1. Biomechanics. 1973. Vol. 6. pp. 79-92. Pagamon Rss. Printed in Great IMain

BIOMECHANICAL ANALYSIS OF KNEE FLEXION AND EXTENSION*

GARY L. SMIDT:

Abstract- For the knee, mathematical analyses for the sagittal plane were performed on data obtained from roentgenograms and a load cell incorporated in a specially constructed force table. Normal knees from 26 subjects were examined. Some of the results were: (I) the axis of rotation for the knee displaced 3.2cm through a 90 range and is located at the level of the lateral femoral epicondyle, (2) the moment arm is greater for knee extensors than the flexors. (3) the maximum average torque values of nearly I200 kg-cm result from eccentric (lengthen- ing) and isometric (static) muscle contractions while the isometric (static) flexor contraction near complete extension produced a maximum average torque of 636 kg-cm. and 14) during muscular efforts against external load the contact forces at the patello-femoral and tibial- femoraljoints varied from 0.38 to 340 times body weight.

INTRODUCTIOX

THE KNEE joint has been studied from many different vantage-points, but much of the interest has focused on the importance of the role of the quadriceps in knee function. To more clearly define this role, numerous in- vestigators (Berger. 1966: Campney and Wehr. i 965: Clarke and Bailey, 1950: Clarke et al., 1950; Lindahl et af.. 1969; Mendler, 1967 and Williams and Stutzman, 1946) have related angular position of the knee joint and force generated by the quadriceps mechanism. Others have compared force generated by the quadriceps with respect to different types of muscular contraction (MoEroid et al., 1969; Peterson, 1960 and Thistle et al., 1967) and normal versus abnormal (Mendler, 1967: Bender and Kaplan, 1963 2nd ODonoghue et nl., 1951). The force engendered by the flexors of the knee have also been studied (Clarke and Bailey, 1950: Clarke et al., 1950: Limdahl et al., 1969: Mendler, 1967; Williams and Stutzman. 1956 and ,Moffroid et al., 1969; Thistle et al.. 1967). Still other investigators (Courvoisier, 1968: Hallen and Lindahl, 1966: Lieb and Perry, 1968: Lindahl and Movin, 1968: Lindahl and Movin, 1967;

Maquet, 1969: Morrison, 1968: Morrison, 1969: Morrison, 1970: Shinno, 196 1: Shinno, 1961 and Shinno, 1961) have concentrated on the mechanics of the knee joint. For example, iMorrison, 1969 and 1970, has identified joint reaction force at the knee during walking, but relative to active and resistive exercise of the knee no studies have been located which report forces at the patello-femoral and tibia&femoral joints, nor torque generated by the knee flexors and extensors for eccentric (lengthening), con- centric (shortening), and isometric (static) muscle contractions.

PROCEDURE

Twenty-six normal men served as subjects for this study. Their mean age was 28 yr, mean height 176 cm and mean weight 82 kg. The data acquisition and reduction was accomplished in two phases: (1) serial lateral radiographs of the knee joint, and (2) measure- ment of torque generated by the knee flexors and extensors.

All measurements were obtained for the right side.

Received lOJuly 1972. _ -Director and Assistant Professor of the Graduate Degree Program for Physical Therapy, College of [Medicine.

University of Iowa, lowa City. Ioua 52230. U.S.A.

79

80 G. L. SMIDT

Serial X-rays of knee The center of rotation or instant center

does not change position for a ball and socket type of joint. In the case of the human, the femoral head might be the best example of such a body. However. when the outline of the joint surfaces is non-spherical, changes in location of the instant center occur with respect to the change in angular relationship at the joint. Superincumbent on the tibia1 plateau are two menisci which could effec- tively allow articulation with a circular object and not require a change in position of the instant center, but a change might be expected at the knee joint due to the non-spherical construction of the femoral condyles.

To determine the instant centers for the knee for the sagittal plane, seven different X-rays were taken from lateral views at 15 intervals from complete extension to 90 flexion. On the X-ray table the subject lay on his right side, with his left lower extremity flexed and supported by several sandbags. During the X-ray procedure the changes in angular position at the knee were accom- plished exclusively by movement of the shank and foot.

