*Correspondencing author. Tel.: 913-588-4565; Fax: 913-4565-4568.E-mail address: wliu@kumc.edu (W. Liu).

Journal of Biomechanics 33 (2000) 871}879

The e!ect of hamstring muscle compensation for anterior laxityin the ACL-de"cient knee during gait

Wen Liu!,*, Murray E. Maitland"

!Department of Physical Therapy Education University of Kansas Medical Center, 3901 Rainbow Boulevard, Kansas City, USA"Sports medicine Centre, Faculty of Kinesiology, The University of Calgary, Calgary, Canada

Accepted 29 October 1999

Abstract

The hamstring muscles have been recognized as an important element in compensating for the loss of stability in the ACL-de"cientknee, but it is still not clear whether the hamstring muscle force can completely compensate for the loss of ACL, and the consequencesof increased hamstring muscle force. A two-dimensional anatomical knee model in the sagittal plane was developed to examine thee!ect of various levels of hamstring muscle activation on restraining anterior tibial translation in the ACL-de"cient knee during levelwalking. The model included the tibiofemoral and patellofemoral joints, four major ligaments, the medial capsule, and "ve muscleunits surrounding the knee. Simulations were conducted to determine anterior tibial translation and internal joint loading at a singleselected position when the knee was under a peak external #exion moment during early stance phase of gait. Incremental hamstringmuscle forces were applied to the modeled normal and the ACL-de"cient knees. Results of simulations showed that the ACL injuryincreased the anterior tibial translation by 11.8 mm, while 56% of the maximal hamstring muscle force could reduce the anteriortranslation of the tibia to a normal level during the stance phase of gait. The consequences of increased hamstring muscle forceincluded increased quadriceps muscle force and joint contact force. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Knee; Anterior cruciate ligament; Instability; Gait

1. Introduction

The e!ect of ACL-de"ciency on tibiofemoral jointstability during gait has not been well established, al-though the abnormal anterior shift of the tibia on thefemur has been well documented by passive stability tests(Torg et al., 1976; Daniel et al., 1985; Reuben et al., 1989;Gross et al., 1993; Haimes et al., 1994). Very few studieshave been able to quantify the translations of the ACL-de"cient knee during gait. Morrison (1970) used a math-ematical modeling approach to provide indirect evidencethat the ACL is necessary to normal knee motion duringgait. Hasan et al. (1998) measured a signi"cantly greateranterior}posterior displacement during stance phase ofgait cycle for the ACL injured knees than for the controlnormal knee. Marans et al. (1989), however, found a sig-ni"cantly greater anterior translation of the tibia relativeto the femur only during swing phase of the gait.

The hamstring muscles have been recognized as animportant element in compensating for the loss of stabil-ity in the ACL-de"cient knee (Solomonow et al., 1987;Ka> lund et al., 1990). Ciccotti et al. (1994) concluded froman electromyographic study of the ACL-de"cient andACL-reconstructed subjects that increased activity of thebiceps femoris muscle might indicate a protective mecha-nism. It has also been reported that subjects with anACL-de"cient knee returned to high level of sports afterattaining su$cient hamstring strength (Giove et al., 1983;Walla et al., 1985). In another study, subjects complainedof fewer episodes of the knee giving way after a trainingprogram designed to decrease the reaction time of ham-string muscle action (Ihara and Nakayama, 1986). Des-pite the literature suggesting that hamstring muscleactivation can prevent abnormal anterior motion of thetibia on the femur in the ACL-de"cient knee, the extent ofsuch compensation has not been studied extensively.

Due to di$culties in measuring in vivo internal forcesof the knee joint, mathematical models have been widelyused in biomechanical studies of the knee (Andriacchiet al., 1983; Yasuda and Sasaki, 1987; Kaufman et al.,

0021-9290/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 1 - 9 2 9 0 ( 0 0 ) 0 0 0 4 7 - 6

Fig. 1. Schematic of the musculoskeletal knee model showing the bonystructure and the muscles included in the model at a single selectedposition during early stance phase of gait.

