Journal of Biomechanics 31 (1998) 963—967

Technical Note

An analytical model of the knee for estimation of internal forcesduring exercise

Naiquan Zheng, Glenn S. Fleisig*, Rafael F. Escamilla, Steven W. Barrentine

American Sports Medicine Institute, 1313 13th Street South, Birmingham, AL 35205, U.S.A.

Received in final form 13 March 1998

Abstract

An analytical model of the knee joint was developed to estimate the forces at the knee during exercise. Muscle forces were estimatedbased upon electromyographic activities during exercise and during maximum voluntary isometric contraction (MVIC), physiologicalcross-sectional area (PCSA), muscle fiber length at contraction and the maximum force produced by an unit PCSA under MVIC.Tibiofemoral compressive force and cruciate ligaments’ tension were determined by using resultant force and torque at the knee,muscle forces, and orientations and moment arms of the muscles and ligaments. An optimization program was used to minimize theerrors caused by the estimation of the muscle forces. The model was used in a ten-subject study of open kinetic chain exercise (seatedknee extension) and closed kinetic chain exercises (leg press and squat). Results calculated with this model were compared to thosefrom a previous study which did not consider muscle length and optimization. Peak tibiofemoral compressive forces were3134$1040 N during squat, 3155$755 N during leg press and 3285$1927 N during knee extension. Peak posterior cruciateligament tensions were 1868$878 N during squat, 1866$383 N during leg press and 959$300 N for seated knee extension. Nosignificant anterior cruciate ligament (ACL) tension was found during leg press and squat. Peak ACL tension was 142$257 N duringseated knee extension. It is demonstrated that the current model provided better estimation of knee forces during exercises, bypreventing significant overestimates of tibiofemoral compressive forces and cruciate ligament tensions. ( 1998 Elsevier Science Ltd.All rights reserved.

Keywords: Knee model; Muscle force; Cruciate ligament; Knee exercise

1. Introduction

Understanding the biomechanics of the knee duringexercises, such as knee extension, leg press and squat, isvery important for therapists to design rehabilitationprograms and for trainers to strengthen and conditionathletes. Muscle force, ligament tension and joint contactforces between the femur and tibia during exercise are themain issues to evaluate exercises. For example, therapiststry not to overload the anterior cruciate ligament (ACL)for ACL injured and reconstructed patients during theirrehabilitation programs. It has been controversial whichexercise should be utilized for a patient since the ACLtension during different exercises are not fully under-stood (Wilk et al., 1997).

*Corresponding author. Tel.: 001 205 918-2138; fax: 001 205 918-0800; e-mail: glennf@asmi.org

Recently, resultant forces of the knee during differ-ent exercises have been studied (Lutz et al., 1993; Stuartet al., 1996; Wilk et al., 1996). Some efforts were madein these studies to estimate muscle forces and to com-pare muscle activity; however, further study of tensionsof cruciate ligaments and the bone-to-bone contactforce of the tibiofemoral joint was prevented by limitsof the biomechanical models utilized. Muscle forceswere estimated based only on the physiological cross-sectional area and electromyographic activities, notmuscle fiber length at contraction. Thus, muscle forceswere overestimated at full extension for the knee exten-sors without taking into account the shorter muscle fiberlength.

The purpose of this study was to develop an analyticalmodel of the knee in the sagittal plane during quasistatic(i.e. isometric or low speed) exercise. The tibiofemoralcompressive force and tensions of the anterior and poste-rior cruciate ligament were presented during the kneeextension, leg press and squat.

0021-9290/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 0 5 6 - 6

2. Materials and methods

Resultant force and moment at the knee were the sumsof those due to individual ligament, muscle and jointcontact:

F3%4"

n.+i/1

F.i#

nl+j/1

Flj#

n#+k/1

F#k, (1)

M3%4"

n.+i/1

M.i#

nl+j/1

Mlj#

n#+k/1

M#k, (2)

MAi"r

Ai]F

Ai(A"m, l, c; i"1, 2, 3,2, n

A) , (3)

where F3%4

and M3%4

were the resultant force and momentat the knee; n

., nl and n

#were the number of muscle

groups, ligament bundles and bone contact interfacesused in the model; F

.iand M

.iwere the force and

moment of the ith muscle group; and Fljand Ml

jwere the

force and moment of the jth ligament bundle; F#k

andM

#kwere the force and moment of the kth bone contact

element; and rAi

was the vector of a moment arm.To simplify the model some assumptions were made:

(a) collateral ligaments were not included since they havelittle effects on the mechanics of the knee in the sagittalplane (Crowninshield et al., 1976), (b) the flexion—exten-sion moment of the cruciate ligaments were ignored sincethey were located close to the rotation center of the knee,(c) the tibiofemoral force was assumed to be applied inthe line of knee rotation center, therefore, the contribu-tion of the tibiofemoral force to the resultant extensiontorque was ignored; and (d) the tibiofemoral joint andpatellofemoral joint were assumed to be frictionless dueto the synovial fluid.