The instant centers for the seven different angular positions were obtained from the X-ray film by a method similar to that reported by Frankel and Burstein (1970). Two points, 11 and 13 cm proximal and anterior to the distal femoral condyle, were marked on each film. For each series of X-rays for each sub- ject, the tibia and fibula were superimposed and the locations of the two points were marked for each of the seven different angular positions. With the aid of a flexible piece of plastic, two curved lines were drawn through the series of points. From the lines at locations corresponding to each of the angular positions of the knee, perpendiculars were erected by the use of a front surface mirror. The instant centers for each position were considered the points at which the perpendicular lines intersected (Fig. 4).

For each position, the moment arm distance

for the knee extensor mechanism was the perpendicular distance from a line between the patella and the tibia1 tubercle, indicative of the long axis of the patellar ligament, and the appropriate instant center. The line of action for the knee flexors were assumed to be parallel to the long axis of the femoral shaft at the level of the fibular head, so the perpendicular distance from this line to the appropriate instant center was considered the moment arm for the knee flexors. The mea- surements for the moment arms and other geometrical considerations used to obtain forces at the patello-femoral and tibial- femoral joints were also obtained from X-ray film (Fig. 4). All measurements from ihe X-rays were corrected for magnification.

Methodfor obtaining force measurements Modifications were made on the force table

used in previous studies (Jensen et al,, 1971 and Olson et nl.. 1972) so that force mea- surements could be obtained for eccentric (lengthening), isometric (static), and concen- tric (shortening) muscle contractions of the knee flexors and extensors. The subjects side-lying position on the table was similar to the position assumed when X-rays were taken, so the influence of gravity was elimin- ated. The hip was in the neutral position. The right thigh was stabilized with two posterior padded supports and two anterior, while the lateral femoral condyle was placed over the center of the table. A cuff attached to a cable was placed distal from the tibia1 plateau at a point 75 per cent of the length of the shank and foot (Fig. 1). The cable was aligned at right angles to the long axis of the shank and was afhxed to a load cell incorporated in a motor-driven radial arm. The direction of movement for the radial arm was controlled by a hand switch and the velocity of the arm was constant at 13 per sec. The load cell and the hand switch were connected to a dyno- graph and tape recorder, where force data and signals indicative of radial arm movement were obtained.

i-:iL,,i,, 7. ,I))

BlOMECHA\NICAL ANALYSIS OF KNEE FLEXION AND EXTENSION 81

With knowledge of the velocity of the radial arm and the paper speed of the dynograph recorder, it was possible to simultaneously determine the angular position of the knee and the external force produced by the flexors and extensors of the knee. Each subject was given the same instructions, the opportunity to warm up. and approximately one minute of rest between contractions, which were re- quested to be voluntarily maximal. The cable and radial arm were placed posterior to the shank for knee extensor contractions and for contractions of the knee flexors and radial arm and cable were placed anteriorly (Figs. 1 and 2). Force generated by isometric contrac- tions were obtained at 15 intervals from com- plete extension to 90 fiexion. Continuous force measurements over the same range were

procured for concentric and eccentric con- tractions (Fig. 3). During each testing session the subject performed a total of 1.8 separate efforts exclusive of warm-up, and the order of eccentric, isometric, and concentric contrac- tions was randomized to help eliminate bias of the data.

The data were acquired on magnetic tape and reduced by an IBM I800 computer. Peak force was obtained for each isometric con- traction and the force produced by the eccen- tric and concentric contractions was sampled at approximately three degree intervals.

Calculation of muscle tension andjoint forces The symbols and definitions associated with

the distances and forces for this study appear in Fig. 4. Exclusively for isometric contrac-

b

conc.ntric *Idon

Fig. 3. Schematic drawing to illustrate the three types of muscle contraction.

a = Direction of knee motion. b = Starting position. c = Point and line of application of external force. d = Angular position.

Bhi-VoL6.No.I-F

82 G. L. SMIDT

Fig. 4. Forces and moment arms for knee flexion and extension (N = 26).

0 = Instant center.

F, = Tension along quadriceps.

F,I = Tension along patellar ligament. D,, = Moment arm for patellar ligament. F,e = External force for knee extension. D , = Moment arm for external force. F,, = Tension along hamstring muscles. Dh = Moment arm for hamstrings. F rh = External force for knee flexion. F,, = Compression force at patello-femoral joint. F, = Compression force at tibial-femoral joint. F, = Anterior-posterior shear force at tibial-femoral

joint.

tions at 15 intervals, mathematical calcu- lations were performed to determine tension in the knee flexors and extensors, compressive and shear forces at the tibial-femoral joint, and compressive force at the patello-femoral joint. The force applied by the cable was assumed to be parallel to the shear component offered by the patellar ligament and the ham- strings.

