and were normalizedto a sarcomerelength (L,) of 2.5pm to physiologicaltendonstrain)indicatedthat the variability compensatefor length changesoccurringduring fixation. in the torque profile was not greatly influencedby moTendonsfrom eachMTU wereloadedto their respect- ment arm variability with joint rotation. If the coefficient ive POto determineindividual tendon complianceunder of determination were lower, we would infer a greater physiologicaltensions(Loren and Lieber, 1995). influenceof momentarm on the torque profile shape. Comparisonsamongpeakjoint torque, momentarms, Wrist torque model and wrist anglesin the various forearm rotations were The musclemodel wasbasedon an arbitrary muscle performedusingtwo-way analysisof variance(ANOVA), with variable architectural andfiber type properties,sim- usingMTU and forearmrotation asgrouping variables. ilar to that of Zuurbier and Huijing (Zuurbier and Huij- Multipie paired comparisonswere performed ~usf-~oc ing, 1992). Briefly, each muscle was modeled as half usingFisher’sprotected least-squares differencemethod. a parallelogram,with sidesof fiber length (Lr), muscle Data were analyzed using StatView software (Version length(L,), and aponeurosislength(La). The areaof this 4.0, Abacus ConceptsInc., Berkeley, CA). Significance triangle was held independentof musclelength, which level (a) was selectedas 0.05. Data are expressedas requirespennationangle,the anglebetweenaponeurosis mean&-S.E.

G. J. Loren er al.

334 RESULTS

Flexor

(Nm)

Arms (mm)

Moment

Moment arms

Torques

Forcei (N,

Moment arms varied considerably throughout the range of joint motion. The extensionmoment arms of ECRB and ECRL were smallestin wrist flexion and increasednearly linearly with progressivewrist extension (Fig. 2). Radial moment arms were maximum with the wrist deviated toward the ulna (Fig. 4). ECU extension moment arm was greatestnear neutral wrist extension (Fig. 2).With forearm rotation from supinationto pronation, maximum ECU ulnar moment arm significantly increased(p < 0.01)whilemaximumECU extensionmoment arm significantly decreased(p < 0.01). Flexor moment arms were greatest with the wrist flexed and decreasedwith extension(Fig. 3). Maximum FCR radial moment arm occurred in ulnar deviation (Fig. 4). Conversely,maximumFCU ulnar moment arm wasnoted in radial deviation (Fig. 5). Predicted muscle forces

Wrist extensorMTUs werepredictedto operateprimarily on the plateau of the sarcomerelength-tension curve (Fig. 6). Only the ECRB waspredicted to operate at sarcomerelengthscorrespondingto lessthan 80% PO in the normal rangeof motion. Wrist flexor MTUs were predicted to operate predominantly on the ascendingand steepascendinglimbs of the length-tensioncurve. Peakmuscleforcesfor both Extensor Moment Arms (mm)

Torques

(Nm) Forces lea

(N)

-40 Flexion

-20 Joint

0 Angie

20

40 Extension

Fig, 3. Determinants of flexion torque. Torque profiles of extensor MTUs are shown enlarged with moment arm-joint angle relations and muscle force-joint angle relations provide as insets. 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 asin flexion-extension, the wrist flexors achievedmaximal muscleforcesat joint anglescorrespondingto long musclelengths,i.e.the FCR in ulnar deviation and the FCU in radial deviation (Figs 4 and 5). Determinants

of wrist joint torque

Wrist extensortorque varied markedly asa function of joint angle (Fig. 2). Extensor torque profiles were domin-