1991; Lutz et al., 1993; Pandy and Shelburne, 1997).A few modeling studies have examined the function ofknee ligaments during gait (Morrison, 1970; Harrington,1976; Biden et al., 1990; Berchuck et al., 1990; Andriacchi,1993; Mikosz et al., 1992). However, these models havenot examined the e!ect of hamstring muscle force on theanterior tibial translation during gait in the ACL-de"-cient knee. Up to date, it is not clear whether the ham-string muscle force can completely compensate for theloss of ACL, and what are the consequences of increasedhamstring muscle force.

In the present modeling study, it was hypothesized thathamstring muscle force in a knee model with certain setsof parameters could completely compensate for the lossof ACL ligament in restraining the anterior tibial transla-tion during gait. This compensation, however, wouldincrease quadriceps muscle force and joint contact force.

2. Methods

A two-dimensional knee model in the sagittal planewas developed in this study (Fig. 1). All parameters of themodel are summarized in Tables 1 and 2. The modelincluded the tibiofemoral and patellofemoral joints andthree rigid bodies: the femur, tibia and patella. Body-a$xed coordinate systems were attached to the femur (u,v) and the tibia (x, y) (Fig. 2(a)). The origin of the femoralsystem was located at the intercondylar notch with the

v-axis increasing in the proximal direction along the longaxis of the femur and the u-axis directed posteriorly. Theorigin of the tibial system was located at the center ofmass of the tibia. The y-axis increased in the proximaldirection along the long axis of the tibia and the x-axiswas directed posteriorly. The position and orientation ofthe femur relative to the tibia was described in the tibialsystem by a vector (x

0&, y

0&) that de"ned the origin of

femoral system and the knee angle (a). Full extension ofthe knee was de"ned as the neutral position when bothx0&

and a were zeros.Knee motion and internal loads were studied at

a single selected position when the external #exion mo-ment on the knee reached a peak value during earlystance phase of gait (Winter, 1990). It was assumed thatthe quadriceps muscle force was at its peak value at thisposition in order to balance the peak external #exionmoment. Examination of the hamstring muscles in com-pensation for joint instability at this speci"c position wasimportant because of a large potential anterior transla-tion of the tibia due to the shear force from the patellartendon.

The contour of the femoral condyle in the tibiofemoraljoint was represented by an ellipse described byYamaguchi and Zajac (1989). The contour of the tibialplateau was represented by a straight line sloped 83posteriorly relative to x-axis of the tibia coordinate sys-tem (Nisell et al., 1986).

The coordinates of the contact point on the femoralcontour were described by a function of the coordinatesof the origin of the femoral system (x

0&, y

0&) and knee

angle (a) (Abdel-Rahman and Hefzy, 1993). Furthermore,a single tangent to both the femoral and the tibial con-tour at the contact point, a geometric compatibility con-dition, was assumed in order to exclude the possibility ofpenetration or overlap between the femur and the tibia(Abdel-Rahman and Hefzy, 1993). The contact force (F

#)

at the tibiofemoral joint was then calculated along thedirection of a co-linear normal vector at the contactpoint. By combining both the contact and geometriccompatibility conditions, the y-coordinate of the originof the femoral system (y

0&) was represented by a function

of its x-coordinate (x0&

) and knee angle (a). Since the kneeangle was a predetermined parameter, x

0&was the only

unknown position variable.Four major ligaments: the anterior cruciate ligament

(ACL), posterior cruciate ligament (PCL), medial collat-eral ligament (MCL), and lateral collateral ligament(LCL), and the medial capsule were included in themodel. Each of the two cruciate ligaments was represent-ed by anterior and posterior bundles. The positions ofinsertion sites of the ligaments were adapted from Beyn-non et al. (1996) for cruciate ligaments and from Abdel-Rahman and Hefzy (1993) for collateral ligaments. Thepositions of insertion sites of the medial capsule wereadapted from Blankevoort et al. (1991). A straight line