Tension in the flexors and extensors was obtained by solving moment equilibrium equations about the axes of the knee joint for

F,, and F,,. These equations are:

( Fh X &) - ( Fti X D,) = 0 flexion (1)

(F,, X DP1) - (F,, X D,.) = 0 extension. (2)

Compressive and shear forces at the tibial- femoral joint were obtained by the use of the following equilibrium equations:

F, = F,, - F,[ sin 8 + F, = 0 extension (3)

Fy=Fplcosd-F,=O (4)

where 8 equals the angle between the long axis of the tibia and the line of pull for the patellar ligament and:

F, = Fh sin 4 - F, = 0 flexion (5)

F,= F,,cos~--Fc=O (6)

where 4 is the knee angle as measured from complete extension.

A concurrent force system was assumed at the patello-femoral joint since the lines of force for the patellar ligament, quadriceps tendon, and patello-femoral joint must inter- sect at a common point or movement (rocking) of the patella would occur. The magnitude of the force for the patellar ligament and quadri- ceps tendon was assumed to be equal. The origin of the x-y coordinate system was placed at the intersection of the forces and the x axis was oriented to coincide with the line of force for the patello-femoral joint. The y components of the quadriceps and patellar ligament offset each other so the following equation was used to obtain the compression force at the patello-femoral joint:

F,= (Fg~~~y)+(Fy~~~~y)--Fp,=O (7)

where y is the angle between the lines of force for the (1) patellar ligament and patello- femoral force, and (2) quadriceps tendon and the patello-femoral force.

BIOMECHANICAL ANALYSIS OF KNEE FLEXIOK .4ND EXTEBSION 83

A nthropometric measurements Proximal from the medial knee joint line.

thigh circumference measurements were obtained at levels 30 and 60 per cent of the femoral length. The estimated center of the femoral head (Johnston and Smidt, 1969) and the knee joint line were the landmarks used to ascertain the femoral length. The tibia1 length was measured from the knee joint line to the tip of the medial malleolus (Table 1).

Table I. Data for subjects (N = 26)

.2 (cm) S.D. Range

Thigh circumference (cm) 46.7 4-44 39-56 distal

Thigh circumference (cm) 56.6 5-29 46-67 proximal

Femoral length (cm) 43.6 2-43 38-50 Tibia1 length (cm) 39.1 2.35 3244 Weight t kg) 82.4 13.71 i7- 107 Height (cm) 176.4 7.01 157-191 Age tyr) 28.4 7.32 19-47

RESULTS

Reliability of the data For seven subjects the reliability of the

method used to obtain force measurements was determined by the test-retest technique on the same day. The coefficient of correlation (r) was used as the index for reliability. The TS for peak force were O-69 for eccentric

extension. O-61 for eccentric Aexion, 0.72 for concentric extension. and 0.69 for con- centric flexion (Table 3). The ranges of TS were 064-0.97 for isometric extension and 060-0~80 for isometric flexion for the various knee angles. The most reliable measurements for isometric flexion and extension were obtained with the knee flexed at 75.

From the X-ray film measurements for the knee flexion and extension moment arms were procured on a test-retest basis of differ- ent days. The majority of the TS were 0.80 or above (Tables 3 and 4).

Table 3. Reliability for obtaining extension moment arm distances (cm) in data reduction

process (.Y = 16)

Angle at knee (0) hlean S.D. r

0 T 4.1 0.51 R -1.4 0.51

0.94

15 T 4.7 0.51 R 1-8 0.52

0.95

30 T -1.9 0.53 R 4.9 0.50

0.96

40 T 4.9 046 0.91 R -1.9 0.43

60 T -1.7 0.48 R 4.7 0.37

0.80

75 T 4.3 044 0.67 R 4-3 0.44

90 T 3.8 0.50 0.70 R 3.8 0.53

Table 2. Reliability of force measurements (kg). t.y = 7)