ated by the moment arm-joint anglerelations(Table 1). For example,in a simulatedECRB torque motor with invariant muscleforce, variability in the moment arm with joint rotation correlated highly with the torque CRB profile (r’ = 0.990).Similar relationshipswere noted for the ECRL (r* = 0.996) and ECU (r’ = 0.981). Muscle CRL force-joint angle curves generally contributed lessto extensor torque profiles (Table 1). An ECRL or ECU torque motor with constant moment arm produced cu a torque-joint anglerelation that correlatedpoorly with -40 -20 0 20 40 the predicted torque profile (r2 = 0.482 or r* = 0.322, Flexion Joint Angle Extension respectively),indicating a smallinfluenceof muscleforce Fig. 2. Determinants of extension torque. Torque profiles oi variability onjoint torque. Given the largeECRB muscle extensor MTUs are shown enlarged with moment arm-joint force variability with wrist rotation, an expectedlylarger angle relations and muscle force-joint angle relations provided influenceof muscleforce on torque output wasapparent arm as insets. Note the considerable influence of ECRBmoment (r* = 0.960).The influenceof tendon complianceon exvariability with joint rotation on the torque profile. (Data presented for neutral forearm rotation. Shaded area represents tensor torque profiles was negligible(r2 range, 0.9540.991;Table 1). SEM of five cadaveric specimens.)

Biomechanical determinants of wrist joint strength Radial

Deviation

Torque

335

(Nm) Force(N) *I

Momem Am (mm) 3

’ Sarcoiere Lew$h

4

5

(pn)

ECRL

FCA -10

-5

0 Joint Angle

Radial

5

10

15

Ulnar

Fig. 4. Determinants of radial torque. Torque profiles of extensor 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

I 4

FCU

y”.5t

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

-5

0 Joint Angle

5

10

15

Fig. 5. Determinants of ulnar torque. Torque profiles of extensor MTUs are shown enlarged with moment arm-joint angle relations and muscle force-joint angle relations provided as insets. (Data for neutral forearm rotation.)

Fig. 6. Operating ranges of the wrist motors on the isometric sarcomere length-tension relation, Extensors operated primarily on the plateau region while the flexors operated predominantly 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 rotation.) Mean sarcomere operating ranges were determined independently 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.

extensionwere attenuated by decreasingmoment arms (Fig. 3). Consequently,torque profiles were dominated by neither muscle force nor moment arm relations (Table 1).Tendon compliancehad a greatereffect on the flexor torque profiles compared to those of the extensors. This resulted from the relatively greater influence of muscleforce on the flexor torque profile with tendon compliancepermitting muscleforce.changes. Moment arm variability with radial-ulnar deviation comparedto flexion-extension was lessinfluential on torque profiles of the radial extensors (Table 1). The muscleforce-joint angle relation, though, with radial deviation in the constant moment arm simulationconsiderably influencedthe ECRB and ECRL torque profiles (r’ = 0.875 and r2 = 0.808, respectively).Furthermore, muscleforce variability substantially influenced FCR radial torque profile (r* = 0.839)although the moment arm curve was the primary determinant (r2 = 0.999).Given the relatively constantECU moment arm with ulnar deviation, the ECU ulnar torque profile wasdominatedby the muscleforce-joint anglerelation (r2 = 0.999). The FCU ulnar torque profile, like the flexion profile, wasinfluencedby both momentarm and muscleforce variability (r2 = 0.994 and r2 = 0.934, respectively). DISCUSSION

The purpose

of this investigation

was to define the

biomechanicalbasisof the isometricjoint torque profiles of the humanwrist motorsand to usethis information to Flexor torque demonstratedcomparatively lessvaria- provide guidelinesfor tendon transfers.We incorporated bility with wrist rotation. The virtually constant flexor the angulardependenceof momentarm and muscleforce torque was not due to invariant muscleforce and con- and accountedfor tendon complianceto illustrate the stant momentarm. Rather, increasingmuscleforceswith interactionsbetweenthesefactors.Torque output for the

33h

G. J. Loren et al. Table 1. Wrist motor torque profile coefficients of determination (r’)*

ECRB ECRI.‘ ECU FCR FCU

__--

Radial-ulnar deviation

Flexion-extension Constant moment arm

Invariant muscle force

Inelastic tendon

Constant moment arm

Invariant muscle force

Inelastic tendon

0.960 0.482 0.322 0.307 0.943

0.990 0.996 0.981 0.123 0.790

0.954 0.991 0.976 0.415 0.444

0.875 0.808 0.999 0.839 0.933

0.684 0.288 0.000 0.999 0.994

0.998 0.985 0.922 0.999 0.990

*Data presented for neutralforearmrotation.

