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Faculty of mathematics, physics and informatics
Comenius University
Department of Biophysics and Chemical physics
Diploma work
Matej Daniel
Bratislava
2001
Faculty of matematics, physics and informatics
Comenius University
Department of Biophysics and Chemical physics
Department of Orthopaedic Surgery, Clinical center, Ljubljana, Slovenia
Diploma work
Gradient of contact stress as a parameter determining
biomechanical status of human hip
Matej Daniel
Bratislava
2001
Čestné prehlásenie
Týmto čestne prehlasujem, že rigoróznu prácu na tému „Biomechanické parametre určujúce stav ľudského bedrového kĺbuÿ som vypracoval samostatne a s použitím uvedenej
literatúry.
Dobšinská ľadová jaskyňa, 31. júla, 2001
Podpis:
Acknowledgments
This work was made on the Department of Orthopaedic Surgery, Clinical Center, Ljubljana, Slovenia in collaboration with Faculty of Electrical Engineering, Ljubljana, Slovenia
financed by student exchange program CEEPUS, No. A-0103. The radiographs used in this
work were taken from the archive of the Department of Orthopaedic Surgery.
I would like to thank doc. Veronika Kralj-Iglič for her many suggestions and constant
support during this research. I am also thankful to doc. Aleš Iglic for his guidance through the biomechanics and doc. Jozef Vojtaššák, M.D. who showed a clinical aspects of
biomechanics.
I thank prof. Franjo Pernuš from the Faculty of Electrical Engineering and doc. Vane
Antolič, M.D. from the Department of Orthopaedic Surgery for fruitful suggestions.
I am grateful for collaboration to the colleagues from the Department of Orthopaedic
Surgery, Ljubljana: to mag. Rok Vengust, M.D., mag. Oskar Zupanc, M.D., Borut Pompe, M.D., Boštjan Kersnič, M.D., Matej Drobnič, M.D., Blaž Mavčič and to mag. Anton
Jaklič from the Institute for Material Technology – IMT, Ljubljana. I also thank the head
of the Institute of Biophysics Saša Svetina, Faculty of Medicine for his support.
Abstract
Bedrový kĺb je jedným z hlavných nosných kĺbov v tele. Pretože je tento kĺb často postihnutý degeneratívnymi procesmi, ktoré vedú k imobilizácii pacienta, skúmajú sa faktory,
ktoré ovplyvňujú jeho vývoj. Predpokladá sa, že dlhodobo zvýšený tlak na kĺbovú chrupku
urýchľuje vývoj koxartrózy [22, 51]. Preto sa v predchádzajúcich štúdiách používala na
popis stavu kĺbu maximálna hodnota tlaku. Objavujú sa aj názory [8], že okrem tejto hodnoty je dôležité aj rozloženie tlaku. Predpokladá sa, že vysoká hodnota gradientu tlaku
na laterálnom okraji acetabula (ďalej gradientu tlaku) môže byť dôležitejším parametrom
popisujúcim stav kĺbu ako samotný tlak. Preto by bolo zaujímavé použiť na ohodnotenie
stavu kĺbu gradient tlaku a taktiež ďalšie parametre, ktoré zahrňujú kombináciu tlaku a
gradientu tlaku.
V prezentovanej práci sme zaviedli nové biomechanické parametre – gradient tlaku,
index tlaku a funkčný uhol nosnej plochy. Pre určenie významnosti týchto parametrov sme
uskutočnili štúdie stavu populácii kĺbov.
Ak chceme rozumieť funkcii bedrového kĺbu, musíme poznať jeho štruktúru. Preto sa
v prvej časti práce zaoberáme anatómiou bedrového kĺbu. Ďalej prezentujeme prehľad
literatúry v oblasti biomechaniky bedrového kĺbu. Tieto práce delíme do dvoch skupín:
práce, ktoré sa zaoberajú silou pôsobiacou v bedrovom kĺbe a práce, ktoré sa zaoberajú
rozložením tlaku na kĺbovú chrupku. Podrobnejšie sa venujeme metódam, ktoré používame
v ďalšej analýze.
Na odvodenie metódy pre výpočet biomechanických parametrov sme použili dva nedávno vyvinuté matematické modely. Jeden pre výpočet sily pôsobiacej v kĺbe pri stoji na
jednej nohe [27] a druhý pre určenie rozloženia tlaku v kĺbe [30]. V tejto práci je okrem iného
prezentovaný aj nový jednoduchší spôsob odvodenia rovníc druhého modelu, ktorý spočíva
vo voľbe alternatívneho súradnicového systému (Obr. 3.2). Vstupom do týchto modelov
sú niektoré geometrické parametre panvy a proximálneho femuru určené zo štandardného
antero-posteriórneho röntgenového snímku (Obr. 3.4).
Presnosť určenia biomechanických parametrov závisí od presnosti modelu a od presnosti určenia vstupných parametrov modelu. Tieto sú ovplyvnené zväčšením röntgenového
snímku. Preto sme prispôsobili metódu na získavanie geometrických parametrov panvy a
proximálneho femuru tak, že sme zobrali do úvahy zväčšenie röntgenového snímku. Na zá-
klade nameraných dát sme zistili, že priemerné zväčšenie snímkov je väčšie ako priemerne
uvažovaných 10%. Zistili sme, že najmenej je ovplyvnený zväčšením funkčný uhol nosnej
plochy a najviac index tlaku. Ak je zväčšenie snímkov neznáme a predpokladáme, že je
rôzne, najvhodnejším parametrom na ocenenie stavu kĺbu je funkčný uhol nosnej plochy.
Pomocou matematickej simulácie sme odhadli chybu, ktorej sa dopúšťame pri štúdiách veľkého počtu pacientov a vypracovali sme metódu korekcie tejto chyby, ktorú sme následne
použili v štúdii normálnych a dysplastickych bedrových kĺbov. Zistili sme, že rozdielnosť
geometrických parametrov panvy vplýva na variáciu hodnôt biomechanického parametra
viac ako rozdielnosť vo zväčšení snímkov. Navrhujeme, aby sa v budúcnosti pri rádiografickom výšetrení umiestnil štandard známych rozmerov na úroveň veľkého trochantra. To
by umožnilo znížiť šum spôsobený nerovnakým zväčšením snímkov.
Účinok jednotlivých biomechanických parametrov na vývoj a stav kĺbu sme skúmali
na populácii normálnych a dysplastických bedrových kĺbov, kĺbov po Salterovej osteotómii
a kĺbov postihnutých vývinovou dyspláziou. Skúmali sme nielen nami definované biomechanické parametre ale aj parametre používané v predošlých štúdiách – maximálny tlak,
kumulatívny tlak a parameter, ktorý sa používa v klinickej praxi – Wibergov uhol.
Dysplázia bedrového klbu sa považuje za stav, ktorý v dôsledku nepriaznivých biomechanických pomerov v kĺbe vedie k jeho degeneratívnym zmenám. Zo štúdie normálnych
a dysplastických bedrových kĺbov vyplýva, že všetky nami skúmané parametre sa signifikantne líšia medzi normálnymi a dysplastickými bedrovými kĺbmi na úrovni významnosti
menšej ako 0,001. Ukazuje sa, že nízky Wibergov uhol a funkčný uhol nosnej plochy, vysoký maximálny tlak, gradient tlaku a index tlaku sú biomechanicky nepriaznivé, čo je v
súlade s predošlými štúdiami [35, 45, 70]. Zistili sme, že pri maximálnom tlaku, gradiente
tlaku a indexe tlaku je vplyv iných geometrických parametrov panvy ako Wibergov uhol
vyšší pri nižších hodnotách tohto uhla. Gradient tlaku a index tlaku sú u väčšiny dysplastických kĺbov kladné a u väčšiny normálnych kĺbov záporné. Pozorovali sme, že táto
zmena znamienka nastáva pri Wibergovom uhle rovnom približne 22◦ , čo je v súlade s
klinickými štúdiami [46]. Na vysvetlenie pozorovanej skutočnosti sme vychádzajúc z Pauwelsovej teórie kauzálnej histogenézy mezenchymálneho tkaniva [57] navrhli novú hypotézu
na vysvetlenie vplyvu zaťaženia na chrupku. Vychádzame z toho, že jedným zo stimulov
ovplyňujúcich metabolickú aktivitu chondrocytu môže byť zmena jeho tvaru podmienená
deformáciou chrupky. Gradient tlaku nám potom vyjadruje rýchlosť výtoku intersticiálnej
tekutiny a teda rýchlosť deformácie chrupky. Znamienko gradientu súvisí so smerom toku
intersticiálnej tekutiny vzhľadom k acetabulu.
Ďalej sme skúmali vývoj kĺbu počas dlhšieho časového obdobia. Keďže v archívoch nie
sú viacnásobné snímky zdravých kĺbov, zvolili sme si pacientov, ktorí v detstve podstúpili
Salterovu osteotómiu a preto boli následne sledovaní. Bolo by zaujímavé sledovať súvislosť
medzi stavom kĺbov bezprostredne po operácii a následný vývoj týchto kĺbov. Bohužiaľ
model na určenie sily pôsobiacej v bedrovom kĺbe sa ukázal byť nepoužiteľný pri malých
deťoch. V dôsledku značného rozdielu v zväčšení snímkov sme na ocenenie stavu kĺbov
použili funkčný uhol nosnej plochy. Zistili sme, že v priemere väčší Wibergov uhol po
operácii vedie z dlhodobého hľadiska k biomechanicky priaznivejšiemu výsledku.
V štúdii kĺbov postihnutých vývinovou dyspláziou sme skúmali súvis medzi klinickým
skóre, ktoré zahrňuje subjektívne pocity pacienta, a biomechanickými parametrami. Tu
prezentujeme len čiastkové výsledky, pretože tento výskum stále pokračuje. Hoci štatistická významnosť tejto štúdie je nízka, ukazuje sa, že biomechanické parametre môžu byť
vhodnejšie na popis stavu kĺbu ako Wibergov uhol.
Súčasťou tejto práce bolo aj prispôsobenie počítačového programu HIPSTRESS pre
c
c
a tabuľkový kalkulátor MS Excel
. To umožnilo použitie
operačný systém MS Windows
tohto programu v prezentovaných biomechanických štúdiách a uľahčí jeho používanie v
klinickej praxi.
2
Table of Contents
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Part 2. The aim of the work . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Part 3. Material and methods . . . . . . . . . . .
3.1. Theory . . . . . . . . . . . . . . . . . . . . . . .
3.1.1. Derivation of the model equations . . . .
3.1.2. Gradient of stress and functional angle of
3.2. Determination of the biomechanical parameters
3.3. Analysis of data . . . . . . . . . . . . . . . . . .
3.3.1. Statistical analysis . . . . . . . . . . . .
3.3.2. Effect of magnification in biomechanical
patients . . . . . . . . . . . . . . . . . .
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Part 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1. Structure of the hip joint . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1. Bones of the hip joint . . . . . . . . . . . . . . . . . . . . . .
1.1.2. Articular capsule of the hip joint . . . . . . . . . . . . . . .
1.1.3. Articular cartilage of the hip joint . . . . . . . . . . . . . . .
1.1.4. Movements within the hip joint . . . . . . . . . . . . . . . .
1.2. Loads on the hip . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1. Direct measurements . . . . . . . . . . . . . . . . . . . . . .
1.2.2. External measurements combined with mathematical model
1.3. Stress in the hip joint articular surface . . . . . . . . . . . . . . . .
1.3.1. Direct measurements . . . . . . . . . . . . . . . . . . . . . .
1.3.2. External measurements combined with mathematical models
1.4. Evaluation of hip status by biomechanical parameters . . . . . . . .
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the weight-bearing area .
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studies of large group of
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Part 4. Results on populations . . . . . . . . . . . . . . . .
4.1. Normal and dysplastic hips . . . . . . . . . . . . . . . . . .
4.1.1. Dysplasia of the hip . . . . . . . . . . . . . . . . .
4.1.2. Patients . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3. Results . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Evaluation of the hips after the Salter osteotomy . . . . .
4.2.1. Salter innominate osteotomy . . . . . . . . . . . . .
4.2.2. Patients . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3. Results . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Evaluation of the hips subject to developmental dysplasia .
4.3.1. Harris hip score . . . . . . . . . . . . . . . . . . . .
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3
Table of Contents—Pokračovanie
4.3.2. Patients and method . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Part 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1. Magnification of the radiographs . . . . . . . . . . . . . . . . . . . .
5.2. Results on populations . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1. Biomechanical status of normal and dysplastic hips . . . . .
5.2.2. Biomechanical evaluation of hip joint after Salter osteotomy
5.2.3. Evaluation of the hips subjected to developmental dysplasia
5.3. Effects of stress on the hip . . . . . . . . . . . . . . . . . . . . . . .
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Part 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Appendix A. Computer system for determination of contact stress .
I
Appendix B. Computer system for determination of geometrical parameters of the hip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IV
Appendix C. Radiograph of the hip . . . . . . . . . . . . . . . . . . . . . . .
VI
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII
4
List of Symbols
acceleration of the segment of
the body
A
size of the weight bearing area
Ai
relative cross-sectional area of
the i-th muscle
b
x-coordinate of the weight of
~L
the loaded leg W
B
constant describing the linear
rise of stress on the plane per~
pendicular to R
c
x-coordinate of the origin of
the ground reaction force
C
pelvis width
constant describing the linear
D
rise of stress on the plane per~
pendicular to R
d
time-weightened exponent
fi
average tension in the i-th
muscle
~
intersegmental force
FI
~
F
resultant force of the muscles
~
force of the i-th muscle
Fi
gradpm stress gradient at the lateral
margin of the weight-bearing
area
H
pelvis height
moment of inertia of the segJ
ment of the body
~k
positional vector from the
center of the femoral head to
the center of gravity of partial
~ L)
~B −W
body weight (W
l
distance between the centers
of the femoral heads
~a
L
L0
m
M
~I
M
p
p0
peff
pc
pd
pI
pmax
r
~rCG
~rCI
~ri
~ri 0
~
R
~si
length of the metal lamella
length of the metal lamella in
the radiograph
mass of the segment of the
body
magnification factor
intersegmental rotational moment
contact stress
value of stress in the pole
effective stress
cumulative stress
stress damage threshold
stress index
maximum contact stress
radius of the femoral head
positional vector from the
center of rotation of the segment the center of gravity of
the segment
positional vector from the
center of rotation to the origin
of the intersegmental force
positional vector from the
center of rotation to the proximal attachment point of the
i-th muscle
positional vector from the
center of rotation to the distal attachment point of the ith muscle
hip joint resultant force
unit vector in the direction of
the force of the i-th muscle
5
List of Symbols—Pokračovanie
sµ
sM
S
~
W
~B
W
~
WL
x0
x
Yi0
Yi
z
α
~
γ
δ
θ
Θ
ϑAC
ϑACM
standard deviation of the
magnification of the biomechanical parameters
standard deviation of the
magnification factor
size of the projection of the
weight-bearing area onto the
~
plane perpendicular to R
weight of the segment of the
body
body weight
weight of the loaded leg
length of the femur
vertical distance from the center of the femoral head to the
effective muscle attachments
point on the greater trochanter
magnified value of the biomechanical parameter
true value of the biomechanical parameter
horizontal distance from the
center of the femoral head to
the effective muscle attachments point on the greater
trochanter
angular acceleration of the
segment of the body
angle between the stress pole
and the chosen point at the
articular sphere
magnitude of displacement of
the femoral head
angle of inclination of the bor~
der plane with respect to R
angle of the displacement of
the pole from vertical axis
the acetabular angle
the ACM angle
ϑF
ϑL
ϑL 0
ϑM
ϑM 0
ϑR
ϑU S
µ
ν
σ
Φ
ϕ
ϕR
functional angle of the weightbearing area
angle of the inclination of the
lateral intersecting plane with
respect to x = 0
angle of the inclination of the
lateral intersecting plane with
respect to x = 0 plane in rotated coordinate system
angle of the inclination of the
medial intersecting plane with
respect to x = 0 plane
angle of the inclination of the
medial intersecting plane with
respect to x = 0 plane in rotated coordinate system
angle of the inclination of the
~
resultant hip joint force R
with respect to vertical
the angle of the inclination of
the acetabulum
magnification of the biomechanical parameter
angle describing the rotation
of femur around the axis through the femoral head
angle of the inclination of the
force F~ with respect to the horizontal
angle of the displacement of
the pole in the horizontal
plane from x-axis in the counterclockwise direction
angle describing the rotation
of the pelvis in the frontal
plane
angle of the inclination of the
~ in
resultant hip joint force R
the horizontal plane
6
1.1 Structure of the hip joint
Part 1
Introduction
1.1
Structure of the hip joint
In order to understand the function of the hip joint it is necessary to know its structure.
However in the biomechanical analysis we have to involve also the other parts of the body
which are connected to the movements in the hip joint. Therefore in the following part,
the morphology of the hip joint and the muscles, that are relevant for the hip joint, will
be described.
The hip joint forms the articular connection between the lower limb and the pelvic
girdle (bony ring formed by the hip bone and the sacrum) (Fig. 1.1). It is strong and
stable multiaxial ball-and-socket (spheroidal) type of synovial joint, where the articulating
bone surfaces are covered with articular cartilage. The joint is surrounded by the articular capsule. A synovial membrane underlying the interior of the joint capsule secretes a
lubricant known as synovial fluid. In this part we will describe bones, articular cartilage,
articular capsule and muscles of the hip joint in more details.
Sacrum
Ilium
Acetabulum
Femoral
head
Pubis
Ischium
Femur
Figure 1.1. Bone structure of the hip. Adapted from Hall, 1995.
7
1.1 Structure of the hip joint
Iliac crest
Tubercle of crest
Anterior superior
iliac spine
Anterior inferior
iliac spine
Superior ramus of pubis
Crest of pubis
Head of femur
Greater trochanter
Body of pubis
Intertrochanteric line
Lesser trochanter
Inferior ramus of pubis
Pubic symphysis
Obturator foramen
Ischiopubic ramus
Femur
Figure 1.2. Bones of the hip joint, anterior view. Adapted from Agur, 1991.
1.1.1
Bones of the hip joint
As mentioned above, the hip joint is a ball-and-socket type of joint. The femoral head
represents the ball and the acetabulum, which is a part of the pelvis, represents the socket
(Fig. 1.1).
The pelvis forms the body connection between the trunk and the lower limb. The
mature hip bone (os coxae) is a large flat bone formed by the fusion of the ilium, the
ischium and the pubis (Fig 1.1). The two hip bones, which form together with the sacrum
most of the bony pelvis, are united anteriorly by the pubic symphysis (Fig. 1.2).
The ilium, that contributes the largest part to the hip bone, forms the superior part
of acetabulum (Fig. 1.6). The ilium has winglike posterolateral surfaces, the ala, that
provide attachment for muscles (Fig. 1.11). Anteriorly, the ilium has an anterior superior
iliac spine (Fig. 1.2) and inferior to it an anterior inferior iliac spine (Fig. 1.2). From the
anterior superior iliac spine the long curved superior border of the ala of the ilium, the
iliaca crest (Figs. 1.2, 1.3), extends posteriorly, terminating at the posterior superior iliac
spine (Fig. 1.3). Inferior to it there is posterior inferior iliac spine (Fig. 1.3). A prominence
on the external part of the crest, the tubercle of the iliac crest (Figs. 1.2 1.3), lies 5 to 6
cm posterior to the anterior superior iliac spine [1]. The lateral surface of the ala of the
ilium has three rough curved lines, the posterior, anterior and inferior gluteal lines, that
separate the proximal attachments of the three large gluteal muscles (Fig. 1.3).
8
1.1 Structure of the hip joint
Iliac crest
Posterior gluteal line
Posterior superior iliac spine
Posterior inferior iliac spine
Greater sciatic notch
Ischial spine
Lesser sciatic notch
Ischial tuberosity
Spiral line
Tubercle of crest
Anterior gluteal line
Inferior gluteal line
Neck of femur
Greater trochanter
Intertrochanteric crest
Lesser trochanter
Gluteal tuberosity
Linea aspera
Femur
Figure 1.3. Bones of the hip joint, posterior view. Adapted from Agur, 1991.
The ischium composes the posteroinferior part of the hip bone. The superior part of the
body of the ischium fuses with the pubis and the ilium, forming posterioinferior aspect of
the acetabulum (Fig. 1.3). The ramus of the ischium joins the inferior ramus of the pubis
to form a bar of the bone, the ischiopubis ramus (Fig. 1.2). The rough bony projection at
the junction of the inferior end of the body of the ischium and its ramus is the large ischial
tuberosity (Fig. 1.3). The posterior border of the ischium forms the lower margin of a deep
indentation, the greater sciatich notch (Fig. 1.3). The large triangular ischial spine at the
inferior margin of this notch (Fig. 1.3) separates the greater sciatich notch from a smaller
indentation, the lesser sciatich notch (Fig. 1.3).
The pubis composes the anteromedial part of the hip bone and contributes to the
anterior part of the acetabulum. The pubis is divided into body and two rami, superior
and inferior (Fig. 1.2). Its symphyseal surface unites with the pubis of the opposite side to
form the pubic symphysis (Fig. 1.2). The anterosuperior border of the united bodies and
symphysis forms the crest of pubis (Fig. 1.2). A large aperture in the hip bone is formed
by pubic and ischial rami – obturator foramen (Fig. 1.2).
All three parts of the hip bone join to form the acetabulum. The acetabulum is the
large cup-shaped cavity on the lateral aspect of the hip joint. The margin of acetabulum
is deficient inferiorly at the acetabular notch. The rough depression in the floor of the acetabulum extending superiorly from the acetabular notch is the acetabular fossa (Fig. 1.5).
