An analytical model of the patello-femoral joint
by Jack Stephen Hagelin
A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE
* in Mechanical Engineering
Montana State University
© Copyright by Jack Stephen Hagelin (1978)
Abstract:
The intent of this research was to develop an analytical model of the human knee joint for the purpose
of aiding orthopedic surgeons in determining the optimum corrective operation to alleviate the
condition of recurrent patellar dislocation.
A kinematic model was developed defining the relative positions of the femur, tibia, and patella for all
angles of flexion. The patellar - surface was approximated by a cosine curve rotated through a specified
arc length and the patella was modeled as a cylinder. The assumption was made that the quadriceps
muscle group could be approximated by a single muscle in which the resultant force acting on the
patella was equivalent to the total muscle group. The accuracy of this model is contingent upon the
accurate determination of the proximal attachment for the effective quadriceps muscle.
Nineteen nonlinear simultaneous equations were developed to relate the nineteen unknown kinematic
parameters and were solved using Newton-Raphson's Method on the computer. The computer output is
in the form of a video display indicating the position of the patella in the patellar groove. Surgeons may
simulate operations on the computer model and determine visually the path of the Datella throughout
flexion. Trial operations are simulated until the desired patellar path is reached. STATEMENT OF PERMISSION TO COPY
In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l l m e n t o f t h e r e q u i r e ­
ments f o r an advanced degre e a t Montana S t a t e U n i v e r s i t y , I a g re e t h a t
t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r i n s p e c t i o n .
I further
a g r e e t h a t pe rm is si o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y
purpos es may be g r a n t e d by my major p r o f e s s o r , o r , in h i s a b se n c e , by
the Director of L ib ra rie s .
I t i s unde rs too d t h a t any copying o r p u b l i ­
c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a in s h a l l not be allowed w it h o u t my
w r itt e n permission.
Signature ^
'
v
Date
O
'll
dot M n *
AN ANALYTICAL MODEL OF THE PATELLO-FEMORAL JOINT
by
JACK STEPHEN HAGELI N
A t h e s i s sub mit te d in p a r t i a l f u l f i l l m e n t
o f t h e require me nts f o r t h e degree
of
MASTER OF SCIENCE
*
in
Mechanical Engineering
Approved:
C h ai r p e r s o n , Graduate Committee
Head, Major Department
MONTANA-STATE UNIVERSITY
. Bozeman,..Montana
October, 1978
iii
ACKNOWLEDGEMENT
The a u t h o r wishes to thank Dr. D, 0, B l a c k k e t t e r f o r h i s a s s i s ­
t a n c e and guidance t h r o u g h o u t t h i s r e s e a r c h e f f o r t w i t h o u t which t h i s
r e s e a r c h could not have been accomplished..’
The a u t h o r would a l s o l i k e t o thank t h e I n t e r c i t y R ad i o lo g ic a l
L a b o r a t o r i e s , In c . in Bozeman, Montana f o r t h e i r d ona tio n o f t h e x - r a y s
e s s e n tia l to t h i s research.
The encouragement and p r a y e r s o f t h e a u t h o r ' s f a m il y and f r i e n d s
were v i t a l t o t h e completion o f t h i s work and were g r e a t l y a p p r e c i a t e d .
TABLE OF CONTENTS
Page
V I T A .....................................................
ACKNOWLEDGEMENT
ii
.................................................................................................
LIST OF TABLES....................................................................................................
LIST OF FIGURES
...........................................................................
ABSTRACT ....................... ..................................................................
iii
v
vi
. . . . .
vii
CHAPTER I .
INTRODUCTION................................... .... . . ........................... '
I
CHAPTER 2.
ANALYTICAL MODEL . . . . . . :
2
CHAPTER 3.
RESULTS
...........................................................................
CHAPTER 4.
SUMMARY
..................................................................
APPENDIX I .
REFERENCES
........................................
PROGRAM KNEEPLOT FLOW CHART ..........................
. . .
19
. . . . .
30
. . . .
33
34.
V
LIST OF TABLES
Page
Variables
8
vi
LIST OF FIGURES
Page
! . - - A n a l y t i c a l model pa ra m e te r d e f i n i t i o n
3
2 . —Quadriceps muscle group. V ....................
3 . -- Q u a d r ic e p s proximal muscle a tt a c h m e n ts
............................... ....
5
, , , , ....................
6
4 . - - P a t e l l a r s u r f a c e m o d e l ........................ ....
10
5 . " F r e e body diagram o f p a t e l l a ...........................................................
15
6 . —Actual v id e o d i s p l a y o u t p u t . : .......................................................
20
7 . —Comparison o f model w ith a c t u a l x - r a y d a t a .............................
21
8 . - - S e n s i t i v i t y o f p a t e l l a r path t o e f f e c t i v e q u a d r ic e p s o r i g i n
9 . --Sample run o f o p e r a t i o n s i m u l a t i o n
. •........................
23
25
ABSTRACT
The i n t e n t o f t h i s r e s e a r c h was t o develop an a n a l y t i c a l model o f
t h e human knee j o i n t f o r th e purpose o f a i d i n g o r t h o p e d i c surgeons in
d e t e r m i n i n g t h e optimum c o r r e c t i v e o p e r a t i o n t o a l l e v i a t e t h e c o n d i t i o n
of recurrent p a te lla r dislocation.
A k in e m a tic model was developed d e f i n i n g t h e r e l a t i v e p o s i t i o n s o f
t h e femur, t i b i a , and p a t e l l a f o r a l l a n g le s o f f l e x i o n . The p a t e l l a r
s u r f a c e was approximated by a c o s in e curve r o t a t e d through a s p e c i f i e d
a r c l e n g t h and t h e p a t e l l a was modeled as a c y l i n d e r . The assumption
was made t h a t t h e q u a d r ic e p s muscle group could be approximated by a
s i n g l e muscle in which t h e r e s u l t a n t f o r c e a c t i n g on t h e p a t e l l a was;',
e q u i v a l e n t t o : t h e t o t a l muscle group. The a cc ura cy o f t h i s model i s
c o n t i n g e n t upon t h e a c c u r a t e d e t e r m i n a t i o n o f t h e proximal a tt a c h m e n t .
f o r t h e e f f e c t i v e q u a d r ic e p s mu s cl e. Nineteen n o n l i n e a r sim ul ta neo us e q u a t i o n s were developed t o r e l a t e
t h e n i n e t e e n unknown kin e m a ti c pa ra m et e rs and were s o lv e d u s i n g , NewtonRaphson's Method on th e computer. The computer o u t p u t i s in t h e form
o f a video d i s p l a y i n d i c a t i n g th e p o s i t i o n o f t h e p a t e l l a i n ' t h e . p a t e l ­
l a r groove. Surgeons may s i m u l a t e o p e r a t i o n s on the computer model and
de te rm in e v i s u a l l y th e path o f t h e p a t e l l a th ro u g h o u t f l e x i o n . T r i a l
o p e r a t i o n s a r e s im u l a t e d u n t i l t h e d e s i r e d p a t e l l a r path i s reached., ,
CHAPTER .1 .