872 W. Liu, M.E. Maitland / Journal of Biomechanics 33 (2000) 871}879

Table 1Parameters! used in the model including geometry of bones, coordinates of insertion sites of ligamentous elements, coordinates of the origin andinsertion sites of muscle units, and coordinates of some relevant points

a5*"&

b5*"&

a1!5&

b1!5&

a1!5

b1!5

¸1!5

s h35.4 21.8 28.6 19 39.4 16.3 65.2 8.2 83

Ru!ACL

Rv!ACL

Ru1ACL

Rv1ACL

Ru!PCL

Rv!PCL

Ru1PCL

Rv1PCL

RuMCL

29 !5 28 5 21.5 14 10.5 !0.5 5

RvMCL

RuLCL

RvLCL

Ru.#!1

Rv.#!1

Px!ACL

Py!ACL

Px1ACL

Py1ACL

5.7 15 1.7 5 !1.3 !5 175.5 7 177.5

Px!PCL

Py!PCL

Px1PCL

Py1PCL

PxMCL

PyMCL

PxLCL

PyLCL

Px.#!1

18 173 30 164 0 137 25 138.7 !0.4

Py.#!1

X56"

Y56"

OxRF

OyRF

OxVAS

OyVAS

OxHL

OyHL

153.2 !32 110 29.8 !31.1 !29 192.4 124.4 !100.1

OxHS

OyHS

Ox'!4

Oy'!4

IxHL

IyHL

IxHS

IyHS

Ix'!4

5 211.1 12.7 7.1 8.1 108.2 10.1 108.6 36.8

Iy'!4

Of t9

Of t:

Ohf9

Ohf:

P59

Pty P&6

P&7

!28.8 0 !250 !74 478.1 0 175 0 !21.8

!Parameter used in the model:a5*"&

, b5*"&* The semi-major and semi-minor axes of the ellipse of femoral contour in the tibiofemoral joint (mm).

a1!5&

, b1!5&* The semi-major and semi-minor axes of the ellipse of femoral contour in the patellofemoral joint (mm)

a1!5

, b1!5* Length and width of the patella (mm).

¸1!5* Length of the patellar tendon (mm).

s * The o!set along v axis between the centers of two ellipses of femoral contours (mm).h * The angle of tibial plateau slope relative to anterior}posterior axis.Ru

*, Rv

** The coordinates of insertion sites of ligamentous elements (anterior and posterior bundles of the ACL and PCL, MCL, LCL, and medial

capsule) on the femur in the femoral coordinate system (mm).Px*, Py

** The coordinates of insertion sites of ligaments on the tibia in the tibial coordinate system (mm).

X56"

, >56"* The coordinates of the center of tibial tubercle (mm).

OxRF

, OyRF* The coordinates of the origins of muscle units including rectus femoris de"ned in the pelvic coordinate system originated at the hip joint

(u"!7.4 cm, v"47.8 cm), the vasti unit de"ned in the femoral coordinate system, hamstring (long head) de"ned in the trunk coordinate system,hamstring (short head) de"ned in the femoral coordinate system (mm), and gastrocnemius de"ned in foot coordinate system (mm).

IxHS

, IyHS* The coordinates of origins of muscle units including hamstring (long head) and hamstring (short head) de"ned in the tibial coordinate

system (mm), and gastrocnemius de"ned in femoral coordinate system (mm).Of t

x, Of t

y* The coordinates of the origin of the foot in tibial coordinate system (mm).

Oh fx, Ohf

y* The coordinates of the center of the hip in tibial coordinate system (mm).

P5x

, P5y* The coordinates of the point P

5in tibial coordinate system (mm).

P&u

, P&v* The coordinates of the point P

&in femoral coordinate system (mm).

represented the path of each ligament bundle and themedial capsule between insertion sites. Each ligamentwas modeled as an elastic element with a non-linearforce}strain relationship at low strain level and a linearforce}strain relationship at higher level (Wismans et al.,1980; Blankevoort et al., 1991). The initial strain at fullextension position of the knee (a"03, and x

0&"0) and

sti!ness of each ligamentous element were adapted fromBlankevoort et al. (1991).

