Some investigations of the accuracy of knee joint kinematics

SOME INVESTIGATIONS OF KNEE JOINT

OF THE ACCURACY KINEMATICS*?

RICHARDP. DUKE,$ JAMESH. SOMFXSET~

and PAUL BLACHARSKI/~

Department of Mechanical and Aerospace Engineering, Syracuse University, Syracuse, NY 13210, USA and DAVID G. MURRAY Department of Orthopedic Surgery, Upstate Medical Center, Syracuse, NY 13210, U.S.A. Abstract-A

previous paper on the kinematics of the human knee established the loci of points which were the intersections of the instantaneous axis of rotation and sagittal planes. This paper is a study of the effects and magnitudes of three possible errors in examining those kinematics. The sources of error are: (1) the influence of the preservative on the tissue, (2) the effect of error in locating the points from photographic data, (3) the effect of mechanical data reduction techniques. In order to remove the influence of the preservatives, a fresh cadaver joint was used in this study, eliminating error of the iirst type. Care was taken to minimize &ror of the second type. A computer program was written to replace the mechanical construction technique, thereby eliminating the possibility of error of the third type. This program, used as a sub-routine, facilitated analysis of the magnitude of errors of the second type. As a final step another computer program was used to evaluate the construction technique used in the previous paper.

INTRODUCTION

A previous study of the kinematics of the human lmee (Ha&s, 1941) established loci of the intersections of the instantaneous axis of rotation and the sagittal plane of the medial and lateral sides of the knee. The previous study utilized preserved cadaver joints and a combination of photographic and mechanical construction techniques to produce the loci. The technique proceeds by connecting the corresponding consecutive images of the two reference points with a line segment Perpendicular bisectors of the segments are then constructed until they intersect, the intersection being the intersection of the instantaneous axis and the sagittal plane in question. The use of preserved joints and the use of a mechanical analytical technique together with the inherent accuracy of a photographic process, induce three sources of possible error, viz: (1) preserved joints might acquire a permanent deformation of the soft tissue and a change in the elastic properties of the ligaments and tendons. This factor, first discussed in an article by Haines (1941) and further discussed by Blacharski and Somerset (1975) is shown to be insignificant @ the results of this paper.

(2) The technique used requires the location of a point of finite size (otherwise it would be invisible to a photographic process), therefore a certain error must necessarily be inherent in the loci of those points. (3) The data reduction technique provides additional error in the location of the instantaneous centers. Elimination of error of the first type was accomplished in this study by using a fresh (non-preserved) knee joint in the analysis. The technique described in the original paper was followed except for the final data reduction technique. Error of the second type could only be minimized, not eliminated Care was taken to locate the points, necessarily of finite size. Error of the third type was eliminated (at least to the sixteen decimal places carried by the computer). A computer program was written which, from the reference point locations, calculates the “instantaneous centers” through a vector algorithm. As a final step another computer program was written to relate the effect of the construction variables on the final loci of points.

ALGORITHM FOR LOCATING CENTERS

* Received 5 June 1974. 7 This work was supported

OF ROTATION

jointly by the Hendricks Fund for equipment, by the National Institute of Health, and by the Fannie E. Rippel Foundation. $ Undergraduate Research Assistant. (j Professor of Mechanical and Aerospace Engineering. ((Former Graduate Assistant.

The original analysis by Blacharski and Somerset (1975) was three-dimensional, and involved the use of a three-dimensional analytical technique to determine the instantaneous axis of rotation for various 659