In this model, resultant forces and torques at the kneewere assumed to be known. Forces of the muscles wereestimated using surface EMG data from eight musclegroups (rectus femoris, vastus medialis, vastus lateralis,vastus intermedius, biceps femoris, semitendinosus,semimembranosus, and gastrocnemius) and the followingequation:

F.i"c

ikiA

ip.i

EMGi

MVCi

, (4)

where ki

was a muscle force—length factor defined asa function of knee and hip flexion angle; A

iwas the

physiological cross-sectional area (PCSA) of the ithmuscle; p

.iwas maximum voluntary isometric contrac-

tion force per unit PCSA of the muscle; EMGi

andMVIC

iwere window averages of ith muscle EMG during

exercise and maximum voluntary isometric contraction;coefficient c

iwas the weigh factor which was adjusted in

an optimization program to minimize the errors inmuscle force estimation.

The muscle force—length factor k was determined bythe knee joint angle and hip joint angle based on the

cross-bridge model of muscle. The muscle length wasdetermined by the knee joint angle, hip joint angle andmuscle line of action and geometry of the lower ex-tremity (Pierrynowski, 1991), which determined themuscle fiber length and sarcomere length (Herzog et al.,1990). For single joint knee muscles, the musclelength factor k was determined as a segmental linearfunction of knee flexion angle since the isometric tensionof the muscle is a segmental linear function of the sar-comere length (Gordon et al., 1966). For two jointmuscles, k was a segmental linear function of both kneeand hip flexion angles (Fig. 1). Since the model wasdeveloped for quasistatic exercise, no force-velocity fac-tor was included.

PCSA data from Wickiewicz et al. (1983) were used todetermine the ratios of PCSA between muscle groups.According to Narici et al. (1988), the total PCSA of thequadriceps (PCSA

Q) was approximately 160 cm2 for

a 75 kg man. Total PCSA of the quadriceps was scaledup or down by individual body mass (PCSA

Q"

160 *BW/75 cm2). The maximum voluntary contractionforce per unit PCSA was assumed to be 40 Ncm~2 forthe quadriceps and 35 N cm~2 for the hamstring andgastrocnemius muscles (Narici et al., 1988; Cholewicki etal., 1995; Narici et al., 1992; Wickiewicz et al., 1984).

Moment arms of muscle forces and angles of the line ofaction for muscle and ligaments were represented aspolynomial functions of the knee flexion angle using datafrom Herzog and Read (1993) (Tables 1 and 2). Since linesof action and moment arms for muscles and ligamentsfrom literature were limited to the sagittal plane, only theforces in the sagittal plane were analyzed in this model.Using the equilibrium equations the cruciate ligamentforces (F

PCL, F

ACL) and tibiofemoral force (F

TF) were

Fig. 1. Force—length factor k for two joint muscle of quadriceps vs kneeand hip flexion angle. Muscle fiber length has significant effect on themuscle force output.

964 N. Zheng et al. / Journal of Biomechanics 31 (1998) 963—967

Table 1Muscle moment arm lengths (¸, m) as a polynomial function of knee angle (h, degree) based on Herzog and Read (1993)

Function ¸"B0#B

1h#B

2h2#B

3h3#B

4h4

Coefficient B0

B1

B2

B3

B4

Patellar tendon 4.71E-02 4.20E-04 !8.96E-06 4.47E-08 0.00Biceps femoris 1.46E-02 !9.26E-05 8.55E-06 !8.78E-08 2.38E-10Semimembranosus 2.84E-02 !1.61E-04 6.81E-06 !8.80E-08 2.77E-10Semitendinosus !4.11E-03 5.86E-04 6.90E-06 !5.31E-8 0.00Gastrocnenius 1.99E-02 !3.50E-04 9.20E-06 !1.03E-07 4.07E-10

Table 2Lines of action for muscles and ligaments (/, degree, 0 to the anterior, 90 to the inferior) as a polynomial function of knee angle (h, degree, 0 for fullextension) based on Herzog and Read (1993)