Type of contraction

Concentric extension

Concentric Rexion

Eccentric extension

Mean peak force SD. r

I 30 8.4 R 35 6.8 0.72

T 1 4.3 R 1; 1.8 0.62

7 44 4.6 R 45 2.4 0.69

Eccentric flexion 7 18 4.0 R I6 5.3 0.62

Isometric extension O-90 7 R 064-0.97

isometric flexion O-90 T R

0-60-O-80

Table 4. Reliability for obtaining flexion moment arm: distances (cm) in the data

reduction process (IV = 26)

Angle at knee

(0) Mean S.D. r

0 T 2.5 0.58 R 2.5 o-60

o-93

15 T 3.4 O-56 R 3.4 o-60 0.91

30 T 3.9 0.51 R 3.8 0.53 o*84

45 T 4.1 0.53 R 4-o o-51 o-70

60 T 3.9 046 R 4.0 o-55

0.69

75 T 3.5 0.51 R 3.6 O-61

0.65

90 T 2.6 o-53 R 2.5 O-66

O-72

iz3 4 i 5

11 . I* .6

1 . 7

Pathway of the instant centers The mean locations of the instant centers Fig. 5. Pathway of instant centers with respect to the

for the seven different knee angles fall within tibia and femur (N = 26).

1 = 0 position. 2 = 15 position. 3 = 30 position. 4 = 45 position 5 = 60 position 6 = 75 psotion 7 = 90 position.

a circle which has a diameter of 2.3 cm, and the centers displace through a pathway in a space of 3.2 cm. The centers are located at the level of the lateral femoral epicondyle. The pathway of the instant centers with respect to the tibia form a pattern which opens in the antero-distal direction (Fig. 5).

terminal portions of knee motion, the torque generated by the knee extensors exceeded

Torque generated by knee flexors and ex- the torque associated with the knee flexors tensors (Figs. 7 and 8). A maximum torque of near

Moment arms and forces. Torque is defined 1200 kg-cm occurred during isometric and as the force times the perpendicular distance eccentric contractions of the extensors at the (moment arm) from the line of action for the range of 45-60. Within the range of 3%60, force to the axis of rotation. The axis of rota- approximately 900 kg-cm were produced by tion in this study is the instant center. There the extensors during the concentric contrac- is a moment arm associated with the line of tion. The slope of the curves was much less force for the flexor muscles, the extensor for torque generated by the flexors over the mechanism, the external force exerted at the 90 range. For isometric flexion. maximum distal portion of the shank (Fig. 4). The torque of about 640 kg-cm took place near 0 moment arms for the flexor and extensor (terminal extension), while peak torque of muscles were larger for the mid-portion of about 550 kg-cm occurred from 30-53 for the range of knee motion examined and the eccentric flexion and about 430 kg-cm from moment arm for extension was larger than 20-35 for concentric contraction of the flex- flexion (Fig. 6). ors. In general the mean torque, as a result of

Comparison of torque. Except for the concentric contractions, for both flexors and

84 G. L. SMIDT

BlOMECHANlCAL ANALYSIS OF KNEE FLEXiON AND EXTENSION 85

i5 jo 45 ANGLE Idea) BETWEEN TMA 8 FEMUR

Fig. 6. Moment arms for flexors and extensors: (numbers in brackets are S.D.).

O-xTYrzT 90 Position of Knee (in dograe* 1

Fig. 7. Torque produced by knee extensors. 1v = 26.

EE = Eccentric contraction for extension. IE = isometric contraction for extension.

CE = Concentric contraction for extension.

1200 -

1150 - 1100 -

1050 -

1000 - 950 -

900 - 650 - 600 - 750 -

700 -

150 -

100 -

50 -

OL I I 8 I I IS 30 45 60 75 90

Position of Knee [in degrees1

Fig. 8. Torque produced by knee flexors. IV = 26.

EF = Eccentric contraction for flexion. IF = Isometric contraction for flexion.

CF = Concentric contraction for flexion.

extensors, was less than the torque at corres- ponding positions produced by the eccentric and isometric. Torque by eccentric and iso- metric contractions was similar through much of the range, but lack of smoothness for the torque curve for the eccentric contraction during extension is typified by a stairstep phenomena whereby sudden decreases in torque are followed by sharp increases (Fig. 7).

At 15 intervals comparison between the mean torque measurements for each of the types of muscle contractions for each muscle group was carried out using an analysis of variance statistical technique and the Tukey test. The results of this statistical analysis appear in Table j.