The bony margin of acetabulum forms ”the inlet plane”. In the erect posture the aceta-
9
1.1 Structure of the hip joint
Head
Neck
Greater trochanter
Lateral condyle
(a)
Medial condyle
(b)
Figure 1.4. The frontal view of the proximal end of the femur showing the neck-shaft
angle, denoted by CCD (a) and the top view of the proximal femur showing the angle of
anteroversion, denoted by AT (b). Adapted from Nordin, 1989.
bular inlet plane is not only inclined laterally and inferiorly but also opened anteriorly
(anteversion).
The femur (Fig. 1.1) is the longest and the heaviest bone in the body. The femur consists
of a body and two ends, superior and inferior. The superior end of the femur consists of
a head, neck and two trochanters, greater and lesser (Figs. 1.2, 1.3). The head of the
femur, smooth, rounded proximal end, projects superomedially and slightly anteriorly. It
has approximately spherical shape [71]. There is a little pit on the top, where the ligament
of the head of the femur and the artery supplying the head of the femur are attached. The
head is attached to the femoral body by the neck of the femur (Fig 1.3).
We observe two angles of the femoral neck relative to the femoral shaft: the neckshaft angle, known as CCD angle (centrum-collum-diaphyseal angle) and the angle of
anterotorsion, known as AT angle (Fig. 1.4). The CCD angle is the angle between the
center line of the femoral neck and the centre line of the femoral shaft (see also Appendix C,
Fig. C.1). The AT angle is the angle between the center line of the femoral neck and the
center line of the flexion of the knee. In the newborn the CCD angle is about 150◦ , at the
age of 18 months is usually diminished to 140◦ and in adults it is about 130◦ [71]. In old
people the CCD angle is further reduced to about 120◦ [46]. The AT angle by delivery is
about 35◦ ; after the delivery it decreases and at the end of the bone growth reaches its
final value, usually about 14◦ [46].
At the junction of the femoral neck and the femoral body there are two large elevations –
the trochanters. The lesser trochanter extends medially from the posteromedial part of
the junction of the neck and body (Fig. 1.2). The greater trochanter projects superiorly
and posteriorly (Fig. 1.2). The site of junction of the femoral head and of the femoral
10
1.1 Structure of the hip joint
Lunate surface
Fatpad in
acetabular fossa
Articular cartilage
Head of femur
Acetabular
labrum
Greater trochanter
Transverse
acetabular
ligament
Ligament of the head
of femur (cut)
Artery to head of
femur
Neck of femur
Fibrous capsule
(cut)
Lesser trochanter
Figure 1.5. Hip joint, lateral view. The hip joint was disarticulated by cutting the ligament of the head of the femur and retracting the head from the acetabulum. Adapted from
Moore, 1995.
body is indicated by the intertrochanteric line that runs from lesser to greater trochanter
(Fig. 1.2). A similar but smoother ridge joins the trochanters posteriorly thereby forming
the intertrochanteric crest (Fig. 1.3).
The femur is smoothly rounded, except for a rough line posteriorly, the linea aspera
(Fig. 1.3). There is a roughened area located on the posterior surface of the femur. This
area is called the gluteal tuberosity (Fig. 1.3).
1.1.2
Articular capsule of the hip joint
The stability of the hip joint is provided by a strong fibrous capsule (Figs. 1.5, 1.6) that
envelopes the joint. An increased depth of the acetabular, that is provided by the fibrocartilagineous acetabular labrum and by the transverse ligament (Fig. 1.6), also contributes
to the stability of the joint. The fibrocartilagineous labrum (Fig. 1.5) attaches to the bony
rim of the acetabulum. Therefore more than half of the head is in contact with acetabular
structure. The acetabular labrum wraps around the head of the femur and holds the head
firmly in the acetabulum (Fig. 1.6).
The strong loose fibrous capsule permits free movement of the hip joint. The fibrous
11
1.1 Structure of the hip joint
Ilium
Fibrous capsule
Acetabular labrum
Articular cartilage
Orbicular zone
Acetabular fossa
Retinacula
Ligament of the
head of femur
Acetabular labrum
Synovial membrane
Retinacula
Figure 1.6. Coronal section of the hip joint. Adapted from Moore, 1995.
capsule is attached proximally to the acetabulum and to the transverse acetabular ligament
and distally to the neck of the femur at the intertrochanteric line and to the base of greater
trochanter only anteriorly (Fig. 1.6, 1.7). Posteriorly, the fibrous capsule is attached to the
neck near the intertrochanteric crest. Most capsular fibers take a spiral course from the hip
bone to the intertrochanteric line. They allow flexion of the hip joint but restrict extension
of the joint to 10◦ – 20◦ beyond the vertical position [53]. Some deep fibers pass circularly
around the neck, forming the orbicular zone. These fibers form a collar around the neck that
constricts the capsule and helps in holding the femoral head in the acetabulum (Fig. 1.6).
Some deep longitudinal fibers of the capsule form retinacula (fibrous bands) (Fig. 1.6) that
join with the periosteum . The retinacula contain retinacular blood vessels that supply the
head and the neck of the femur.
Some parts of the fibrous capsule are thicker than the others and are called ligaments.
The fibrous capsule is reinforced anteriorly by a strong, Y-shaped iliofemoral ligament
(Fig. 1.7), which attaches to the anterior inferior iliac spine and to the acetabular rim
proximally and to the intertrochanteric line distally (Fig. 1.7). The iliofemoral ligament
prevents hyperextension of the hip during standing by pulling the femoral head into the
acetabulum.
The fibrous capsule is reinforced inferiorly and anteriorly by the pubofemoral ligament
12
1.1 Structure of the hip joint
Iliofemoral Y
ligament
Iliofemoral
ligament
Pubofemoral
ligament
Ischiofemoral ligament
Anterior view
Posterior view
Figure 1.7. The ligaments of the hip. Adapted from Hall, 1995.
that originates from the crest of the pubic bone and passes laterally and inferiorly to
merge with fibrous capsule of the hip joint (Fig. 1.7). The pubofemoral ligament prevents
overabduction of the hip joint.
The fibrous capsule is reinforced posteriorly by the ischiofemoral ligament that originates from the ischial part of the acetabular rim and spirals superolaterally to the neck of
the femur and to the base of the greater trochanter. The ischiofemoral ligament prevents
hyperextension of the hip joint.
The ligament of the head of the femur is of minor importance in contributing to the
load of the hip joint [1]. Its wide end attaches to the margins of the acetabular notch and
transverse acetabular ligaments, its narrow end attaches to the pit in the head of the femur
(Fig. 1.5). It usually contains a small artery passing to the head of the femur.
Synovial membrane of the hip joint lines the fibrous capsule and covers the ligament
of the femoral head. It covers also the neck of the femur between the attachment of the
fibrous capsule and the edge of the articular cartilage of the head (Fig. 1.6). The synovial
membrane secretes transparent, alkaline, viscous fluid (synovial fluid) into the joint cavity.
The synovial fluid serves as a lubricant, a shock absorber and a nutrient carrier.
1.1.3
Articular cartilage of the hip joint
The round head of the femur is inserted in the cuplike acetabulum of the hip bone. Joint
cartilage covers both articulating surfaces. The head of the femur is covered by the articular
1.1 Structure of the hip joint
13
cartilage, except for the pit of the femoral head where the ligament of the head of the
femur is attached. The cartilage on the acetabulum is thicker around the periphery of the
acetabulum. The central and inferior part of the acetabulum, the acetabular fossa, is not
covered with the cartilage (Fig. 1.5). The articular cartilage attains a horseshoe shaped
structure called facies lunata (Fig. 1.5).
The part of the acetabular fossa, that is not occupied by the femoral head is filled with
fat and is therefore called the fatpad (Fig. 1.5). The malleable nature of the fatpad permits
it to change the shape during the joint movement. The fatpad is covered by the synovial
membrane.
Morphology and function of the articular cartilage
The articular cartilage is a thin layer of fibrous connective tissue on the articular surfaces
of the synovial bones. Articular cartilage consists of cells (5%) and intercellular matrix
(95%). The intercellular matrix consists of structural macromolecules and tissue fluid and
contains 65–80% of water. In the articular cartilage there are two main types of structural
macromolecules: collagen and proteoglycans. Interaction of these substances, which are
produced by the chondrocytes, determine the mechanical behaviour of the cartilage. The
functions of the articular cartilage are: transfer of the forces between the articular bones,
distribute of the forces in the joint and allow of the joint movements with minimal friction.
The molecule that gives the articular cartilage tensile stiffness and strength is the
insoluble fibrous protein – collagen. Due to its fibrous structure the collagen offers only
little resistance to compression, however its tensile stiffness is high (Fig. 1.8). The collagen
fibers in the cartilage twine and bound with other structural macromolecules to create a
relatively stiff and immobile net.
The proteoglycans are a group of glycoproteins formed of subunits of disaccharides
that are joined into a protein core. The proteoglycans are anchored to the network of the
collagen fibers. Due to their molecular structure they are suited to resist the compressive
forces. The sugars may carry negative electrostatic charge and are attracted to water. The
proteoglycans can hold in vitro up to 50 times of their weight of water. The negatively
charged molecules of disaccharides repel each other and attempt to become as far as possible 1.9. In vivo they are restrained by the collagen mesh and exist in as little as 20% of
their volume they would have if free, the proteoglycans are forced together and repulsive
forces between the like charges are increased. The result is a viscoelastic “gel” surrounding
14
1.1 Structure of the hip joint
(a)
(b)
Figure 1.8. Schematic diagram of the collagen fibers in tension (a) and compression (b).
Adapted from Nigg, 1995.
the collagen fibers. The cells of the cartilage are responsible for synthesis and degradation
of the matrix.
The articular cartilage is structurally heterogenous. It changes with the depth from the
joint surface. The constitutive changes are continuous but may be divided into four zones
(Fig. 1.10).
• the superficial zone is the thinnest superficial region of the articular cartilage. A
surface layer of the superficial zone (lamina splendens) consists of random flat tangential bundles of collagen fibrils. The deeper layer of the superficial zone consists
of the collagen fibers lying parallel to the plane of the joint surface. It is structured
to resists the shear stress that develops during joint motion. This deeper layer also
contains some elongated chondrocytes, which are relatively inactive and whose long
axes lie parallel to the joint surface.
• The transitional zone consists of collagen fibers which are parallel to the joint mo-
tion, but less than those in the superficial zone. The chondrocytes are spherical and
metabolically more active.
15
1.1 Structure of the hip joint
(a)
(b)
Figure 1.9. Proteoglycan rich domain. Unstressed in solution: negtively charged groups
repel each other (a) Compressive stress reduces the volume of the aggregate solution, which
increases the density and increases the repulsive forces. Adapted from Nigg, 1995.
Figure 1.10. Collagen fibers and chondrocyte arrangement in the heterogenous layers of
the articular cartilage. Adapted from Nordin, 1989.
1.1 Structure of the hip joint
16
• The deep zone features large numbers of big collagen bundles running perpendicular
to the plane of the hip joint motion. Proteoglycan contents is highest in the deep
zone. The chondrocytes are round and stacked on the top of each other into a column
perpendicular to the joint surface.
• The calcified zone forms a transition from soft articular cartilage to the stiffer sub-
chondral bone. It is characterized by the presence of hydroxyapatite, an inorganic
constituent of bone matrix which provides rigidity. The calcified zone is separated
from the deep zone by the undulating line –“tidemark” (Fig 1.10). The collagen fibers
from the deep zone anchor the cartilage to the bone by anchoring themselves into
the subchondral bone.
1.1.4
Movements within the hip joint
When we spoke about movements involving the hip, we have to consider not only the
movement of the femur but also the movement of the pelvis. The movements of the femur
are primarily due to the rotation occurring at the hip joint. The function of the pelvis is in
positioning the hip joint for effective limb movement. The pelvis can rotate in all directions
that can be described in three planes of movement [23].
The hip movements are described as flexion – extension, abduction – adduction, medial
– lateral rotation and circumduction. The active movement of the hip joint is performed
by a number of muscles. Bellow we describe particular muscles and their functions about
the hip.
Muscles of the hip
Muscles, which cross the hip, allow the locomotion of the body and also contribute to its
stability. The locations of the attachments and the functions of the muscles of the hip are
summarized in Table 1.1.
Flexion and extension
The major flexors of the hip are iliacus and psoas major1 (Fig. 1.11). But also other muscles
are active in flexion [23] : rectus femoris,tensor fascia latae, sartorius and pectineus . These
1
known as iliopsoas because of their common attachment to the femur
17
1.1 Structure of the hip joint
Muscle
Rectus femoris
Iliopsoas
(Iliacus)
(Psoas)
Sartorius
Pectineus
Proximal
attach- Distal attachment
ment
Anterior inferior iliac Patella
spine
Function about
the hip
Flexion
Iliac fossa and adja- Lesser trochanter
cent sacrum
12th thoracic and lum- Lesser trochanter
bal vertebrae and lumbal disc
Anterior superior iliac Upper medial tibia
spine
Flexion
Pectineal crest of pubic ramus
Tensor fascia la- Crest of ilium
tae
Gluteus maximus
Gluteus medius
Gluteus minimus
Gracilis
Adductor magnus
Adductor longus
Adductor brevis
Posterior ilium, iliac
crest, sacrum and coccyx
Between posterior and
anterior gluteal lines
on the posterior ilium
Between anterior and
inferior gluteal lines on
the posterior ilium
Anterior,
inferior
symphysis pubis
Inferior ramus of pubis
and ischium
Anterior pubis – below
pubis crest
Flexion
Assists with flexion, abduction
and lateral rotation
Medial femur
Flexion, abduction and medial
rotation
Iliotibial band
Assists with flexion, abduction
and medial rotation
Gluteal tuberosity of Extension and lathe femur and iliotibial teral rotation
band
Greater trochanter
Abduction, medial rotation
Anterior surface of the Abduction, megreater trochanter
dial rotation
Medial, proximal tibia
Adduction
Entire linea aspera
Adduction, lateral rotation
Adduction, lateral rotation, assists with flexion
Adduction, lateral rotation
Middle linea aspera
Inferior ramus of the Upper linea aspera
pubis
Table 1.1. Muscles of the hip. Adapted from Hall, 1995
18
1.1 Structure of the hip joint
Psoas
minor
Ala of
ilium
Iliacus
Psoas
major
Anterior view
Figure 1.11. The iliopsoas complex. Adapted from Hall, 1995.
Muscle
Proximal
ment
attach- Distal attachment
Hamstrings
Semitendinosus
Medial ischial
berosity
Semimembranosus Lateral ischial
berosity
Biceps femoris
Lateral ischial
(long head)
berosity
Function about
the hip
tu- Proximal, medial tibia
Extension
tu- Proximal, medial tibia
Extension
tu- Posterior lateral con- Extension of thigh
dyle of the tibia, head
of the fibula
The six outward sacrum ilium and is- Posterior greater tro- Outward rotation
rotators
chium
chanter
Table 1.1. continued Muscles of the hip. Adapted from Hall, 1995.
muscles are shown in figure 1.12. Because the rectus femoris is a two-joint muscle (Fig. 1.12)
and is active during the hip flexion and knee extension, it functions more effectively as a
hip flexor when the knee is in extension.
The hip extensors are: gluteus maximus (Fig. 1.14), biceps femoris, semimembranosus,
and semitendinosus. The muscles biceps femoris, semimembranosus and semitendinosus
are called the hamstrings (Fig. 1.13). The gluteus maximus is active only when the hip is
in extension [23]. The hamstrings are two-joint muscles that contribute to the extension of
the hip and flexion of the knee. During the extension the fibrous capsule is tight, therefore
the hip can be extended only slightly beyond vertical. The degree of flexion depends on
the position of the knee. When the knee is flexed and the hamstrings are relaxed, the thigh
19
1.1 Structure of the hip joint
Tensor
fasciae
latae
Pectineus
Sartorius
Rectus
femoris
Anterior view
Figure 1.12. Assistant flexor muscles of the hip. Adapted from Hall, 1995.
Semimembranosus
Semitendinosus
Biceps
femoris
Posterior view
Figure 1.13. The hamstrings, posterior view. Adapted from Hall, 1995.
can be moved toward the abdominal wall.
Abduction and adduction
The gluteus medius (Fig. 1.14) is the major abductor acting in the hip with the gluteus
minimus (Fig. 1.14) assisting. These muscles stabilize the pelvis during the support phase
of walking and running and in the one-legged stance [1]. Abduction is important in the
one-legged stance that is the representative body position considered in this work and will
be mentioned later in more details (Section 1.2.2).
The hip adductors are the muscles that cross the joint medially: adductor longus, adductor brevis, adductor magnus, and gracilis (Figure 1.15). These muscles also contribute
20
1.1 Structure of the hip joint
Gluteus
maximus
Gluteus
minimus
Gluteus
medius
Posterior view
Figure 1.14. The three gluteal muscles, posterior view. Adapted from Hall, 1995.
Adductor
magnus
Adductor
longus
Adductor
brevis
Gracilis
Anterior view
Figure 1.15. Adductor muscles of the hip. Adapted from Hall, 1995.
to the flexion and internal rotation of the hip, particularly when the femur is externally
rotated [23].
Medial and lateral rotation
Although several muscles contribute to the lateral rotation of the femur, there are six
muscles acting solely as lateral rotators. These are : piriformis, gemellus superior, gemellus inferior, obturator internus, obturator externus, and quadratus femoris (not shown).
Outward rotation of the femur occurs in walking to enable the rotation of the pelvis.
The major medial rotator of the femur is the gluteus minimus with the assistance of the
tensor fascia latae, semitendinosus, semimembranosus, and the four adductor muscles. The
1.1 Structure of the hip joint
21
medial rotation of the femur usually does not require substantial muscle force, therefore
the medial rotators are weak in comparison to the lateral rotators.
1.2 Loads on the hip
1.2
22
Loads on the hip
In case of the hip joint, the load is the vector sum of all forces transmitted from the
acetabulum to the head of the femur [46]. We will call this vector sum the resultant hip
~ The load on the hip is expressed in newtons or in body weight units. The load
force R.
can be determined by direct measurements or by external measurements combined with
mathematical models.
1.2.1
Direct measurements
The direct in vivo measurements are performed by implanted instrumented prostheses [3, 4,
6, 61]. The electronic measuring device is integrated into the neck of hip endoprosthesis for
in vivo measurement. Bergmann et al. [4] used electronic measuring device which consists
of a multichannel amplifier with the inductive power supply, a component for wireless
signal transduction from inside the body and strain senzors mounted in the neck of the
prosthesis. From the signals caused by strain in the sensors, the magnitude and direction of
joint force acting on the prosthesis head could be measured by using a complex calibration
procedure [3]. The load of the hip during walking, standing, stair climbing and during other
activities was measured [4].
Instrumented implants potentially offer the most accurate information, but they are
technically complex and offer no direct benefit to the patient [7]. The measurements could
be performed only in the patients with implanted instrumented endoprostheses. The values
from these measurements may not be equal to the values of the forces present in normal
joint [12] due to different properties of endoprosthesis as compared by normal hip. Studies
in several laboratories over many years have resulted in only one or two implantations by
each group and all have encountered many technical problems [7]. However data from these
studies can be used to validate the theoretical models, as shown by Brand et al. [7].
1.2.2
External measurements combined with mathematical model
The major advantage of mathematical modelling is a possibility to study large groups of
subjects with intact hips. Mathematical models of the hip load use some simplifications as
described below and therefore the possibility of mistake is larger than by direct measurements. The input into these models are the geometrical parameters of the body (usually
23
1.2 Loads on the hip
obtained from radiographs) [30, 46, 57], the motion of the body and measured external
reaction forces [6].
By biomechanical modelling of the human motion, the human body is usually divided
into segments connected by joints. One segment is considered as a rigid body with given
mass, center of gravity and moment of inertia. The movement of the segment is caused
by external forces acting on it. The forces acting on the segment are the weight of the
~ and the forces exerted by other segments (i.e. the intersegmental forces F~I ).
segment W
An intersegmental force is a vector sum of all forces acting on the joint surface and all
forces exerted by anatomic structures (i.e. muscles and ligaments). The origin point of the
intersegmental force is taken at the connections between the segments (Fig. 1.16).
The forces acting on the particular segment may cause rotation of this segment but
also rotation of other segments. Rotation of the segment is described by its rotational
moment. The transfer of the rotational moment to other segments is described by interseg~ I . The motion of the segment could be described by Newton’s
mental rotational moments M
equations of motion.
~ +
W
X
F~Ii = m ~a
(1.1)
i
(~rCG
~ ) + (~rCG × m ~a) +
× W
X
i
(~rCIi × F~Ii ) +
X
M~I i = J α
~
(1.2)
i
where m is the mass of the body, ~a is the acceleration of the body segment, J is the moment
of inertia of the segment, α
~ is the angular acceleration, ~rCG is the positional vector from
the center of gravity of the segment to the center of rotation and ~rCI is positional vector
from the center of rotation to the origin of the intersegmental force (Fig 1.16). The index i
runs over all the segments that act on the given segment. From this point of view, human
body is acting as a system of levers.
The intersegmental force between two segments could be expressed as a sum of force
~ and the force exerted by the muscles F~ [6] if other forces are neglected
acting in the joint R
(e.g. forces of ligaments)
~ + F~
F~I = R
(1.3)
~ the intersegmental forces and the forces of
To determine the joint resultant force R,
the muscles must be known. If we include all the forces acting on the segment, the number
of unknown quantities would be higher than the number of equations and we could not
solve this problem. Therefore we add some basic assumptions to simplify the problem. For
24
1.2 Loads on the hip
Figure 1.16. Schematic figure of the forces acting on the body segment, center of rotation
is denoted by C, center of gravity is denoted by G.
example, we can simplify the situation by describing stationary position of the body. Such
~ are equal
models are called statical. Then the acceleration ~a and the angular acceleration α
to zero. In such case the equilibrium of the forces and rotational moments must be fulfilled
for all the segments.
In the contrast to the statical model, the dynamical models allow us to calculate the
dynamic load. Dynamical models are more complex and were developed later.
