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NWERSITY LIBRARI ISE

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This is to certify that the
thesis entitled

Three-dimensional Curvature and Kinematic Analysis
of the Human Spine

presented by

Cheng Cao

has been accepted towards fulfillment
of the requirements for

M. S . . Mechanics
degree 1n

 

 

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Major professor

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MSU Is An Affirmative Action/Equal Opportunity institution
cttckcmaflfi-DJ

 

 

 

 

 

 

 

Three-dimensional Curvature and Kinematic Analysis of
the Human Spine
by
Cheng Cao

A Thesis
Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degrees of

Master of Science

Department of Materials Science and Mechanics

1993

Abstract

Three-dimensional Curvature and Kinematic Analysis of

the Human Spine

by
Cheng Can

The purpose of this study was to develop a methodology to quantify spinal
curvature and kinematics by using a video camera motion analysis technique. The space
curvature and torsion, the projection figures and the three-dimensional angles provided
the spinal curvature information. The kinematic analysis included the gross motion of
thoracic spine, lumbar spine, and thoracic cage. One normal young male subject
performing four static voluntary conditions for spinal curvature analysis and three
dynamic voluntary conditions for spinal kinematic analysis. The angles of each point
were reproducible and provided the pattern of curvature along the spine. The space
curvature and torsion were not as reproducible due to the fact that these parameters
combine the information from all three directions. Coupling of motion in thoracic and
lumbar spine has been found for the axial rotation condition and also matched the range
of motion with other studies. It was concluded that the methodology developed in this
work is valuable since it is simple in testing, fast in data analysis, commercially

available, noninvasive, and produced useful results.

ii

Dedication

To my father Menyuan Cao, mother Dongkai He: Thank-you for your
teaching throughout the past two and a half decades. Please trogive me for
the years I stayed away from you for study.

To my wife Bing Deng: Thank-you for your love and patience, and the

happiness you gave me throughout the period of studying in Michigan State
University.

iii

Acknowledgements
The author wishes to express his sincerest appreciation to the following:

To Dr. Robert Wm. Soutas-Little, his major professor, for the guidance and

encouragement throughout the over two-year studying period.

To Patricia Soutas-Little, LeAnn Slicer, and Bob Wells, for their kindness
and patience in helping me become familiar with everything in the lab.

To my friends in BEL, for the happiness and pleasure I enjoyed with them. A
special thanks to: Javid Iqbal Ahmed, Gordon Alderink, Yasin Dhaher,
Kathy Hillmer, Brock Horsely, David Marchinda, James Patton, and Tammy
Reid.

To my graduate committee: Dr. Soutas-Little, Dr. Robert P. Hubbard, and
Dr. Thomas Pence, thanks for their time in reviewing and commenting.

To Brooks Shoe, Inc., and School of Packaging of Michigan State University,

for their generous funding for this and other research projects.

iv

Table of Contents

 

9.9m Pgfi
Introduction 1
Literature Survey

Part 1: Basic Anatomy, Normal and Abnormal Shape of Human Spinal Column 3
Part 2: Quantifying Methods of Spinal Curvature 12
Part 3: Measurement Techniques of Spinal Curvature and Mobility 29
Mathematical Development

Part 1: Curve Representation and Curve Fitting 40
Part 2: Differential Geometrical Analysis 48
Part 3: Kinematic Analysis 53
Experimental Methods 64
Results and Discussion 69
Conclusions 94
References 96

Introduction

Spinal curvature and kinematics have long been treated as criteria in judging spinal
deformity and treatment. The traditional quantifying method for static spinal curvature is
either Cobb's method or Furguson's method. These methods are based on projections on
planes or the tilting of vertebral end plates. However, the spinal curve is formed in a way
that combines the sagittal plane deviation and frontal plane deviation, and also the coupled
vertebrae rotation. This makes the spinal curve rather complicated, and distortion will
occur no matter what directional plane is projected. Especially, for measuring scoliosis, a
single frontal plane projection may include the curvature of a large amount in the sagittal
plane. White and Panjabi (197 8) have also noted that two different spinal curves might
have the same Cobb's angle. Therefore, it has been realized that a three-dimensional spinal
curve requires a three-dimensional quantification method.