INTRODUCTION
The i n t e n t o f t h i s r e s e a r c h was t o develop an a n a l y t i c a l model o f
t h e human knee j o i n t f o r t h e purpose o f a i d i n g o r t h o p e d i c surgeons in
d e te r m in in g t h e optimum remedial o p e r a t i o n f o r r e c u r r e n t p a t e l l a r sub­
luxation.
R e c u r r e n t d i s l o c a t i o n o f t h e p a t e l l a i s p r i m a r i l y due t o an i r r e g ­
u l a r ge o m e t ri c a l c o n f i g u r a t i o n in t h e l e g anatomy.
Si nc e t h e r e a r e a
number o f d i f f e r e n t a b n o r m a l i t i e s p o s s i b l e , t h e r e a r e a l s o a number o f
o p e r a t i v e t e c h n i q u e s a v a i l a b l e to s u r g e o n s .
This r e s u l t s in a degree
o f u n c e r t a i n t y in t h e i r d e c i s i o n s , which in t u r n r e s u l t s in an u n d e s i r ­
able f a i l u r e r a t e .
To re duce t h i s f a i l u r e r a t e , a model was d e s i r e d which could p r e ­
d i c t t h e e f f e c t o f each o f t h e d i f f e r e n t o p e r a t i o n s performed.
Surgeons
could th e n s e l e c t t h e o p e r a t i o n which would, r e s u l t in a d e s i r a b l e f o r c e
p a t t e r n and a c c e p t a b l e p a t e l l a r p a t h .
To accomplish t h i s , a k in e m a tic model was developed which in c lu d e d
pa ra m et e rs f o r l i g a m e n t d i r e c t i o n s and a d e s c r i p t i o n o f t h e r e l a t i v e ..
p o s i t i o n s o f t h e femur, t i b i a , and p a t e l l a .
The ass umption was made
t h a t a t any given a n g le o f f l e x i o n , t h e r e i s only one p o i n t on t h e pa­
t e l l a which i s in c o n t a c t w it h t h e f e m u r .. The r e s u l t i s a computer
model which o u t p u t s a video, d i s p l a y i n d i c a t i n g t h e pa th o f t h e p a t e l l a
i n t h e p a t e l l a r groove thro ugh f l e x i o n .
CHAPTER 2
ANALYTICAL MODEL
2.1
Introduction
The purpose o f t h i s r e s e a r c h was t o develop an a n a l y t i c a l model o f
t h e p a t e l l o-femoral j o i n t , a c c u r a t e l y d e f i n i n g t h e p o s i t i o n o f th e pa­
t e l l a i n t h e j o i n t f o r a l l a n g le s o f f l e x i o n .
The k in e m a tic model developed i n c o r p o r a t e s t h e use o f v e c t o r s in
d e f i n i n g a l l bone and l ig a m e n t p o s i t i o n s .
A f o r c e a n a l y s i s i s inc lu de d
r e q u i r i n g t h e p a t e l l a to be in s t a t i c e q u i l i b r i u m t h r o u g h o u t f l e x i o n .
2.2
D e f i n i t i o n o f Kinematic Model
Fi g u r e I d e p i c t s t h e model used in t h i s s tu d y .
L i s t e d below i s a
d e s c r i p t i o n o f t h e c o o r d i n a t e system and a l l p o i n t s in t h e m ode l.
A r e c t a n g u l a r c o o r d i n a t e system i s used in t h i s model as d e f i n e d in
F ig ure I - a .
Both t h e l e f t and r i g h t knees i n c o r p o r a t e a l e f t - h a n d e d
c o o r d i n a t e system.
a bo ut t h e femur.
The Y-axis i s t h e a x i s o f r o t a t i o n o f t h e t i b i a
The .X-axis i s p a r a l l e l to th e femoral s h a f t and i s
l o c a t e d l a t e r a l l y in t h e same pla ne as t h e t i b i a ! t u b e r o s i t y .
This a x i s
i s p o s i t i v e in t h e d i s t a l d i r e c t i o n . . The Z - a x is i s p o s i t i v e a n t e r i o r l y .
P o i n t 1O1 i s t h e o r i g i n o f . t h e c o o r d i n a t e system.
Point ' I ' is the
p o i n t o f c o n t a c t between t h e p a t e l l a and t h e p a t e l l a r s u r f a c e . . (The gap
shown in Fi gu re 1-b i s due t o t h e f a c t . t h a t t h e r e i s a l a y e r o f c a r t i 1edge between t h e femur and t h e p a t e l l a . )
The p a t e l l a r s u r f a c e w i l l
h e r e a f t e r be r e f e r r e d t o as t h e cam. , P o i n t ' 2 ' i s t h e t i b i a ! t u b e r o s i t y .
3
Patella
Femur
Patellar
S u r fa c e
T i b ia
T i b ia !
Tuberosity
Fi gu re I . - - A n a l y t i c a l model param ete r d e f i n i t i o n
4
t h e p o i n t o f i n s e r t i o n o f t h e Ligament P a t e l l a e ; and p o i n t ' 3 ' i s t h e
p o i n t o f o r i g i n f o r t h i s l i g a m e n t , l o c a t e d on t h e d i s t a l end o f t h e pa­
tella.
P o i n t ' 4 ' i s t h e p o i n t o f i n s e r t i o n o f t h e q u a d r i c e p s group,
l o c a t e d on t h e proximal end o f t h e p a t e l l a ; and p o i n t ' 5 ' i s t h e e f f e c ­
t i v e p o i n t o f o r i g i n f o r t h i s muscle group , which, i s t o be determined by
t h e su rg e o n .