Statical models of the hip load
Basic assumption included in these models is that the body is in static state of equilibrium.
We can also use these models to estimate load swing phases of gait in which acceleration
is small. Many authors [5, 27, 41, 46, 57, 71] choosen as the most appropriate position, to
assess the load, the stance phase of gait, the one-legged stance (Fig. 1.17) because it also
represents the most frequent body position in everyday activities. The stance phase of gait
corresponds to phase 16 of gait cycle phases after Fisher [43] (Fig. 1.21).
In the one-legged stance the body could be divided into two segments. The first segment
is the loaded leg and the second segment is the rest of the body. In this position, the hip
on the supporting side bears the partial body weight WB − WL (body weight minus the
weight of the loaded leg)(Fig. 1.18). Assuming the equilibrium of the forces (Fig. 1.18) for
the first or for the second segment (Eqn. 1.1) we obtain
~B −W
~ L + F~I = 0
W
(1.4)
25
1.2 Loads on the hip
Figure 1.17. Phases of gait. Stance phase of gait is denoted by arrow. The loaded leg is
marked by shading.
Figure 1.18. Forces acting on the segments of the body in the one-legged stance. Forces
acting on the second segment are shown.
where F~I is intersegmental force exerted by the first segment on the second segment.
In the second segment, the intersegmental force is the sum of the hip joint reaction
~ and the resultant force of the muscles F~ acting on the segment (Eqn. 1.3). It is
force R
~ lies in the center of the femoral head
taken that the origin of the hip joint reaction force R
[6, 27, 46, 57]. According to the action-reaction law, the femoral head exerts an opposite
~ on the acetabulum. Inserting equation (1.4) into equation (1.3) yields:
force −R
~B −W
~ L) − R
~ =0
F~ + (W
(1.5)
The situation regarding the forces in a static one-legged stance is depicted in figure 1.19.
The muscle force acting on the segment F~ is the vector sum of the forces of all the muscles F~i
that are active in the particular body position
X
F~ =
F~i
i
(1.6)
26
1.2 Loads on the hip
T'
Figure 1.19. The forces acting on the hip in the one-legged stance after Pauwels [57].
~ – σ, ϑR , respectively and magnitudes of the levers of
The inclinations of the forces F~ , R
~ L ), F~ – k, r respectively are depicted. The virtual site of the insertion
~B −W
the forces (W
of the muscle force is denoted by T 0 .
where i runs over over all the muscles. Considering the equation (1.5), the equation (1.6)
is written as:
X
i
~B −W
~ L) − R
~ =0
F~i + (W
(1.7)
In the static state of equilibrium the acceleration ~a and the angular acceleration α
~ are
equal to zero, therefore the sums of the forces and rotational moments for both segments
are equal to zero (Eqns. 1.1, 1.2). In our two-segment system, it is taken that rotation
of the segments occurs with respect to the axis through the center of the femoral head.
Therefore the center of the femoral head was chosen for the origin of the coordinate system.
The coordinate system was chosen so that y = 0 plane lies in the frontal plane of the body
and x = 0 plane lies in the sagittal plane of the body (Fig. 1.20). The origin of the
intersegmental forces is in the center of the femoral head (center of rotation), therefore the
intersegmental forces do not produce rotational moments (Fig. 1.18). Considering this in
the equation (1.2) for the second segment (the upper part of the body and free leg), we
can see that the second and the third term on the left side of the equation (1.2) as well as
27
1.2 Loads on the hip
the term on the right side of the equation (1.2) are equal to zero,
~k × (W
~B −W
~ L) + M
~I = 0
(1.8)
where ~k is the positional vector from the center of rotation (center of the femoral head) to
~ I is the intersegmental rotational moment
the center of gravity of the second segment and M
that is needed to maintain balance. In the one-legged stance the equilibrium is ensured by
the activity of the muscles. It is taken that the intersegmental rotational moment is the
sum of the rotational moments of the muscle forces.
~I =
M
X
i
~ri × F~i
(1.9)
where ~ri is the positional vector from the center of rotation to the origin of the muscle
force of the i-th muscle and F~i is the corresponding muscle force. The index i runs over
all muscles that are active in the one-legged stance. Inserting the equation (1.9) into the
equation (1.8) yields:
~k × (W
~B −W
~ L) +
X
(~ri × F~i ) = 0
(1.10)
i
The body weight and the muscle forces act as a system of levers with the center of rotation
in the femoral head.
~B −W
~ L ) is directed
It was estimated [27, 46, 57] that weight of the second segment (W
inward vertically and that its origin lies in the frontal plane.
~B −W
~ L = (0, 0, −(WB − WL ))
W
(1.11)
The center of gravity is shifted toward the nonsupporting side. The magnitude of the
lever arm of the weight of the second segment of the body ~k (Fig. 1.19)
~k = (k, 0, 0)
(1.12)
could be calculated from the geometrical parameters of the body.
It was estimated by Legal [46] that:
k = 0.6 l
(1.13)
where l is the interhip distance (the distance between the centers of the femoral heads). The
intersegmental rotational moment exerted by the first segment on the second segment is
28
1.2 Loads on the hip
of the same magnitude and opposite direction than the intersegmental rotational moment
exerted by the second segment on the first segment. The equilibrium of the rotational
moments of the first and of the second segment yield:
WB c − WL b − MI = 0
(1.14)
−(WB − WL ) k + MI = 0
(1.15)
~ L and c is the x-coordinate
where b is the x-coordinate of the weight of the loaded leg W
~ B (Figure 1.20). Using the equations (1.14)
of the origin of the ground reaction force −W
and (1.15) gives for the lever arm of the partial body weight [52]:
k=
WB c − WL b
WB − WL
(1.16)
The parameters b and c could be expressed by the interhip distance [52].
b = 0.24 l
(1.17)
c = 0.50 l
(1.18)
The weight of the leg could be approximated by the equation [11]:
WL = 0.161 WB
(1.19)
The levers of the muscle forces ~ri and the directions of the muscle forces could be calculated
directly from the geometrical parameters of the body [6, 30, 43] as shown below. So if we
~B −W
~ L and
~ we have to know the force W
want to determine the resultant hip force R,
the resultant muscle force F~ . We have six scalar equations given by three components of
the vector equations (1.7) and (1.10). If all the muscles were included into our equations,
the number of unknown quantities would be much higher than the number of equations.
Mathematically, an infinite number of solutions would satisfy the system of equations.
We can solve this problem by reducing the number of unknowns (reduction method) or
by adding new criteria for muscle forces (optimization method). These methods will be
discussed below.
The aim of the reduction method is to modify an initially indeterminate problem to a
determinate one by reducing the number of unknowns. Pauwels [57] simplified the problem
by taking that all forces that act on the hip lie in the frontal plane (two dimensional
model). Reducing the problem to a two dimension, two equations for two components of
29
1.2 Loads on the hip
~ (Eqns. 1.20, 1.21) and one scalar equation for one component of the
the forces F~ and R
~ is expressed by
rotational moments (Eqn. 1.22) can be written. The resultant joint force R
its magnitude R and by its angle of inclination with respect to the vertical ϑR (Fig. 3.4).
In the Pauwels model a force of one effective muscle F~ is considered. This force represents
the forces of all the muscles (Fig. 1.19). The inclination of the force F~ with respect to the
horizontal is denoted by σ (Fig. 1.19). The virtual site of the insertion of the muscle force
F~ was taken at the junction of the superolateral border of the greater trochanter with it
upper horizontal part [46] (in figure 1.19 denoted by T 0 ). The distance perpendicular to
the muscle force F~ from the virtual site of the insertion of the muscle force to the center
of the femoral head (r) was determined from the position of the greater trochanter in the
radiograph (Fig. 1.19). After determining σ and r from the radiograph and estimating k
three scalar equations with three unknown quantities F, R, ϑR are obtained.
R cos ϑR = WB − WL + F sin σ
(1.20)
R sin ϑR = F cos σ
(1.21)
r F = k (WB − WL )
(1.22)
The angle σ was taken to be 69◦ [57]. The model of Pauwels was used by a number of
authors [5, 22, 41, 46, 51] without or with little changes, as it is very simple. The position
of the greater trochanter is reflected in the magnitude and direction of the the hip joint
reaction force. However, the model is restricted to two dimensions and it does not take into
account other geometrical parameters of the pelvis and femur. Therefore it is unsuitable
for simulating some of the osteotomies. It also does not provide accurate description of the
muscle force because only one muscle forces is included.
Williams & Svensson [71] developed a three dimensional static model. They included
seven muscles and ligaments. Muscles and ligaments were combined into three main muscle
groups according to similar effect. Within each muscle group the direction of the respective
effective muscle and its attachment point was described. In three dimensions there are six
equations (Eqns. 1.7, 1.10) and six unknowns – the three components of the hip joint
~ and the magnitudes of the three effective forces corresponding to the
reaction force R
three muscle groups. By considering the equilibrium of the pelvis and the femur in three
dimensions they obtained solutions for the effective forces of the three muscle groups and
the hip joint reaction force. This model provides a better approximation of the hip joint
reaction force, however due to the simplification (muscle groups rather than individual
30
1.2 Loads on the hip
muscles) it still does not provide an accurate estimation of the muscle forces. The direction
of the muscle forces are fixed, therefore this model could not be easily adapted for different
geometrical parameters of the pelvis. It could not be used for calculation of the change of
the hip joint reaction force by different operative changes of the hip geometry.
In our work we used a three dimensional model in one-legged stance [27]. Accordingly,
we will describe this model in details. The basic assumptions of this model are similar to
the assumptions of the models above (Eqns. 1.7, 1.10).
In this model five muscles are included: piriformis, gluteus medius, gluteus minimus,
rectus femoris and tensor fasciae latae. Since gluteus medius and gluteus minimus are
attached to the pelvis over a rather large area, each of these two muscles is divided into
three parts as presented in Table 1.2. Therefore, there are nine effective muscles in the
model.
The muscles are considered to act in straight line in the direction determined by the
line connecting the points of attachments. Each muscle has one proximal attachment point
and one distal attachment point. We can describe these points by their position vectors:
~ri for the proximal attachment point and ~ri 0 for the distal attachment point. From the
position of the muscle attachment points, the direction of the force of the muscle, given by
the unit vector ~si , is calculated.
~si =
~ri 0 − ~ri
|~ri 0 − ~ri |
(1.23)
The muscle force F~i can be approximately written by the following vector expression [27]:
F~i = fi Ai ~si
i = 1, . . . , 9
(1.24)
where Ai is the relative cross-sectional area of the i-th muscle, fi is the average tension in the
i-th muscle and ~si is the unit vector in the direction of the the i-th muscle. The attachments
points of the muscles are obtained by correcting the reference attachment points according
to data obtained from the standard antero-posterior radiograph. The reference attachment
points are given in the work of Dostal and Andrews [16]. The rotation of the pelvis in the
frontal plane is described by the angle ϕ (Fig. 1.20) while the rotation of femur around the
axis throw the femoral head is described by the angle ν (Fig. 1.20).
We have twelve scalar unknowns (three components of the hip joint reaction forces and
nine magnitudes of the forces of the muscles) but only six equations (Eqns. 1.10, 1.7).
This problem was solved using the reduction method. The muscles were divided into three
1.2 Loads on the hip
Figure 1.20. The model of one-legged stance. After Iglič et al. , 1990.
31
32
1.2 Loads on the hip
Muscle
Group i
gluteus medius-anterior
a
1
gluteus minimus-anterior
a
2
tensor fasciae latae
a
3
rectus femoris
a
4
gluteus medius-middle
t
5
gluteus minimus-middle
t
6
gluteus medius-posterior
p
7
gluteus minimus-posterior
p
8
gluteus piriformis
p
9
F̃i
F~1
F~2
F~3
F~4
F~5
F~6
F~7
F~8
F~9
Ai
0.266
0.133
0.120
0.400
0.266
0.113
0.266
0.113
0.100
fi
fa
fa
fa
fa
ft
ft
fp
fp
fp
Table 1.2. The muscles included in the model of Iglič et al., 1990 are divided into three
groups according to their muscle attachment point positions relative to the frontal plane
of the body : a (anterior), t (middle), p (posterior). The symbol F~i denotes the i-th muscle
force, Ai the relative cross-sectional area of i-th muscle and fi the average muscle tension
in i-th muscle.
groups – the anterior (a), the middle (m) and the posterior (p) group (Table 1.2). It
was assumed that the average tension fi
i = a, m, p is the same for the muscles of the
particular muscle group. The rotation of the pelvis in the frontal plane, described by angle
ϕ, and the rotation of the femur in the frontal plane, described by angle ν, is taken into
account (Fig. 1.20). For the one-legged stance the value of ϕ is taken to be 0.5◦ [27] while
the angle ν is determined by the equations (Fig. 1.20):
sin ν =
b
x0
(1.25)
where the x0 is the length of the femur. The equilibrium equations (Eqns. 1.7 and 1.10) are
i = a, m, p and Ri i = 1, 2, 3, are obtained.
~ lies nearby the frontal plane [27, 64].
The results show that the hip joint reaction force R
then solved and the unknown quantities fi
This model allows adjustment of the muscle attachment points and therefore the muscle
directions for each patient are determined individually. The input into this model are
geometrical parameters of the pelvis and proximal femur obtained from standard AP radiograph (see section 3.2). The attachment points of the muscles are then computed by
correcting the standard geometry of the pelvis [27]. This model could be used for simulation of the biomechanical effect of acetabular osteotomies [27, 28, 30] and of osteotomies
of the femur [29].
Another procedure of solving the problem of unknown muscle forces acting on the joint
33
1.2 Loads on the hip
Author
Year Value of R/WB
Rydell
1966
2.5
Bergmann
1994
3
Pauwels
1976
4
Williams & Swensson 1968
6
Bombelli
1983
3.7
Legal
1987
3.08
Iglič
1993
2.38
Maček-Lebar
1993
2.37
Method
direct measurement
direct measurement
reduction, 2D
reduction, 3D
reduction, 2D
reduction, 2D
reduction, 3D
optimization, 3D
Table 1.3. Predicted values of the magnitude of the hip joint reaction force R with respect
to body weight WB by one-legged stance.
is the approach called optimization method. This technique allows an unique solution by
requiring that the solution not only satisfies the equations of motion but also satisfies
some more or less arbitrary “optimization” (minimization or maximization) criteria [6].
The Newton’s equations of motion (Eqns. 1.1, 1.2) are iteratively solved until a solution
is found. This solution fulfills the optimization criteria. The problem is how to choose the
optimization criteria [7, 13]. We do not know a priori the algorithm that brain uses to
select muscles so the optimization criteria should be based on some physiological principle
[13] that could be expressed by mathematical function – an optimization function.
The optimization functions can be divided into linear and non-linear functions. The
linear optimization function is the sum of linear decision variables, which can be muscle
forces or average muscle stresses. The average muscle stress is defined as the muscle force
divided by the cross-sectional area. These linear optimization functions are for example:
minimization of the sum of the muscle forces, minimization of the sum of the muscle
stresses, minimization of the neuromuscular activation, minimization of the energy [40, 59,
62].
The second type of the optimization functions is more general, it incorporates decision
variables raised to some power and also the products of the decision variables. Non-linear
formulations are not so easily solved. These non-linear minimization functions are for example: minimization of the sum of the muscle forces raised to power two, maximization of
the time of the muscle contraction, maximization of the endurance time of the muscular
contraction [12, 17, 47].
The model mentioned above [27] could not describe the motion of the body in the
sagittal plane because in such case it would involves negative value of some muscle for-
1.2 Loads on the hip
Fa /WB
Ft /WB
Fp /WB
R/WB
34
Optimization Without optimization
0.809
0.458
0.450
1.041
0.363
0.098
2.370
2.383
Table 1.4. Comparison of the values of the magnitudes of the muscle groups forces
Fa , Fp , Ft and the magnitude of the hip joint reaction force R with respect to body weight
WB determined by optimization method or without optimization method.
Figure 1.21. The hip joint resultant force and its components by gait. Adapted from
Pedersen etal., 1999
ces. However, this model could be improved by using the optimization method [47, 48].
The problem of the one-legged stance was solved by using a non-linear optimization function [47]. It was assumed that from mechanical point of view the physiological criterium
is minimal bone loading. So the magnitude of the hip joint reaction force R is taken
as the objective function in the numerical optimization. Nine unknowns muscle tensions
fi
i = 1, . . . , 9 are achieved by numerical minimization of the hip joint reaction force.
Comparison of the values of the muscle forces and hip joint reaction force determined by
using the optimization method or without optimization method are shown in table 1.4. We
can see good agreement of the magnitudes of the resultant hip force, the magnitudes of
the forces in particular muscle groups differs considerably.
1.2 Loads on the hip
35
Dynamical models
Dynamical models allow to calculate the forces in the hip joint during dynamical load.
It means that the acceleration (Eqn. 1.1) and the angular acceleration (Eqn. 1.2) of the
body segments differs from zero. By dynamical models we have to take into account the
displacement history or the motion of the body segments (kinematics), the external reaction
or foot-floor forces (kinetics), and the body segment inertial properties (i.e., the mass, the
location of the center of the gravity and the moment of inertia) to solve the Newton’s
equations of motion (Eqns. 1.1, 1.2). The problem of determination of the intersegmental
forces and moments is called the inverse dynamics problem.
The dynamical model of the gait is well elaborated [6, 7, 12, 58]. The algorithm of the
method is shown in figure 1.22. The pelvis and the involved lower extremity were modeled
as four segments: the pelvis, the thigh , the shank and the foot-plus-shoe segment. These
four segments were considered to be connected by smooth ball-socket joints. By using
the data from kinematics, kinetics of the segments and data of inertial properties of the
segments the intersegmental resultant forces and rotational moments could be determined.
To further determine the hip joint reaction forces, the knowledge of the muscle forces is
necessary (Eqn. 1.3).
For computation of muscle forces a 47-element straight-line three dimensional muscle
model was used. The magnitudes of the muscle forces were obtained by using the optimization technique. The inputs into this model were: the foot-floor reaction force, the movement
history of the body segments and the physical properties of body segments. The foot-floor
reaction force was recorded by using a piezoelectric force plate. Triads of light-emmitting
diods (LED) were attached to the pelvis, thigh and shank of the subject while the motion of
the body segment was recorded with biplanar photography using a videosystem. From the
obtained data, the accelerations and the velocities of the segments of the lower limb were
computed. The hip joint reaction force was determined. The magnitude of the hip joint
reactant force during gait is shown in the figure 1.21. Crownishield et al. [12] investigated
also other activities: climbing and descending stairs, raising from the sitting position, etc.
36
1.2 Loads on the hip
Kinematics
Kinetics
Inertial properties
Equations of motion
˜
˜˜ XXXX
XX
˜˜˜
Intersegmental resultant
Intersegmental resultant
moments
forces
Optimization criteria
Muscle model
Optimization algorithm
Muscle forces
Joint forces
Figure 1.22. Plan for the calculation of muscles and resultant forces. Adapted from
Brand et al., 1994
1.3 Stress in the hip joint articular surface
1.3
37
Stress in the hip joint articular surface
Besides the hip joint force that is described as acting at one point on the femoral head, we
are interested in the distribution of the forces, i.e. in stresses acting in the hip.
We can discriminate between the tensile stress, the shear stress and the compressive
stress. In this work we do not consider the tensile stresses and the shear stresses that
arise in the proximal femur. Further, the shear stress in the hip joint articular surface due
to friction can be neglected for smooth, well lubricated femoral and acetabular articular
surfaces which are spherical and congruent [9, 43]. It means that the articular cartilage
behaves “hydrostatically”, so the forces transmitted across the articular surface are all
normal. Therefore only normal (radial) stress is considered [21, 30, 43]. This compressive
stress is denoted as contact stress p. The contact stress is expressed in pascals (Pa). The
area of the hip where the contact stress differs from zero is called the weight-bearing area.
The value of the contact stress can be estimated by direct measurements or by some
external measurements in combination with mathematical models.
1.3.1
Direct measurements
Direct measurement of the stress are performed in a similar manner as direct measurement
of the load by instrumented endoprostheses. In the direct measurement of the force (section 1.2.1) the load is determined through measured strain within the sensors in the femoral
neck. To measure the contact stress the sensors must be mounted in the articular surface of
the endoprosthesis. Measurement of the hip force refers to the total load on the prosthesis,
whereas the measurement of the stress samples discrete regions of the prosthesis. Thereby
the the local contact stress between the prosthesis and the acetabular cartilage is determined. Carlson et al. [10] developed radio telemetry device which can monitor the magnitude
and distribution of the stress generated between the cartilage of the acetabulum and the
surface of the hip prosthesis. The stress distribution is detected by an array of 14 sensors
integrated in surface of the ball of the prosthesis.
Such prosthesis was implanted in a patient [26]. Stress was measured in walking,
climbing the stairs, rising the chair and during other activities during the rehabilitation
(Tab. 1.5). During gait, the highest measured pressures were located in the superior and
posterior part of the acetabulum.
Advantages and disadvantages of the direct measurements of stress are the same as in
1.3 Stress in the hip joint articular surface
38
~ perpendicular upon the same weight-bearing area A
Figure 1.23. The same force R
cause the different normal stress if different distribution of the stress is considered, (a)
homogeneous distribution of the stress, (b) heterogeneous distribution of the stress.
measurements of the load, as previously described.
1.3.2
External measurements combined with mathematical models
A non-invasive method used to estimate the contact stress in the hip is mathematical
modelling. Stress in the hip depends on several factors that should be included into the
~ acting on the hip, the
model: the magnitude and direction of the hip joint reaction force R
the weight bearing area of the joint and the distribution of the forces on this area.
It is usually assumed that the articular surfaces of the femoral head and the acetabulum
are spherical and are arranged concentrically, when unloaded. The radius of the articular
surface is taken to be the mean of the radii of the articular surfaces of the femoral head
and the acetabulum.