This raises a real challenge to the traditional radiological measurements. Brown
(1976) used a three-dimensional radiological technique taking two projections from
different angles in quantifying spinal deformity. Even this method, providing the most
accurate information, had shortcomings and was not popularly accepted. The difficulties
were: 1. adding extra X-ray exposure which is not good for school screening, dynamic
mourning, and long term treatments; 2. the identification of the space position of each
point on the spinal curve is based on the two projections, thus is very time consuming and

not suitable for computer automation.

Due to these challenges and a trend to reduce the usage of radiology, noninvasive
techniques have been of common interest during the past years. Among all the new
techniques, Moire fringe perhaps is the most successful. This technique is based on using
the back surface contour to predict the actual spinal location. It has been proved that the

Moire fringe can provide clinically useful information, and is commercially available.
However, this method is also time consuming, as a large number of grids must be counted
before analysis. This technique is not reproducible: slight changes of back contorn' will
produce big differences in the results. Therefore, even though researchers have suggested
that it could be used as an aid to radiology, these limits restrict its usage. Other
techniques, such as eletrogoniometer, flexicurve, pantography etc., were less studied than
the Moire fringe method. However, they may be very useful in spinal deformity studies in
the future, (Stokes 1984).

The increasing use of video-camera techniques in motion analysis raises the
possibility that this could be also used in spinal curvature and kinematic analysis. Video-
camera wchniques are usually characteristized by three or four high speed cameras with
computer data acquisition and processing systems. The targets made of reflective
materials and the images are digitized by a pattern recognition method. If two cameras
find the same target simultaneously, its three—dimensional location can be easily
determined. Therefore, comparing this technique with either X-ray or Moire fringe, it is
possible that the spinal static curvature and dynamic movement can be quantified tluee-

dimensionally, noninvasively, fast, and with any of the variety of video-camera systems.

By placing reflective targets on the skin over the spinal processes, a spinal process
curve can be fomd to represent the spinal column. Based on this curve, futher analysis
can be done mathematically. Also, by placing reflective targets on appropriate positions of
the trunk, the motions of either the lumbar or the thoracic spine can be measured. The
purpose of this study is to provide the methodology and mathematics to use a video-

camera motion analysis system in spinal curvature and mobility quantification and analysis.

Literature Survey

Part 1: Basic Anatomy, Normal and Abnormal Shape of Human
Spinal Column

LA. Basic Anatomy

The spinal column [Figure 1] consists of two parts: the vertebrae and the
intervertebral discs. All the vertebrae are constructed similarly, with four parts: body,
arch, articular processes and intervertebral formen, [Figure 2]. However, the vertebrae
are different in different anatomical regions and they are separated into different groups.
There are seven cervical, twelve thoracic, five. lumbar vertebrae and the sacrum and
coccyx, [Tim 1]. The size of each vertebrae increases from the first cervical to the last
lumbar vertebrae, and is a "mechanical adaptation" to the increasing loads of the vertebrae,
(White and Panjabi 1978).

The intervertebral discs make up about 20 to 33 percent of the length of the whole
spinal column. Like the vertebrae, the size of the intervertebral disc increases from the
cervical to the lumbar region, (White and Panjabi 1978). However, the ratio of disc
thickness to the vertebral body height is greater in the cervical and lumbar regions and less
in the thoracic region. The greater the ratio of disc to body thickness, the greater the
mobility; the cervical and lumbar spine have larger mobility than the thoracic region,
(Kapandji 1974).

It is always true that composition determines function and this is true with the
intervertebral discs. "The intervertebral disc acts as a cushion between adjacent vertebral

bodies". There are two predominate types of collagen in intervertebral discs that are

responsible for this cushioning function. The first type, type 1, is mostly found in tissues
that are designed to resist tensile forces, such as skin, tendon and bone, while type 2 is

found in tissues that resist compression, (Ghosh 1988). The different distribution of the

 

 

 

% cervical
F- vertebrae
A; lOTdOSiS (Cl-C7]
A—
| ' 'thoradc
I h ' - vertebrae
2“ k” 08“ (Tl-T12)
| “q “
s“
N) ——
~‘ 52
h‘ ‘7‘— lumbgr
‘ ,6 lordosis 2- verre rae
, ‘\ - (LI-L5)
' .
inter-
vertebral
A P discs
1 80mm and coccyx