660

R. P. DUKE,J. H. SOMERSET, P. BLACHARSKI and D. G. MURRAY

CX, CY ) intersection of the perpendicular bisector

Fig. 1. Sketch of successive positions of body i3 in plane motion. angles of flexion. The reader is referred to the previous paper for details of that analysis. Suffice it to say that the three-dimensional analysis comprises numerous two-dimensional analyses, each of which may be reduced to the following case (see Fig. 1). Given a body, B, moving in the x-y plane, and given the points P and Q, located on the body, and knowing that the x and y coordinates for P and Q at particular times t(i) (i = 1,2,3.. . n) find the successive axes li, which would produce the sequence of positions of the rigid body. The locations of the reference points are taken from a time-lapse photograph of the flexion of a knee joint. Let the location of the first reference point be P(1). Let successive images of the first reference point be P(i) (i = 2,3.. . , n). Let the location of the first image of the second reference point be Q(l), and successive images of the second reference point by Q(i). Because the body, B, defined above, moves in plane motion, the axes of rotation must be perpendicular to the x-y plane and a particular Li is therefore defined by its intersection with the x-y plane. Let that intersection be point C(i). (See Fig. 1.) Figure 2 shows the vector construction attendant to Fig. 1. Denoting V(i) as the vector from P(i) to P(i + 1) and X(i) as the vector from Q(i) to Q(i + l), then, V,(i) = P,(i + 1) - PJi) Vii) = P,(i + 1) - PAi)

Knowing V(i) and X(i), we specify unit vectors a and fi which are perpendicular to the original vectors V(i) and X(i).

The extension of the unit vectors a and /I are LINE 1 and LINE 2 in the diagram in Fig. 2. From our definitions, LINE 1, is the perpendicular bisector of V(i), has a direction defined by a, and has a length L. LINE 2 is the perpendicular bisector of X(i), has a direction defined by /I. and has a length B. Point C(i), the “instantaneous center”, is the intersection of LINE 1 and LINE 2. Now, the following vector equations can be written: P+;+z_.a=c.

X,(i) = QJi + 1) - QJi) X&i) = QQ + 1) - Q,.(i).

Where the subscripts x and y refer to the x and y components of the vectors and the coordinates of the mid-points of V(i) and X(i) become: (bx(i)

+ y),

and

(p,(i) + y))

, Q,.(i) + x+)).

Q&) + x+

((

I(

IO

X-axis

Fig. 2. Vector construction attendant to Fig. 1.

661

Accuracy of knee joint kinematics

those points. Another computer program was written using the previously explained program. METHOD. as a sub-routine to measure the effect of two of these P,i + P,j + y + y + La,i + La,j factors in locating the “instantaneous centers”. The first factor is the angle of intersection of the = C,i + C,j, (2) lines extended through pairs of images of the referand ence points, angle 0. (see Fig. 3.) For the purpose of the analysis the angle, 8. was Q+;+Bb=C, (3) first specified. The locations of the reference points were determined assuming a given distance between which expanded becomes : points P(i) and P(i + I), distance Y on Fig. 3, and a unit distance between sets of reference points, disQxi + Q,j + l&i B&j tance X on Fig. 3. Use of the program METHOD yields the intersection of the perpendicular bisectors. = C,i + C,j, (4) denoted point S, and defined by vector S relative to the origin. Next, an error distance, DEL, was specicollecting i and j components from (2) and (4) above we have four equations in four unknowns, C,, C,, L fied. DEL represents the dimensionless ratio of the error in locating the reference points to the distance and B: between them. (See Fig. 3.) The reference points are assumed to move the disC,-I,L-P,-4;1 i=O tance, Z, in the direction of the perpendicular bi> sector, one point moving the positive amount Z, the C, - R,L - P, - 2 j = 0 other moving the negative amount Z. This direction i > of motion produces a maxi-maximum error denoted by the prime locations in Fig. 3. (5) Cx - BPx - Qx - %)i = 0 Given these new locations for the reference points, c METHOD was used to calculate a new intersection of the perpendicular bisectors denoted S and located C, - B/?, - Q, - 2 j = 0. > by vector S’. The error distance thus becomes 1s - s’l. A second error, called a mini-maximum, was proIn matrix form (5) becomes : duced by reversing the signs on the error distance Z and then calculating a new intersection for the perp,+; 1 0 -i, 0 pendicular bisectors. The new intersection is S” and is defined by the vector S”. The error distance is 0 1 -& 0 P, + 2 = , (6) 1s - S’(. Assuming possible error in the point location to be a circle, with radius the error distance Z. around L 1 0 0 -p, Qx + "i the true location of the point, the loci of the then constructed instantaneous centers will be an oblate figure with symmetry about the y axis (See Fig. 3). B -0 1 0 -&_ Q, + 2 The maxi-maximum-error distance is the largest possible error for a given error ratio DEL and angle, 8. The mini-maximum error distance is the smaller C, and C, being the coordinates of the “instantaneous centers”. From this analysis a computer program was “major radius” of the oblate shape. The sum of the written entitled METHOD and is presented in mini and maxi errors is the largest possible dimension of the oblate shape in which the “instantaneous Appendix A. (The program is in APL). Accuracy of this program was insured by a test center” must lie. Although the actual possible error Z is not always run. In this test the locations of points on the circumin the direction perpendicular to the line connecting ference of two concentric circles were used as input the two reference point images, the largest possible into the program. The resulting output was accurate errors will occur under those conditions. to the sixteen places carried by the computer.