P o i n t s ' 6 ' and 'I ' r e p r e s e n t t h e extreme p o i n t s on the
i n f e r i o r s u r f a c e o f t h e p a t e l l a and a r e col i n e a r w it h p o i n t ' T .
All v e c t o r s a r e d e f i n e d in terms o f t h e p o i n t s l i s t e d above.
Note
t h a t v e c t o r s t and f r e p r e s e n t t h e d i r e c t i o n s o f t h e p a t e l l a r lig am en t
and t h e q u a d r i c e p s group r e s p e c t i v e l y . .
A number o f ass ump tions a r e i n h e r e n t in t h i s model and w i l l be d i s ­
c us se d in t h e f o l l o w i n g s e c t i o n .
2.3
Assumptions
The q u a d r i c e p s group i s made up o f f o u r muscl es:
t h e r e c t u s femor-
i s , t h e v a s t u s l a t e r a l i s , t h e v a s t u s i n t e r m e d i u s , and t h e v a s t u s medi-.
a l i s ( s e e F ig ure 2 - a ) .
All o f t h e s e have as t h e i r i n s e r t i o n t h e p r o x i ­
mal end o f t h e p a t e l l a , b u t t h e i r o r i g i n s d i f f e r g r e a t l y , ranging from
t h e a n t e r i o r i n f e r i o r i l i a c s p i n e f o r t h e r e c t u s f e m o r i s , to p o i n t s a l l
a lon g t h e s h a f t o f t h e femur f o r t h e v a s t u s group.
F i g u r e s 3-a through
3- c i l l u s t r a t e t h e s e proximal p o i n t s o f a t t a c h m e n t .
The assumption was
made t h a t t h e q u a d r i c e p s group c ould be approximated by a s i n g l e muscle
w it h i t s proximal a t t a c h m e n t a t t h e a p p r o p r i a t e p o s i t i o n , as shown in
5
Femur
Vastus Inte rm ed ius
Vastus
Medial i s
Vastus
Lateral is
Rectus
Femoris
yV/T
I
E ffective
Muscle
Patella
Patellar
Ligament
(a)
(b)
Fi g u r e 2 . - - Q u a d r ic e p s muscle group
(a)
actual
(b)
modelled
6
Vastus L a t e r a l i s
Vastus
Medial i s
Vastus In te rm e d iu s
Rectus Femoris
Fi gu re 3 . - - Q u a d r i c e p s proximal muscle a t t a c h m e n t s ,
(a) femur - p o s t e r ­
i o r view, (b) femur - a n t e r i o r view, and (c) r i g h t i l i u m ,
is chi um , and pubis - l a t e r a l s u r f a c e .
7
F ig u r e 2-b .
The e f f e c t i v e proximal p o i n t o f a tt a c h m e n t i s l e f t as a
v a r i a b l e to be determine d by th e surgeon f o r t h e p a t i e n t under c o n s i ­
deration.
For t h e purposes o f t h i s s t u d y , th e p o i n t o f a tt a c h m e n t was
chosen on t h e a n t e r i o r s i d e o f t h e femoral s h a f t , l o c a t e d a ppro xim a te ly
o n e - t h i r d t h e l e n g t h o f t h e femur from t h e d i s t a l end and s l i g h t l y
lateral.
This p o i n t c or re sp on ds to p o i n t ‘ 5' in t h e mo del .
I t was a l s o
assumed t h a t t h e c e n t e r o f r o t a t i o n o f t h e j o i n t does n o t change with
t h e a n g le o f f l e x i o n .
I t i s l e f t to t h e surgeon t o a c c u r a t e l y deter mine
t h i s p o i n t f o r t h e p a t i e n t in q u e s t i o n .
The p a t e l l a was modelled as a c y l i n d e r w ith a r a d i u s equal to the
r a d i u s o f c u r v a t u r e o f t h e p o s t e r i o r s i d e o f th e p a t e l l a . .
This can be
o b t a i n e d e a s i l y from a ' s u n r i s e - v i e w ' x - r a y .
2 .4
Solution
In g e n e r a l , t h e problem c o n s i s t s o f s p e c i f y i n g an a n g l e o f f l e x i o n
and d e te r m in in g t h e p o s i t i o n o f t h e p a t e l l a r e l a t i v e t o t h e femur.
The
components o f a l l seven p o i n t s in t h e model a r e unknown w it h th e excep­
t i o n o f p o i n t '5 >, t h e y-component o f poin t. ' 2 ' , and t h e x-component o f
point ' I ' .
Point '5 ' is the e f f e c t iv e p o int of o rig in f o r the quadri­
ceps gro up , which i s determine d by t h e surg eo n.
The v a l u e o f y^ c o r r e s ­
ponds t o t h e l a t e r a l d is p la c e m e n t o f t h e i n s e r t i o n p o i n t f o r t h e p a t e l ­
l a r li g a m e n t and i s t h e r e f o r e , a c o n t r o l v a r i a b l e in t h e o p e r a t i o n simu­
l a t i o n . . There i s a o n e - t o - o n e co rre sp ond e nc e between a-j and t h e . a n g l e
8
o f f l e x i o n ; t h e r e f o r e , s p e c i f y i n g x-j i s e q u i v a l e n t t o s p e c i f y i n g t h e
angle o f fle x io n .
This le a v e s a t o t a l o f s i x t e e n unknowns as shown in
Table I .
1
Table I . - - V a r i a b l e s
Pt.
unknown
I
2
3
4
*1 z I
X2
z2
7
xI
y2
x3 y 3 z 3
x 4 y 4 z4
5
6
specified
X5 y 5 z 5
*6 y 6 Z6
x? Y7 Z7
The f o r c e a n a l y s i s , t o be d i s c u s s e d in more d e t a i l l a t e r , i n c l u d e s
t h e two f o r c e s e x e r t e d on t h e p a t e l l a by t h e l i g a m e n t s , and a l s o a r e ­
s u l t a n t f o r c e a c t i n g on t h e p a t e l l a a t t h e p o i n t o f c o n t a c t .
The d i r e c
t i o n o f t h i s r e s u l t a n t f o r c e i s normal t o t h e cam a t t h e p o i n t o f con­
tact.
( I t i s assumed t h a t t h e r e a r e no t a n g e n t i a l components o f t h e
r e s u l t a n t f o r c e due t o the. f a c t t h a t t h e r e i s e s s e n t i a l l y no f r i c t i o n
between t h e p a t e l l a and t h e cam.)