Function /"B0#B

1h#B

2h2#B

3h3

Coefficient B0

B1

B2

B3

Patellar tendon !74.40 !5.75E-02 !4.75E-03 3.09E-05Biceps femoris 275.00 !0.8720 !7.12E-04 0.00Semimembranosus 260.00 !0.8880 !8.52E-04 0.00Semitendinosus 255.00 !0.8160 2.63E-04 !6.19E-6Gastrocnenius !90.00 0.00 0.00 0.00Anterior cruciate ligament 227.00 !0.4880 0.00Posterior cruciate ligament !66.00 0.7370 !4.96E-03 0.00

estimated. Eq. (1) can be rewritten as

FTF#F

PCL#F

ACL"F

3%4!

n.+i/1

F.i

. (5)

When the resultant anteroposterior shear force was pos-itive, the tension of the posterior cruciate ligament wasdetermined and tension of the anterior cruciate ligamentwas assumed to be zero. Conversely, when the resultantanteroposterior shear force was negative, tension of theanterior cruciate ligament was determined and tension ofthe posterior cruciate ligament was assumed to be zero.The patellar tendon force was determined by the quad-riceps tendon force and the ratio of the patellar tendonforce to the quadriceps tendon force (Van Eijden et al.,1987).

Since the accuracy of the estimation of the muscleforces depends on the accuracy of the estimation of thephysiological cross-sectional area, maximum voluntaryisometric contraction force per unit PCSA, EMG dataand muscle fiber length, coefficient c

iwas used for each

muscle in estimation of its force. These coefficients weredetermined in an optimization program by minimizingthe difference between the resultant moment of the kneefrom kinetic analysis (M

3%4) and that from the estimation

of the model (+ M.i

) . The following objective functionwas used:

min f (ci)"

n.+i/1

(1!ci)2#jAM3%4

!

n.+i/1

M.iB

2

subject to c-08

)ci)c

)*'), (6)

where c-08

and c)*')

were the lower and upper limit forciand j was a constant. The coefficient c

iwas set at one at

beginning and adjusted by using the Davidon—Fletcher—Powell algorithm (Dennis and Schnabel, 1983).

To test the model, previously collected data were used(Wilk et al., 1996). A three-dimensional motion analysissystem (Motion Analysis Corporation, Santa Rosa, CA,U.S.A.) and force platform (Advanced Mechanical Tech-nology, Inc., Newton, MA, U.S.A.) were used to collectkinematic data, external loads and ground reaction for-ces for ten subjects performing three exercises: seatedknee extension, leg press and squat. The knee flexion orextension angular velocities were found to be approxim-ately 90° s~1 for peak value and approximately 60° s~1

for average, making data acceptable for the quasistaticassumption in the model. The resultant force and mo-ment at the knee were determined by using three-dimensional rigid link models and principles of inversedynamics (Feltner and Dapena, 1989; Wilk et al., 1996).Surface EMG data were collected from rectus femoris,vastus lateralis, vastus medialis, biceps femoris, medialhamstrings and gastrocnemius muscles with a Noraxonsystem (Noraxon, Pheonix, AZ, U.S.A.). The average ofthe rectus femoris, vastus lateralis and vastus medialisEMG data was used to represent the EMG data of thevastus intermedius. EMG data of the medial hamstringswere used for both the semitendinosus and semimem-branosus. The window average of EMG during exerciseswas expressed as a percentage of the window average ofEMG during the maximum voluntary isometric contrac-tion, which was recorded when the muscle force—length

N. Zheng et al. / Journal of Biomechanics 31 (1998) 963—967 965

factor was one, and then the average for three repetitionsof each exercise was taken based on the knee angle. Inorder to consider the electromechanical delay windowaverages were calculated only for past history with a win-dow width of 100 ms. The c

-08, c

)*')and were set to 0.5,

1.5 and 1.0, respectively, for optimization.

3. Results

The tibiofemoral compressive force during leg pressand squat increased with knee flexion angle, while thetibiofemoral compressive force during knee extensionincreased with decrease of knee flexion angle (Fig. 2).Peak tibiofemoral forces were 3134$1040 N duringthe squat, 3155$755 N during the leg press and3285$1927 N during the knee extension.