 

Fig. 1. Lateral View of the Spinal Column. A: anterior, S: Superior, P: Posterior, I:
Inferior. (Gosling et al. 1985)

spawns ' articular
profess Indy process

 

 

 

-- I'
lamina vertebral trauma dde
[cram (norms

Fig. 2. Lateral and Superior Views of Typical Thoracic Vertebrac; (Gosling et a1. 1985)

two types results from factors, such as aging, diseases and curvature of the spine. For
example, there is more type I collagen on the concave side of the curve (which is subject
to compression) than on the convex side (which is subject to tension), (Buckley-Parsons
1984, Ghosh 1988). Brickley-Parsons pointed out that the changes in the distribution of
collagen preceded the development of a curvature and may have led to the formation of
abnormal curvature.

For a standing erect condition of a normal subject, the spinal column is nearly a

straight line in a anteroposterior view and curved in a lateral or sagittal aspect. The curves

viewed laterally [Figure l] are traditionally named lordosis when the concave side of the
curve points backward and kyphosis when it points forward. For a normal person, the

cervical and lumbar regions are lordotic and the thoracic and sacral regions are kyphotic.

LB. Terminology of Spinal Curves

Spinal curves of normal and abnormal subjects are described by direction and
location, and supplementary terms such as structural and non-structural, primary and

secondary, compensatory, simple and compound, major and minor curve, (DeSmet 1985,

Goldstein 1973).

Since most research work and clinical examinations of the spinal curve are based
on projectional X-ray films, the terminologies of spinal curves are given as the sagittal and
frontal plane projections, separately. For sagittal plane or lateral view curves, the terms
indicating direction and location have been introduced. In the frontal plane, the directions
is given by left when the concavity points to left, right when it points to right. Usually, the
curve in the frontal plane is called scoliotic curve. Thus a frontal curve in thoracic region

with its concavity pointing to right can be called right scoliosis in thoracic region.

The terms structural and non-structural refer to the flexibility of the curve. The
structural curve generally is defined as "having a fixed rotation on forward bending and
the presence of a permanent deformation of the vertebrae itself", (Lovctt 1900). The non-
structural or functional curve is a curve that has no structural component. More clearly,
"a curve that is corrected by active or passive bending toward the side of convexity is
considered a non-structural or functional curve while the curve remaining fixed is termed a

structural curve", (DeSmet 1985).

If we consider the curve developing as a function of time, the terms primary and
secondary curves can be easily defined. The primary curve is that curve which developed
first while a secondary curve develops after the primary curve is established. The
secondary curve is mostly a so called "compensatory curve", (Norkin 1992). For example,
the vertebral column of a baby is a long kyphotic curve, maintaining the skeleton as a
mostly upright posture, during growth two lordotic curves develop in the cervical and
lumbar regions while the kyphotic shape is kept in the thoracic and sacral regions. Thus,
cervical and lumbar lordosis can be called secondary curves while thoracic and sacral
kyphosis are called primary curves since they exit at birth. This idea is also used for the
abnormal curves. An abnormal curve that causes another abnormal curve due to-
equilibrium can be called the primary curve and the resultant curve is called the secondary

curve.

The curves are interdependent, and if the head is to remain balanced over the
sacrum. "the region between the head and the pelvis behaves as if it were part of a closed
kinematic chain", (Norkin 1992). Changes in one segment will result in changes in
adjacent superior or inferior segments. If there is a structural curve in the spinal column,
usually there are two compensatory curves above and below developing to maintain the
whole spinal column balance. If they are equal magnitude, the spine is in balance, [Figure
3a]. Spinal imbalance is commonly quantified by using the difference in magnitude
between the upper and lower compensatory cmves, (Leatherman 1988). When the spine
lists to the side of the convexity of the single curve the term 'spinal decompensation' is

used. [Figure 3b].

The complexity of spinal curvature can be described as a simple curve or
compound curve. A simple curve is a single "C" curve. A compound curve consists of

two or more curves. Thus, if there is one curve in the frontal plane, it can be called a

simple fiontal curve and if there is more than one curve, a compound frontal curve

description should be used, such as an "S" curve.