The size of the weight-bearing area is a critical factor : the smaller the weight-bearing
area, the greater stress [57]. The loads on the hip, the geometrical parameters of the acetabulum and the femoral head (i.e. coverage of the femoral head) and the mechanical
properties of the articular cartilage determine the weight bearing area [46]. Geometrical
parameters of the acetabulum and the femoral head could be obtained by external measurements from radiographs [21, 31, 45].
When we compute stress in the hip joint, we assume that the hip joint reaction force is
~ = (Rx , Ry , Rz ). If it is assumed that the hip joint reaction force lies in the frontal
known, R
~ by its magnitude and by its inclination with respect to vertical ϑR .
plane, we can define R
~ is inclined medially with respect to the vertical (Fig. 3.4).
The angle ϑR is positive if R
Compressive stress is defined as uniformly distributed force per area A [25]. The direction of the force coincides with the normal to A [25]. However we must take into account
that the normal to the weight-bearing area of the hip joint A is not everywhere parallel to
39
1.3 Stress in the hip joint articular surface
z
z
R
R
x
(a)
(b)
x
~ on the hemispheric surface according to Legal, 1987. If
Figure 1.24. Action of the force R
the force is directed to the top of the hemisphere, the distribution of stress is homogenous
(a). If the force is eccentric, stress raises toward the margin (b).
~ Therefore, the hip joint reaction force must be resolved into a
the hip joint reaction force R.
~ that are perpendicular to the area elements dA. Stress acting upon
set of partial forces dR
the segment of the weight-bearing area dA integrated over the area A gives the resultant
~
hip force R.
Z
~ = R
~
p dA
(1.26)
A
~ = ~n dA
dA
(1.27)
where ~n is unit vector parallel to the normal to dA. If we want to determine the stress
distribution, we should add some basic assumptions about the stress distribution function
on the weight-bearing area. Otherwise, infinite number of solutions of stress distribution
could satisfy the equation 1.26 (Fig. 1.23).
The simplest model considers a homogeneous distribution of the stress over the weightbearing area [43]. Let us assume that the weight-bearing area is a hemisphere (Fig. 1.24).
The coordinate system is chosen so that z = 0 plane is identical with the base of the
hemisphere, center of the coordinate system coincides with the center of the hemisphere and
~ points into the center
z-axis is pointing toward the hemisphere. We assume that the force R
~ = (0, 0, −R) and that the normal stress p is homogenously
of the hemisphere so that R
distributed over the weight bearing area. In the spherical coordinates (r, ϑ, ϕ), the partial
40
1.3 Stress in the hip joint articular surface
Figure 1.25. Effective homogeneous stress distribution. Adapted from Legal, 1987.
~ is expressed as
force dR
~
~ = p dA
dR
(1.28)
~ = −(sin ϑ cos φ, sin ϑ sin φ, cos ϑ) r2 sin ϑ dϑ dφ
dA
(1.29)
where
r is the radius of the hemisphere and the unit vector ~n (Eqn. 1.27) is oriented as to point
into the center of the hemisphere. Considering equation (1.26), z-component of the force
R is
Rz = −
Z Z
p cos ϑ r2 sin ϑ dϑ dφ
(1.30)
The integration is performed ever the weight-bearing area (ϑ ∈ [0, π/2], ϕ ∈ [0, 2π]) and
the value of stress p can be expressed as:
p=
R
π r2
(1.31)
In our case π r2 is a projection of the weight-bearing area onto the plane perpendicular
~ If the weight-bearing area is considered as a section of sphere and if the hip joint
to R.
~ is centered over this section of sphere, then it can be shown that for
reaction force R
homogeneously distributed stress
R
(1.32)
S
where S is size of the projection of the weight-bearing area onto the plane perpendicular
~
to R.
p=
~ is eccentric with respect to a hemisphere, stress increases toward the edge of the
If R
~ [57](Fig. 1.24). It is possible to
surface that is closest to the inclination of the resultant R
41
1.3 Stress in the hip joint articular surface
-R
CE
R
Figure 1.26. Schematic diagram of the spherical segment used to calculate effective stress
and its projection onto a plane perpendicular to R (the area used to to calculate maximum
pressure). Adapted from Legal et al., 1977.
use the assumption of the homogeneous distribution of stress if we replace the actual stress
distribution over the weight-bearing area with homogeneous distribution over a spherical
~ (Fig. 1.25) [46]. Such spindle-shaped sphesegment that is symmetrical with respect to R
rical segment is bounded by two planes. The lateral plane is the plane of the acetabular
margin which is determined by ϑCE angle (center-edge angle of Wiberg, Appendix C). The
angle θ between the force R (Fig. 1.26) and this plane equals
θ = ϑCE + ϑR
(1.33)
The medial border of this segment is the plane that is inclined for an angle θ with respect to
~ in the medial direction (Fig. 1.26). The surface area used to calculate the effective stress
R
is obtained by projecting the spindle-shaped spherical segment onto a plane perpendicular
~ (Fig. 1.26). The projection is an ellipse, so that the effective stress peff is according
to R
to equation (1.32)
R
(1.34)
π sin θ
The assumption of homogenous distribution of stress is not realistic. Studies of radiogpeff =
r2
raphic bone density indicates that stress in the hip joint increases toward the acetabular
margin [57]. Therefore the model of linear pressure rise of stress on the plane perpendicu~ was developed [44]. The weight-bearing area was considered to be a part of the
lar to R
1.3 Stress in the hip joint articular surface
42
-R
Figure 1.27. A schematic figure of the model of the linear rise of stress on the plane
~ After Legal et al.,1978.
perpendicular to the hip joint reaction force R.
femoral head covered by the acetabulum and lying over the plane that is perpendicular to
~ and crosses the femoral head center (Fig 1.27). For the sake of simplicity the coordinate
R
~ while the y = 0 plane
system is chosen so that its z-axis is pointing in the direction −R
~
is in the frontal plane of the body. The value of stress on the plane perpendicular to R
~ It is assumed that
(z = 0 plane) contributes to the z-component of the partial force dR.
stress p(x, y) on the z = 0 plane decrease in the frontal plane linearly with x,
p(x, y) = B − D x
(1.35)
where unknowns B and D are obtained by solving the equation (1.26) and B is the value
~ crosses the articular surface. The model was improved by
of stress at the point where R
taking into account the acetabular anteversion and the width of the articular gap [45]. This
model was used to estimate the effect of some osteotomies on stress in the hip joint [46]
and also to estimate the chronic stress tolerance level [22, 51].
Model of Brinckmann et al. [9] is based on the deformation of the articular cartilage. It
was considered that the femoral head could be described as a sphere and acetabulum as a
congruent spherical shell separated by a soft cartilagineous layer (Fig. 1.28). After loading
the femoral head is displaced relative to the acetabulum. Thus the intermediate layer will
be strained. Results of the measurements indicate that the displacement is small compared
to the radius of the sphere [9]. So, strain of the intermediate layer is considered within the
limits of elasticity. In other words, it is taken that the articular cartilage obeys Hooke’s
1.3 Stress in the hip joint articular surface
43
Figure 1.28. A scheme showing squeezing of the cartilage due to loading of the hip:
unloaded hip (a), loaded hip (b).
law, so that local stress is proportional to local deformation of the cartilage. Therefore,
stress at given point of the sphere will be proportional to the displacement of the articular
sphere at this point. The magnitude of the displacement of the center of the sphere will be
assigned by δ. Due to symmetry of the sphere, the minimal distance between the loaded
sphere and the spherical shell will be in the direction of the displacement of the center of
the sphere. This point of the minimal distance and is called the pole of stress distribution.
Using the cosine law for triangle depicted in figure 1.28 we obtain:
r2 = r02 + δ 2 − 2 r0 δ cos γ
(1.36)
where r is the radius of the articular sphere, r0 is the distance from the original center of
the sphere to the chosen point at the articular sphere and γ is an angle between the stress
pole and this point (Fig 1.28). Because δ œ r, δ 2 could be neglected in equation (1.36)
and r could be after using equation (1.36) expressed as:
r
δ
0
r = r 1 − 2 0 cos γ
r
After expanding the root up to the terms of the first term order in δ,
r = r0 − δ cos γ
(1.37)
(1.38)
δ is constant. The for displacement of the chosen point on the articular surface is
r0 − r ∝ cos γ
(1.39)
We assume that the contact stress at any point of the weight-bearing area p is proportional
to the deformation given by equation (1.39),
p = p0 cos γ
(1.40)
1.3 Stress in the hip joint articular surface
44
where p0 is the value of the stress in the pole.
The weight-bearing area is determined by the condition that the value of the contact
stress must be positive; it means that the the angle γ is smaller or equal to 90◦ . If the
direction and the magnitude of the hip joint reaction force is known, we can compute the
value of stress at any point of the weight-bearing area after equation (1.26). The system
~ (Eqn. 1.26)
consists of three equations for three components of the hip joint reaction force R
and can be solved numerically. By using this model the stress on the articular surface of the
hip joint in healthy persons and persons with idiopatic osteoarthrosis was examined [9]. No
statistically significant difference was found in maximal contact stress calculated relative
to body weight (pmax /WB ). The statistically significant difference was found in maximum
contact stress (pmax ), because the body weight of the patients with idiopatic coxarthrosis is
on average higher than the body weight of the healthy persons. Within the healthy as well
as within the group of patients with coxarthrosis the mean values of pmax and pmax /WB
are higher in female population than in male population.
Basic assumptions of the cosine distribution of the stress (Eqn. 1.40) are included also
in the model of Iglič et al. [30, 35]. Using the spherical coordinate system originating in
~ is given by the
the center of the articular sphere (see page 26), the resultant hip force R
vector
~ = (R sin ϑR cos ϕR , R sin ϑR sin ϕR , R cos ϑR )
R
(1.41)
where R is the magnitude of the resultant hip force, ϑR is the inclination of the direction
of the resultant hip force with respect to vertical axis and ϕR is the angle of rotation of
the direction of the resultant hip force in the horizontal plane (from the positive x-axis in
the counterclockwise direction). Cosine of the angle γ can be written as
cos γ = sin Θ sin ϑ cos Φ cos ϕ + sin Θ sin ϑ sin Φ sin ϕ + cos ϑ cos Θ
(1.42)
where the polar angle Θ determines the angular displacement of the pole from the vertical
axis, while the azimuthal angle Φ describes the angular displacement of the pole in the
horizontal plane from the x-axis in the counterclockwise direction.
The weight bearing area is defined as a part of the articular sphere constrained by
the acetabular geometry as well as the position of the stress pole. The lateral border of
the weight-bearing area, determined by the acetabular geometry, may be visualized as an
intersection of the articular sphere with the plane passing through the center of the sphere
and being inclined by the ϑCE angle with respect to the vertical body axis. Since only the
1.3 Stress in the hip joint articular surface
45
Figure 1.29. Schematic presentation of the weight-bearing area. The rectangular Cartesian coordinate system is oriented so that x and z axis lie in frontal plane of the body
through the centers of both femoral heads. Symbol P denotes the pole of the stress distribution determined in spherical coordinates by angle Θ and Φ. The angle ϑR describes the
inclination of the hip joint resultant force R with respect to x = 0 plane.
positive values of stress have a physical meaning, the medial border of the weight-bearing
area, which is dependent on the position of the pole of stress, is determined as the line
where stress (Eqn. 1.40) vanishes, so that
cos γ = 0
(1.43)
The medial border of the weight-bearing area determined by the condition (1.43) consists
of all points that lie π/2 away from the pole of the stress and may likewise be visualized
as an intersection of the articular sphere with the plane passing through the center of the
sphere, the inclination of this plane being determined by the location of the stress pole.
As both intersection planes which confine the weight-bearing area are passing through the
center of the sphere they both form circles of radii r at the intersection of the plane and
the articular sphere (Fig. 1.29).
The stress distribution in a given body position is calculated by solving the three
components of the vector equation (1.26) where equations (1.40), (1.42) and (1.43) are
46
1.3 Stress in the hip joint articular surface
taken into account,
ϑR + Θ ∓ arctan
cos2 (ϑCE − Θ)

€π
π ∓ 2 − ϑCE + Θ − 12 sin(2 (ϑCE − Θ))
p0 =
!
=0
3R
cos(ϑR + Θ)
€

π
2 r2 π ∓ 2 − ϑCE + Θ − 12 sin(2 (ϑCE − Θ))
Φ = ϕR or Φ = ϕR ± π
(1.44)
(1.45)
(1.46)
Here the upper sign stands for the case when the pole lies on the lateral side of the contact
hemisphere or outside the contact hemisphere and the lower sign stands for the case when
the pole lies on the medial medial side of the contact hemisphere or outside the contact
hemisphere in the medial direction. The value of Θ corresponding to the equation (1.44) was
determined by Newton iteration method. The value of p0 is expressed from equation (1.45)
using the obtained Θ. If Θ is negative, Φ should be in the interval between −π/2 and π/2
while if it is positive, Θ should be in interval between π/2 and 3π/2.
By knowing the magnitude and the direction of the resultant hip force, center-edge
angle of Wiberg and radius of the articular sphere, the value of the stress at the pole and
the position of the pole can be determined from the equations (1.44)–(1.46). The stress
distribution on the weight-bearing area is then calculated by equations (1.40) and (1.42). If
the pole of the stress distribution is located within the weight-bearing area, the location of
the maximum stress (pmax ) coincides with the location of the maximum stress, and in this
case pmax equals p0 . However, when the stress pole lies outside the weight-bearing area,
the stress on the weight-bearing area is maximal at that point on the weight-bearing area
which is closest to the pole.
For the sake of simplicity, all the above models assume spherical femoral head and
acetabulum and spherical articular area. The method that may including a deformed shape
of the femoral and specific shape of the articular surface (facies lunata) was presented by
Genda et al. [21]. The femoral head and the articular surface of the acetabulum were divided
~ is applied to the center of the femoral head,
into small flat segments. When the load R
the femoral head is displaced relative to acetabulum. Because the femoral head behaves as
a rigid body, the displacement of every segment is the same. Assuming that only contact
stress occurs and deformation of the cartilage is in the limits of elasticity (Hooke’s law), the
force for every segment was computed. The segments with negative stress were eliminated
~
and the sum of all the forces was required to be equal to R.
47
1.3 Stress in the hip joint articular surface
Author
Hodge
Year pmax [MPa]
1989
2.5
10.2
18
Legal
1980
1.2–1.34
Brinckmann 1981
1.1–1.7
Iglič
1990
1.6
Genda
1995
2.0–2.45
Comment
direct measurement
gait
stair-climbing
rising from the chair
one-legged stance
stance phase of gait
one-legged stance
gait
mathematical
mathematical
mathematical
mathematical
model
model
model
model
Table 1.5. Predicted values of the hip joint peak contact stress.
Usually many years are required before the degenerative changes in the hip become
evident. Hadley et al. [22] therefore introduced the time-dependent cumulative pressure
exposure parameter pc of the general form,
pc =
Zt1
(p − pd ) td dt
(1.47)
t0
where the p denotes the contact stress at the specific point in the time, pd represents the
stress damage threshold below which no adverse effects occurs, while d is a time-weightened
exponent. Only positive values of the quantity (p − pd ) contribute to pressure overdosage
accumulation; if p < pd then (p − pd ) is taken to be zero. In the praxis the stress changes
in the time are not known. However, p can be determined for the series of discrete points
corresponding to different times at which archival radiographs are available. Then, the
integral form (Eqn. 1.47) can be approximated by discrete summation,
n
X
pc =
(p − pd )i (∆ti )d
(1.48)
i=1
where the index i denotes the data taken at a specific clinical visit and ∆ti denotes the
elapsed time between subsequential visits i and i − 1. The time-weightened exponent d is
usually taken to be 1. The parameter pc is useful for follow-up study, where longer time
period is involved. This methodology can be used to determine the chronic stress tolerance
level as shown by Maxian et al. [51]. They estimated the damage threshold of the cartilage
to be 2.0 MPa.
1.4 Evaluation of hip status by biomechanical parameters
1.4
48
Evaluation of hip status by biomechanical parameters
The distribution of the contact stress on the weight-bearing area importantly influences
the development of the hip, therefore knowing the biomechanical status of the hip can be
used for predicting the development of the hip. It was suggested that the excessive hip
stress acting over a long period accelerates development of coxarthrosis [22, 51]. On the
other hand Brinckmann et al. [9] report that patients with coxarthrosis have on average
equal normalized peak stress as normal persons. In clinical praxis, it would be convenient
to determine the stress distribution in order to decide for the treatment and plan optimal
postoperative hip and pelvis geometry. It is acknowledged that the biomechanical status
of the hip can be estimated by the centre-edge angle of Wiberg ϑCE [46, 70] and also by
some other geometrical parameters of the pelvis and proximal femur e.g. inclination angle
of the acetabulum ϑU S , ACM angle, acetabular angle ϑAC (Appendix C) or the combination of these parameters – hip index, Severin’s index. These parameters were introduced
to represent physical quantities such as forces and stresses in the hip joint and the size
of the weight bearing area. The main physical parameters previously used to determine
~ [2, 5, 8, 12] and the
biomechanical status of the hip are the hip joint reaction force R
contact stress p in the hip joint articular surface [7, 9, 21, 22, 30, 35, 51].
However, it has been suggested [8] that high stress gradient on the lateral border of
the weight-bearing area could be even more important than the stress itself. Therefore
it would be of interest to evaluate the hip status also by the stress gradient and some
additional parameters obtained by combining stress and stress gradient. Also the size of
the weight-bearing area could be used to evaluate the hip status. In this work we will
derive the mathematical formulation of the stress gradient and its connections with other
biomechanical parameters and radiographical parameter. Using these parameters we will
evaluate the biomechanical status of the hip on a population of the hip for which the data
will be taken from archives. We will try to construct the hypothesis that elucidates the
effect of the biomechanical stimuli on the cartilage.
The accuracy of the calculated biomechanical parameters is influenced by the accuracy
of the measured values of the geometrical parameters of the pelvis and proximal femur.
Therefore it is important to accurately determine of the magnification factor of the standard
antero-posterior radiograph [66]. However not all the biomechanical parameters are equally
sensitive to the accuracy of the measurements due to magnification. For example the center-
1.4 Evaluation of hip status by biomechanical parameters
49
edge angle does not depend on the magnification at all while the resultant hip force and
the stress, that are calculated by using geometrical parameters such as interhip distance,
are more sensitive.
Although progress has been made in understanding and predicting development of the
hip by biomechanical parameters, there is no decisive answer yet regarding the effect of the
stress. Further, stress gradient has previously not yet been systematically studied. In order
to clarify these issues, a relevant mathematical model should be chosen and retrospective
studies should be made on large populations. The presented work is an attempt to make a
step forward in this direction.
50
2 The aim of the work
Part 2
The aim of the work
It is the aim of the work to introduce stress gradient as an important biomechanical parameter for evaluation of the hip status and test its relevance on the populations of the
patients.
Specific aims were to derive an analytic expression for the stress gradient, to define
the biomechanical parameters: stress gradient at the lateral border of the weight-bearing
area gradpm , stress index pI and functional angle of the weight-bearing area ϑF that reflect
the stress gradient, to develop a user-friendly computer program for assessment of hip
stress distribution, to improve the method for determination of the input data to these
programs by considering the effect of the magnification of the radiographs and to analyze
the populations of the hips: the dysplastic hips in comparison with normal hips, the hips
subject to Salter innominate osteotomy in a long-term followup and the untreated hip
subject to developmental dysplasia of the hip in the long-term followup. The results of this
work should contribute to the development of objective evaluation of the status of the hip
and therefore help in optimal decision for the treatment of the disordered hips.
51
3.1 Theory
Part 3
Material and methods
3.1
3.1.1
Theory
Derivation of the model equations
The model for determination of stress distribution used in this work [35] was briefly described before. Here we present a new derivation that is considerable more transparent and
simple than the one presented previously [30, 35]. This simplification is in the application
of an alternative coordinate system.
The weight-bearing area is the spindle-shaped spherical surface delimited by two planes,
the lateral and the medial intersecting plane. The lateral intersecting plane is inclined for
ϑL with respect to x = 0 plane and the medial intersecting line is inclined for ϑM with
respect to x = 0 plane . The position of the pole of stress distribution is denoted in
spherical coordinates by angles Θ and Φ, respectively (Fig. 1.29). The hip joint reaction
force is considered to be lying in the frontal plane [27, 64] and is described by its magnitude
and inclination ϑR with respect to x = 0 plane.
~ = (−R sin ϑR , 0, −R cos ϑR )
R
(3.1)
Because the weight-bearing area is symmetric with respect to y = 0 plane and the force
lies in this plane, the pole of the stress also lies in y = 0 plane, i.e. Φ is equal to zero
or π [30].
The center of the coordinate system is taken to be at the center of the spherical surfaces.
For the sake of simplicity the coordinate system is chosen so that z-axis is crossing the
pole. To achieve this, the coordinate system must be rotated for an angle −Θ. Therefore
the inclination angles of the intersecting planes and the inclination angle of the hip joint
0
reaction force are described by angle ϑL0 , ϑ0M , ϑR
(Figure 3.1).
ϑ0L = ϑL − Θ
(3.2)
ϑ0M = ϑM + Θ
(3.3)
0
ϑR
= ϑR + Θ
(3.4)
52
3.1 Theory
z
'=0
z
R
'
'
R
R
L
L
Pxx
xxxxxxxxxxx
M
xxxxxxxxxxx
xxxxxxxxxxx
xxxxxxxxxxx
xxxxxxxxxxx
xxxxxxxxxxx
xxxxxxxxxxx
(a)
Px
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
x
R
(b)
'
M
x
Figure 3.1. (a) The original coordinate system is rotated in the y = 0 plane for Θ so that
in (b) the rotated coordinate system the z-axis points toward the pole of the stress (P).