External forces applied on the tibia at the single se-lected position during early stance phase of gait includedthe weight of the shank and the resultant forces andmoment of the ankle joint (Fig. 2(b)). A set of normal gaitdata (Winter, 1990) was used in the simulations. Thesubject's body weight was 56.7 kg with a calculatedshank weight of 3.5 kg. The peak #exion moment on the

knee occurred at 13% of the gait cycle starting from heelstrike. At this position, the tibia was aligned almostvertically relative to the ground (0.33 tilting backward)while the knee was #exed at 15.63. The resultant forces onthe ankle joint were 595.7 N in the vertical direction and97.1 N in the horizontal direction pointing posteriorly.The resultant ankle dorsi#exion moment was 0.2 Nm.

Three major muscle groups included in the model werethe quadriceps, hamstring and gastrocnemius muscles.The quadriceps muscles were represented by two units:rectus femoris and the vasti (a combination of vastuslateralis, vastus intermedius, and vastus medialis). Thehamstring muscles were also simpli"ed as a short headunit (i.e. biceps femoris (short head)) and a long head unitwith the same origin and insertion sites as the bicepsfemoris (long head) but a summed maximal isometric

W. Liu, M.E. Maitland / Journal of Biomechanics 33 (2000) 871}879 873

Table 2Parameters! used in the model including limb orientation, external loading, force ratio, initial strain and sti!ness of ligamentous elements

a5*"

a&%.

a6

F!x

F!y

M%95

r3v

r4l

!0.33 !15.93 03 97.1 595.7 0.2 0.172 0.194

e!ACL

e1ACL

e!PCL

e1PCL

eMCL

eLCL

e.#!1

k!ACL

0.06 0.1 !0.24 !0.03 0.004 !0.07 !0.06 5000

k1ACL

k!PCL

k1PCL

kMCL

kLCL

k.#!1

5000 9000 9000 8250 6000 2000

!Parameters used in the model:a5*"* Tibial angle measured between tibial long axis and a vertical line.

a&%.* Femoral angle measured between femoral long axis and a vertical line.

a6* Upper body angle measured between upper body long axis and a vertical line.

F!x

, F!y* Resultant force vector on the ankle joint (N).

M%95* Resultant moment of force on the ankle joint (N m).

r37* Force ratio of quadriceps muscles de"ned as ratio of the maximal isometric forces of rectus femoris unit versus the vasti unit.

r4-* Force ratio of hamstring muscles de"ned as ratio of the maximal isometric forces of hamstring short head unit versus long head unit.

e** Initial strain of the ligamentous elements (anterior and posterior bundles of the ACL and PCL, MCL, LCL, and medial capsule).

ki* Sti!ness of the ligamentous elements (anterior and posterior bundles of the ACL and PCL, MCL, LCL, and medial capsule) (N).

Fig. 2. (a) Simpli"ed two-dimensional tibiofemoral and patellofemoral joints in the current model. Body-a$xed coordinate systems were attached tothe femur (u, v) and the tibia (x, y). Seven ligamentous elements were included: (1) aACL: anterior bundle of the ACL; (2) pACL: posterior bundle of theACL, (3) aPCL: anterior bundle of the PCL, (4) pPCL: posterior bundle of the PCL, (5) MCL: the medial collateral ligament, (6) LCL: the lateralcollateral ligament, (7) mcap: the medial capsule. The quadriceps muscle forces were simpli"ed as two units: rectus femoris (F

3&) and the vasti unit (F

7!4).

Knee angle (a), angle of patellar tendon relative to y-axis (a1-

), and the angle of patella relative to the y-axis (a1) were also illustrated. Point P

&was the

intersection point of the v-axis and the femoral contour. (b) The forces applied on the tibia include resultant moment and forces of ankle joint (M!,

,F!9

, F!:

), weight of the tibia (=5), the contact force of tibiofemoral joint (F

#), and the forces from patellar tendon (F

1-), hamstrings long head (F

)-), short

head (F)4

), ACL anterior bundle (F!ACL

), posterior bundle (F1ACL

), MCL (F.#-

), and medial capsule (F.#!1

). The forces from posterior cruciate andlateral collateral ligaments were not illustrated since they were not loaded in the current study.