wrist motors wasnot constantthroughout the physiological range of motion, reflecting varying influencesof momentarm, muscleforce, and tendon strain. Moment arm magnitudescomparedfavorably to those previously reported for the wrist (Horii et al., 1991; Tolber et al., 1985; Youm et al., 1976).As suggestedby Brand (1985),we found considerablemomentarm variability throughout the range of joint motion. Previous studieswhich have reported constantor linearly varying momentarms (Horii et at., 1991;Ketchum et ai., 1978; Youm et al., 1976)may not, therefore,accurately represent normal wrist kinematics.Forearm rotation only significantly afteredECU momentarm; an anatomicalbasis hasbeenpreviously described(Brand, 1985;Youm et al., 1976). Predicted muscleforce changeswith wrist rotation variably influencedflexor and extensor torque profiles. Muscle force generationreflectedthe operating rangeof each muscleon the sarcomerelength-tension relation. Sarcomereexcursionwith joint motion wasdetermined by musclefiber length and moment arm magnitude. Fiber lengths and moment arm magnitudes,however, varied amongthe wrist motorsso muscleand sarcomere length changeswith joint rotation wereunique(Table 2). For example, the musclewith the shortest fibers was FCW. Of the prime wrist motors. FCU had a peak flexion-extension moment arm of intermediate magnitude. Consequently,the FCU fiber length to maximum momentarm ratio (L,: r) wasthe lowest(Table2). As the wrist wasflexed, sarcomerelength changeper degreeof joint rotation (d&/d& wasgreatestfor FCU, resultingin substantialsarcomerelength variation and, therefore, muscle force change over the physiological range of motion. It follows that maximalmuscletensionwasgenerated over a limited angular range.Conversely,ECRL had the longestfiber length of the prime wrist motors,an intermediate extension moment arm, and the highest L,: r (Table2). As the wrist extended,dLs/dc#wasminimal. Muscle force generation was virtually constant as sarcomeresremainedon the plateauof their length-tensionrelation. High &:r wasthereforeindicative of small sarcomerelength changesper degreeof joint rotation and consequently minimal muscle force variability throughout the range of motion. Conversely, low Lf:r

resulted in large sarcomere excursions and greater muscleforce changeswith joint rotation (Lieber and Brown, 1993). The mostcommonaction of the humanwrist is rotation from extension-radialdeviation to flexion-ulnar deviation. Moment arm magnitudes,though, for this combinedmotion of the wrist motors werenot definedin this study. Utilizing the vector sumof the ffexion-extension and radial-ulnar deviation moment armsat 0” of wrist rotation, combinedmoment arm magnitudeswere estimated. Of the radial extensors,ECRB had a greater combinedmomentarm. Lf:r and dLs/d$ for ECRB and ECRL were comparable to values obtained for wrist extension,The combinedFCU moment arm, Lf:r, and dLs/dQ, were intermediateto valuesmeasuredfor pure flexion or ulnar deviation (Table 1). Tendon strain under physiologicalloads varied substantially among wrist MTUs and was significantly greater for wrist flexors than extensors (Loren and Lieber, 1994, 1995).Muscle fiber shorteningat the expenseof tendonlen~hening (maths, 1989;Hoffer et nl., 1989)skewsthe sarcomere-lengthtensioncurve and alters the MTU operating range(Lieber et al., 1991, 1992a; Xajac, 1989).In FCR andFCU, with shortfiber lengthsand limited angular rangesof maximal muscleforce generation, skewof the sarcomerelength-tensionrelationsdue to tendon compliance increased muscle tensions at greaterdegreesof wrist extensionby allowing sarcomere shortening from the descendinglimb of the isometric force curve. If the flexors werein serieswith a noncompliant tendon, maximalmu&e forcesfor FCR and FCU would occur at approximately 19”and 27” lessextension, respectively;resultingjoint torque would be maximum for FCR at 12” and for FCU at 26” wrist extension (comparedto 35” and 39” extension for the compliant tendon). It is not clear how or if such a shift in peak torque anglewould alter strengthandcoordination 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 lengthsto joint motion have reported operating ranges on the ascendinglimb (Herzog et al., 1991;Rack and Westbury, 1969),the plateau(Lieberet al., 1992~;Rome