S in c e p o i n t ' 1 1 i s unknown, th e
d i r e c t i o n c o s i n e s o f t h e normal f o r c e a r e a l s o unknown, adding t h r e e
more unknown t o t h e problem:
px$ p^, and pz .
There a r e now n i n e t e e n
9
unknowns and hence, n i n e t e e n e q u a t i o n s a r e n e c e s s a r y t o o b t a i n a s o l u ­
tion.
F ig u r e s 4 -a and 4-b i l l u s t r a t e t h e r e l a t i o n s h i p between t h e t h r e e
components o f p o i n t ' I ' t o d e s c r i b e t h e shape o f th e cam.
The two-
dimensional cu rve in t h e X-Z p l a n e , Sq , i s approximated by t h e r e l a t i o n ­
ship:
R0 = a90 + b
where:
Rq i s t h e d i s t a n c e from t h e o r i g i n to t h e c urv e Sq , and Gq i s
t h e a n g le o f t h e R-axis w it h t h e X - a x i s , p o s i t i v e c o u n t e r - c l o c k w i s e .
The c o n s t a n t s a and b a r e obtained, e a s i l y from x - r a y s .
To ap proxim ate t h e shape o f t h e groove in th e cam, a c o s i n e curve
i s used as shown in Fi g u r e 4 - b .
The f o l l o w i n g r e l a t i o n s h i p r e s u l t s :
R ^ R0 + .5Dll-cos(.Try-|/P)]
where:
R i s t h e d i s t a n c e from t h e Y -axis t o t h e cam and y-j i s th e y -
.component o f p o i n t 1T
o n . t h e cam.
The c o n s t a n t D i s t h e aver age depth
o f t h e p a t e l l a r groo ve , and P i s o n e - h a l f t h e width o f t h e groove, (oneh a l f the period o f the cosine curve).
Noting t h a t R^ - x^ t Z y and cos 90 - jc-| /( x ^ t z ^ ) 1^ 2 , th e s u r f a c e
e q u a t i o n re duc es t o :
10
Z
R
Fi g u r e 4 . - - P a t e l l a r s u r f a c e model
11
( x f ^ ) 172 = (a COS-1Cx1Z ( X f t z f ) I ^ j + b] + .BDri-COS(Try1ZP)J.
I t w i l l be n e c e s s a r y , f o r t h e purpose o f o p e r a t i o n s i m u l a t i o n , to
i n t r o d u c e an a n g le o f r o t a t i o n a bout t h e X-a xis i n t o t h i s e q u a t i o n .
This i s accomplished usi ng a c o o r d i n a t e t r a n s f o r m a t i o n w it h t h e a ngle of
r o t a t i o n bei ng p o s i t i v e in th e c o u n t e r - c l o c k w i s e d i r e c t i o n .
This r e ­
s u l t s in t h e f o l l o w i n g e q u a t i o n :
S1(X^y-J5Z1) B Ixf+Cz-jCose + y ^ i n e ) 2] 172 Ctrpfcos-1I x 1/ (X2-Kz1cose + y ^ i n e ) 2 ) 1/ 2]']. BDIl-cosIir(y1Cose-Z1s in e ) /P l] - O
where:
0)
9 i s t h e a n g l e o f r o t a t i o n , o = a , and p = b / a .
A second r e l a t i o n s h i p i s t h a t t h e normal v e c t o r t o t h e cam a t t h e .
p o i n t o f c o n t a c t i s t h e g r a d i e n t o f t h e s u r f a c e f u n c t i o n , S1 , a t t h a t
point.
This r e s u l t s in t h e f o l l o w i n g t h r e e e q u a t i o n s :
Px ■" 95V axI
Py p 9Y
a^ l
Pz - SS1Zaz1
■
(2)
CS)
(4 )
In t h e computer program, t h e s e d e r i v a t i v e s - a r e a r r i v e d a t n u m e r i c a l l y .
12
I t was assumed t h a t t h e p a t e l l a r lig a m e n t does n o t s t r e t c h , and
hence, t h e magnitude o f v e c t o r t i s a c o n s t a n t .
This may be w r i t t e n a s :
Cx2-X3 ) 2 + (y 2- y 3 ) 2 + C z 2-Z 3) 2 = r 2
where:
Cs)
r 3 i s t h e l e n g t h o f t h e l i g a m e n t , determined from x - r a y s .
Vecto rs J and t have c o n s t a n t magnitudes th r o u g h o u t f l e x i o n .
This
magnitude i s approximated as o n e - h a l f t h e r a d i u s o f t h e p a t e l l a , hence:
(X3-X6 ) 2 + (y3- y 6 ) 2 +• ( Z 3- Z 6) 2 = ( r / 2 ) 2
(6)
(X 4 - X 7) 2
(7)
+ (y4 - y 7 ) 2 + Cz4 -Z7 ) 2; = ( r / 2 ) 2
where 4 i s t h e r a d i u s o f t h e p a t e l l a .
The sum o f t h e magnitudes o f v e c t o r s ft and T i s a c o n s t a n t e q u a li n g
t h e l e n g t h o f t h e p a t e l l a , Cp . ' T h e r e f o r e :
(X6-X7) 2 + (y6-y 7) 2 + (Z6-Z7) 2 = Cp2
(8)
Based on t h e assumption t h a t t h e c e n t e r o f r o t a t i o n o f t h e t i b i a
( p o i n t 1O-1) remains s t a t i o n a r y , i t can be s t a t e d t h a t t h e d i s t a n c e from
t h e Y-axis t o t h e p o i n t o f i n s e r t i o n o f t h e p a t e l l a r li g a m e n t a l s o i s a .
f i x e d c o n s t a n t th r o u g h o u t f l e x i o n .
Thi s may Be w r i t t e n a s :
Jx22 + z 22 ^ r 22 .
'
(91
The v a l u e o f r 2 , t h e d i s t a n c e from t h e Y-axis t o t h e p o i n t o f i n s e r t i o n
13
o f t h e l i g a m e n t , i s o b t a i n e d from t h e f o ll o w i n g r e l a t i o n s h i p :
r9 - ( r 9 2 -f 62 + Zr9 Scoso))1/ 2 ,
d
-O
*0
where r9
i s t h e d i s t a n c e from t h e Y-axis t o t h e t i b i a ! t u b e r o s i t y ,
which can be o b t a i n e d from x - r a y s ; 6 i s t h e l o n g i t u d i n a l dis p la c e m e n t o f
t h e p a t e l l a r l ig a m e n t al ong t h e t i b i a ! s h a f t , measured p o s i t i v e l y in t h e
d i s t a l d i r e c t i o n ; and w i s t h e a n g le between t h e v e c t o r from th e o r i g i n
t o t h e t i b i a ! t u b e r o s i t y and t h e v e c t o r p a r a l l e l to t h e t i b i a ! s h a f t
p a s s i n g thr oug h t h e o r i g i n .