forces of biceps femoris (long head), semimembranosus,and semitendinosus. The origin and insertion sites, andthe maximal isometric forces of muscles were adaptedfrom Friederich and Brand (1990) and Delp (1990). An

action circle was used for the hamstrings and gastrocne-mius when they wrapped around the head of the femur(Shelburne and Pandy, 1997). The radius of the circle foreach muscle was the sum of the major semi-axis of the

874 W. Liu, M.E. Maitland / Journal of Biomechanics 33 (2000) 871}879

ellipse of femoral contour and the average thickness ofthe muscle. The average thickness of hamstring long headunit, short head unit, and gastrocnemius was taken as 12,3 and 3 mm, respectively, estimated from the data ofShelburne and Pandy (1997). The activation levels of thetwo hamstring muscle units were assumed to be equiva-lent based on the observation from recorded muscleEMG signals during gait (Kadaba et al., 1989). Forcesfrom the two hamstring muscle units were assumed to beproportional to each other with a force ratio (r

4-) de"ned

as the ratio of maximal isometric forces of hamstringshort head unit (402 N) relative to long head unit(2075 N) (Delp, 1990).

The approach used to model the patellofemoral jointin this study was similar to that used by Tumer andEngin (1993) and Shelburne and Pandy (1997). The fem-oral contour of the patellofemoral joint was adaptedfrom Yamaguchi and Zajac (1989) as an ellipse centeredat (0, 8.2 mm) in the femoral coordinate system (Fig. 2(a)).The patella was approximated as a rectangle (Yamaguchiand Zajac, 1989; Shelburne and Pandy, 1997). The twoquadriceps muscle units (rectus femoris and vasti) in-serted on its proximal side, and the patellar tendon at-tached on its distal side. Muscle forces acted on thepatella along the lines that connected their origin sites onthe pelvis and/or the femur and insertion site on thepatella. The patellar tendon was assumed to be non-extensible. The force of patellar tendon was along a lineconnected by its insertion sites on the patella and thecenter of tibial tubercle. The internal edge of the patellawas assumed to be a plane articulated with the femoralgroove. The contact surfaces were assumed to be rigidand frictionless. The contact force was along the normaldirection of the contact plane of the patella. The quasi-static equilibrium of the patellofemoral joint was repre-sented by the resultant forces balanced in u and v direc-tions, and moments balanced around contact point. Theforces from the two quadriceps muscle units (rectusfemoris and vasti) were assumed to be proportional toeach other with a force ratio (r

37"0.172) de"ned as the

ratio of maximal isometric forces of rectus femoris(779 N) relative to vasti unit muscles (4520 N) (Delp,1990). In the "nal equation, patellar angle (a

1in Fig. 2(a))

was the only unknown independent variable for a givenknee angle (a). After resolving the equation using aniteration method, the force of the patellar tendon (F

1-)

was represented by a function of vasti unit muscle force(F

7!4).

The inertial forces and moments of the tibia and thefoot were ignored in the quasi-static equilibrium equa-tions of the system since they were small during thestance phase of gait (Harrington, 1976). All forces appliedon the tibia and moments about a point P

5, the crossing

point between the y-axis and tibial contour (Fig. 2(b)),were assumed to be balanced. Three equations includingforces in x and y directions and moments about the point

(P5) were obtained. Three independent unknown vari-

ables in the equations were the position variable x0&

, thejoint contact force (F

#) and the vasti unit muscle force

(F7!4

).A program coded in Matlab (The MathWorks, Inc.

Natick, Mass.) was used to resolve the quasi-static equi-librium conditions of the tibiofemoral joint to determinethree unknown independent variables (x

0&, F

#and F

7!4).