Basic assumptions of the cosine distribution of the stress (Eqn. 1.40) are included also
in this model.
p = p0 cos γ 0
(3.5)
where γ 0 is angle between the point on the articular surface of the hip and the position of
the stress pole in rotated spherical coordinates (z-axis).
To simplify derivation of the model equation, a new coordinate system is introduced.
The transformation equations from this alternative coordinate system to Cartesian rectangular coordinate system [x, y, z] are:
x = r cos ϕ sin ϑ
(3.6)
y = r sin ϕ
(3.7)
z = r cos ϕ cos ϑ
(3.8)
where r is the radius of the articular sphere and ϑ and ϕ are the coordinates (Fig. 3.2).
If we determine position of any point on the articular surface by spherical coordinates
ϑ, ϕ in coordinate system (3.6)–(3.8) then the contact stress in this point is according to
equation (1.40),
cos γ 0 = cos ϑ cos ϕ
(3.9)
The unknown quantities: the value of the stress at the pole and the coordinate of the pole
53
3.1 Theory
Figure 3.2. Schematic representation of the weight-bearing area in alternative coordinate
system. The rectangular Cartesian coordinate system is oriented so that x and z axis lie
in frontal plane of the body through the centers of both femoral heads. Symbol P denotes
the pole of the stress distribution.
Θ and Φ can be obtained by solving vector equation
Z
~ =R
~
p dA
(3.10)
A
~ could be in the coordinate system (3.6)–
where the segment of the weight-bearing area dA
(3.8) expressed as:
~ = −(cos ϕ sin ϑ, sin ϕ, cos ϕ cos ϑ) r2 cos ϕ dϑ dϕ
dA
(3.11)
The negative value of the segment of the weight-bearing area means, that the unit vector
~n (Eqn. 1.27) is oriented into the center of the articular sphere. This orientation of the
~ is oriented into the center of the
vector ~n was chosen, as the hip joint reaction force R
weight-bearing area.
The integration of the components of the vector equation (3.10) is performed over the
weight-bearing area. The lateral intersecting plane is determined from the hip geometry
(ϑL is equal to center-edge angle of Wiberg, ϑCE angle) by using equation (3.2). The medial
intersecting plane is determined by condition that stress (Eqn. 3.5) vanish, so ϑM is for
54
3.1 Theory
π/2 direction from the pole of the stress distribution In the rotated coordinate system,
ϑ0M = π/2 (Eqn. 3.3). The integration bounds are
h
πi
ϑ ∈ −(ϑCE − Θ),
2
h π πi
ϕ ∈ − ,
2 2
(3.12)
(3.13)
Considering equation (3.10), the components of the hip joint reaction force are:
π
Z2
−(ϑCE −Θ)
π
Z2
π
Z2
− π2
p0 cos ϑ cos ϕ r2 cos2 ϕ sin ϑ dϕ dϑ = R sin(Θ + ϑR )
(3.14)
π
Z2
p0 cos ϑ cos ϕ r2 cos ϕ sin ϕ dϕ dϑ = 0
(3.15)
−(ϑCE −Θ) − π2
π
Z2
π
Z2
p0 cos ϑ cos ϕ r2 cos2 ϕ cos ϑ dϕ dϑ = R cos(Θ + ϑR )
(3.16)
−(ϑCE −Θ) − π2
A major advantage of the application of the coordinate system (3.6)–(3.8) over previously
used spherical system [30, 33, 35] is therein that the boundaries are fixed. Left term of the
equation (3.15) equals to zero due to symmetry of the weight bearing area with respect
R
R
to the y = 0 plane. Using that cos x sin x dx = 21 sin2 x + const,
cos2 x dx = 21 (x +
R
cos x sin x) + const,
cos3 x dx = sin x − 13 sin3 x + const and (3.1) it follows from
equations (3.14) and (3.16)
π
2
p0 r2
+ ϑCE
3
2
2
p0 r2 cos2 (ϑCE − Θ) = R sin(Θ + ϑR ) (3.17)
3
‘
− Θ + sin(ϑCE − Θ) cos(ϑCE − Θ) = R cos(Θ + ϑR )) (3.18)
By dividing (3.17) by (3.18) we obtain a nonlinear equation for Θ,
tan(Θ + ϑR ) −
π
2
+ ϑCE
cos2 (ϑCE − Θ)
=0
− Θ + sin(ϑCE − Θ) cos(ϑCE − Θ)
(3.19)
This equation is solved numerically by the Newton’s iteration method and the value Θ is
obtained. The value of the stress at the pole p0 is then obtained from equation (3.18):
p0 =
3R
2 I r2
(3.20)
55
3.1 Theory
where
π
2
+ ϑCE − Θ + sin(ϑCE − Θ) cos(ϑCE − Θ)
(3.21)
cos(ϑR + Θ)
As previously states, if the pole lies inside the acetabular shell, then maximal contact stress
I=
pmax is equal to p0 . If the pole lies outside the weight-bearing area, the point of maximal
stress is located on the rim of the acetabulum. The value of the maximum contact stress
can be expressed after equation (1.40).
pmax = p0 cos(ϑCE − Θ)
3.1.2
(3.22)
Gradient of stress and functional angle of the weight-bearing area
The maximum contact stress describes the value only in one point and does not describe
how the stress varies with direction. Therefore we tried to define some new parameters
that can be connected to the shape of the distribution function.
Let as assume rotated coordinate system with the z-axis pointing to the pole of the
stress at the top of the sphere (Fig. 3.1). In spherical coordinates [r, ϑ, ϕ] (Fig. 3.3) stress
is taken to be proportional cosine of the angle between the pole of the stress (z-axis) and
the chosen point at the articular surface (Eqn.1.40)[9].
p = p0 cos ϑ
(3.23)
where p0 is the value of the stress at the pole. The gradient of stress is expressed by [42],
gradp = 5p =
∂p
1 ∂p
1 ∂p
~er +
~eϑ +
~eϕ
∂r
r ∂ϑ
r sin ϑ ∂ϕ
(3.24)
where ~er , ~eϑ , ~eϕ are the orthogonal unit vectors in direction r, ϑ, ϕ respectively as illustrated
in figure 3.3 and r is the radius of articular sphere. Considering (3.23) and (3.24) we obtain
for the gradient of stress:
p0
sin ϑ ~eϑ
(3.25)
r
The gradient of stress has direction of tangent to the articular sphere and points towards the
gradp = −
pole of stress. It was observed from the radiographs that usually the degenerative changes
occur at the margin of the acetabulum [57]. To test the hypothesis that high gradient
oriented outward from the acetabular shell is biomechanically unfavorable we introduce
the parameter gradpm which gives the value of stress gradient at the lateral margin of the
weight-bearing area. By using the equations (3.2) and (3.25) we obtain:
p0
gradpm = − sin(ϑCE − Θ)
r
(3.26)
56
3.1 Theory
Figure 3.3. The unit vectors basis in spherical coordinates
If the pole of the stress lies outside the weight-bearing area then Θ > ϑCE and gradpm
is positive. It means that the gradient of stress is pointing away from the weight-bearing
area. If the pole of the stress lies inside the weight-bearing area then Θ < ϑCE and gradpm
is negative. We suppose that positive gradpm is biomechanically unfavorable while negative
gradpm is biomechanically favorable.
As it has also been suggested [8] that unfavorable distribution of the stress together
with its high value is especially unfavor. Therefore we introduce a new parameter – stress
index pI , which takes into account both quantities. Stress index is defined as the value of
the gradient of stress at the acetabular margin multiplied by the value of the contact stress
at the acetabular margin. By using equations (1.40), (3.2) and (3.26), we obtain for the
stress index
pI = p(ϑCE )gradpm
p2
pI = − 0 cos(ϑCE − Θ) sin(ϑCE − Θ)
r
(3.27)
(3.28)
We introduce another parameter called the functional angle of the weight-bearing area
ϑF ,
π
+ ϑCE − Θ
(3.29)
2
This parameter, which does not strongly depend on the size of the radiograph, is actually
ϑF =
the size of weight bearing area A divided by the square of the radius of the articular surface
r2 ,
ϑF =
A
2 r2
(3.30)
57
3.1 Theory
The parameters gradpm (3.26) and pI (Eqn. 3.28) could be expressed by the functional
angle of the weight-bearing area as
p0
sin ϑF
r
p20
sin 2ϑF
=
2r
gradpm =
(3.31)
pI
(3.32)
3.2 Determination of the biomechanical parameters
3.2
58
Determination of the biomechanical parameters
To determine the above defined biomechanical parameters (pmax , gradpm , pI and ϑF ), the
external measurements combined with mathematical model were used as described above [31].
Data of geometrical parameters of pelvis and proximal femur were obtained from standard
antero-posterior radiographs (Appendix C). Only standard antero-posterior radiographs
with clearly visible entire pelvis and both proximal femurs with no sign of aseptic necrosis
or chondrolysis of the femoral head were included in the final analysis.
As mentioned in the page 32 the reference values of the attachment points of the muscles
~ must be rescaled in order
involved in the model for computation of the resultant hip force R
to adjust configuration of the hip and pelvis to the individual person. The input parameters
of the model for determination the hip joint resultant force are (Fig. 3.4): the distance
between the centers of the femoral heads, i.e. the interhip distance l, the vertical distance
between the center of the femoral head and the highest point on the crista iliaca (PH ), i.e.
the pelvis height H, the horizontal distance between the center of the femoral head and
the most lateral point on the crista iliaca (PC ), i.e. the pelvis width C, the vertical and the
horizontal distances from the center of the femoral head to the effective muscle attachment
point (T ) on the greater trochanter (x and z respectively) and the body weight WB . The
point T is determined by the intersection of the contour of the greater trochanter and the
normal to thought midpoint of the straight line connecting the most lateral point (point 1)
and the highest point (point 2) on the greater trochanter . The values of these geometrical
parameters are determined in a relative frame where the straight line connecting the centers
of both femoral heads defines the horizontal. The lines determining the values of H and
x are perpendicular to the horizontal, while the lines determining the values of C and z
are parallel to the horizontal. If the body weight is unknown, then the magnitude of the
resultant force R could be expressed only relative to the body weight R/WB .
The input parameters of the model for determination of the contact stress distribution
in the hip joint are (Fig. 3.4): the magnitude of the hip joint resultant force R, its direction
represented by the inclination angle with respect to vertical ϑR , the center-edge angle ϑCE
and the radius of the femoral head r.
The contours of the bony structures in the antero-posterior radiograph were digitized
by using a digital graphic board. The geometrical parameters of the pelvis and proximal
femur mentioned above were determined automatically by the computer program HIJOMO
3.2 Determination of the biomechanical parameters
59
Figure 3.4. The geometrical parameters of the pelvis and the proximal femur needed for
determination of the maximal stress on the weight-bearing area. The stress distribution
~ are shown schematically.
and the the resultant hip force R
(Appendix B).
To determine the contact stress distribution in the hip joint we used the computer program HIPSTRESS (Appendix A). The computer program HIPSTRESS is based on the three dimensional model for determination of the resultant hip force in the one-legged stance
(described in details in section 1.2.2) and on the mathematical model for determination
of the stress distribution on the weight-bearing area (described in details in section 3.1.1).
The results of the calculation are the position of the pole of the stress Θ and the value of
stress at the pole pmax .
The maximal stress on the weight-bearing area pmax , the gradient of the stress gradpm ,
the stress index pI and the functional angle of the weight-bearing area ϑF could then be
expressed by using the equations (3.22), (3.26), (3.28) and (3.29) respectively. If the body
weight is unknown then the maximal stress on the weight-bearing area pmax , the gradient
of the stress gradpm and the stress index pI can be given relative to the body weight
pmax /WB , gradpm /WB and pI /WB2 .
60
3.3 Analysis of data
3.3
Analysis of data
3.3.1
Statistical analysis
Processing the radiographs of the population of the hips yields sets of values of the biomechanical parameters. These sets of data were analyzed by statistical methods. Data are
represented numerically by the average value and by the standard deviation. Some of the
sets of data are represented also graphically by histograms.
To compare the differences between different groups parametrical and nonparametrical
statistical tests were used. If we obtained normal distribution of the data, we used the
two-tailed pooled t-test. If the sample was small nonparametrical Wilcoxon’s W two-tailed
test was used [42].
c
For statistical analysis we utilized the software package Analyse-it
for Microsoft
c
c
and the statistical software Statistica
.
Excel
To study dependencies between the quantities we used the regression analysis. If we
did not know the theoretically predicted dependence, we used simple linear regression.
When the correlation between the center-edge angle and other biomechanical parameters
was examined, it was taken into account that the center-edge angle and biomechanical
parameters are not independent (see section 3.1). Further, the dependence between the
center-edge angle and the biomechanical parameters is not linear and could be expressed
only numerically. Therefore we fitted the dependencies numerically by using the model
equations. For example, the maximum contact stress pmax (Eqn. 3.20) depends on the
magnitude and inclination of the load (R and ϑR respectively), on the radius of the femoral
head (r) and on the centre-edge angle of Wiberg (ϑCE ) (Section 3.1.1). If R, ϑR and r are
given, we can determine the normalized contact stress pmax for ϑCE of every hip in the
population. To find the best fit for the population the sum of squares α was minimized by
variation of these three variables by using a simple iteration algorithm [42],
α=
n
X
i=1
(pmax (ϑCE i , R, ϑR , r) − pmax i )2
(3.33)
where i runs over all data. To obtain the minimum of the sum of the squares we tried different starting values of variables R, ϑR and r and we took into account the solution which
reflects the relations contained in the model. The correlation coefficient is the measure of
the influence of the input parameters of the model (excluding center-edge angle) on the
examined biomechanical parameter.
61
3.3 Analysis of data
3.3.2
Effect of magnification in biomechanical studies of large group of patients
Variation in results is the main characteristic of all experiments [18]. In our measurements
we can also observe such variation. There is inherent variability between the measured
subjects, i.e. the variability of the geometrical parameters that are needed to compute the
biomechanical parameters. There is also the variation associated with lack of uniformity
in the conditions and execution of the experiment. By determination of the biomechanical
parameters in the hip we are interesting only in the first kind of variation (for example the
variation of stress in the normal hips due to different shape of pelvis within the population).
The second source of variation can occur due to inaccurate measurement of the geometrical
parameters from the antero-posterior radiograph and due to different magnifications of
the radiographs. According to Jaklič [36] we assumed that our measurements from the
radiographs were exact enough and this source of variation was taken to be negligible.
Therefore in the following text the role of the magnification will be discussed.
Usually, standard magnification is taken to be 10% [66]. The theoretical prediction of
the influence of the magnification factor on the biomechanical parameters was obtained
by mathematical modelling. The standard values of the geometrical parameters [16] were
taken and the effect of the magnification from 0% to 50% was calculated. The relative
changes in the parameters are shown in the figure 3.5.
It can be seen from the figure 3.5 that the parameter that is most sensitive to the
magnification is the stress index pI . On the other hand the parameter in which the influence
of the magnification is low is the functional angle of the weight-bearing area ϑF .
Problem how to estimate the role of magnification in large group of patients where
the magnification factor of each radiograph is unknown. Let us describe the effect of the
magnification on the biomechanical parameter Y by
Yi0 = Yi + Yi µj (Yi )
(3.34)
where Yi0 is the magnified value of the biomechanical parameter, Yi is the true value of this
biomechanical parameter and µj (Yi ) is the magnification of this biomechanical parameter.
For example, if we made many radiograph of chosen subject (the same geometrical parameters and therefore the same Yi ) we could estimate the average value of the magnification
62
Relative biomechanical parameter
3.3 Analysis of data
ϑF
pmax
gradpm
pI
Magnification
Figure 3.5. The effect of the magnification on the biomechanical parameters
of the biomechanical parameter µ(Yi ) as:
0
Y
µ(Yi ) = i − 1
Yi
(3.35)
The standard deviation of the magnification of the biomechanical parameter µ (sµ 0 can be
expressed by
sµ2 =
1 2
s 0
Yi2 Yi
(3.36)
where sYi0 is standard deviation of the biomechanical parameter influenced by magnification. As a first approximation we can assume that the µ is the same for every value of the
biomechanical parameter Yi . If the group of patients was large enough, then the average
0
value of the biomechanical parameter Y would be
0
Y =Y +Y µ
(3.37)
where Y is the true average value of the biomechanical parameter (without effect of the
magnification). If we know the average value of the biomechanical parameter influenced by
0
magnification Y , we can estimate the average true value of this parameter by using the
equation:
0
Y
Y =
1+µ
(3.38)
63
Number of patients
3.3 Analysis of data
Magnification
Figure 3.6. Histogram of the magnification factor M determined from the antero-posterior
radiographs.
According to error propagation equation [42] the value of the standard deviation of the
biomechanical parameter without the effect of the magnification sY can be estimated as:
’ 2
“
sµ2
sY 0
2
2
sY = Y
(3.39)
02 −
(1 + µ)2
Y
So if we want to estimate the real value of the biomechanical parameter and its standard
deviation, we have to estimate the average value of the magnification of the biomechanical
parameter µ and its standard deviation sµ as will be shown bellow.
We have shown the importance of the accurate determination of the magnification factor
in determination of contact stress distribution from antero-posterior radiographs in clinical
practice [66]. In the group of twelve patients the magnification factor for each subject was
determined by using the metal lamella of known length L0 which has been placed at the
level of the femoral head centers before making the radiograph. From the length of the
lamella in the antero-posterior radiograph L we calculated the magnification factor M as
L
.
L0
The average magnification determined from these data was 21.3%, while the standard
deviation was 5.13%.
In this study we used data for the magnification factor as in [66] and also some older
radiograph of the same persons. When we used the radiographs of the same person that
were taken several years apart, we scaled the magnification by the radii of the femoral head.
We could do this only for the radiographs taken at the skeletal maturity. The femoral head
after which the magnification factor of the older radiograph was determined was chosen
randomly. The magnification factor M was computed as
r
,
r0
where r is the radius of the
64
3.3 Analysis of data
femoral head in the older radiograph and r0 is the known value of the radius of the femoral
head. The average magnification factor determined form the older radiographs was 16.5%,
while its standard deviation was 6.88%. The difference between the magnification factor
determined form the recent radiograph and magnification factor determined from the older
radiograph is statistically significant, probability of the two-tailed Wilcoxon’s statistics is
less than 0.001.
The final sample consisted of 24 subjects. Figure 3.6 shows the distribution of the magnification factor M . The average magnification M of all the radiographs was 18.9%, while
the standard deviation sM was 6.45%. We consider that the distribution of the magnification factor is normal and we created a set of data with approximately normal distribution
2
]. We simulated the effects of the magnifications on the geometrical parameters of
N[M , sM
the pelvis and proximal femur and processing data with standard magnification 10% for
three different values of ϑCE and the standard values of the parameters of the pelvis [16].
The results of the simulation are shown in table 3.1. The magnifications of the biomechanical parameters and its standard deviations were computed by using equations (3.38) and
(3.39) respectively and the results are shown in table 3.2.
ϑCE
◦
10
30◦
50◦
2
pmax /WB [m−2 ] gradpm /WB [m−3 ]
pI /WB
[m−5 ]
0
0
0
Y
Y
Y
Y
Y
Y
8
5939
6798
164 052 197 937
9.85 .10
1.35 .109
2611
3006
-22 179
-28 246 -5.72 .107 -8.03 .107
1963
2269
-56 400
-78 066 -7.16 .107 -1.01 .108
ϑF [◦ ]
0
Y
Y
53.3 53.9
103.1 103.6
140.5 146.5
Table 3.1. Average values of the biomechanical parameters obtained by modelling the
0
effect of the magnification (Y ) and the true average values of these parameters (Y ), for
different center-edge angles (ϑCE ).
ϑCE
10◦
30◦
50◦
2
pmax /WB
gradpm /WB
pI /WB
ϑF
µ
sµ
µ
sµ
µ
sµ
µ
sµ
-0.126 0.0795 -0.171 0.1066 -0.270 0.1635 -0.011 0.0069
-0.132 0.0827 -0215 0.1329 -0.287 0.1862 -0.004 0.0028
-0.135 0.0848 -0.278 0.1104 -0.284 0.1780 -0.042 0.0015
Table 3.2. Values of the average magnifications factors of the biomechanical parameters
µ and standard deviations of these factors sµ for different center-edge angles (ϑCE ) .
The results of the effect of the magnification show that by taking into account the error
caused by the magnification, the values of the biomechanical parameters are lower than
3.3 Analysis of data
65
their true values. It means that by computing the parameters we have a systematic error.
This error is largest in the gradient of the stress gradpm and smallest in the functional angle
of the weight-bearing area ϑF (Tab. 3.2). This could also be seen in figure 3.5. Therefore,
the functional angle of the weight-bearing area ϑF is more appropriate for evaluation of
the biomechanical state of the hip if the magnification is not known.
The radiographs of dysplastic hips and the hips subject to the Salter osteotomy contained no length standard, so that the magnification could not be determined for each
radiograph. In analyzing the dysplastic hips we used the above described correction while
in analyzing the hips subjected to the Salter osteotomy we used a standard value 10%.
In analyzing the hips subject to developmental dysplasia magnification of the radiographs
was known.
In the analyzing the dysplastic hips, the average values of the biomechanical parameters,
that are not affected by different magnification were estimated by using equation (3.38)
while its standard deviations were estimated by using equation (3.39). Average magnification factors µ of the biomechanical parameters were taken as the average of the computed
values µ for different center-edge angle (Tab. 3.2). The standard deviation of the magnification factor sµ was taken as the average of the computed value sµ for different center-edge
angle shown in table 3.2.
4.1 Biomechanical status of normal and dysplastic hips
66
Part 4
Results on populations
After derivation of new biomechanical parameters it is of interest to test their relevance on
the population of the hips. We have evaluated following population: dysplastic hips, hips
subject to the Salter osteotomy and hips subject to developmental dysplasia.