Tension in patellar ligament and flexor muscles

Probably because of change in the moment

86 G. L. SMIDT

Table 5. Results of statistical comparison between torque measurements at 15 intervals: (N = 26) (P = O-05)

Angle at Comparison among types of contraction* knee (0) relative to amount of torque generated

5 Iso Flex = Iso Ext > Ecc Flex > Cone Ext = Ecc Ext = Cone Flex I5 Iso Ext = Ecc Ext- > Iso Flex = Cone Ext = Ecc Flex = Corm Flex 30 Ecc Ext = Iso Ext > Cone Ext > Iso Flex = Ecc Flex = Cone Flex 45 Ecc Ext = Iso Ext > Cone Ext > Ecc Flex = lso Flex = Cone Flex 60 Ecc Ext = Iso Ext > Cone Ext > Ecc Flex = lso Flex = Cone Flex 75 lso Ext = Ecc Ext > Cone Ext > Iso Flex = Ecc Flex = Cone Flex 90 lso Ext Z= Ecc Ext > lso Flex = Cone Ext = Cone Flex = Ecc Flex

* Ecc Ext = Eccentric Extension Iso Ext = Isometric Extension

Cone Ext = Concentric Extension Ecc Flex = Eccentric Flexion Iso Flex = Isometric Flexion

Cone Flex = Concentric Flexion > is greater than

arm through the 90 range the patterns for the mean tension in the patellar ligament and flexor muscles did not conform to the patterns for mean torque generated by the knee extensors and flexors during isometric contractions. The tension in the patellar ligament was lowest (I42 kg) near complete extension, and the magnitude of the tensions progressively increased through the range to a peak value of 277 kg at the 90 position.

Through the 90 range the maximum tension values relative to the flexor muscles form a parabolic spectrum with a peak of 279 kg near complete extension and minimum ten- sion of 124 kg at the 45 position.

Forces at tibia&femoral joint The_ magnitude of the compressive and

shear forces existent at the tibial-femoral joint for knee extension were dependent on the external load and the angular relation- ship between the long axes of the patellar ligament and the tibia. With respect to flexion, the determinants were the external load and the knee angle.

At 1.5 intervals from the extended to flexed positions the compressive force resulting from extension steadily declined while the

force from flexion gradually increased. The peak forces for extension and flexion were 275 and 269 kg respectively (Fig. 9).

At different positions for extension both anterior and posterior shear forces of the tibia

I~:261

is il ANGLE ldep) BETWEEN TIBIA 8 FEMUR

Fig. 9. Compression force at tibial-femoral joint during maximum isometric contractions: (numbers in brackets are S.D.).

BIOXIECHANICAL ANALYSIS OF KNEE FLEXION AND EXTENSION 87

on the femur occurred. The maximum an- terior shear force was 34 kg at the 15 position and a maximum posterior shear force of 3 1 kg took place at the 90 position. All shear forces for flexion were in the posterior direction with a peak value of 15 1 kg at the flexed position (Fig. 10).

[N = 261

; ;5 30 45 60 75 90

ANGLE (de&) BETWEEN TIBIA 8 FEMUR

-100 .- ow IO Eztnr*s o--- ha IO FhlU%

y-i,, 5 -.ps,

s, Fig. 11. Compression force at patelio-femoral joint for

\ maximum isometric contractions of knee extensors.

ANGLE (de@ BETWEEN TIBIA 8 FEMUfi

Fig. 10. Shear force at tibial-femoral joint for maximum isometric contractions.

Positive shear indicates a tendency of tibia to move anteriorly on the femur.

Negative shear indicates a tendency of the tibia to move posteriorly on the femur.

Relationships between parameters Particular interest was focused on the

relationships between torque produced by the muscles and (1) their moment arms, and (2) the thigh circumference. Low correlations were found (rs from 0 to 0.12) between thigh circumference and torque generated by the extensors at each of the seven knee angles, but the torque by the flexors correlated considerably higher (rs from 0.12 to 0.49).

Force at patello-femoraljoint Low positive correlations (rs from O-08 to

The location of the force at this joint 0.29) occurred between the moment arms for

changed with relative change of position of the patellar ligament and torque produced

the patella in relation to the articular groove while torque-moment arm relationship for

on the distal femur. The mean total displace- the flexors tended to be in the low negative

ment of the patella relative to the femur was range.