Simulations were conducted on the modeled normal andACL-de"cient knees. The ACL-de"cient knee was de-"ned here as a knee with a completely ruptured ACL. Inthe simulations, the gastrocnemius muscle was assumedto be inactive at the single selected position of stancephase (Tibone et al., 1986). Incremental hamstring muscleforces ranging from 0 to 56% of the maximal isometricforce were applied at 2% increments to the modeledknee. Simulations were "rst conducted on the normalknee. The ACL-de"cient knee was then simulated withthe same joint con"guration and external loads to exam-ine the compensatory e!ect of hamstring muscle forces.

In order to compare the results of the current studywith in vivo measurements in the literature (Lafortune etal., 1992), the displacement of a point P

&(Fig. 2(a)) along

the x-axis in the tibial coordinate system was derived.Point P

&was de"ned to be at the intersection of the v-axis

and the femoral contour. When the knee was in neutralposition (full extension), the point (P

&) was located near

the joint contact point and its x-coordinate was zero. Atknee #exion of 15.63, the point (P

&) was located 5.9 mm

anterior to the origin of the femur. The displacement ofthe point P

&in the x-direction was obtained by subtract-

ing 5.9 mm from the value of the x0&

. The displacement ofpoint P

&is reported as the resultant anterior tibial trans-

lation throughout this paper.

3. Results

ACL injury increased the anterior tibial translation by11.8 mm, while hamstring muscle force could compensatefor the loss of the ACL constrains (Fig. 3(a)). Results ofsimulations at a single selected position with a peakexternal #exion moment on the knee during gait showedthat the tibia moved 1.4 mm posteriorly relative to thefemur in the normal knee, and 10.4 mm anteriorly in theACL-de"cient knee without hamstring muscle forces(Fig. 3(a)). Hamstring muscle force at 56% of its maximalisometric strength moved the tibia posteriorly by only1.1 mm in the normal knee, but restored near-normaltibiofemoral joint contact position in the ACL-de"cientknee.

Ligamentous loading was a!ected by the ACL injury,and hamstring muscle force reduced the load on second-ary constraints (Fig. 3(b)). In the normal knee, the an-terior shear force was resisted by the two bundles of theACL (Fig. 3(b)) with minimum loading on the MCL and

W. Liu, M.E. Maitland / Journal of Biomechanics 33 (2000) 871}879 875

Fig. 3. (a) The anterior tibial translation relative to the femur versushamstring muscle force for the normal and ACL-de"cient knees; (b) thecomponents of tension forces of ligaments directed posteriorly versushamstring muscle forces.

Fig. 4. (a) The component of patellar tendon force in anterior directionversus hamstring muscle force for the normal and ACL-de"cient knees;(b) The change of patellar tendon orientation (a) due to the change inhamstring muscle force for the normal and ACL-de"cient knees.

zero loading on the LCL and medial capsule (not shown).When the ACL was completely ruptured the MCL andmedial capsule resisted anterior shear force. However,the total anterior shear force sustained by the MCL andmedial capsule in the ACL-de"cient knee was substan-tially smaller than that resisted by two bundles of theACL in the normal knee. The reduction of anterior shearforce was probably due to a change of the patellar tendonangle (a

1-in Fig. 2(a)). The hamstring muscle forces

unloaded the ACL ligament in the normal knee, and theMCL ligament and medial capsule in the ACL-de"cientknee. The PCL ligament was not loaded in the normalknee or the ACL-de"cient knee either with or withouthamstring muscle force.

Anterior shear force on the tibia caused by the patellartendon decreased due to ACL de"ciency, but increasedwith hamstring muscle force (Fig. 4). The patellar tendonexerted an anterior shear force of about 169 N on thetibia in the modeled normal knee without hamstringmuscle force. This force increased to 345 N at 56% of themaximal hamstring muscle force (Fig. 4(a)). A complete

rupture of the ACL changed the patellar tendon angle(a

1-) from 14 to 4.53 and reduced the anterior shear force

from 169 to 41 N (Fig. 4(b)). The hamstring muscle forceson the ACL-de"cient knee increased the angle (a

1-) and

anterior shear force of the patellar tendon.An increase in hamstring muscle force was accom-

panied by an increase of quadriceps muscle force in boththe normal and ACL-de"cient knees (Fig. 5). Withouthamstring muscle force, muscle force in vasti unit washigher in the normal knee (639 N) than in the ACL-de"cient knee (466 N). This was mainly due to the an-terior translation of the tibia in the ACL-de"cient kneethat changed moment arm of the patellar tendon. Withapplication of 56% of the maximum isometric hamstringforce, the vasti unit muscle force increased to 1228 N inthe normal knee, and 1192 N in the ACL-de"cient knee.