137

Biomechanical determinants of wrist joint strength Table

2. Wrist MTU properties and torque determinants* ECRL

ECRB PtAW

Mmm)t Tendon E (%)t -L: Lft Flex&-Extension: rmal (mm)

58.8 i 5.0 70.8 * 1.7 1.99 4_ 0.20 2.89 + 0.11

2.7 5.6 0.14 0.18

FCR

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

51.9 59.8 2.48 3.86

-t- 3.7 * 1.5 & 0.45 + 0.12

FGU

89.1 * 8.4 41.9 2 I.6 3.68 & 0.31 4.96 + 0.18

1.2 2 (E) 5.0 1 (E) 0.11 3 (E) 0.24 1.1

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

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

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

f6.8 & 1.4 20 f 17 (F) 87.5 f 8.5

rffe r: a?T,., I”) d&&P (nmi”)

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

27.5

0+3(u) 3.62 + 0.16 11.5 + 0.6

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

29.3 + 1.8 1 k 3 CR) 51.4 * 3.5

;m.%x(“)

16.9 + 0.8 1 rt 3(U) 58.8 If: 5.0 1.01 14 + 0.10 1 (U)

11 ?; 1 (R) 1.57 t- 0.16 22.4 rt 1.0

21.7 f 0.9 2.65 & .13

19.5 4 1.2 2.17 F .14

Angleat rmar1”)

F,,, (N) Angle at F,,, (“) rmar.(Nm) Angle at z,,, (“) L,:r

dLdd, (nmi”) Radial-Ulnar rmar (mm)

:““‘“(ii nax ;.fe$t

Combined r, (mm)

Lr:r,

19.3 + 29 + 58.8 + 11 f 1.07 + 17 + 3.21 + 11.4 rt

31.9 f 127.3 + 1.78 f 2.10 f

ECU

45' 1.19 If 0.18 39 + 4 (E)

2.54 It 0.23 14.2 + 0.6

Deivation rmar

(“1

Extension-Radial

Deviation

22.6 f 2.0 2.78 & 0.26

and Fkxion-Ulnar

14.6 + 1.6 8.19 & 1.6

6+0(R)

1.51 rf: 0.11 1 + 3(R) 1.65 + 0.14 24.6 f 1.3

+ 2.1

9?;2@)

68.6 i 5.6 12 &0.0(R) 2.40 & 0.17

Deviation:

10.2 5 1.3 5.01 * .75

*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- were performed using independentmethodologiesand scendinglimb (Herzog and ter Keurs, 1988;Lieber and different specimens, they provide strong support for the Boakes,1988;Lieber and Brown, 1993)of the sarcomere validity of our model. length-tension relation. Our data support suchspecialWrist motor torque profiles illustrated a complex inization within the muscle-joint systemof the human teraction betweenskeletalmuscleand articular motion. wrist. Wrist flexors operatedprimarily on the ascending Several authors have demonstrated that maximum limb with higher sarcomerelength changesper degreeof muscleforceand peakjoint torque occur at differentjoint wrist rotation while extensorscontracted predominately angles(Herzog et al., 1991;Hay et al., 1990;Lieber and along the plateau region with smallersarcomereexcur- Boakes,1988;Lieber and Shoemaker,1992)and that the sions. The average slope (d&/dqb) of the sarcomere angleof maximumjoint torque isnot necessarilythejoint length-joint anglerelationsof ECRB (11.4nmf) and the anglewherethe musclegeneratesmaximumforce nor the predicted ECRB sarcomere operating range anglewherethe moment arm is maximized.Wrist exten(2.18-3.33pm) were almost identical with recent in- sor torque profiles paralleled the moment arm curves traoperative human musclesarcomerelength measure- with both peak torque and rn~irn~ momentarm ocments(Lieber et al., 1994)[Fig. 7(A)]. Data obtainedon curring in wrist extension.The momentarm was potenthe FCU during wrist rotation also demonstratedan tiated by maximal muscletension in wrist extension, identical d&/d+ asthat predictedby the model(Fridkn functionally maximizing musclestrength at the expense and Lieber,unpublisheddata) [Fig. 7(B)] but are offsetto of restricting the rangeof wrist motion over which peak longer sarcomerelengthsby about 1pm. This probably torque wasproduced.In contrast, the opposingangular reflectedthat sarcomerelengthswere predictedwith the dependence of the flexor momentarmsand muscleforces elbow flexed to 90” but intraoperative measurements produced a torque profile with a broad range of nearweremadewith the elbowextended.As thesetwo studies maximal torque and limited the torque magnitude.