7.4cm. The force as a result of isometric contractions was least (65 kg) near complete DISCUSSION extension and in excess of 200 kg for the 45, The instant centers for 90 of knee motion 60, 75 and 90 positions (Fig. 11). A sum- traversed a distance of 3.2 cm and formed an mar-y of force measurements appears in involute which opened in the antero-distal Table 5. direction with respect to the verticaily oriented

88 G. L. SMIDT

Table 6. Summary of measurements for isometric contractions for knee extension: (N = 26)

Knee angle between tibia F, Torque DP1 F,,( e F, F, PL, F,,

and femur (kg) (2, (kg-cm) (cm) (kg) (0) (kg) (kg) (kg) (kg)

5 19.4 31.7 614.7 4.35 141.8 19.6 132.6 27.3 46.7 15 26.5 854.7 4.72 181.4 19.9 168.1 34.7 61.2 64.5 30 33.8 1073.8 4.87 221.3 18.4 208.0 33.0 70.8 128.8 45 38-l 1209-s 4.89 247.9 15.4 236.6 28.0 66.7 176.3 60 37.6 1195.0 4.67 257.6 11.5 251.2 19.3 52.0 208.8 75 36.8 1170-o 4.33 272.7 5.8 268.6 -9.5 30.4 210-o 90 32-7 1039.0 3.80 276.8 0.9 274.8 -31.0 24.0 216.8

F, = External force for knee extension. D, = Moment arm for external force. Dti = Moment arm for patellar ligament. Fpl = Tension in patellar ligament.

6 = Angle between patellar ligament and long axis of tibia. F, = Compressive force at tibial-femoral joint. F, = Shear force at tibial-femoral joint.

PL, = Shear force contribution by patellar. F,, = Compressive force at patello-femoral joint.

Table 7. Summary of measurements for isometric contractions for knee Rexion: (N = 26)

Knee angle between tibia F,h

and femur (kg) (2, Torque D,, (kg-cm) (cm) (2)

H, (kg)

5 20.0 31.7 636.4 2.50 270.2 5 268.8 -3.2 23.2 15 17.7 573.6 3.38 177.2 15 170.2 -27.5 45.6 30 17.9 568.7 3.87 151.5 30 131.2 -58.1 75.9 45 15-7 495.1 4.08 123.8 45 87.6 -70.8 87.8 60 15.9 505.5 3.94 129.9 60 65.2 -96.5 112.5 75 14.4 457.0 3.52 132.2 75 34.5 -113.5 127.8 90 11.4 361.4 2.56 150.6 90 0 -150.6 150.6

F rll = External force for knee flexion. D r = Moment arm for external force. Dh = Moment arm for hamstrings. Fh = Tension in hamstring muscles. & = Annie between long axis of tibia and femur.

F, = Compression force-at tibial-femoral joint. F, : : Shear force at tibial-femoral joint. H, = Shear force contribution by hamstrings.

ment of musculo-skeletal disorders. By integrating data from this study and Morrisons work ( 1969- 1970) regarding forces, tension, and motion at the knee joint for certain activi- ties and isometric contraction of the flexors and extensors under load, some interesting comparisons result (Table 8). How does the magnitude of the forces at the knee joint en- countered during resistive exercise compare

shank. The configuration for the pathway is somewhat different from the results of Frankel and Burstein (1970). who depict the opening in the anterior direction.

The use of resistive exercise was clearly indicated and widely utilized in the treatment of poliomyelitis, but currently there seems to be diminishing use of organized and super- vised resistive exercise programs for treat-

BIOMECHANICAL ANALYSIS OF KNEE FLEXION AND EXTENSION 89

90 G. L. SMIDT

with walking and stairclimbing? With the knee in slight flexion, the peak tension requirement for the knee extensors was 0.93-l-48 times body weight for walking. The values for iso- metric exercise are greater at 1.73-2.20 times body weight. The highest forces at the tibial- femoral joint during walking also occur with the knee in an attitude of 5-15 flexion and compression is 263.4 times body weight while the force during exercise is considerably less. The maximum tension along the quadri- ceps was reported to occur with the knee flexed at 45 when climbing stairs. At this position, the muscle tension for exercise was somewhat larger compared to stairclimbing. However, similar to walking, the joint com- pression force for stairclimbing was much greater than the force for exercise. Probably the main contributing factor for the larger joint compression load is the floor reaction force experienced during ambulatory ac- tivities.