Contact forces in both tibiofemoral and patellofemoraljoints increased as hamstring muscle force was applied(Fig. 6). Contact force of the tibiofemoral joint was higherin the ACL-de"cient knee (1546 N) than the normal knee(1410 N) (Fig. 6(a)). The increased joint contact force wasdue to a change in the patellar tendon angle (a

1-) that led

to a higher vertical component of patellar tendon force.When a 56% of the maximal isometric hamstring force

876 W. Liu, M.E. Maitland / Journal of Biomechanics 33 (2000) 871}879

Fig. 5. The vasti unit force versus hamstring muscle force for thenormal and ACL-de"cient knees.

Fig. 6. The joint contact forces versus hamstring muscle force for thenormal and ACL-de"cient knees in (a) the tibiofemoral joint, and(b) the patellofemoral joint.

was applied, the contact force increased dramatically to3143 and 3075 N in the normal and ACL-de"cient knees,respectively. Injury of the ACL ligament unloaded thepatellofemoral joint with the contact force decreasingfrom 435 N in the normal knee to 235 N in the ACL-de"cient knee (Fig. 6(b)). Increased hamstring forces pro-duced higher contact force in the patellofemoral joint.

4. Discussion

At a single selected position at which the knee wasunder a peak external #exion moment during earlystance phase of gait, simulation of the normal knee cal-culated a small posterior displacement (1.4 mm) of thetibia on the femur compared to the extended knee posi-tion. In a study by Lafortune et al. (1992), they found anaveraged maximal posterior displacement of 3.6 mm in"ve healthy subjects. The #exion angle of the knee (203)corresponding to the maximal posterior displacement intheir study was greater than the angle (163) used in thepresent study. This di!erence may contribute to di!er-ence in posterior displacements of the tibia on the femurbetween two studies.

The current study calculated an increase of 11.8 mm inthe anterior displacement of the tibia relative to thefemur for the ACL-de"cient knee compared to the nor-mal knee at the single selected position of early stancephase during gait. Measurements of joint motion aftersectioning the ACL in cadaveric knees have demon-strated signi"cant increases in anterior shift of the tibiaunder various loads (Fukubayashi et al., 1982; Sullivan etal., 1984; Haimes et al., 1994; Torzilli et al., 1994). Resultsof the current study generally agreed with these experi-mental "ndings though a direct comparison is di$cult.For instance, Torzilli et al. (1994) found that at 153 ofknee #exion the anterior translation of the tibia relativeto the femur under 133 N simulated quadriceps force was5.9 mm and a compressive loading of 444 N generatedadditional anterior translations of 9 mm for the ACL-transected knees. Increased anterior translation was alsoreported in many in vivo measurements on the ACL-de"cient patients using an arthrometer (Torzilli et al.,1984; Maitland et al., 1995) and during gait (Hasan et al.,1998).

The current study was limited by simplifying thethree-dimensional anatomical structure of the knee jointto a two-dimensional model. Rotations of the knee,which in#uences anterior tibial translation (More et al.,1993), could not be examined because the model allowedonly motions in the sagittal plane. Other limitations ofthe current model included representing the surface of thetibial plateau as a linear element, not representing theproperties of the menisci, and simplifying the bony con-tact to be rigid in nature. A more precise knee joint modelwould include more accurate bony geometry, it wouldinclude more precise modelling of soft tissue elementssuch as menisci, and would include compliance of articu-lar structures. Therefore, predictions of the current studyare likely to overestimate anterior translation of the tibiarelative to the femur.