The v a r i a b l e 6 can be a d j u s t e d t o s i m u l a t e
the o perative technique of d isplacing the in s e r tio n of the p a t e l l a r
li g a m e n t l o n g i t u d i n a l l y .
Four a d d i t i o n a l e q u a t i o n s a r e d e r i v e d from th e assumption t h a t vec ­
t o r s J and t a r e p a r a l l e l to t h e normal v e c t o r a t t h e p o i n t o f c o n t a c t .
Thus,
*3 - *6
pX
Oo)
=3 - =6 " Pz
Z
^3 * /6 _ &
-
cm
=3 - =6 " Pz
*4 " x7 _ Px
z4 " z7
Pz
021
14
y 4 ~ ^ 7 _ Py
03)
z4 - =7 " Pz
Vect or s
and t a r e r e q u i r e d to be col i n e a r r e s u l t i n g in t h e f o l l o w ­
ing two e q u a t i o n s :
_ 7 ___ ZL _ I ~ 6
zy - z I " z I - z 6
04)
y? - ^ i
05)
^i - %
z7 - zI ~ zI - z6
I t i s r e q u i r e d t h a t t h e l i n e 6 -1 - 7 l i e in th e pla n e t a n g e n t t o t h e
cam a t t h e p o i n t o f c o n t a c t .
Thi s may be s t a t e d in t h e f o ll o w i n g equa­
tion:
Cx7 - X1Ipx + Cy7 - y-|)Py + Cz7 - Z1Jpz = o
06)
The f i n a l t h r e e e q u a t i o n s r e s u l t from t h e f o r c e b a l a n c e on t h e pa­
tella.
Fi g u r e 5 d e p i c t s t h e f o r c e s a c t i n g on t h e p a t e l l a , F1 and
r e p r e s e n t t h e t e n s i l e f o r c e s e x e r t e d on t h e p a t e l l a by t h e qu a d ri c ep s
group and t h e p a t e l l a r lig a m e n t r e s p e c t i v e l y .
The normal f o r c e ft i s
t h e r e s u l t a n t f o r c e between t h e p a t e l l a and t h e femur a t t h e p o i n t o f
contact.
These f o r c e s a r e w r i t t e n in v e c t o r n o t a t i o n as f o l l o w s :
=. Fj
t (Z5-Z4 )Cl
15
Fi gu re 5 . - - F r e e body diagram of p a t e l l a
F2 = F2E(X2-X3 ) 1- + ( y 2- y 3 ) j + ( z 2- z 3) k ]
N - N[px i t py j t p z k]
where F-j, F2 , and N a r e r e - n o r m a l i z e d f o r c e magnitudes.
The d is p l a c e m e n t v e c t o r s R1 and R2 shown in Fig ur e 5 may be w r i t ­
ten as:
= (X4-X 1 )T + ( y 4- y 1 ) J + (Z4- Z 1 )Ic
16
R2
= (X 3- X 1 ) i
+ (yg-y-jjj
+ ( Z 3- Z 1 )R
I t i s r e q u i r e d t h a t th e p a t e l l a be in s t a t i c e q u i l i b r i u m th r o u g h ­
o u t f l e x i o n , t h e r e f o r e , t h e sum o f t h e f o r c e s in a l l t h r e e d i r e c t i o n s
and t h e sum o f t h e moments a bou t a l l t h r e e axes must be z e r o .
Balancing
the forces y ie ld s the following m atrix equation:
X5 - X 4
X2 - X 3
Px
' / 5 -^ 4
y 2~y 3
pY
z 5~z 4
z 2" z 3
Pz
0
>i"
<
F2
>p< 0 L
N
0
For a s o l u t i o n to e x i s t , t h e d e t e r m i n a t e o f th e c o e f f i c i e n t m a tr ix must
equal z e r o , hence:
PxC(Zg-Z3) (yg-y^j-fyg-ysifzg-z^)] +
Py E(X2 - X 3 ) ( Z 5- Z 4 ) - ( Z 2- Z 3 ) (X 5-X4 ) J +
(17)
PzE(Y2-Y3) U 5-X4 )-(X2-X3)(Y5-Y4)J p 0
Balancing t h e moments a bou t a l l t h r e e axes y i e l d s t h e fo l l o w i n g :
F-j ((y^-y-j.) (Xg-X4)1- U 4-X1) (.Y5-Y4)] +
F2I U 3^ 1) (X 2 - X 3 H
x 3 - X 1 Ky 2-Y4)J
- 0
F 1E(X 4 - X 1 ) ( Z 5- Z 4 ) - ( Z 4 - Z 1 ) ( X 5-X4 ) ] +
F 2 E(X 3- X 1 ) ( Z 2- Z 3 ) - ( Z 3- Z 1 ) ( X 2- X 3 ) J = 0
17
F1 [ ( Z 4 - Z 1 ) (y5- y 4 ) - Cy4 -X1 ) Cz5 -Z4 ) ] +
F2E(Z3-Z1) (Y2- ^ M y s-V i H z2~z3)J p 0
1
Combining t h e f i r s t two o f t h e s e e q u a t i o n s r e s u l t s in t h e fo l l o w i n g :
C fy 4 - ^ 1 ) (X 5-X 4 ) - C x 4 - X 1 ) Cy5 -Y4 I l I C x 3 - X 1 1( Z 2- Z 3 ) - C z 3- Z 1 K x 2 -JC3 ) ] [( X 4-X 1) (Z 5-Z4 ) - ( Z 4-Z 1 )
(X5-X4 )IE C y3-Y 1)(X 2-X3)-C x 3-X 1X y 2 -Y3 ) ] = 0
(1 8 )
F i n a l l y , combining t h e f o r c e and moment e q u a t i o n s y i e l d s :
CPz^x 5"x4 ^ px t z 5"z4 ^ ^ x3™x l ^ z 2"z 3 ^ t z 3 - z i ) Cx2-X3 ) ] [ P z ( X 2- X 3 ) - P x ( Z 2- Z 3 ) I I ( X 4 - X 1 ) ( Z 5- Z 4 ) - ( Z 4 - Z 1 ) ( X 5-X4 ) ] F
0
(1 9 )
Thi s s e t o f n i n e t e e n n o n l i n e a r si m u lt a n eo u s e q u a t i o n s was so lved
us in g Newton-Raphson' s Method.on t h e computer.