which expanded becomes :

x$+x+ +

+

P

LX

c,

COMPUTER ANALYSIS OF ERROR

FRESH JOINT STUDY

IN KNEE STUDY

Assuming a certain experimental error in the location of points P(i) and Q(i) (a “type two” error as described in the introduction), there are several factors which affect the magnitude of the error in location of the instantaneous centers calculated from

The procedure used in the previous study by Blacharski and Somerset (1975) on preserved knee joints was followed in this study, but with a fresh (non-preserved) knee joint. Time lapse photographs were taken of the joint with a transponder (a pair of reference points) attached. A second set of photographs

662

R. P. DUKE,J. H. SOMERSET, P. BWCHARSKI and D. G. MURRAY Y-axis

Due to similar triangles error increases a5 X increases Y = 0.5 DEL = Z/Y X = unity

te locus of maximum

Fig. 3. Construction of error analysis

X, 2.24

3.04

2.64

3.44

dimension of axis,

3.64

4.24

in

4.64

5.04

5.44

5.64

I

6.24

I

Fig. 5(a). X, I.60

2.20

’

’ “I

L-

260

”

”

’

3.00

”

dimension of axis, 3.40

““”

3.60

‘I’

““”

in.

4.20

”

4.60

”

‘I’

5.00

560

5.60

111111111‘11111(

-2.64I

I

I

I

I

Fig. S(b).

I

I

I

I

I

I

-_..._..-l”“._

1

I

Fig. 4. Locations of instantaneous

.

I

centers from previous paper.

665

Accuracy of knee joint kinematics was taken after cutting the anterior cruciate ligament. A final set was taken after cutting the remaining cruciate ligament. After processing, the photographs were placed on a large piece of tracing paper. The reference points were then transferred to the tracing paper by sticking a needle through the center of each reference point. The origin and axes were similarly transferred. The coordinates arrived at from the data described above were the input for the computer program to establish the loci of the intersection of the instantaneous axis of rotation and the sagittal planes. RESULTS The results of this study on fresh knee joints indicate that the influence of the preservative on the joint has little effect on the motion of the knee. Figure 4 shows the data from the previous study by Blacharski and Somerset (1975) while Fig. 5 shows simifar data from the fresh knee study. The variation

X,

noted on the fresh joint falls within the limits found in the previous study. The results of the “error analysis” computer program were plotted in Figure Sets 6 and 7. Set 6 shows plots of location error vs nondimensionalized experimental error (DEL) for various values of angle 0. The second set of graphs shows non-dimension location error vs the angle 0 for various values of experimental error (DEL). From this graphic data it is evident that the angle f3is a major factor in determining the error inherent in the “instantaneous center”. Since this analytical technique is so sensitive to the angle 6, one tries to ,conduct experiments wherein 0 is made large. Care should be taken in using these graphs since some plotting error is created by the fact that a terminal with preset spacing was used to print the graphs. To put this data into proper perspective, actual values of the experimental dimensions are now stud-

dimension

I.60

1.20

m

of OXIS,

200

2.40

2.80

3.20

360

I

I

4.00

I

Fig. S(c).