In the present simulations, the location of the tibiarelative to the femur did not change substantially withincreased hamstring muscle forces in the normal knee. Itindicated a limited role of hamstring muscle force in

W. Liu, M.E. Maitland / Journal of Biomechanics 33 (2000) 871}879 877

restraining anterior translation of the tibia when theACL was intact. Hamstring EMG activities detectedduring late swing and early stance have been suggested torepresent a function of the muscle as an extensor of thehip joint (Tibone et al., 1986) and a sti!ening force of theknee against anterior shear force (Baratta et al., 1988).The latter function of the hamstring muscle may not benecessary for the normal knee.

The hamstring muscle forces, however, might be veryimportant for the ACL-de"cient knee during level walk-ing. Several authors have suggested that the hamstringmuscles can compensate for instability of the ACL-de"-cient knee during functional activity (Solomonow et al.,1987; Ka> lund et al., 1990; Ciccotti et al. 1994). Limbird etal. (1988) found that during late swing and early stance,ACL-de"cient individuals demonstrated signi"cantlygreater hamstring activity as compared to healthy indi-viduals. A similar "nding was reported by Ciccotti et al.(1994) and suggested that increased activity of the bicepsfemoris in the ACL-de"cient knee may indicate a protec-tive mechanism. In the current study, the anterior trans-lation of the tibia in the ACL-de"cient knee decreasedwith application of the hamstring force. When hamstringforce reached 56% of its maximal isometric force, thetibial translation was reduced to a near-normal level.

On the other hand, there are disadvantages associatedwith a strategy to use hamstring muscle forces to com-pensate for increased laxity in the ACL-de"cient knee.Simulations showed that increased hamstring muscle for-ces produced an additional knee #exion moment thatmust be balanced by a corresponding increase in a quad-riceps muscle force. In the normal knee, the quadricepsmuscle force of 14% of maximum isometric force wasrequired to balance the external #exion moment withouthamstrings muscle force. For the ACL-de"cient knee,this force increased to 26% of its maximal isometricforce with 56% of hamstrings muscle force. Increasedco-contraction of antagonistic muscles would decreasemotion e$ciency, and increase energy expenditure andtibiofemoral contact force. Also, learning hamstring co-contraction to compensate for increased anterior tibialtranslation during gait may not be trivial.

Alternate strategies may also decrease anterior tibialtranslation during gait. Gait studies have found thatabout 75% of subjects with chronic ACL-de"cient kneesshowed an adapted gait pattern to avoid peak quadricepsmuscle force (Berchuck et al., 1990; Andriacchi, 1993). Anadapted gait pattern would alter the position and theloading of the knee during gait, which has not beenaccounted for in the current study. Future studies areneeded to examine in detail the dynamics of the knee withadapted gait pattern.

A change of patellar tendon angle of the ACL-de"cientknee due to anterior tibial translation was demonstratedin the current study. If a large enough external anteriorshear force was applied to the tibia, one would expect

that the patellar tendon might shift further, even orientedsuch that shear component of the patellar tendon forcewould be directed posteriorly. This may explain the re-sults of an in vitro study by Aune et al. (1997), in whichthey loaded knee specimens to failure with and withoutsimulated quadriceps forces and found that the quad-riceps muscle force protected the anterior cruciate liga-ment from injury. Therefore, a twofold functional role ofthe quadriceps muscles is proposed here: (1) Withinsmall anterior translation of the tibia on the femur, con-traction of the quadriceps produces an anterior shearforce; (2) if the knee is under an external anterior shearforce and anterior translation of the tibia exceeds a cer-tain limit, the patellar tendon will start to resist anteriorshear force and protects the anterior cruciate ligamentfrom injury.

In summary, a two-dimensional knee model was de-veloped for analysis of the knee joint motion and internalloading during gait. Hamstring muscle force was shownto be able to compensate for laxity of the ACL-de"cientknee for a set of model parameters used in the currentstudy. However, such compensation required a relativelyhigh level of co-contraction of antagonistic muscles lead-ing to a higher joint compressive load and energy expen-diture. These disadvantages may explain, in part,alterations of adapted gait pattern in many individualswith ACL-de"cient knees, rather than simply increasingco-contraction of the hamstring muscles.

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