6. J. toren

et af.

l

a.01 -90

, -60

z

x

-30

Wrist

0

Angie

30

60

l

/,

-&I

90

-40

(degrees)

-20

Wrist

l

0

Angle

2b

40

20

40

sb

(degrees)

,__, -30

Wrist

0

Angle

30

60

EXt3”SsoOn

(degrees)

ISO FlSWil

-40

-20

Wrist

0

Angle

(degrees)

so

Extension

Fig. 7. (A) Comparison between predicted ECRB sarcomere length-joint angle relation (solid line) and that measured in- Fig. 8. Simulated FCU to ECRL tendon transfer torque profile. traoperatively as reported by Lieber et al. 1994 (filled cimles) (A) Muscle force as a function of wrist joint angle for sarcomere Sarcomere lengths werecorrectedfor tendoncompliance during lengths 2.1 m(O), 3.Ojun (A), and 3.9 pm(m) at neutral wrist muscleactivation.Predictedslopeandsarcomere lengthrange rotation.(B) Wrist joint torque resulting from tendon transfer at comparefavorably to experimental data. (B) Comparison be- different resting sacrcomere lengths: 2.1 pm (*I, 3.0 pm (A), and tween predicted FCU sarcomere length-joint angle relation 3.9pm (a). Note the greater disparity in muscle force-joint (solid line) and that measured intraoperatively (filled circles; anglerelations compared to torqueprofiles.Thisis because the Friden and Lieber,unpublished data).Notethat thepredicted ECRL momentarm dominates the torqueprofile.Combined slopecompares favorablywith experimental databut sareomereextensor moment for ECU f ECRB$ ECRLis provided{O) length range is offset by about 1 pm. This is probably due to for comparison. differences in elbow flexion-extension angle for predicted compared to experimental data (see discussion.)

The summatedwrist extensor torque was dominated by ECRB, given its largest moment arm and greatest force generatingcapacity. Although the isometricmuscle force of ECU was comparable, the poor mechanical advantagelimited its functional contribution as a wrist extensor. Peak extensor torque was 1.8Nm at 18” of wrist extension. Although FCR generally maintained a larger flexion momentarm, the summedflexor torque profile reflected a greater FCU contribution given the higher muscleforcesgenerated.Summatedflexor torque was 1.7Nm near the limit of wrist extension. Thesedata provide isolation relevant to the selection of a donor musclefor tendon transferusedto restore lost function. A comparabler *PCSA product isdesirable to restoremaximum torque capacity and a similar Lf:r ratio shouldbe soughtto replicatejoint excursion(Zajac, 1992)and the torque profile. Physiolo~ca~tendon complianceand tendon length to fiber length ratio may also beconsidered.The relative import of suchdeterminants, however, is variable among motors (Table 1) and does not take into accountthe possibilitythat the MTUs may adapt (seebelow). Becausethe extensortorque profilesare dominatedby the momentarm-joint anglerelation,surgicalreconstruc-

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 muscleand 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 influenced by its operating range on the sarcomere length-tension relation. The tensionimposedduring reconstructionestablishes the resting sarcomerelength and thus the active sarcomere operating range. Adjusting the tension of the tendon transfer may profoundly affect the resulting torque profile. A slack tenorrhaphy (e.g. sarcomere length = 2.1pm at neutral wrist rotation) is predictedto yield an FCU-ECRL motor operating primarily on the amendinglimb of the sarcomerelength tension curve with peak muscleforce in wrist flexion [Fig. g(A)] and consequently a markedly compromisedjoint torque [Fig. 8(B)]. However, tenorrhaphy of the FCU-ECRL tendon under tension(e.g.sarcomerelength = 3.0pm at neutral) may more accurately replicate the pre&ted ECRL sarcomereoperating range on the plateau and approximatethe extensortorque profile [Fig. 8(A)]. Fur-