4.1
Biomechanical status of normal and dysplastic hips
It was suggested that unfavor biomechanical conditions in the dysplastic hips rises causes
the development of the secondary osteoarthritis [22]. Therefore we are interesting in the
values of the biomechanical parameters in dysplastic hips in comparison with normal hips.
4.1.1
Dysplasia of the hip
Dysplasia of the hip refers to mechanical deformations and deviations in the size and shape
or mutual proportions between femur and acetabulum [19]. The dysplasia of the hip is in the
clinical praxis important as indication for the operation in order to stop the development
of the pathological processes.
The dysplasia of the hip can be diagnosed according to anatomical changes in the hip,
that are visible at the radiograph (e.g. coxa vara, the presence of the osteofyts, shape and
density of the trabecular net in the femur) [19, 46, 57] (Fig. 4.1). The estimation of the
biomechanical status and the decision about the operation has often been based on the
RTG parameters and clinical status of the hip.
The main RTG parameter, that is used for assessment of the hip dysplasia, is the
center-edge angle of Wiberg ϑCE [70] (Appendix C). The size of ϑCE gives a numerical
value of lateral coverage of the femoral head. The range from 20◦ to 25◦ is considered
to be lower limit for normal hips, while the value below 20◦ is pathognomic for the hip
dysplasia [46]. However, it was suggested, that besides ϑCE , also the radius of the femoral
head should be taken into account in assessment of the hip dysplasia [43]. It was shown
that in the normal hips the ϑCE angle correlates with the femoral head radius. Hips with
large heads were found to have smaller ϑCE [46]. Other RTG parameters used to diagnose
4.1 Biomechanical status of normal and dysplastic hips
67
Figure 4.1. Antero-posterior radiograph of young adult with bilateral dysplasia, greater
on left than on right. Adapted from Daugherty et al., 1998.
the dysplasia are the ACM angle (determines the deep of the acetabulum), angle of SharpUlman, (determines the latero-inferior inclination of the acetabulum) (Appendix C) and
others.
The RTG parameters were introduced to represent the physical quantities such as forces,
stresses in the hip joint and size of the weight-bearing area. In order to take into account
complex interactions and pursue a more realistic description, relevant mathematical models
that directly give these quantities were developed [6, 57, 64]. Such models would also enable
the study of the influence of individual geometrical parameters of the hip and could be
used in predicting an optimal configuration of the hip after the operation [6, 30].
4.1.2
Patients
In our study the group of the dysplastic hips and group of normal hips were examined.
The group of dysplastic hips consists of the hips that were afterwards operated at the
Department of the Orthopaedic surgery in Ljubljana due to diagnosis Dysplasia coxae. The
diagnosis of dysplasia was assigned on the basis of clinical and radiographic evaluation.
All the patients were operated due to dysplasia and the analyzed radiographs were taken
prior to the operation. Our sample included 20 subjects with unilateral dysplasia and 18
subjects with bilateral dysplasia. In total we had 56 dysplastic hips. In this group 9 hips
belonged to male persons and 47 belonged to female persons, 32 hips were right and 24
4.1 Biomechanical status of normal and dysplastic hips
68
were left.
The normal hips belong to 146 persons who were subject to the X-ray examination of the
pelvic region for reasons other than degenerative diseases of hip joints. These radiographs
showed no sign of the hip pathology. In these persons, both hips were healthy however, only
one hip was taken into account because both hips had approximately the same geometrical
parameters. The particular hip was chosen randomly. In this group, 33 hips belonged to
male and 113 belonged to female persons, 71 hips were right and 75 hips were left. Body
weight of the patients of both groups was not known.
4.1.3
Results
The correlation between the center-edge angle and the normalized peak contact hip joint
stress pmax /WB is shown in figure 4.2. In the previous studies it was suggested to describe
the dependence pmax (ϑCE ) by an exponential trial function [35]. The correlation coefficient
for our fit is higher (R2 = 0.992) comparing to correlation coefficient obtained by fitting
with an exponential function (R2 =0.687) and the numerically obtained curve is consistent
with the used mathematical model. Low ϑCE correlates with high pmax /WB and vice versa.
Scattering of the data in figure 4.2 shows that in the determining peak contact stress the
parameters other than ϑCE are also important. For example, in two hips with approximately
the same center-edge angle (6◦ ) the peak stress normalized by the body weight was shown
to differs for about 4000 m−2 . The influence of the parameters other than center-edge angle
on the normalized hip joint contact stress is larger for smaller ϑCE .
The correlation between the center-edge angle and the normalized gradient of the
stress gradpm /WB is shown in figure 4.3, while the correlation between the center-edge
angle and the normalized stress index pI /WB2 is shown in the figure 4.4. For lower ϑCE
the values of the gradient of stress and the stress index are positive and differ several
time, while for higher ϑCE the values of these parameters become negative. The quantities
(gradpm (ϑCE ), pI (ϑCE )) change sign when ϑCE is approximately 22◦ . The influence of parameters other than ϑCE , that is noticeable on the scattering of the data, is high when the
ϑCE is lower than 22◦ .
Figure 4.5 shows the correlation between the centre-edge angle and the functional angle
of the weight-bearing area ϑF . There is a positive correlation, high values of ϑCE correlate
with high values of ϑF . The influence of the parameters other than ϑCE on the functional
angle of the weight-bearing area exhibited by scattering of the data is low.
69
pmax /WB
[m−2 ]
4.1 Biomechanical status of normal and dysplastic hips
ϑCE
[◦ ]
Figure 4.2. The correlation between the center-edge angle ϑCE and the normalized peak
contact joint stress pmax /WB . The values for the normal hips are denoted by ♦ and the
values for the dysplastic hips are denoted by 4. The curve represents numerical fit after
model equations by these parameters: R/WB = 2.82, ϑR = 12.93◦ , r = 24.55 mm.
The correlations coefficients and their statistical significance between the sets of the
data of the biomechanical parameters and the corresponding theoretical curve obtained by
numerical fitting are shown in table 4.1. The correlation coefficients are a numerical expression of the influence of the parameters other than ϑCE on the biomechanical parameters.
It can be seen from table 4.1 and from figures 4.4 and 4.5 that this influence is the highest
in the stress index and the lowest in the functional angle of the weight-bearing area.
Correlation
pmax /WB – ϑCE
gradpm /WB – ϑCE
pI /WB2 – ϑCE
ϑF – ϑCE
R2
0.850
0.897
0.773
0.983
P
< 0.001
< 0.001
< 0.001
< 0.001
Table 4.1. Correlation coefficients between the parameters determining biomechanical
state of the hip and the centre-edge angle ϑCE and their statistical significance.
Next, the statistical significance of the difference in radiographic and biomechanical
parameters between the normal and the dysplastic hips was calculated by the two-tailed
pooled t-test. The average values and the standard deviations of the sets of the data of
70
gradpm /WB
[m−3 ]
4.1 Biomechanical status of normal and dysplastic hips
ϑCE
[◦ ]
pI /WB2
[109 m−5 ]
Figure 4.3. The correlation between the center-edge angle ϑCE and the normalized gradient of the stress gradpm /WB . The values for the normal hips are denoted by ♦ and the
values for the dysplastic hips are denoted by 4. The curve represents numerical fit after
model equations by these parameters: R/WB = 2.72, ϑR = 10.65◦ , r = 25.85 mm.
ϑCE
[◦ ]
Figure 4.4. The correlation between the center-edge angle ϑCE and the normalized stress
index pI /WB2 .The values for the normal hips are denoted by ♦ and the values for the
dysplastic hips are denoted by 4. The curve represents numerical fit after model equations
by these parameters: R/WB = 2.14, ϑR = 12.25◦ , r = 21.55 mm.
71
ϑF
[◦ ]
4.1 Biomechanical status of normal and dysplastic hips
ϑCE
[◦ ]
Figure 4.5. The correlation between the center-edge angle ϑCE and the functional angle
of weight-bearing area ϑF . The values for the normal hips are denoted by ♦ and the values
for the dysplastic hips are denoted by 4. The curve represents numerical fit after model
equations by these parameters: R/WB = 2.60, ϑR = 8.95◦ .
the particular biomechanical parameters and results of t-test are presented in table 4.2.
For every parameter parameter the null hypothesis assuming equal average values [18] is
rejected at the level of significance lower then 0.001 and it may be concluded that the each
of the parameters is different in normal hips than in the dysplastic hips.
Next, the effect of the magnification in the group of normal and dysplastic patients was
estimated (section 3.3.2). The differences in the biomechanical parameters between the
groups of the normal and the dysplastic hips were tested by the two-tailed pooled t-test.
Results are shown in the table 4.3.
Normal
Dysplastic
Parameter
Average StDev Average StDev
◦
ϑCE [ ]
37.678
8.595
13.214
8.329
pmax /WB [m−2 ]
2691.6
650.3
5274.1
2398.105
gradpm /WB [m−3 ] −44 493.5 25 332.9 148 105.5 191 177.3
pI /WB2 [m−5 ]
−9 107
5.8 107 1.22 109 2.09 109
◦
ϑF [ ]
117.020
16.055
66.787
22.158
Difference
t
P
18.22 < 0.001
11.96 < 0.001
11.95 < 0.001
7.60 < 0.001
17.81 < 0.001
Table 4.2. Differences between normal and dysplastic hips
72
4.2 Evaluation of the hips after Salter osteotomy
Normal
Parameter
Average StDev
ϑCE [◦ ]
37.678
8.595
−2
pmax /WB [m ]
3097.3
688.44
−3
gradpm /WB [m ] −57 141
31 393.9
pI /WB2 [m−5 ]
−1.25 108 7.454 107
ϑF [◦ ]
119.28
16.359
Dysplastic
Average StDev
13.214
8.329
6069.0
2699.0
190 205 243 871.4
1.69 109 2.874 109
68.08
22.585
Difference
t
P
18.22 < 0.001
12.34 < 0.001
12.04 < 0.001
7.67 < 0.001
17.82 < 0.001
Table 4.3. Differences between normal and dysplastic hips after estimation of the effect
of the magnification.
4.2
Biomechanical evaluation of hip joint after Salter innominate
osteotomy
Besides the current values of biomechanical parameters in the hip, it is of interest to study
the development of the hips in the longer follow-up study. The radiographs of healthy
persons taken at different times are not available. Therefore the analysis of patients that
have undergone Salter osteotomy in the childhood and were therefore followed for many
years was made.
4.2.1
Salter innominate osteotomy
The Salter innominate osteotomy was introduced by Canadian orthopaed Robert Salter for
treatment of the congenital dislocation of the hip [63]. The indication for this osteotomy
is the developmental dysplasia in the children.
The goal of this operation is to improve the coverage of the femoral head, thereby
increasing the weight-bearing area of the joint. The main part of the Salter osteotomy
is the osteotomy of the ilium from the spina iliaca anterior inferior to the greater sciatic
notch. The bone wedge, that was taken from the crista iliaca at the same side, is inserted
into the place of the osteotomy (Fig. 4.6). Osteotomy is held open anterolaterally by this
wedge and thus roof of the acetabulum is shifted more anteriorly and laterally.
The advantages of the Salter osteotomy of the acetabulum are that the lateral coverage
of the femoral head is provided by hyaline cartilage and that the triradiate cartilage stays
intact.
4.2 Evaluation of the hips after Salter osteotomy
73
Figure 4.6. Schematic representation of the Salter innominate osteotomy. Adapted from
Daugherty et al., 1998.
4.2.2
Patients
At the department of Orthopaedic surgery, Ljubljana, 63 patients (70 hips) underwent
Salter osteotomy due to developmental dysplasia of the hip in the period from 1974 to 1983.
44 hips met our enrollment criteria: no radiographic sign of aseptic necrosis or chondrolysis
of the femoral head preoperatively and postoperatively, antero-posterior radiographs of the
pelvis and both proximal femurs available after the operation, no operation on the hip in
the followup period.
The mean age of the patients at the operation was 54 months (from 18 months to 10
years) while at the latest control it was 14 years (from 9 to 23 years). The center-edge
angle of Wiberg ϑCE was determined from the radiographs shortly after operation, once in
the course of the first postoperative year, once in the 2nd or 3rd postoperative year, once
3 to 7 years postoperatively and once 7 to 13 years postoperatively. Only patients having
followup period greater then 7 years were included.
Our sample of the operated hips consists of 38 hips while our final sample of the
contralateral nonoperated hips consists of 21 hips that fulfilled all the above criteria. The
operation was performed on one hip in 32 patients and on both hips in 3 patients.
Body weight of the patients was not known. The magnification factor of the radiographs
was not known and therefore standard magnification 10% was used.
74
pmax /WB
[m−2 ]
4.2 Evaluation of the hips after Salter osteotomy
ϑCE,followup
[◦ ]
Figure 4.7. The correlation between the centre-edge angle ϑCE,followup and the normalized
peak contact joint stress pmax /WB , both determined from the radiographs of the latest
control. The curve represents numerical fit after model equations by these parameters:
R/WB = 2.60, ϑR = 8.2◦ , r = 25 mm.
4.2.3
Results
First we present the biomechanical parameters obtained from the radiographs taken at the
latest control. These biomechanical parameters were computed for hips that have undergone the operation. Figure 4.7 shows the correlation between the center-edge angle and
the normalized peak contact stress pmax /WB , both determined from the radiograph of the
latest control. The correlation coefficient is 0.43 and is statistically significant (P< 0.001).
Figure 4.8, figure 4.9 and figure 4.10 show the correlation between the center-edge
angle determined from the radiographs of the latest control ϑCE,followup and the following
parameters: the normalized gradient of the stress gradpm /WB and the normalized stress
index pI /WB2 and the functional angle of the weight-bearing area ϑF , respectively. The
correlation coefficients and their statistical significance are shown in table 4.4.
The scattering of the data in these figures shows the influence of factors other than ϑCE
on the respective biomechanical parameters, therefore the correlation coefficients (Tab. 4.4)
are the measure of the influence of the parameters other than ϑCE on the biomechanical
parameters. It can be seen from table 4.4 that variation is the highest in the normalized
75
gradpm /WB
[m−3 ]
4.2 Evaluation of the hips after Salter osteotomy
ϑCE,followup
[◦ ]
pI /WB2
[108 m−5 ]
Figure 4.8. The correlation between the centre-edge angle ϑCE,followup and the normalized
gradient of the stress gradpm /WB , , both determined from the radiographs of the latest
control. The curve represents numerical fit after model equations by these parameters:
R/WB = 2.72, ϑR = 7.85◦ , r = 25.45 mm.
ϑCE,followup
[◦ ]
Figure 4.9. The correlation coefficient between the centre-edge angle ϑCE,followup and
the normalized stress index pI /WB2 , both determined from the radiographs of the latest
control. The curve represents numerical fit after model equations by these parameters:
R/WB = 2.70, ϑR = 7.85◦ , r = 24.95 mm.
76
ϑF
[◦ ]
4.2 Evaluation of the hips after Salter osteotomy
ϑCE,followup
[◦ ]
Figure 4.10. The correlation between the centre-edge angle at ϑCE,followup and the functional angle of the weight-bearing area ϑF , , both determined from the radiographs of the
latest control. The curve represents numerical fit after model equations by these parameters: R/WB = 2.60, ϑR = 8.00◦ .
Correlation
pmax /WB – ϑCE,followup
gradpm /WB – ϑCE,followup
pI /WB2 – ϑCE,followup
ϑF – ϑCE,followup
R2
0.433
0.872
0.745
0.938
P
< 0.001
< 0.001
< 0.001
< 0.001
Table 4.4. Correlation coefficients between biomechanical parameters and the centeredge angle ϑCE,followup both determined from the radiographs of the latest control and their
statistical significance.
4.2 Evaluation of the hips after Salter osteotomy
77
maximum contact stress. As in the previous section, the sources of variation are: the variations in geometrical parameters between the patients and the variation in magnification
factor. Because the radiographs of the patients were taken under different conditions, in
different ages and the variation in the size of the radiographs was observed we can assume that the magnification factor highly differs between the radiographs. It was shown in
section 3.3.2 that in case when the magnification is unknown and, or differs considerably
within the population, most appropriate biomechanical parameter that describes the status
of the hip is functional angle of the weight-bearing area.
Next, we studied the change of the centre-edge angle during the followup period ∆ϑCE
(∆ϑCE is equal to ϑCE at the latest control minus ϑCE shortly after operation). If ∆ϑCE is
positive, the center-edge angle increased while if it is negative, the center-edge angle decreased during this period. Figure 4.11 shows the histograms corresponding to the operated
hips (a) and to the contralateral nonoperated hips (b).
The average ∆ϑCE in the population of the operated hips is 3◦ while the average
∆ϑCE in the population of the contralateral nonoperated hips is 9◦ . In the population
of the nonoperated hips, there were only 3 (15 %) in which ϑCE decreases, while in the
population of the operated hips, 11 (27 %) underwent a decrease in ϑCE . The difference
in ∆ϑCE the group of operated and nonoperated hips is statistically significant at the
significance level 0.01 (Wilcoxon’s W two-tailed statistic). In the operated hips we found a
statistically significant positive correlation between the postoperative ϑCE and ϑCE at the
latest control (R2 = 0.29) at the significance level 0.001 (R2 = 0.26) (Fig. 4.12). Dashed
line shows the value of ϑCE in case of ∆ϑCE = 0. For the points that lie above this line
∆ϑCE is positive while for points that lie below this line ∆ϑCE is negative.
Finally we present the influence of the postoperative pelvic geometry on the long-term
effect on the biomechanical status of the hip. Figure 4.13 shows the correlation between the
postoperative ϑCE and the functional angle of the weight-bearing area ϑF determined from
the radiographs of the latest control. The correlation is statistically significant (R2 =0.27)
at the significance level 0.001 (R2 = 0.26).
4.2 Evaluation of the hips after Salter osteotomy
78
Number of hips
(a)
∆ ϑCE
[◦ ]
Number of hips
(b)
∆ ϑCE
[◦ ]
Figure 4.11. The histograms of the change of the centre-edge angle during the followup
time ∆ϑCE corresponding to operated hips (a) and to the contralateral nonoperated hips
(b).
79
ϑCE,followup
[◦ ]
4.2 Evaluation of the hips after Salter osteotomy
ϑCE,shortlypostop.
[◦ ]
ϑF
[◦ ]
Figure 4.12. The correlation between the postoperative center edge angle ϑCE,shortlypostop.
and the center-edge angle taken at the latest control ϑCE,followup .
ϑCE,shortlypostop.
[◦ ]
Figure 4.13. The correlation between the postoperative center-edge ϑCE,shortlypostop. angle
and the functional angle of the weight-bearing area determined from the radiographs of
the latest control ϑF . The curve represents numerical fit after model equations by these
parameters: R/WB = 2.60, ϑR = 11.84◦ .
4.3 Evaluation of the hips subject to developmental dysplasia
4.3
80
Evaluation of the hips subject to developmental dysplasia
Long-term studies of the nonoperative treatment of the developmental dysplasia of the
hip have taken note of relatively high incidence of unsatisfactory outcome [22]. The reason
for an elevated incidence of late radiographic and clinical abnormalities is unclear. It was
suggested [51] the the unsatisfactory outcome is caused by excessive contact stress. Hadley et al. [22] suggest that the clinical hip score is result of the long-term acting stress on
the hip. The concept of this study is to test the relationship between the functional and
biomechanical criteria.
4.3.1
Harris hip score
The Harris hip score is a worldwide distributed method for clinical evaluation of the status
of the hip. This score was introduced by Harris [24] in 1969. The appointment of the
score consists of two parts. The first part is filled in by the patient and it consists of
questions associated with patient’s opinion on pain, ability of gait (limp, distance of the
gait, support by the gait) and functional activities (stair climbing, putting on shoes, sitting,
public transportation). The maximum rate of this part of the Harris hip score is 91 points.
The second part is filled in by the physician after clinical examination (range of abduction,
adduction, flexion, internal rotation, external rotation and absence of deformities). The
maximum available rate of this part is 9 points. Maximum Harris hip score is 100 points.
4.3.2
Patients and method
We studied 11 patients (2 men, 9 women) that had undergone nonoperative treatment
due to developmental dysplasia of the hip at the Department of Orthopaedic Surgery in
Ljubljana in the years 1926-62. The average age of patients at the latest control was 50
years (from 38 to 74 years). The radiograph of the hips was taken with a standard of known
length mounted at the tip of the greater trochanter. The Harris hip score was evaluated
by a physician. Fourteen hips met our enrolment criteria: radiographs that contain all the
necessary bony structures, no diseases that could influence the Harris hip score and no
operation on the hip. For each hip minimum two radiographs were available: the older one
taken at the age of the skeletal maturity and the recent one taken at the appointment of
the Harris hip score.
81
Number of hips
4.3 Evaluation of the hips subject to developmental dysplasia
Harris Hip Score
Figure 4.14. Histogram of the determined Harris hip scores
Magnification of the recently taken radiographs was computed from the known length
of the standard (section 3.3.2). The magnification of the older radiographs was scaled
according to radius of the femoral head as a parameter of the stable length. In contrast
with previous studies (sections 4.1 and 4.2), the body weight of the patients was known
and the biomechanical parameters are expressed by their absolute values (not relative to
body weight). The cumulative exposure pressure parameter pc was computed by using
equation (1.47). The threshold limit was taken to be 2 MPa and the time-weightened
exponent d was taken to be one, according to Maxian [51]. It was assumed that stress in the
hip changes linearly from the value determined at the first control to the value determined
at the most recent control. Other parameters were computed by using equations described
above (section 3.1). The relation between the relevant quantities was described by linear
regression.
4.3.3
Results
The average Harris hip score was 90.5 and its standard deviation was 9.78. The frequency diagram of the Harris hip score, determined in our group of patients, is presented in
figure 4.14.