2.5
O pe ra t io n S im u la ti o n
■
To be u s e f u l t o s u r g e o n s , t h i s model must i n c o r p o r a t e t h e a b i l i t y
t o s i m u l a t e a spectrum o f t h e most common c o r r e c t i v e o p e r a t i o n s in te n d e d
.
to a l l e v i a t e the condition of re c u rre n t p a t e l l a r subluxation.
o p e r a t i o n s a r e s i m u l a t e d in t h i s s t u d y .
Two
The f i r s t i s t h e movement in
t h e l a t e r a l and l o n g i t u d i n a l d i r e c t i o n s o f t h e i n s e r t i o n p o i n t f o r t h e
p a t e l l a r ligament.
These movements a r e a cc o m p li s h e d .b y a d j u s t i n g th e
v a r i a b l e s y 2 and 6 r e s p e c t i v e l y .
T his o p e r a t i o n i s t h e most common as
i t i s t h e e a s i e s t t o pe rfo rm , and has t h e . s h o r t e s t r e c o v e r y tim e.
The
second o p e r a t i o n i s t h e a x i a l r o t a t i o n o f t h e d i s t a l end o f t h e femur.
18
Thi s has a much g r e a t e r e f f e c t than t h e pre v io u s o p e r a t i o n , but has a
much l o n g e r r e c o v e r y r a t e .
This o p e r a t i o n i s accomplished by a d j u s t i n g
t h e v a r i a b l e 9 , which i s t h e a n g le o f r o t a t i o n a bout t h e X-axis.
.
CHAPTER 3
RESULTS
3.1
Introduction
The f o r m u l a t i o n o f t h i s model r e s u l t s in a computer model o f t h e
P a te ll o - F e m o r a l j o i n t .
The o u t p u t i s in t h e form o f a v id e o d i s p l a y
i n d i c a t i n g t h e p o s i t i o n o f t h e p a t e l l a in t h e X-Z and Y-Z p l a n e s .
F ig ure 6 i s a sample o f t h e o u t p u t from a g r a p h i c s t e r m i n a l .
The f i g u r e a t t h e top i s a mot ion view o f t h e j o i n t in t h e X-Z
p la n e i l l u s t r a t i n g t h e X and Z components o f t h e k in e m a tic c o n f i g u r a t i o n
f o r t h e system.
I t s main purpose i s to i n d i c a t e t o surgeons how high
t h e p a t e l l a i s r i d i n g in t h e cam.
The f i g u r e on t h e r i g h t i s a view o f
t h e cam in t h e Y-Z pla ne i n d i c a t i n g t h e pa th o f t h e p a t e l l a in the
groove o f t h e cam.
3 .2
C o m p a t i b i l i t y o f Model wit h X -r a y
.
F ig ure 7 i s a f u l l scale- view d e m o n s t r a t in g t h e a c c u r a c y o f t h e
model in t h e X-Z p l a n e .
The s o l i d l i n e s r e p r e s e n t a c t u a l .x-ray d a t a ,
and t h e dashed l i n e s r e p r e s e n t t h e model i n c o r p o r a t i n g d a t a from th e
same x - r a y .
T h e / p o s i t i o n o f t h e modelled p a t e l l a i s t h e c r i t i c a l f a c t o r
i n t h i s view, and as shown, matches c l o s e l y t h e a c t u a l p o s i t i o n .
3.3
S e n s i t i v i t y o f Model t o E f f e c t i v e ! M u s c l e P o s i t i o n
More im p o r t a n t than t h e p o s i t i o n o f t h e p a t e l l a in t h e X-Z pla ne i s
i t s p o s i t i o n in t h e Y-Z p l a n e .
All o f t h e i n p u t d a t a f o r t h i s model i s
J
20
Z
P a t e l l a r Path
Fi g u r e 6 . -- A c tu a l video d i s p l a y o u tp u t
Actual
Model
Fi gu re / . - - C o m p a r i s o n o f model w ith a c t u a l x - r a y d a ta
22
f a i r l y s t r a i g h t f o r w a r d e x c e p t f o r t h e e f f e c t i v e q u a d ri c e p s group p o i n t
o f o r i g i n U 5 ^y51 Z5 ) .
F ig u r e s 8-a thro ugh 8-c i n d i c a t e t h e s e n s i t i v i t y
o f t h e p a t e l l a r path t o t h i s p o i n t .
Fi g u r e 8-a de m o n s t ra t e s th e e f f e c t o f a l t e r i n g t h e x-component o f
point ' 5 ' .
There i s a s l i g h t s h i f t inward as Xg i s i n c r e a s e d .
The
‘ g e n e r a l shape and tendency o f t h e path does no t change as Xg i s changed,
and hence, t h e s e l e c t i o n . o f Xg i s n o t a c r i t i c a l f a c t o r i f i t i s chosen
in the c o r r e c t ra n g e '(+ 2 c e n tim ete rs).
The y-component o f p o i n t ' 5 ' i s v a r i e d in Fi gure 8 - b .
I t is evi­
d e n t t h a t t h i s va lu e i s c r i t i c a l l y im p o r t a n t as t h e whole path can be
r a d i c a l l y a f f e c t e d in a f i f t e e n c e n t i m e t e r ra n g e .
I t is therefore
. n e c e s s a r y t h a t a very a c c u r a t e va lu e be used f o r y g.
I t i s recommended
t h a t i t be t o t h e n e a r e s t c e n t i m e t e r .
The change in t h e p a t e l l a r pa th with a f o u r c e n t i m e t e r change in Zg
. i s shown in F ig u r e 8 - c .
Thi s v a l u e i s n o t as c r i t i c a l as y g , but should
be found t o t h e n e a r e s t 2 c e n t i m e t e r s . ,
; 3.4
O pe ra t io n S im u la ti o n
F ig u r e s 9-a thr ou gh 9 - f de m onst ra t e t h e e f f e c t o f . a s e r i e s of
o p e r a t i o n s performed on t h e model as an example o f t h e proc ed ure to be
used by t h e surgeon and does no t r e p r e s e n t an a c t u a l p a t i e n t .