Biomechanicaldeterminants of wrist joint strength ther manipulationof the restingsarcomerelengthduring tendon transfer by intraoperative laser diffraction may optimize the magnitude of the torque product. A sarcomerelength of 3.9 pm at 0” flexion-extension of the FCU-ECRL motor is predictedto produce a maximum torque in wrist extensioncomparableto the summated torque of ECRB, ECRL, and ECU, yet the transfer torque profile differsmarkedly from the normal relation [Fig. 8(B)]. With transferthen, the lost eleganceof ECRL motor designis overcome by the greater FCU muscle force. Manipulation of the donor operating range by setting tenorrhaphy length at surgery, then, may establish the desiredtorque profile andfavorably influencethe ultimate clinical result. A generalrecommendationin tendon transfersurgery is to perform tenorrhaphy under slight traction (Beasley, 1970;Hovius, 1993;Omer, 1968),although the physiological rationale for this is not clear.It is believedand even tacitly stated that muscleswill adapt their sarcomere number (and fiber type distribution?) to the new biomechanicalrequirementsimposedupon the transferred unit. However, this may be an overstatementof muscle’s ability to adapt in general.First, the magnitudeof muscle adaptation to alteredusevarieslargely betweenmuscles [reviewed in Chap 4 and 5 of Lieber (1992);raw data in Lieber et al. (1988, 1986a,b1989);Simard et al. (1982); Spectoret al. (1982)]and it would not be clear which of thewrist MTUs would adaptto whichextent. Second,since transferaltersmuscleuseandlengthin a way whichis not completelyunderstood,it is not clear exactly how strong a ‘remodelingstimulus’will beenprovided to the muscle. Since the subtletiesof such adaptationsamong human wrist MTUs have not been experimentallydetermined, we cannot incorporate them into the presentstudy. Application of these results to actual experimental measurements of humanwrist torque mustbe madewith caution basedon the assumptionsbuilt into the study designand biomechanicalmodel.First, we have modeled the muscleas an amplified sarcomerewith ideal sarcomerelength-tensionproperties(Gordon et al., 1966). Evidenceexistswhich demonstratesthat the descending limb of the length-tension curve may be modified in isolated single fibers due to intersarcomeredynamics (Altringham and Bottinelli, 1985; Lieber and Baskin, 1983;ter Keurs et al., 1978)but it isnot clearwhetherthis extendsto the whole muscle,whereendomysialconnections betweenfibers are present.This uncertainty would only affect our model at sarcomerelengthsabove 2.8pm where thesephenomenaoccur. Second,the prime wrist motorsare biarticular with definablemomentarmsat the elbow. Our study excludedthis parameterby fixing the elbow 90“ of flexion where most manipulative activity occurs.Given the availabledata on elbow momentarms (Brand, 1985),we anticipate that the effects of elbow motion on the torque profile are greatestfor ECRL with diminishingeffectsfor FCR, FCU, and ECRB and ECU. If elbow momentarm wereconsidered,the ECRL would operateat sarcomerelengthscloserto the plateauof the length-tension curve with elbow flexion and at even longer sarcomerelengths with elbow extension(ECRB