The relationship between the center-edge angle and the Harris hip score is shown in
figure 4.15. The relationships between the the maximum contact stress, the gradient of
the stress, the stress index, the functional angle of the weight-bearing area, the cumulative pressure and the Harris hip score are shown in the figures 4.16 – 4.20, respectively.
82
Harris Hip Score
4.3 Evaluation of the hips subject to developmental dysplasia
ϑCE
[◦ ]
Figure 4.15. The correlation between the center-edge angle ϑCE and the Harris hip score.
The above biomechanical parameters were determined from recently taken radiographs.
The correlation coefficients are shown in table 4.5. The maximum contact stress and the
cumulative stress show statistical significant correlation with the Harris hip score at the
significance level 0.01 (R2 =0.467), however no statistical coefficient was found between
other biomechanical parameters and the Harris hip score. No statistical significant correlation was found between the Harris hip score and the body weight of the patient (P=0.610)
too, not shown.
Figures 4.15 – 4.20 show the Harris hip score decrease with increasing pmax , gradpm , pI
and pc while the Harris hip score increase with increasing ϑCE and ϑF . It is in agreement
with our expectations and previous studies [22, 50] that high pmax , gradpm , pI is biomechanically unfavorable while high ϑCE and ϑF is biomechanically favorable. The statistical
significance of this study is small due to small amount of data. Only preliminary results
are presented as the gathering of the data is in progress.
83
Harris Hip Score
4.3 Evaluation of the hips subject to developmental dysplasia
pmax
[106 Pa]
Harris Hip Score
Figure 4.16. The correlation between the maximum contact stress pmax and the Harris
hip score.
gradpm
[107 Pa.m−1 ]
Figure 4.17. The correlation between the gradient of the stress gradpm and the Harris
hip score.
84
Harris Hip Score
4.3 Evaluation of the hips subject to developmental dysplasia
pI
[1013 Pa2 .m−1 ]
Harris Hip Score
Figure 4.18. The correlation between the stress index pI and the Harris hip score.
ϑF
[◦ ]
Figure 4.19. The correlation between the functional angle of the weight-bearing area ϑF
and the Harris hip score.
4.3 Evaluation of the hips subject to developmental dysplasia
Parameter
ϑCE
pmax
pmax /WB
gradpm
gradpm /WB
pI
pI /WB2
ϑF
pc
pc /WB
R2
0.053
0.442
0.289
0.233
0.149
0.386
0.223
0.078
0.451
0.286
85
P
0.431
0.009
0.047
0.080
0.172
0.018
0.088
0.332
0.008
0.049
Harris Hip Score
Table 4.5. Correlation coefficients between radiographic, biomechanical parameters and
the Harris hip score obtained by linear regression and their statistical significance.
pc
[107 Pa.year]
Figure 4.20. Correlation between cumulative pressure pc and the Harris hip score
86
5 Discussion
Part 5
Discussion
The objective of this study was to define new biomechanical parameters, derive their mathematical formulations and use these parameters by evaluation of the biomechanical state
of the hip. Also the effect of the magnification factor of the standard antero-posterior
radiograph on biomechanical studies of large groups of patients was investigated. New
biomechanical parameters like gradient of stress gradpm , stress index pI , and functional
angle of the weight-bearing area ϑF have been compared with a radiographic parameter –
center-edge angle of Wiberg ϑCE and biomechanical parameters: maximum contact stress
in the hip joint pmax and cumulative pressure pc in the studies of the biomechanical status
of the populations of hips.
Determination of the biomechanical parameters is based on the recently developed model that enables the determination of the hip stress distribution and analytical expression
for gradpm , pI and ϑF by using data obtained from standard antero-posterior radiographs.
The method is based on the mathematical models for determination of the hip joint re~ in one-legged stance [27] and other for determination of the contact stress
sultant force R
distribution in the articular surface [30]. In these models several simplifications were introduced that influence the accuracy of this method.
In the model for calculation of the hip joint resultant force only five muscles were included and the problem of the muscles forces required in order to maintain balance was solved
by using reduction method [27]. The attachment points of the muscles were corrected for
an individual subject according to the geometry of the pelvis and proximal femur determined from the standard antero-posterior radiograph. For more exact calculation of the
muscle force, it would be convenient if more muscles of the lower extremity were included.
In such case, the problem of unknown muscle forces should be solved by the optimization
method [58]. The more muscles are involved, the more attachment points should be known.
If all of the muscles of the lower extremity were involved, then it would be impossible to
determine the attachment points of the muscles by scaling the standard antero-posterior
radiograph of the pelvis, because we cannot extract the data on the distal part of the lower
extremity from the radiograph. The best way to estimate the positions of the attachment
5 Discussion
87
points of the muscles would be to obtain their directly from tomographic scan (CT or
NMR). However the computerized tomography could not be widely distributed for this
purpose because of its technical complexicity and higher costs compared to the standard
radiograph. Also the radiation dose received by the patient is in CT examination higher
than in standard RTG examination [38]. By using standard antero-posterior radiograph
we cannot compute the hip forces during dynamic activities. External measurements that
are needed for computation of the hip force in these activities [6, 12, 58] are complicated
and could not be used in clinical praxis. Special examinations of the patient to collect
kinematic and kinetic information are required. On the other hand, the method that use
standard antero-posterior radiographs enables to use the data from archives and requires
no additional examination of the patient.
The magnitude of the muscle forces can be determined by the reduction method or
by the optimization method. The method of reduction was used in our study. It follows
from the work of Maček-Lebar et al. [48] that the hip joint reaction force calculated by the
model used in this work is almost the same as when the optimization procedure is applied
(Tab. 1.4). Therefore, mistake in determining the quantities used in this work due to use
of reduction method instead of optimization method is small.
In derivation of the calculation of the biomechanical parameters (pmax , gradpm , pI , ϑF ),
we used the cosine radial stress distribution [35]. It is based on the assumption that the radial stress in the hip joint articular surface can be calculated according to Hooke’s law [45],
i.e. the radial strain in the articular surface of the hip is assumed to be proportional to
the radial strain within the cartilage layer. As the cartilage was assumed to behave as
ideally elastic body, stress was taken to be proportional to the strain. The experimental
results [65] imply, that the stress/strain relationship for the human articular cartilage could
approximately be described by a linear function, i.e. for the cartilage Hooke’s law can be
used. In addition, the femoral head and the acetabular surface are taken to be spherical. In
the normal hips the femoral head and the acetabulum are out-of-round [35]. This deviation
from sphericity is even larger in some abnormalities of the hip. The articular surfaces of
the femoral head and the acetabulum in normal hips were found to have a shape of rotational conchoid [39]. Since the validity of the cosine stress distribution function is based
on the assumption that the acetabulum and the femoral head have spherical geometry, the
deviation from this situation would lead to the stress distribution different from a cosine
function. The Hooke’s law implies that stress in the cartilage is proportional to strain and
5 Discussion
88
to thickness of the cartilage. We assumed constant thickness of the cartilage layer before
deformation. In reality the articular cartilage of the acetabulum is slightly thinner at the
edges [56]. The same strain therefore causes higher stress in thinner than in thicker regions of the cartilage and also higher gradient of stress and stress index at the edges of the
acetabulum. The model could be further upgraded by considering special corrective coefficients of the cosine function [34], that describe deviation from sphericity and by assuming
different thickness of the cartilage on the weight-bearing area.
In the model for the calculation of the maximum contact stress the weight-bearing area,
is assumed to be a spindle-shaped spherical segment [30]. The position and the size of the
weight-bearing area is determined by the inclination of the lateral and medial intersecting
planes with respect to vertical (Fig. 1.29). The shape of the weight-bearing area is overestimated because it does not take into account the acetabular anteversion and the acetabular
fossa that decrease the actual size of the weight-bearing area. However, the region of the
acetabular fossa would not be expected to actually distribute much load [35]. It was noted
earlier that the plane of the acetabulum is not only inclined laterally, but is also inclined
anteriorly (acetabular anteversion). This means, that the anterior acetabular margin does
not extend as far over the femoral head as the posterior margin. The anteversion may be
important in determination of the contact stress, because the contact stress depends on it
as well as on the lateral coverage of the femoral head [9, 21, 46]. For more exact determination of the stress distribution it would be convenient to upgrade the model by taking into
account the acetabular anteversion, especially in dysplastic hips. In dysplastic hips, the
anteversion varies more widely than in normal hips [21]. Due to a smaller weight-bearing
area in dysplastic hips its influence on the stress distribution is higher. It was reported [38]
that three-dimensional acetabular coverage of the femoral head can be estimated by using
antero-posterior radiograph of the hip.
5.1
Magnification of the radiographs
In the section 3.3.2 the role of magnification of standard antero-posterior radiographs in the
biomechanical studies was tested. The effect of the different magnification of the radiograph
was studied in one patient (Fig. 3.5) and also in the group of patients (Tabs. 3.1, 3.2). The
results of the theoretical modelling were applied in the study of normal and dysplastic hips
(Tab. 4.3).
5 Discussion
89
First, the effect of the magnification from 0% to 50% on the biomechanical parameters
was estimated. The parameter that is most influenced by magnification is stress index pI
and the parameter that depends least on the magnification is the functional angle of the
weight-bearing area ϑF . This was expected from the model assumptions for the particular
biomechanical parameters. The functional angle depends on the center-edge angle ϑCE and
on the coordinate of the pole of the stress Θ (Eqn. 3.29). If the magnification of the whole
radiograph is the same, then the center-edge angle is not influenced by the magnification.
It can be seen from the model equations that the coordinate of the pole Θ depends only
on the sum of the inclination of the hip joint reaction force ϑR and the center-edge angle
ϑCE (Eqn. 3.19). Therefore the variation of the functional angle of the weight-bearing area
due to magnification reflects the variation of the inclination of the hip joint reaction force
ϑR due to magnification of the radiograph.
The value of the contact stress depends on the position of the pole Θ, magnitude
of the load R and radius of the femoral head r squared (Eqn. 3.20). These parameters
are influenced by magnification. The equations for the computation the gradient of stress
(Eqn. 3.28) and for the stress index (Eqn. 3.26) are more complex. The theoretical basis
for higher influence of the magnification on these parameters is the error propagation
equation [42].
Next, corrections of the obtained parameters due to different magnifications of the
radiographs were calculated. The calculated magnification factors can be considered as
rough estimates because several simplification were introduced. We assumed that average
magnification of the biomechanical parameter (µ) is constant for a given biomechanical
parameter, i.e. µ is function of Yi . But different geometrical parameters of the pelvis and
proximal femur could redound to the same biomechanical parameter. Then the effect of
the magnification on the biomechanical parameters is different because of different importance of the individual geometrical parameters by the computation of the biomechanical
parameters [14]. For better estimation of the average value of the magnification of the
biomechanical parameters (µ), the variation of the geometrical parameters together with
the variation of the magnification of the radiographs in the population should be included
in the model.
Also, it was assumed that the average magnification of the biomechanical parameter
µ is constant for a given biomechanical parameter. It is not constant as could be seen in
table 3.2, but the differences are small. Therefore, as first approximation its average value
5 Discussion
90
was assumed.
The sample of radiographs with known magnification, used for determination of the
average magnification factor and its standard deviation, was small – consisted of twentyfour radiographs. Therefore the values of the average magnification factor and its standard
deviation may not be a satisfactory representative. In the future larger sample should be
analyzed.
The magnification of the radiograph significantly differs between the recently taken and
older radiographs. The differences may be caused by different technique of X-ray examination, for example different X-ray devices. It has practical importance in the followup
studies. When we will take the average magnification 10% and there is no real differences
in the biomechanical factors, computed values of the biomechanical factor from the older
radiographs will be higher as the computed values from the recently taken radiographs
due to different magnification of the radiographs. Therefore we suggest that by taking the
radiographs in the future it would be convenient to mount a standard of known dimension
at the level of the femoral head center. Then we could calculate also the magnification of
the older radiographs as shown above.
The correction of the average values of the biomechanical parameters and of their
standard deviations due to varied magnifications, is included in the study of normal and
dysplastic hips. Comparing the results obtained by considering the correction and the
results obtained without considering the correction (Tabs. 4.3 and 4.2, respectively), we
can state that the value of t, obtained by t-test, is approximately the same taking into
account the effect of the magnification than without it. According to this we can state that
the variation in the parameters due to differences in the geometrical parameters of the
pelvis and proximal femur is much higher than the variation in the results due to different
magnification of the radiograph.
5.2
Results on populations
The next step after creation of the model is its verification. Previous studies on populations
suggest that distribution of contact stress in the hip may be an important factor which
affects the state of health or disease of the adult hip [22, 37, 41, 51]. In our study, new
biomechanical parameters were defined and their relevance was tested. In test of new
biomechanical parameters on the population of the hips different populations were studied:
5 Discussion
91
normal and dysplastic hips, hips subjected to Salter osteotomy and the untreated hips
subject to developmental dysplasia of the hip.
5.2.1
Biomechanical status of normal and dysplastic hips
The differences in biomechanical parameters (pmax , gradpm , pI , ϑF ) and in radiographic parameter (ϑCE ) between groups of normal and dysplastic hips are shown in table 4.2. All
differences between both groups are considerably and statistically significant at the significance level lower than 0.001. To compare the differences between the parameters, t-values
obtained by t-test are presented in in table 4.2. The parameter, that shows the highest
difference, is the center-edge angle of Wiberg, although it is not a physical quantity. It
could be explained by fact that dysplastic hips are often estimated by using this criterium. Similarly, high difference is exhibited in the functional angle of the weight-bearing
area, because it depends mainly on the center-edge angle. Further, the influence of the
other geometrical parameters on it is lower than in other biomechanical parameters. The
difference in the peak contact stress and in the gradient of stress between both groups is
approximately the same, the difference in stress index is even lower. This could be explained by the influence of the other geometrical parameters that increase standard deviation
assigned to the biomechanical factor and therefore decrease the statistical significance of
the difference.
The relationship of small lateral coverage of the head of the femur by the acetabulum to
the pathogenesis of the coxarthrosis is well explored [22, 63, 70]. Our results also show that
dysplastic hips have higher peak contact stress than the normal hips. This is in accordance
with previous studies [21, 30, 35]. In dysplastic hips we also obtained higher (positive)
gradient of stress, higher stress index and lower functional angle of the weight-bearing
area, than in the normal hips. This is in accordance with our assumptions.
The analysis of the antero-posterior radiographs of healthy and dysplastic normal hips
by using a simple two-dimensional mathematical model of the hip articular stress [43]
showed that in the healthy human hip (corresponding to large enough ϑCE ) the calculated
peak stress varied slowly in a large interval of values and directions of the resultant hip
force (R and ϑR , respectively), while in dysplastic hips its value changed considerably upon
the change of R and ϑR . The variation in R and ϑR represents the variation in size and
shape of the pelvis and of the proximal femur (section 3.2). As it could be seen in figure 4.2
the variation in values pmax /WB is higher in dysplastic hips. Also in the gradient of stress
5 Discussion
92
and the stress index, the influence of R and ϑR in dysplastic hips is much higher. The
values of gradpm and pI are approximately the same in a wide range of values of ϑCE in
normal hips (Figs. 4.3 and 4.4).
In the dependence of gradpm and pI on ϑCE we obtained the change of the sign of the
biomechanical parameters. The change of the sign is caused by the change of the position
of the pole of stress (Eqns. 3.26 and 3.28). If the pole of the stress is located inside the
weight-bearing area then the gradient of stress and the stress index are negative. If the pole
is located outside the weight-bearing area, they are positive. The gradient of stress and the
stress index are zero, when the pole of the stress lies on the acetabular rim. The change of
the sign of the gradpm and pI occurs at the center-edge angle of approximately 22◦ . It can
be seen from the figures 4.3 and 4.4 that this value of the center-edge is approximately a
“border” between the normal and the dysplastic hips. In the clinical praxis the range of
ϑCE from 20◦ to 25◦ is considered to be lower limit for normal, while the value below 20◦
is pathognomic for the hip dysplasia [46]. Our results indicates that the border value could
be related with the shape of the the stress distribution and with consecutive flow of the
intersticial fluid in the articular cartilage as will be discussed below.
In some hips from the group of dysplastic hips, a noticeable deviation from spherical
shape of the femoral head due to degenerative changes was noticed. As mentioned above,
in this case the assumption of the cosine distribution is not valid. Despite of this, these
hips were included into our study. It was observed that in dysplastic hips the shape of
the femoral head is changed into higher radius. According to equations (3.20), (3.26) and
(3.28), the higher the radius of the femoral head, the smaller peak contact stress, gradient
of stress and stress index. However, according to equations (3.19), (3.29) the value of the
functional angle of the weight-bearing area is not influenced by the change of the radius of
the femoral head.
It looks that the increase of the radius of the femoral head in dysplastic hips occurs in
order to reduce stress, gradient of the stress or stress index. Validation of this statement
requires further investigation of the development of biomechanical parameters in the dysplastic hips. For exact determination of stress in the dysplastic hips a three-dimensional
mathematical model of the stress distribution that includes also the anteversion and different shapes of the femoral head and of the acetabulum is needed.
It is well established [46] that ϑCE lower than 20◦ is related to the development of the
degenerative changes, while the value of ϑCE larger than 30◦ is found in healthy patients. If
5 Discussion
93
the ϑCE lies between these values, the center-edge angle alone is not sufficient to predict the
development of the hip joint. A hip joint with a larger ϑCE sometimes develops osteoarthritis more rapidly than one with smaller ϑCE [21]. Thus, the biomechanical parameters
that contribute to the development of the hip should be considered. These parameters are
excepted to correlate with the center-edge angle by its larger and smaller values. All the
biomechanical parameters studied in this work show such correlation. For their further
validation it will be of interest to study the development of the hips with ϑCE from 20◦ to
30◦ and the influence of the particular biomechanical parameter on it.
5.2.2
Biomechanical evaluation of hip joint after Salter osteotomy
It is of interest to study the development of the hips in the longer follow-up study. Therefore
an analysis of the patients that have undergone Salter osteotomy was made. It would
be relevant to study the correlation between the biomechanical parameters shortly after
the operation and at the latest control. Unfortunately, the mathematical model for the
calculation of the resultant hip joint force used in this work does not apply to young
children. Therefore the status of the hip shortly after operation was estimated by centeredge angle.
First, the correlation between the center-edge angle and the biomechanical parameters
was tested to show influence of the parameters other than ϑCE on the biomechanical
parameter. In comparison with the previous section, we found higher correlation coefficient
for gradpm (ϑCE ) and pI (ϑCE ) dependencies than for the pmax (ϑCE ) dependence (Tab. 4.4).
It is caused by low variation of gradpm and pI for larger ϑCE in our group, where the
majority of the hips had ϑCE larger than 20◦ . The functional angle of the weight-bearing
area ϑF was found to be the most relevant parameters, because by using this angle the
noise due to different magnification of the radiographs was suppressed.
We have found (Fig. 4.11) that in the group of the operated hips more underwent a
decrease of the center-edge angle during the followup period than in the group of nonoperated hips. The cause of different development of the operated hips could be: modification
of the blood supply to the acetabulum at the operation or unsatisfactory biomechanical
conditions that influence the development of the hip (e.g. high peak contact stress). In the
first case the change of the center-edge angle would be random, while in the second case
it would depend on the unfavorable values of the biomechanical parameters. Then there
would be a correlation between the biomechanical parameters in the hip and the change
5 Discussion
94
of the center-edge angle. We were not able to determine the biomechanical parameters
shortly after the operation. We could only determine the relation between the change of
the center-edge angle and the biomechanical parameters at the most recent control.
We have studied the correlation between the postoperative center-edge angle ϑCE and
the functional angle of the weight-bearing area ϑF at the latest control. We found on the
average a larger postoperative angle would yield a larger ϑF , which is biomechanically
favorable. A smaller postoperative center-edge angle yields a smaller ϑF , which is biomechanically unfavorable. Our results support the hypothesis that a procedure yielding a
larger weight-bearing area results in a biomechanically more favorable outcome. However,
in order to make a definite answer, more studies, including the improvements of the model
specific for the state of Salter osteotomy, as well as for other osteotomies for treatment of
the dysplastic hips would be required.
5.2.3
Evaluation of the hips subjected to developmental dysplasia
For evaluation of the status of the hip clinical, radiographic and biomechanical criteria are
used. The concept of this study was to test the relationship between the biomechanical
criteria and the clinical score. The Harris hip score reflects the subjective feeling of the
patient (e.g. pain) and ability to practice some activities (e.g. walk, range of motion, etc.).
The biomechanical parameters are based on mathematical models and describe physical
quantities acting on the hip (e.g. stress, gradient of the stress, size of the weight-bearing
area). The evaluation of the score is more or less subjective, while the computed values
of the parameters are more objective. However, there are uncertainties that rise from the
simplifications included in the model. For verification of the biomechanical parameters,
it is relevant to study the correlation between the biomechanical parameters (determine
biomechanical status of the hip) and scores that are used for clinical evaluation. Because
our sample of patients was small the statistical significance of this study is low. Gathering
of the data is in process, however we give preliminary results.
The maximum contact stress and the cumulative stress show statistical significant correlation with Harris hip score at the significance level 0.01. However, the correlation coefficient
for the stress index is close to threshold values. The results indicate a strong dependencies
and therefore a larger sample would be excepted to improve the statistical significance.
From the biological point of view, the value of the biomechanical parameter determined
at a certain time alone may not accurately describe the biomechanical status of the hip
5 Discussion
95
joint because usually, a long period of time is required before the degenerative changes
appears [51]. Therefore in this study the parameter that includes the time exposure was also
calculated. Hadley et al.[22] tested the relationship between the excessive contact pressure
and the long-term outcome in hips subjected to unilateral congenital dislocation of the
hip. These hips were treated conservatively. In contrast with our results no correlation was
found between the clinical scores and the peak contact stress pmax determined at the most
recent control and relatively good correlation was found for cumulative pressure exposure
pc similarly as in our study. Hadley used a modified two-dimensional model of Pauwels [57]
for calculation of the hip joint reaction force and the uniform stress distribution model for
calculation of the peak contact stress [46]. With respect to the models used in our analysis
(sections 3.1.1 and 1.2.2) these models are simplier and less adaptive to the geometry of an
individual hip. The magnification of the radiographs in the study of Hadley [22] was also
not taken into account. It would be interesting to determine the biomechanical parameters
in the sample of Hadley [22] by using our models and then repeat the analysis.