From these comparisons we see that tension in the quadriceps mechanism appears to be greater, and the tibial-femoral force less for exercise than measurements of the same factors for performing basic activities. These findings should encourage clinicians to con- sider resistive exercise in preparation for stairclimbing and walking or in conjunction with assisted ambulation as part of the treat- ment for dysfunction of the knee.

The moment arm for the patellar ligament and the knee flexors was greatest at the mid- range (45-60) of knee motion. It has already been mentioned that the peak torque values for extension are also in the mid-range, so for extension of the knee, it would appear that the mechanical factor of moment arm is of greater significance than the length of the muscle at the time of contraction. It must be recognized, however, that at the greatest position of flexion (90) for this study, the contractile elements of muscle were active, but passive contribution of force by the series and parallel elastic elements (connective tissue) in the musculo-tendenous unit was

probably negligible because of insufficient stretch on the unit. Bear in mind that more stretch of the extensor muscles could ob- viously be accomplished with the knee in complete flexion. Contrary to the findings for extension, the torque resulting from the knee flexors seemed to be more affected by the physiological phenomena of length-tension relationship of the musculo-tendenous unit than moment arm, which is a biomechanical factor.

The torque generated by the eccentric and isometric muscle contractions was about the same in this study, while other investigators (Olson et al., 1972 and Doss and Karpovich, 1965) have reported larger torque values for the eccentric contraction. Positive work is done by the muscle on the load during the concentric contraction and the eccentric con- traction represents an illustration of negative work whereby the external forces are doing work on the muscle. It has been shown in isolated muscle that the lengthening contrac- tion under load consumes less energy or is more efficient than other types of contraction (Abbott er al., 1952 and Hill, 1960). Ap- parently a chemical reversal associated with the contractile process during lengthening accounts for the low energy loss, and this mechanism has obvious implications for activity which places a premium on endurance. This rationale does not explain the increased force produced during the eccentric contrac- tion, and there seems to be no evidence with respect to the contractile element which in- cludes the length of the sarcomere. However, some mechanism in the contractile element could be responsible for increased force for the lengthening because the passive con- tribution of force from the parallel and series elastic elements should be the same for any given muscle length irrespective of type of muscle contraction. Also larger numbers of motor units may be operational during the lengthening contraction.

The stairstep phenomena finding for the eccentric torque curve can probably be

BIOMECHANICAL ANALYSIS OF KNEE FLEXION AND EXTENSION 91

explained by the function of the control REFERENCES system for the skeletal muscular system. The sudden decrease in torque generated at various intervals probably results from an inhibitory feedback signal from the golgi tendon oqans which are sensitive to tension (Guyton, 197 1).

SUMMARY

(1) For knee motion in the sagittal plane, the pathway of instant centers forms an in- volute which opens in the antero-distal direc- tion and is located in the region of the lateral femoral epicondyle.

(2) The moment arm is greater for knee extensors than flexors and appears at the 45 position.

(3) Maximum average torque values of nearly 1200 kg-cm result from eccentric and isometric muscle contractions while the isometric flexor contraction near complete extension produced maximum average torque of 636 kg-cm.

(4) The torque generated by the musculo- tendenous units is approximately the same for eccentric and isometric contractions which are both greater than concentric.

Relative to maximum isometric contrac- tions over the 90 excursion of the knee, the following results apply.

(5) The mean shear force at the tibial- femoral joint ranged from 0*38-l-84 times body weight.

(6) The mean force at the patello-femoral joint was 0.79-3.64 times body weight.

(7) The required muscle tension in the flexor and extensor muscles ranged from I.39 to 3.35 times body weight.

(8) The compression force at the tibial- femoral joint ranged from 0.42 to 3.40 times body weight.

Acknowfedgemenrs-The author wishes to express his gratitude to Dr. A. Burstein for his aid in appropriately applymg the method used to determine the instant centers for the knee: fo Mrs. Beverly Luchinske and Mr. Charles Simpson for their invaluable assistance in the data acquisition. reduction, and analysis for this study. and fo Mrs. Alice Gill for typing the manuscript.

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