339

and ECU with negligibleelbow momentarmswould be lesseffected).The oppositesituation would betrue for the wrist flexors-FCR and FCU would operate at sarcomerelengthscloserto the plateau with elbow flexion and at even shorter lengthswith elbow extension.Obviously,further experimentationis requiredto quantify the magnitudeof this effect. Third, sincethe length-tension relationshipis only strictly valid for maximally activated musclefibers,the modelmay not predict the shapeof the torque profile during submaximal activation. Since force-length propertiesof submaximallyactivated units generally showoptimal length at longer musclelengths (Heckmanet al., 1992;Rackand Westbury, 1969)consideration of this factor would skew operating rangesto longer sarcomerelengths. Finally, torque magnitudes predictedin this study appearlow comparedto experimentally measuredvalues(Griersonet al., in press).This probably reflects the use of cadaveric specimensfrom elderly subjectsand a relatively low specificmuscletension (0.25MPa) comparedto that measuredfor intact human muscle(Fukunaga et al., 1992; Schantz et al., 1983).It shouldalsobenoted that experimentalmeasurement of wrist flexion and extensiontorque inherently include a substantialtorque contribution of the digital motors. In summary,this investigationintegratedmomentarm determinationswith previous reports detailing muscle architecture (Lieber et nl., 1990) and tendon biomechanics(Loren and Lieber, 1995)to understandthe determinantsof wrist strength. Extensor strength was primarily dependenton the moment arm-joint angle relation while flexor torque was influenced both by musclearchitectureand tendon compliance.The contributions of a MTU’s biomechanicaldesignto the joint strength profile therefore deservesemphasiswhen consideringrestoration of extremity function. Furthermore, surgicalmanipulationof sarcomerelengthduring tendon transfermay assistin replicating the desiredtorque profile of the recipient (nonfunctional)motor and favorably influencethe functional result in the reconstructedlimb. Acknowledgements-This work was supported by the Veterans Administration and NIH grant AR35192. The authors acknowledge Drs Paul Brand, Michael Botte, Reid Abrams, David Pierotti and Richard Braun for helpful suggestions and discussions, Paul Yeatman and Brett Sokoloff for data analysis, and Christian Giangreco and John Butler for technical assistance. We thank Drs Scott Delp and Thomas Buchanan (Northwestern University) for accessto 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.

G. J. Loren et al.

340

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 variation in isometric tension with sarcomere length in vertebrate 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. Biomechanits (in press). Griffiths, R. I. (1989) The mechanics of the medial gastrocnemius muscle in the freely hopping wallaby (Thylogale billardierii). _I

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.

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

244-250.

Lieber, R. L., Friden, J. O., Hargens, A. R.. Danzig. L. A. and Gershuni, D. H. (1988) Differential response of the dog quadriceps 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.

Surg. 18A, 83-90.

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 musculoskeletal model of the human lower extremity: the effect of muscle, tendon, and moment arm on the moment-angle relationship 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.

Physiol.

205-210.

ter Keurs, H. E. D. J., Iwazumi, T. and Pollack, G. H. (1978) The Relation Mechanism

in Skeletal in Muscle

Muscle: Reoisited in Contraction (Edited

by Pollack, G. H. and Sugi, H.). University Park Press, Baltimore. 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. Williams 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.

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.

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

261, C86-C92.

Lieber, R. L., Loren, G. J. and Friden, J. (1994) In viuo measurement of human wrist extensor muscle sarcomere length changes. 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. Res. 7, 408-412.

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

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. Biomechanics

28: 791-799.

Mayer, L. (1916) The physiological method of tendon transplantation-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, Length-Tension Cross-Bridge

91, 435-448.

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 transfer. .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. Contractile 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.

Orthop.

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

Lieber, R. L., Brown, C. G. and Trestik. C. L. (1992a) Model of muscle-tendon interaction during frog semitendinosis hxedend 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,

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. R. (1984) Predictability of skeletal muscle tension from architectural determinations in guinea pig hindlimbs. J. Appl. Physiol. 57, 1715-1721. 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 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 swimming. Am J. Physiol. 163, 281-29s. Roy, R. R., Baldwin, K. M. and Edgerton, V. R. (1991) 7’he Plasticity ity. In

of Skeletal

Muscle:

Effects

Exercise and Sport Sciences Williams and Wilkins, Baltimore.

of Neuromuscular Reviews, pp.

Activ-

269-312.

Biomechanical determinants of wrist joint strength

341

ation on the lon~tudi~l

growth of striated muscle fibers. J.

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 humans. Acta Physiol. Stand. 117, 219-226. Simard, C. P., Spector, S. A. and Edgerton, V. R. (1982) Contractile properties of rat hindlimb muscles immobilized at different 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 hindlimbs 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-

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

Joint Moment at Different Forearm Rotations (Nm)* ~exion-Extension Motion Pronated Neutraf