In t h i s /
case, point fiv e is ( - 1 5 ,7 .5 ,5 .) in the u n its of centim eters.
This
r e p r e s e n t s a r i g h t l e g , t h e r e f o r e , t h e l e f t s i d e o f t h e groove in t h e
23
Z
Z
y 5 = 7.5
*5 = - 8 -0
y
y
y 5 = 0 .0
Xg = - 1 0 . 0
y
y
Xg
y 5 = -7.5
= -15.0
y
y
(a)
(b)
Figu re 8 . - - S e n s i t i v i t y o f p a t e l l a r pa th t o e f f e c t i v e q u a d r i c e p s o r i g i n
(a)
X-component (y5=7.5;Zg=5.0)
(b)
Y-component ( x g ^ - l S . *,Zgs5.0)
24
Z
Z
Z5 = 7 .0
Z5 - 3 .0
y
y ------------------------(c)
Figu re
.- - C o n t i n u e d
(c) Z-component (x5= - 1 5 . ; y 5=7.5)
\
25
Z
0 = .3
6 = 0.
F ig ure 9 . --Sample run o f o p e r a t i o n s i m u l a t i o n
26
Fi gu re 9 .- -C o n ti n u e d
y
27
Fig ur e 9 .- - C onti nued
28
figures is the la te r a l side.
F ig u r e 9-a r e p r e s e n t s th e h y p o t h e t i c a l p a t i e n t ' s c o n d i t i o n b e fo re
o p e r a t i o n s a r e pe rf o rm ed .
Notic e t h a t t h e l a t e r a l s i d e o f t h e groove is
a lm o s t f l a t and t h e pa th i s halfway up t h e l a t e r a l c ondyle .
This con­
f i g u r a t i o n would c o n c e i v a b l y r e q u i r e o n ly a small e x t e r n a l l a t e r a l f o r c e
t o d i s l o c a t e t h e p a t e l l a , and an o p e r a t i o n would pro ba bl y be recommended.
The medial d i s p l a c e m e n t o f t h e p a t e l l a r li ga m e nt i s t h e e a s i e s t
o p e r a t i o n t o perform and should t h e r e f o r e be t h e f i r s t o p e r a t i o n simu­
l a t e d on t h e program.
F ig u r e 9-b r e p r e s e n t s t h e pa th a f t e r t h e li ga m e nt
has been d i s p l a c e d two c e n t i m e t e r s m e d i a l l y .
c l o s e r t o t h e c e n t e r o f t h e groove.
This bro u g h t t h e path much
Moving th e l i g a m e n t an a d d i t i o n a l
.
c e n t i m e t e r m e d i a l l y b r i n g s t h e path even c l o s e r to t h e c e n t e r o f t h e .
groove as shown in Fi gu re 9 - c .
In many i n s t a n c e s , su rge ons a r e tempted -
t o s t o p a t t h i s p o i n t s i n c e the. l i g a m e n t can no t be d i s p l a c e d f u r t h e r
m e d i a l l y , and t h e only a l t e r n a t i v e i s to t w i s t t h e femur.
The r o t a t i o n
o f t h e femur i s a m a j o r o p e r a t i o n and i s u s u a l l y a void ed i f a t a l l
possible.
However, in t h i s c a s e , t h e l a t e r a l condyle i s r a d i c a l l y low,
and a l a r g e enough l a t e r a l f o r c e co uld s t i l l d i s l o c a t e t h e p a t e l l a .
Hence, t h e r o t a t i o n o f t h e femur i s p r ob a bly n e c e s s a r y in t h i s c a s e .
With t h e l ig a m e n t r e t u r n e d t o i t s o r i g i n a l p o s i t i o n . Figu re 9-d
de m o n s t ra t e s a r o t a t i o n o f t h e femur, by .2 r a d i a n s ( a p p r o x im a te ly 11
degrees).
This d id n o t improve r a d i c a l l y t h e path from Fi g u r e 9-a and
i t i s t h e r e f o r e concluded t h a t an o p e r a t i o n i n v o l v i n g . b o t h t h e d i s p l a c e -
29
merit o f t h e li g a m e n t and t h e a x i a l r o t a t i o n o f t h e femur i s n e c e s s a r y .
R o t a t i n g t h e femur .3 r a d i a n s (a b o u t 17 d e g r e e s ] and d i s p l a c i n g th e
l i g a m e n t 3.54 c e n t i m e t e r s ( 1 . 4 in c h e s ) m e d i a l l y r e s u l t s in Fi gure 9 -e .
The l a t e r a l condyle i s now even w it h t h e medial c on d y le , and pro v id e s a
s a t i s f a c t o r y r i s e on t h e l a t e r a l s i d e .
The path i s f a i r l y c l o s e to th e
c e n t e r o f t h e groo ve , and t h e p a t e l l a would r e q u i r e a much g r e a t e r f o r c e
to d islo c a te .
F ig ure 9-e t h e r e f o r e would r e p r e s e n t t h e f i n a l s t a t e
a f t e r the indicated operation.
F ig ure 9 - f shows a pa th which i s c l o s e r to t h e c e n t e r than Figure
9 - e , y e t i s an u n d e s i r a b l e c o n f i g u r a t i o n .
In t h i s f i g u r e , t h e lig am en t
a t t a c h m e n t i s moved two c e n t i m e t e r s p r o x im a ll y and would t h e r e f o r e
l e s s e n t h e t e n s i o n in th e li g a m e n ts r e q u i r i n g a s m a l l e r l a t e r a l f o r c e to
d is lo c a te the p a t e l l a .
I t shou ld be noted t h a t t h e w r i t e r i s an e n g i n e e r and i s t h e r e f o r e
w r i t i n g from t h a t p e r s p e c t i v e .
The above sample run may have implied
c o n c l u s i o n s which surgeons would n o t r e a c h .
I t should a l s o be s t a t e d
t h a t i f t h i s model i n d i c a t e s an o p e r a t i o n t h a t i s r a d i c a l l y d i f f e r e n t
from what a surgeon would have done o t h e r w i s e , then f u r t h e r i n y e s t i g a ' t i o n i s recommended.
CHAPTER 4
SUMMARY
4.1
Introduction
The purpose o f t h i s r e s e a r c h was t o develop a ma thematical model o f
the Patello-Femoral j o i n t .