There were no significant tension in the anterior cruci-ate ligament during the leg press and squat (Fig. 2). Thetension in the anterior cruciate ligament increased whenthe knee approached full extension during the seatedknee extension with a peak value of 142$257 N. Ingeneral, tension in the posterior cruciate ligament in-creased with knee flexion. Peak PCL tensions were1868$878 N during the squat, 1866$383 N during theleg press and 959$300 N during the seated knee exten-sion. Optimization produced significant adjustments onmuscle forces (Fig. 3).

4. Discussion

A planar quasistatic model of the knee in the sagittalplane was developed, which can be expanded to three-dimensional when lines of action and moment arms ofligaments and muscles are determined in three dimen-sions. The model was used for a study of 10 subjectsperforming the knee extension, leg press and squat(Zheng et al., 1996). The model took into account muscleactivities, especially the co-contraction of the extensorsand flexors of the knee. Although co-contraction of thehamstrings and quadriceps may have no effect on theextension torque of the knee joint, it has a significanteffect on tibiofemoral compressive force and anteropos-terior shear force. Co-contraction of the hamstrings andquadriceps is an important factor in determining tensionsin the cruciate ligaments. The co-contraction has moreeffect on the tension in the posterior cruciate ligamentwhen the knee flexed 90°, and has more effect on thetension in the anterior cruciate ligament when the knee isnear full extension. This is due to varying lines of actionfor the patellar tendon and hamstrings muscle. Thus, it isimportant to take into account the co-contraction of thehamstrings and quadriceps.

The model provided better estimation of internal for-ces of the knee during exercises than those in previous

Fig. 2. Tibiofemoral compressive force and PCL/ACL tension vs kneeflexion angle during knee extension (circle), leg press (triangle) andsquat (square). There were no significant difference in peak tibiofemoralcompressive forces among three exercises, but there were significantdifference in peak ACL and PCL tensions.

Fig. 3. Muscle force estimation of quadriceps before and after optim-ization for a squat of one subject. Optimization had significant adjust-ments on muscle forces.

studies which did not consider muscle length and optim-ization. Linear muscle activity—force relationship wasassumed at a given knee flexion angle (Bean et al., 1988;Lawrence et al., 1983). The muscle force—length factork was about 0.25 for vastus lateralis, medialis and inter-medius when the knee was in full extension. By ignoringthe force—length factor the muscle forces would be esti-mated as high as four times. When only EMG and PCSAwere considered, the tibiofemoral compressive forceswere overestimated with peak values of 6139$1709 Nduring the squat, 5762$1508 N during the leg press and4598$2547 N during the knee extension (Wilk et al,,1996), much higher than those calculated with this model.By ignoring the force—length factor of the muscles, highertibiofemoral compressive force (3453$1313) at 30° kneeflexion was reported than that (2198$805) at 90° during

966 N. Zheng et al. / Journal of Biomechanics 31 (1998) 963—967

isometric leg press (Lutz et al., 1993). No cruciateligament tensions during exercises have been previouslyreported. Tibiofemoral shear forces were reported tocompare different exercises (Wilk et al., 1996; Lutz et al.,1993). Wilk et al. reported maximal posterior shear forceof 1783$634 N during the squat, 1667$556 N duringthe leg press and 1178$594 N during the knee extension.After considering the orientation of the PCL at the kneeflexion angle when these maximal shear force occurred,PCL tensions would be about 2318 N during the squat,2167 N during the leg press and 1531 N during the kneeextension, which were respectively 24, 16 and 60% higherthan those calculated with this model.

The optimization program was used for adjusting themuscle force calculations, based upon knee resultantforces and torques. EMG assisted optimization has beenrecently introduced in muscle force estimation(Cholewicki and McGill, 1994). The objective functionallowed the existence of co-contraction of quadriceps andhamstring muscles. Various objective functions from theliterature (Pedersen et al., 1987) were tried and found tobe unable to include the co-contraction of the extensorsand flexor of the knee. Smaller range of lower and upperlimit for c

iwill allow less adjustment for muscle force

estimation and may lose some effect of optimization.Smaller (0.1, 0.01) should be used if the accuracy of theknown resultant extension torque of the knee (M

3%4) is

lower. Further study is needed to improve the muscleforce—length relationship during knee exercises, espe-cially to specify the difference between concentric andeccentric contractions. The model does not includea force-velocity relationship for muscle force estimationsince it was developed for low speed exercise. Force-velocity relationships should be included in the modelbefore it can be used for high-speed knee activities. Theimprovement of the model could be achieved with in-creased knowledge of the assumptions used in the modelincluding PCSA, EMG—force relationship, lines of actionof muscles and other data needed to expand to a three-dimensional model.

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