40
40

20 25

a b

. Fig. 3 Measuring spinal balance (compensation). The sum of the
upper and lower compensatory curves is always equal to the size of
the structural curve.

a. When the spine is in perfect balance then the upper and lower
compensatory curves are of equal magnitude.

b. When the spine lists to the side of the convexity of the curve
(deeompensation) the lower compensatory curve rs bigger than
the upper. ln this case the spine is deoompensated by 10 degrees.

(Leatherman 1988)

According to "the degree of deviation from midline", the severity of spinal
ctuvattu'e can be described by terms like "major", "minor" or "smaller" curve, (Schafcr
1987). The major crave is one of a compound curve which is the farthest away from the

midline in contrast to the minor or smaller curves of the compound curve.

LC. Spinal Deformity and Scoliosis

When a normal spine is viewed from the front or back, it will be shown as a
"straight line or subtle, right physiologic curve", (White and Panjabi 197 8). The term

scoliosis originally meant any abnormal deviation of the spine from the midline, but for

many years the term has been restricted to lateral deviation when viewed from anterior or
posterior. In the sagittal plane, the cervical and lumbar curves are normally lordotic and
the thoracic curve is normally kyphotic, the terms for "exaggeration of these curves" are

referred to as "hyperlordorsis and hyperkyphosis", (Schafcr 1987).

Scoliotic curves should be described by their direction and location and by the
supplementary terms described above. Traditionally, scoliotic curves have been described I -
as being either primary or secondary, or alternatively major or minor, but the terms of L-

structural and compensatory curves are now used, (Schafcr 1987).

According to the probable initial cause, scoliosis can be separated into three
subgroups: idiopathic, neuromuscular and congenital. Idiopathic scoliosis means that a
lateral cm'vature of the spine developed and we know of "no congenital spinal anomaly or
associated musculoskeletal condition". Among all the different kinds of scoliosis, 80% to
90% of the cases are idiopathic, (DeSmet 1985). Congenital and neuromuscular scoliosis
are also important because they are often "the most severe and intractable cases",
(DeSmet 1985). Congenital scoliosis is lateral spinal curvatme caused by anomalous
asymmetric development of one or more vertebrae and neuromuscular scoliosis is defined

as scoliosis due to a neurologic or muscular disorder.

Scoliosis is also fi'equently classified according to the skeletal age of the patient.
Curves should be differentiawd by patient age because of the different prognosis and
treatment options in the different age groups, (DeSmet 1985). Infantile scoliosis refers to
spinal curvature beginning during the first 3 years of life, juvenile scoliosis is defined as
scoliosis which happened between 3 years of age and the beginning of puberty and
adolescent scoliosis refers to those at or after the beginning of puberty but before

"completion of skeletal maturity", (DeSmet 1985). Due to the rapid curve progression

10

occurring during accelerated growth, "most patients with scoliosis present clinically during

adolescence".

In order to know how common idiopathic scoliosis is and how many children will
develop severe deformities or problems in adulthood, school screening program for the
early detection of scoliosis have been widely used in the United States and abroad, (Ascani
1977, Asher et a1. 1980, Lonstein et a1. 1982, Segil 1974). In some countries and in more
than 15 states in America, the program is legislated so that the screening is a mandated

53...... 3

school program. In these states approximately three million children are screened
annually, (Lonstein 1988). The Commission on Chronic Illness defined "screening" as
"the presumptive identification of unrecognized disease or defect by application of tests,
examination or other procedures which can be applied rapidly." Since scoliosis screening
examines "well people with no disease, it is not diagnosis, and refers the positive findings
for further evaluation", and this program meets the definition of screening, (Lonstein
1988).

"Scoliodc curves are always accompanied by axial rotation of the vertebrae",
(Stokes 1984). It has been shown both on anatomic specimens (Roaf 1966) and with
computer modeling (Schultz 197 2, 197 6) that the spine rotates when scoliosis develops.
Because the anterior portion of the spine grows faster, if the total length of the trunk
remains comparatively unchanged, the anterior part of the spine has to "deviate sideways
and deviate more than the posterior part", (Roaf 1966). Rotation occurs and the greater
the difference between the anterior and posterior components, the greater must be the
rotation. This rotation permits "the posterior structures to remain the same length while .
the anterior elements elongate", (DeSmet 1985). This is also true for frontal plane curves,
that is, to form a side direction curve one lateral side has to develop faster and rotation

occurs as a result.