By computation of the cumulative contact stress, it was assumed that stress in the hip
is changing linearly from the value determined at the first control to the value determined
at the most recent control. This assumption is not realistic. For better determination of
the cumulative stress pc we should determine the peak contact stress at more controls.
The Harris hip score varied over a small range (68–100) in our sample of almost healthy
people. This score was developed to evaluate the state of diseased hips and of the hips
after operations [24]. For example in our sample the maximum score 100 was obtained in 2
subjects and the minimum score was 68. Therefore we suggest to use in the future another
type of the hip score that reflects reflects the state in almost healthy hips.
5.3
Effects of stress on the hip
To understand the effect of the load on the hip, it is necessary to take into account mechanisms that influence behavior and development of the hip. In the first step were considered
the radiographical parameters (i.e. ϑCE , ϑU S , ϑAC – see Appendix C), that are determined
by long term clinical experiences and were compared to the physical parameters, that describe processes in the hip (i.e. deformation of the cartilage, flow of the intersticial fluid).
Next step is to better understand regulation mechanisms acting on the cellular and molecular level and connect them with the biomechanical state of the hip. As we are interesting
5 Discussion
96
in the biomechanics of the cartilage, in the following text the effect of the biomechanical
parameters on the regulation mechanisms on the cartilage level is discussed.
The cartilage, tendons and bones develop from a pluripotent mesenchymal tissue. According to Pauwels [57] the differentiation of the cells from the pluripotent mesenchymal
cells depends on the kind of stress applied to the cells. Stretch is the mechanical stimulus
for the formation of the collagenous fibrils. It occurs for example in tendons. The hydrostatic pressure causes the differentiation of the cell into the cartilage cell. There is no specific
exciting mechanism for bone formation, but bone tissue can develop only from the osteogenic cells in an environment where the cells are protected from intermittent stretching
by a rigid anchor (fibrils of calcified ground substance). This theory is assigned as a theory
of the “causal histogenesis of mesenchymal supportive tissue”. By considering this theory,
the specific shape of the cartilage in the hip joint – facies lunata can be explained [46, 49].
The cartilage in the hip is present where the compressive stress varies between the upper
and lower physiological threshold of magnitude. If the stresses rise above or fall below
these toleration limits, the result is a loss or degeneration of the cartilage with consequent
subchondral remodelling and development of coxarthrosis [22, 57].
Pauwels [57] states that the bone is even more sensitive to the magnitude of the stress.
The physiological magnitude of stress is the stimulus to continuous bone transformation
whereby the formation and resorption are balanced. An increased stress stimulates formation of bone and a decreases stress stimulates resorption. The bone condensation in the
roof of the acetabulum is the proof of this fact. This feature is called “sourcil” (eyebrow).
With a normal distribution of joint stress, the sourcil appears narrow and even (Fig. 5.1 a).
As joint stress becomes concentrated due to a lack of head coverage (e.g. by subluxation of
the femoral head), the area of dense bone thickness increases towards the acetabular margin and assumes a more triangular shape – Pauwel’s triangle (Fig. 5.1 b). The similarity
between the shape of the dense bone in the acetabular roof is of practical significance. It
indicates directly both the magnitude and the distribution of the articular stress.
In the normal hip the sourcil is even and narrow. Maquet [49] observed that the centre
of the sourcil does not coincide with the direction of the force and is shifted more laterally.
He suggests the explanation of this fact by the projection of the weight-bearing area by
the radiography. We suppose that this fact can be explained also within the model of
Brinckmann et al. [9] or Iglič et al. [31] that are described above. In these models, the
point of maximal contact stress, the pole of the stress, does not coincide with the direction
5 Discussion
97
~
Figure 5.1. Subchondral sclerosis in the normal hip (a) and in the subluxated hip (b), R
denotes force acting on the hip
of the force. It follows from the model equations that the pole is positioned laterally with
respect to the force. It means that these models support the ideas proposed by Pauwels.
The determination of the position of the center of the acetabular sourcil together with
the position of the pole calculated by the model can be used to further understand the
relevance of the model assumptions.
Mechanical signals do not influence only differentiation of the mesenchymal cells but
are also believed to be significant factors in the initiation and progression of the joint degeneration processes [55, 72]. Deformation of the cartilage is related not only to the fluid
flow and deformation of the collagen–proteoglycans matrix but also with the deformation
of the chondrocytes. Although chondrocytes are not believed to be sensory cells, it is evident that they have ability to respond to a diverse array of biophysical phenomena which
are associated with extracelular matrix. Due to charged nature of the cartilage matrix,
mechanical loading of the joint exposes the chondrocyte population to a complex array of
biophysical signals such as fluid flow, fluid stress, osmotic pressure, electric potential gradients and changes in interstitial pH [20]. Deformation of the chondrocytes (i.e. changes in
shape or volume) may also be involved in the process of mechanical signal transducing [72].
The pathways, through which chondrocyte deformation is transduced to an intracellular
signal which regulates cellular activity, are not know. It has been proposed that mechanical signalling across membrane may be transduced via cytoskeleton. Guilak [20] used
confocal scanning microscopy to perform in vivo three-dimensional morphometric analysis
of the nuclei of the viable chondrocytes during compression of articular cartilage explants.
He found significant decrease in the height and shape of the cells and of the nuclei after
applied compression. Disruption of the actin cytoskeleton using cytochalasin D altered the
5 Discussion
98
relationship between matrix deformation and the changes in nuclear height and shape but
not in volume. He also suggested that the actin cytoskeleton plays important role in the
link between compression of the extracellular matrix and deformation of the chondrocyte
nuclei. Wu et al. [72] developed a theoretical model of the behavior of the chondrocyte
deformation by the compression of the cartilage.
The change in the shape of the chondrocyte under compression is similar to the change of
the shape of the chondrocyte by strain, the chondrocytes are stretched [57]. In the cartilage,
the region of highest strain of the cartilage and therefore of the highest deformation of the
chondrocytes is the superficial zone of the cartilage. This zone is also characteristised by
low contents of proteoglycans, high contens of thin collagen fibrils and by the presence
of metabolically relatively inactive elongated chondrocytes [54]. It is assumed that the
collagen fibrils in the cartilage are oriented into direction of the highest strain [55]. The
collagen in the superficial zone of the cartilage appears to provide the joint with a cartilage
wear resistant protective skin. The superficial region of the cartilage seems to be similar to
the tendons. One of the mechanisms that may contribute to this are the similar mechanical
stimuli acting on the cells.
The transitional and deep zones consist of collagen fibers with larger diameter than
those in the superficial zone and the contents of the proteoglycans in them is higher than
in the superficial zone.
Hypothetically, high deformation of the cartilage causes the production of the collagen.
Collagen serves as a protection against splitting of the structure of the cartilage and as a net
which hold the proteoglycans. In the cartilage where the deformation is higher than normal
the collagen will be produced also in the regions where normally mainly proteoglycans
are produced. The compressive stiffness of articular cartilage is smallest at the cartilage
surface and largest in middle zones. It was suggested that the zones of the cartilage with
high contents of collagen cannot play role in resisting compression [54].
In the models for the deformation of the chondrocytes it was assumed that the gradient
of the stress is zero [72]. As indicates in this work, the gradient of stress in the human hip
cartilage differs from zero. From the hydrodynamics is known, that the gradient of stress
is equal to the force acting on the unit volume of a fluid [25]. It follows from the second
Newton’s law that the acceleration is proportional to the force acting on the body. Although
articular cartilage is a porous viscoelastic material [55], and the flow of a intersticial fluid
is different than the flow of an ideal liquid [39], it can be assumed that the velocity of the
5 Discussion
99
efflux of the intersticial fluid is proportional to the gradient of stress in the pores of the
cartilage. The interstitial fluid flows from the regions of higher stress to the regions of lower
stress in the direction opposite to the gradient of stress. The velocity of the deformation
of the cartilage is then proportional to the efflux of the interstitial fluid and therefore
proportional to the gradient of stress.
In study of dysplastic hips it was observed that gradient of the stress is higher the
dysplastic hips than in the normal hips (section 4.1). The gradient of stress differs not
only in its absolute value but also in the sign (positive in dysplastic hips and negative in
diseased hips). In the following, we will try to explain the effect of the gradient on the
articular cartilage.
In the most of the normal hips the pole of the stress is located inside the weight-bearing
area. The gradient of stress (Eqn. 3.26) on the pole is zero and the flow of the interstitial
fluid on the pole is slow. According to the theory of causal histogenesis, mainly hydrostatic
pressure is acting on the chondrocytes and they product mainly proteoglycans [57].
In the dysplastic hips, the pole is located outside the acetabular shell. The value gradient of the stress is positive over the whole weight-bearing area. Therefore the interstitial
fluid flows more rapidly. The deformation of the chondrocytes occurs and therefore they
probably product mainly collagen that provides resistance to the tension. The collagenproteglycan matrix is changing and therefore the intrinsic mechanical properties of the
tissue are changed. Because the collagen exert little resistant to compression (Fig. 1.8),
the chondrocytes are deformed more and more and the degenerative process develops. The
development of the structural changes occurs, if the deformation of the cartilage crosses
the threshold value. According to this hypothesis, a different representation of the types of
collagen would be found in the normal and diseased hips. It wold be interesting to study
a presence of types of collagen in normal and diseases hips.
There is a difference between normal and dysplastic hips also in the sign of the gradient
of stress at the lateral acetabular margin. According to equation (3.26) the sign of the
gradient of stress expresses a direction of the flow of the interstitial fluid. If the pole of the
stress is located inside the weight-bearing area (gradient of the stress is positive), the fluid
flows laterally. At the edges of the weight-bearing area, where stress diminishes we observe
the bulging of the cartilage due to volume preservation of the interstitial fluid flowing from
the stressed areas (Fig. 5.2). This bulging of the cartilage helps to hold the femoral head in
the acetabulum and increases the weight-bearing area. In the dysplastic hips, the interstitial
5 Discussion
100
Figure 5.2. Schematic representation of the fluid flow and bulging of the cartilage under
a compressive load. Adapted from Nigg, 1995.
fluid flows inside the acetabular shell (pole is located outside), therefore the consecutive
bulging of the cartilage extrudes the femoral head from the acetabulum. This difference
between the normal and dysplastic hips was observed also in our study (Sec. 4.1) and a
possible explanation with respect to the position of the pole could be as written above.
In a similar way a possible role of the acetabular labrum could be elucidated. The
cartilage of the acetabular labrum serves as a barrier against high gradient of stress at the
edge of the acetabulum. Because the labrum is an elastic structure, it could together with
the bulging of the cartilage reduce high gradient of the stress at the lateral border (i.e.
reduce the strain of the chondrocytes).
In the hips with arthritis a decreased viscosity of the synovial fluid [39] and an increased
permeability of the cartilage [55]. Low viscosity together with the higher permeability
facilitates the flow of the fluid and therefore the cartilage deforms quicker.
The homogenously distributed stress causes the deformation of the chondrocyte from
the spherical shape to the ellipsoidal shape [20]. If the gradient of stress in the fluid differs
from zero, that the deformation of the cell will be asymmetric. This may also influence the
metabolism of the cell.
It is known that the incidence of the osteoarthrosis rises with the age. From the theoretical observations it is known that the elastic modulus of the cartilage decreases with
the age [65]. Therefore the same force causes higher deformation of the cartilage.
From this point of view, stress describes the deformation of the cartilage and the gradient of the stress describes the velocity of this deformation. The stress index includes both
5 Discussion
101
these quantities and seems to be also relevant factor. To describe the effect of the gradient
of the contact stress on the cartilage, further studies based on the cellular level are needed.
Also other mechanisms should be considered such as: reduction of the fluid film lubrication
between the articular surfaces, loosening of the collagene network, disruption of the collagene fibers, loss of the proteoglycans [55]. We also did not consider redistribution of stress
in the cartilage during dynamical loading [54] and lateral tensions in the cartilage [39, 55].
102
6 Conclusions
Part 6
Conclusions
The presented biomechanical analysis includes the derivation of new biomechanical parameters – gradient of the contact stress, stress index and functional angle of the weightbearing area. To evaluate the relevance of this parameters studies on the population of
patient have been performed. Also the effect of different magnifications of the radiographs
on the biomechanical parameters was tested. A new, simpler method for derivation of the
equations of the model for calculation of the contact stress distribution was introduced.
For calculation of the biomechanical parameters the computer system HIPSTRESS was
c
c
.
and table calculator MS Excel
adapted under operation system MS Windows
The biomechanical parameters that are the most and the least influenced by magnification are the stress index and the functional angle of the weight-bearing area, respectively.
Therefore for the studies in which the size of the radiographs is unequal or the magnification factor is unknown or largely scattered, it is relevant to use the functional angle of
the weight-bearing area. In this study, a method is proposed to correct the biomechanical
parameters according to the predicted scattering caused by unknown magnification. In the
study of normal and dysplastic hips, it was shown that the variation in results due to
magnification is much smaller than variation in results due to variation in the geometrical
parameters of the pelvis and proximal femur in the population, so that our conclusions
regarding the populations are sound. However, to assess correct values of biomechanical
parameters to an individual hip, we suggest that it would be necessary that while taking
radiographs in the future, a standard of known dimensions were mounted at the level of
the femoral head centers – at the tip of trochanter.
From the study of normal and dysplastic hips results we can conclude the difference in
all biomechanical parameters is considerable and statistically significant at the significance
level lower than 0.001.
The study of the hips subject to Salter osteotomy indicates that the hip stress distribution is important in development of the hip. Because of considerable variation of the size
of the radiographs, the functional angle of the weight-bearing area was proved to be the
relevant parameter. We found that on the average, larger postoperative center-edge angle
6 Conclusions
103
yields a better biomechanical outcome.
In the study of untreated hips subject to the developmental dysplasia of the hip, the
correlation between the Harris hip score and the biomechanical parameters was examined.
These results are only preliminary. The statistical significance of the study is low, because
only a small sample of patients was involved.
The features involving stress gradient are discussed within a frame on the microscopic
level. A hypothetical explanation is proposed that links the processes of the development
of the hip with the motion of the interstitial fluid, deformation of the chondrocytes and the
biomechanical quantities. Our results indicate that the gradient of stress is an important
quantity that affects the development of the hip.
I
Appendix A
Computer system for determination of contact
stress
The computer system HIPSTRESS allows determination of the contact stress distribution
in the hip joint for known pelvic and hip geometrical parameters. These parameters can
be determined directly from the antero-posterior radiograph. HIPSTRESS was developed
in the Laboratory of Applied Physics, Faculty of Electrical Engineering in collaboration
with Medical Faculty at the University of Ljubljana.
This system consists of two parts: one for determination of the load of the hip and the
other for determination of the hip joint contact stress distribution.
~ is calculated by the program based on the mathematical model of
The load of the hip R
the hip joint in the one-legged stance [27, 30]. The hip joint contact stress can be calculated
after solving a relatively simple non-linear algebraic equation [30, 35].
The calculation of the hip joint force requires as input data (Fig. 3.4): the distance
between two femoral head centers l (interhip distance), the vertical coordinate of the
trochanter x, the horizontal coordinate of the trochanter z, the inclination of the femoral
neck ϕ, the height of the pelvis H and the horizontal distance between the most lateral
point on the crista iliaca and the femoral head center C. The reference values of the muscle
attachment points are then adapted for every patient individually. In the one-legged stance
~ lies almost in the frontal plane of the body [27, 64]. Therefore the output of
the load R
the calculation of the hip load in one-legged stance is the magnitude of the load R and
its inclination with respect to vertical ϑR . If the body weight is unknown the load can be
given relative to the body weight R/WB .
The part of the HIPSTRESS for determination of the hip joint contact stress requires
as input: the magnitude and the direction of the resultant hip force which can be obtained
from the first part of program, the radius of the femoral head r and and the center-edge
angle of Wiberg ϑCE (Fig. 3.4). The output of the calculation is the position of the pole
Θ and the value of the maximum contact stress pmax . If the body weight is unknown the
value of the maximum contact stress can be given relative to the body weight pmax /WB .
II
Figure A.1. The input form of the program HIPSTRESS
Figure A.2. The output form of the program HIPSTRESS
III
c
[31]. It
The original version of the HIPSTRESS was written in TURBO PASCAL
consists of two programs. The advantage of this version is that it does not have requirement
c
to operation system and runs on any computer with installed TURBO PASCAL
.
Because of the necessity of the user-friendly system for clinical praxis the new version
c
was developed within this work. This version offers to user
in the Microsoft Visual Basic
a user-friendly graphical interface and the graphical representation of the results. It also
allows the input of the personal data of the patient and manipulation with the data – i.e.
printing the results, saving the results and importing the data from the archives. The data
are saved in text form, so they can be imported to other programs. Using of the program
is described in more details in the help file that is distributed together with the program.
c
This version of HIPSTRESS requires operating system Microsoft Windows 95
or higher
c
version of this operating system. It was also tested on Microsoft Windows 98
. The input
form is shown in figure A.1 and the results of the computation are shown in figure A.2.
For extensive clinical studies the algorithms of the program were included as a macro in
c
the Microsoft Excel
yielding the possibility of easy examination of large groups of patients
and postprocessing of the results using this table editor (e.g. creating graphs, statistical
calculations, manipulation of the data).
~ can be also obtained by other methods [4,
The value of the hip joint reaction force R
6, 7] . In order to calculate the stress with the values of the load the HIPSTRESS2 was
developed. It consists of the second part only as described above and still offers user-friendly
interface.
The computer system HIPSTRESS is available from the authors free of charge only to
be used for scientific purposes and according to ethical principles as described in README
file that is distributed together with the program.
IV
Appendix B
Computer system for determination of
geometrical parameters of the hip
The hip joint morphology can be expressed by many geometrical parameters, such as point,
distances and angles, which can be extracted from standard antero-posterior radiographs.
To determine numerous of parameters by hand is time consuming and subjective. Therefore computer system HIJOMO (Hip Joint Morphometry) was developed to provide the
possibility of objective and easy analysis of a large number of radiographs [36]. HIJOMO
was developed at the Faculty of electrical engineering, University of Ljubljana by A. Jaklič
and F. Pernuš.
Figure B.1. Demonstration of the fitting the femoral head radius with the computer
program HIJOMO
The input for the program HIJOMO are the contours of the pelvis and femur obtained
from the standard antero-posterior radiographs. The radiographs are digitalized using a
graphical table. Then the main geometrical parameters of the proximal femur and pelvis are
automatically determined. This computer system determines some characteristic points of
the pelvis and of the femur (e.g. the margin of the acetabulum, the top of the trochanter,
the most lateral and vertical point of the pelvis) and then computes the characteristic
distances (e.g. the interhip distance, the height of the pelvis, the width of the pelvis)
V
and the characteristic angles (e.g. the center-edge angle, the CCD angle). The curve that
represent the femoral head and the acetabulum are fitted by circles using the least squares
method (Fig. B.1).
The accuracy of the results depends on the accuracy of drawing and digitizing the
contours of the hips. It was estimated [36] that then the distances are determined with the
precision of ±1 mm and the angles with the precision of ±1◦ .
Minimum system requirements: PC (processor 80486, graphical card SVGA 800×600,
color monitor), graphical table NUMONICS. Program runs under operating system Micc
rosoft DOS
5.0.
VI
Appendix C
Radiograph of the hip
The most common radiographic view of the hip is standard antero-posterior view (AP radiograph). This radiograph is taken in normal or intermediate position with the legs in
the neutral rotation and a transverse condylar axis of the knee (confirmed by placing the
patient supine with the lower legs hanging over the table edge) [46].
Figure C.1. Standard anteroposterior view of the hip with denoted characteristic points.
From the standard AP radiograph the following geometrical parameters (points, distances and angles) can be obtained:
• point C – the center of the femoral head (Fig. C.1)
• point T – the site of insertion of the hip abductors on the greater trochanter (Figs. 3.4, C.1)
VII
• point E – the superior rim of the acetabulum (Fig. C.1)
• point F – the inferior rim of the acetabulum (Fig. C.1)
• point D – the lowest and more lateral point of the teardrop (Fig. C.1)
• point L – the lowest point of the ilium at the triradiate cartilage (only by children),
not shown
• point K – the lowest point of the sclerosis in the acetabular roof (Fig. C.1)
• point M – the center of the line from the point E to the point F (Fig. C.1)
• point N – the deepest point in the acetabulum, it lies on the line that is perpendicular
to the line from the point E to the point F and crosses the point M (Fig. C.1)
• r – the radius of the femoral head (Figs. 3.4, C.1)
• s – the radius of the acetabulum (Fig. C.1)
• l – the interhip distance (Fig. 3.4)
• C – the width of the pelvic (Fig. 3.4)
• H – the height of the pelvis (Fig. 3.4)
• γ – the projected CCD angle (Fig. C.1)
• ϑCE – the center-edge angle of Wiberg, the angle between the line from the point E
to the point C and vertical1 line (Figs. 3.4 C.1)
• ϑU S – the Sharp-Ulman angle, the angle of the inclination of the acetabulum, US
angle, the angle between the line from the point D to the point E and the horizontal2
line, not shown
• ϑAC – the acetabular angle, the angle between the line from the point L to the point
E and the horizontal line, not shown . In adults, the point K is taken instead of the
point L, not shown
• ϑACM – the ACM angle, the angle between the line from the point M to the point N
and the line from the point N to the point E, not shown
1
2
vertical line – parallel to longitudinal body axis
horizontal line – perpendicular to vertical line
VIII
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