X, 0 28

0 68

I 08

148

dimension 188

228

of OXIS, 268

I”. 308

348

363

I

Fig. 5(d).

I

4 28

I

R. P. DUKE, J. H. SOMEMET,

666

X, 3.36

2.96

376

4.16

P.

BLACHARSKI and

drmenslon of axis,

4 56

4.96

D. G. MURRAY

in.

5 36

5.76

6.16

6.56

696

I

I

,,11’1,11’,,,,‘,,~,‘11,,‘,,,,‘,,,,’,,,1’l,,,’,l,lJ

.-0.;

e

-2.32I

I

I

I

I

I

I

I

I

Fig. 5(e). X, 044 .g

0.64

“1 ““I

”

-1.26

1.24

1.64

dimension of axis,

204

2.44

” “““”

’ ”

”

in.

2.84

3.24

‘I”

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3.64

404

4.44

““““““““’

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6 15

-246-

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Fig. S(f).

e

1.5-

1.0 -

5.5 -

LO-

!S-

!.O

5-

.o -

b.5 -

0

I 005 Maxi-maximum

5, IO, 15,20,25,

I

I 0 IO

error

0 I5

DEL vs DEL

and x)

for 8 equal

degrees

Fig. 6(a).

A4

I 0 20 tithe

0.25 bIbwing.

0

Maxi-manmum 35,40.45,50,55,

error

DEL VI DEL for 8 equal and

60 degrees

Fig. 6(b).

to

the fo4bwlng’

Accuracy of knee joint kinematics

667

668

R.

P. DUKE,J. H.

P. BLACHARSKI and D. G. MURRAY

SOMERSET,

ied to determine acceptable limits for experimental error DEL. The following values are typical, not only for work in this paper and work by Blacharski and Somerset (1975), but of other knee joint studies. For example: 0 = 25 degrees

0.25;i 5 020-

0

005 Mrkncwmum 65.?0,75.60,65,

0 I5

0 IO armr vs DEL

for 8 equal

0.20 to ihe

0 25

following:

and 90 deqnes

Fig. 6(f). Fig. 6. Plots of error vs DEL for various values of angle 8. The values of error larger than five have been omitted

from these graphs.

distance between images Y = 1 in. distance between sets of reference points X = 3 in. (since the values of S in the graphs were calculated using unity for X, the error value must be multiplied by the X distance). NOW, if a final error of less than one-half inch is desired, the value of DEL must be less than 0.008. If experimental error DEL in conjunction with the angIe B yields a possible experimental error greater than one-half in., then those results are highly suspect since the loci only moves cu. 1 in. through the range of flexion. On the other hand, results by Blacharski and Somerset (1975) have a DEL of approx. 0.005 and has values of 0 of 30 degrees. This yields a maximum error of cu. 0.180in. in the location of the instantaneous centers. It should be noted that these results are contingent upon using an exact method of location of the instantaneous centers from the reference points. A mechanical construction technique would add additional error.

60

Maxi-maxlmum error vs 8 far DEL equal to the following: 0.002,0.004,0.006.0.008,0.010.0.020

Fig. 7(a).

70

60

90

669 Accuracy

0

of knee

joint

kinematics

-.e followinq: Maxi-moxlmum error vs 8 for DEL equal to ttuc 030,0.040,0.050,0.060.0.070 ond 0.000

0

Fig. 7(b).

51

4 5

4

3

30

b z w

25

2'.O

15

I0

015

0

10 Moxl-moxlmum 0 090,O. 100,O

20

30

40

50

error vs 8 for DEL equal to the I IO, 0.120,O 130, ond 0 140

Fig. (7~).