ECU

3.3 E-7 x2+ 254 E-7 x3

FCU

- 1.0 -0.012x + l.lE-4x2 - 0.90 - 8.4E-3x + 8.2E-5 x2 + 3.78E-10x5 1.05 + 0.010 x - 2.0E-4 x2 - 4.OE-6 x’ 0.94 + 8.1E-3 x - 8.6E-5 x2 - 3.2E-6 x3 - 4SE-8 x4 - OS2 - 9.9E-7 x3 + 3.2E-12 x6 -0.61 - l.lE-3x + 7.lE-5x2 - 7.6E-7 x3 0.43 -I- 6.OE-3 x - 2.4E-6 x2 0.35 + 6.OE-3 x - 2.38-6 x3 - 2.OE-6 x3 - 2.4E-8 x4 - 4.6E-12 x6

ECRB FCU ECRL

ECU FCU ECRB FCU ECRL

0.36 - 1.2E-4x’ - 2.9E-10x5

1.4 - 4.lE-3 x -8&E-4 x2 1.16 -0.020x -0.78 -0.014x + l.lE-3x2 -0.32 -0.013x - 0.65 - 3.1E-3 x + 4.OE-4 x2

Radix-~nar Deviation Motion 1.4 - 6.2B3 x - 7.4E-4 x2 - 1.1E-6 x3 1.5 - 0.037 x - 8.OE-10x’ -0.93 - 3.lE-3x + 4.1E-6x4 -0.19-0.013x+2.0~7x5 - 0.736 - 2.3E-3 x + S.OE-4x2

Muscle

Pronated

Muscle Force at Different Forearm Rotations (N)* Flexion-Extension Motion Neutral

ECU FCU ECRB FCU ECRL

50 62.0 + 0.76 x - 3.3E-3 x2 58.0 - 6.2E-3 x2 40.0 f 0.28 x - 7.68-7 x4 31

49.0 - 0.068 x 62.0 + 0.69 x - 3.8E-3 x2 59.0 - 5.5E-3 x2 40.0 + 0.33 x - 2.2E-3 x2 32

ECU FCU ECRB FCU ECRL

50.0 - 0.34 x - 0.018 x2 62.0 - 0.80 x 59 40.0 +0.17x 32

Radial-Ulnar Deviation Motion 50.0 - 0.33 x - 0.018 x2 62.0 - 0.98 x 59 40.0 + 0.11 x 32

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

Supinated

0.40 i- 2.9E-3 x - f.7E-4 x2 - 1.6E-6 x3 + LOE-12 x6 - 0.80 - 4SE-3 x + 9.1E-5 x2 - 3.OE-7 x3 0.87 + 7.68-3 x - 8.7E-5 x2 - 2.8E-6 x3 - 9.4E-9 x4 - 0.63 - 2.7E-3 x + 8.2E-5 x2 - 8.6E-7 x3 + 8.OE - 14 x7 0.32 + 2.8E-3 x - l.OE-6x”

0.98 - l.OE-4 x3 1.4 -0.044x -0.95 - 1.9E-3x + 7.lE-4x2 - 0.25 - 0.011 x + 4.58-4 x’ - 0.78 + l.OE-3 x2 - 2.lE-5 x3

Supinated 50.0 62.0 + 59.0 40.0 + 32

0.096 x - 1.4E-3 x2 0.65 x - 4.3E-3 x2 5.OE-3 x2 0.32 x - 1.5E-3 x2

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

G. J. Loren et

342

al.

__-----_

Pronated

MLiSCk ECU FCU ECRB

Moment Arm at Different Forearm Rotations (mm) Fiexion-Extension Motion Neutral

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

FCU ECRL

- 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

ECU FCU ECRB FCU ECRL

28.0 + 0.082*x 19 0 - 9 4E-4*x3 -’ 13.0 10.24*x f 0.018*x2 - 8.7 - 0.21*x - 20 *0 - 0 .095*x -I- 0.014*x2

Radial-Ulnar Deviation Motion 28 23.0 - 0.24*x - 4.0*&6*x5 - 16.0 + 5.3&5*x4 - 5.7 - 0.27*x -t 8.OE-5*x4 - 23.0 - 0.072*x f 0.017*x2

--6.IE-5%3

_.-. .-. -.

Supj~ated 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

21 22.0 - 0.45*x - 16.0 + 0.010*x2 - 7.3 - 0.26*x + 0.017*x2 - 24.0 + 0.033*x2 - 6.3E-4*x3