. The model was developed w it h a view to a i d ­
ing surgeons in de terming t h e optimum o p e r a t i o n t o a l l e v i a t e t h e c o n d i ­
tion of recurrent p a te lla r dislocation.
• V ect or s were used t o r e p r e s e n t t h e p o s i t i o n s o f t h e femur, t i b i a ,
p a t e l l a , q u a d r i c e p s group , and t h e p a t e l l a r li g a m e n t .
A s e t of nineteen
n o n l i n e a r si m u lt a n eo u s e q u a t i o n s were g e n e r a t e d r e l a t i n g t h e n i n e t e e n
unknowns.
These e q u a t i o n s were s ol ved usi ng Newton-Raphson1s Method on
t h e computer, r e s u l t i n g in an a n a l y t i c a l model o f t h e knee j o i n t with
t h e a b i l i t y to s i m u l a t e s u r g i c a l o p e r a t i o n s .
The o u t p u t i s in t h e form o f a vide o d i s p l a y i n d i c a t i n g t h e path o f
t h e p a t e l l a in t h e j o i n t t h r o u g h o u t f l e x i o n .
Sim ulated o p e r a t i o n s a r e
performed on t h e model u n t i l t h e d e s i r e d p a t e l l a r pa th i s rea ch e d.
4.2
Recommendations
This.model a c c u r a t e l y d e f i n e s t h e p o s i t i o n o f . t h e p a t e l l a in th e
j o i n t w it h t h e assumption t h a t t h e q u a d r i c e p s group e f f e c t i v e proximal
a t t a c h m e n t can be l o c a t e d a c c u r a t e l y .
Following a r e recommendations
which would r e s u l t in a more a c c u r a t e model.
. The m a jo r assumption used in t h e model i s t h e a p p ro xi m a tio n o f t h e
q u a d r i c e p s group w it h a o n e - s t r a n d m u s c l e . . This r e q u i r e s t h e de te r m in a ­
31
t i o n o f an e f f e c t i v e p o i n t o f o r i g i n .
In t h e c u r r e n t model, t h i s valu e
i s i n p u t by t h e surgeon and remains f i x e d f o r a l l a n g le s o f f l e x i o n .
There a r e two problems i n h e r e n t with t h i s a ss um pti on.
One i s t h a t t h i s
e f f e c t i v e p o i n t o f o r i g i n i s d i f f i c u l t t o de ter mi ne a c c u r a t e l y , and th e
second i s t h a t t h i s p o i n t changes as t h e f l e x i o n a n g le c h an ge s .
In
view o f t h i s , two recommendations a r e made.
1.
An a c c u r a t e r e l a t i o n s h i p should be developed between t h e e f f e c t i v e
p o i n t o f o r i g i n and t h e a ngle o f f l e x i o n .
2.
The second recommendation s uper ce des t h e f i r s t .
Vecto rs should be
i n c o r p o r a t e d in t h e model t o r e p r e s e n t a l l f o u r m us cl es in th e
q u a d r i c e p s group.
This would add no new unknowns t o t h e problem,
■
e n a b l i n g t h e same b a s i c program to be u s e d , b u t would a l t e r th e
f o r c e a n a l y s i s and th u s Equations (17) through (19) would change.
I t would s t i l l be n e c e s s a r y to de te r m in e t h e e f f e c t i v e p o i n t of
o r i g i n f o r each o f t h e f o u r m u s c l e s , but t h i s i s n o t n e a r l y as
d i f f i c u l t as d e te r m in in g t h e e f f e c t i v e o r i g i n f o r t h e t o t a l group.
3.
' The magnitudes o f v e c t o r s J and. t. were assumed to be o n e - h a l f th e
Z
■
'
radius of the c y lin d ric a l p a te l la fo r a ll p a tie n ts .
This i s not
n e c e s s a r i l y t r u e and t h e s e v a l u e s c ould be i n p u t . v a r i a b l e s e a s i l y
. o b t a i n e d from x - r a y s .
This would r e q u i r e a l t e r i n g Equations (6)
and ( 7 ) .
4.
The f i n a l recommendation i s t o i n c o r p o r a t e a v a r i a b l e in t h e o p e r a ­
t i o n s i m u l a t i o n t o r e p r e s e n t t h e movement o f t h e p a t e l l a r li ga m e nt
32
in t h e d i r e c t i o n normal t o t h e t i b i a ! s h a f t .
As i t now s t a n d s , t h e
model only allow s f o r movements in t h e l a t e r a l and l o n g i t u d i n a l ■
directions.
The a n t e r i o r s i d e o f t h e t i b i a ! s h a f t i s V-shaped, and
hence, a p u r e l y l a t e r a l d is p l a c e m e n t i s not r e a l i s t i c .
APPENDIX I
PROGRAM KNEEPLOT FLOW CHART
,Start)
Function D e f i n i t i o n
/ In p u t X-ray Data /
/ In p u t 0 , 6 , y 2 / ■
— ^ Do M =
to 1 0 0 ) -
------------- — - ( ^End^ )
\ P l o t Total S o l u t i o n y
Calculate
I n i t i a l Guess
- ( do X 1 -
to
V a r i a b l e s = Guess
Guess = L a s t S o lu t io n
Calculate .
C o e f f i c i e n t Matrix
Vector Matrix
Increment V a r i a b l e s
Call Sim ult
----------------- ^2-<z^ n vergence?x >-^
S t o r e S o lu t io n
REFERENCES .
Curnahans L u t h e r , and W ilk es, Applied Numerical Methods^ Wiley, 1969,.
pp. 319.
1
Landau, Barbara R ., E s s e n t i a l Human Anatomy and P h y s i o l o g y , S c o t t ,
Foresman, 1976.
Li eb , F r e d r i c k J . , M.D., and J a c q u e l i n P e r r y , M.D., "Quadriceps Func­
t i o n , " Jqurnal_2 L B o n e _ a j ^
Vol. 50-A, No.' 8,
December, 1968.
Rasch, P h i l i p J . , and Roger K. Burke, K in esi ol ogy and Applied Anatomy,
3rd E d i t i o n , P h i l a d e l p h i a , Lea and F e b a g e r , 1967.
MONTANA STATE UNIVERSITY LIBRARIES
762 100
i:
66 O
N378
Hl2
cop.2
Hagelin, Jack S
An analytical model
of the patello-femoral
joint