60 followinq.

70

90

R. P. DUKE, J. H. SOMERSET,P. BLACHARSKI and D. CL MURRAY

i.0 -

41.5 -

4..o -

38.5 -

31.0-

b t w

2‘.5 -

2 .Oc

I

I

0

Maxi-maximum error vs 8 for DEL equal to the following: 0.150,0.160,0.l70,0.180,0.190, and 0.200

Fig.

7(d).

5.ol-

45-

4.0 -

35-

3.0 -

z-o-

1.5 -

to-

0.5-

Maxi-maximum error vs 8 for DEL equal to the fdbwing: 0.210,0.220,0.230.0.240,and 0.2!50

Fig. 7(e).

67

Accuracy of knee joint kinematics

Mini-maximum 0 002,O 004,O

ermr YS fl for DEL 006,O QO0,0.010,

equal to the and 0.020

followmg:

Fig. 7(f).

4

.o -

3

5-

3

.0 -

2 ‘5-

b k w

2

o-

,

5-

1

O-

0 ‘5-

0

60 50 10 20 30 40 Mini-maximum error vs tl for DEL equal to the followmg’ 0.030.0040,0.050.0.060,00?0,and OOBO

Fig. 7(g).

70

60

90

672

R. P. DUKE, J.H.SOMERSET,

4

P. BLACHARSKI

and D. G.MURRAY

.5

4 0 F 3 .5-

3 .0-

2 .5z L w 2!.O-

I.5-

I.o-

CI.5-

0 Mini-maximum ent~ vs 8 for DEL ~wI in the following : 0.090,0.100,0.~~0,0.~20,0.~~,and 0.140

Fig. 7(h).

>-

5.1

4.5-

4.O-

3.5-

3.o-

b2 iz w

.5-

2 O-

I.5-

I.o-

Cb.5-

0

I IO

I

20

I

30

I 40

I

I

50 60 . ..~ Mini-maximum error vs 8 DEL equal to TM WlQwinP: 0.1~,0.~,0.170,0.180,0.190,and 0.200

Fig. 7(i).

I

I

I

70

80

90

673

Accuracy of knee joint kinematics

5.0-

45-

40-

3.5-

30-

z = 2.5w 20-

I.51

l.O-

0.5-

I

I

I

I

I

I

I

1

20

30

40

50

60

70

80

90

I

0

IO

Mmi-maxbmum error vs 8 for DEL eqwl 0.210,0.220,0.230,0.240, and 0.250

to the following :

Fig. 7(j). Fig. 7. Plots of error vs angle 0 for various values of DEL. The values of error larger than five have been omitted from these graphs REFERENCES

Blacharski, P. and Somerset, J. (1975) A three-dimensional study of the kinematics of the human knee. J. Biomechanics

Haines, W. R. (1941) A note on the actions of the cruciate ligaments of the knee joint, J. Anat. 75, 373.

8, 375-384.

APPENDIX V

A

METJOD; MlJ;MD;

PP;QQ

Cl1

MU+4

C23

if+1

C31

MUCl

2

;31+(1.

-1)x-PPt(+/((PP+PTStd+l;

C41

ivuc3 4

;41+(1,

-1)x-QQ+(t/((Q.~cPTSCn+l; Ir 33-PTSCiI; 4 31)*2))*C.5

c51

MD+bpPTSC;I; 1

163

:IDC3

c71

YD+ 4 1 PM3

CR1

X+(BVLJ)+.xMLI

191

X

[lOI.

+3x~(~+/~+i)<(pPTS)[ll n

4+4

2 p10

4l+PTSCif;

0

PIt(PTSCRt1; 3

11001

1

21-P”St-H;

~It(PTSCift1; 3

41-PTSlJ;

2 ll-PTS13; 2 11)*2))*0.5

1

21)12 3 41)+2