11

Spinal rotation brings one more term into the two-dimensional definitions
(scoliosis, kyphosis and lordorsis) and three-dimensional deformities are better described
using the inter-relationship of rotation to the two-dimensional definitions. Thus, none of
the two-dimensional definitions accurately describe the condition and magnitude of the

curve.

l.D. Treatment of Scoliosis

In the 18th century, physicians and surgeons tried to treat most adolescent
idiopathic scoliosis with exercises. However, there was no difference between the results
of exercise treatment in one group of patients and of a "simultaneous control group" of
similar patients without any treatment, (Lonstein and Winter 1988). In 1945, Walter
Blount and Al Schmidt (Blount and Moe 1973) first found a truly successful treamrent
orthosis for this disease, called the Milwaukee brace. However, the Milwaukee brace is
used generally to "prevent progression of a mild-moderate curve until maturity or until a
more appropriate age for surgery, or to prevent postoperative regression or for some
cases of nonoperative scoliosis that do not exceed 40 degrees", (Turek 1977). After this
first success, an enormous number of braces have been developed. There are basically
two types: the higher braces used for thoracic curves, and the lower braces used for
thoracolumbar and lumbar curves. The traditional Milwaukee brace is still the best choice
for "the classic T5-T12 right thoracic curve", but thoracolumbar and lumbar curves are
best treated by "the low-style underarm braces ('I'LSO)", (Lonstein and Winter 1988).
Electrical stimulation treatment (surface stimulation) for reducing the progression of
idiopathic scoliosis has become quite popular in recent years, (Brown et a1. 1984,
Axelgaard et al. 1983). During one study, when electrodes were placed over "the lateral

trunk musculature rather than on the paraspinals", the scoliosis reduction improved

12

threefold, (Shultz et al. 1981). This method can often be used as a good alliterative to
bracing.

Part 2: Quantifying Methods of Spinal Curvature

2.A. Based on X-ray Films

2.A.l. Projectional Plane Curvature Analysis

The measmement of the magnitude of the deformity in scoliosis is usually
performed by either the Cobb (1948) method or the Ferguson (1945) method. "The Cobb
method is probably the most widely used and has been selected by the American Scoliosis
Research Society as the standard method of measurement", (Kittieson 1970, McAlister
1975). Cobb(l948) suggested that "the angle of the scoliotic curvature be measured by
drawing lines parallel to the superior surface of the proximal end vertebrae and the inferior
surface of the distal end vertebrae of the curve". The angle between these lines, or the
angle between perpendiculars of these lines ( if the curve is of small magnitude and the
two lines do not easily meet) is the angle of the scoliotic curvature according to Cobb,
[Figure 4]. The top vertebrae is defined as "the highest one whose superior surface tilts
maximally to the concavity of the curve" while the distal end vertebrae is defined as the
lowest one whose "inferior surface tilts maximally to concavity of the curve", (Cobb
1948). Usually, the proximal and distal end vertebrae of the curve have the least vertebral
rotation. On the condition that the superior or inferior surfaces of the end vertebrae are
not clear, Cobb (1948) suggested "the superior or inferior pedicle surfaces" could be used
instead. Thoracic kyphosis and lumbar lordosis can be measured in a similar fashion,
(McAlister 1975) [Figure 5].

13

C)
Q

Q
‘

  
  

Cl
C

(‘3
2 fig. 4. Measuring spinal curve magnitude in frontal plane (scoliotic angle) by Cobb's
method. (McAlister r975)

Ferguson's method is "a more direct method of curve measurement not that
dependent on the inclinations of the surfaces of the end vertebrae", but is more difficult to
use especially in curves with Cobb angle greater than 50 to 60 dcgrecs,(Georgc 1961).
Not the proximal or distal vertebrae, but the apex vertebrae of the curve usually has the
greatest rotation. The center of this vertebrae is determined by "the intersection of the
lines connecting each superior corner of one side of the vertebrae with the inferior comer
on the opposite side", (Ferguson 1945) [Figure 6]. The centers of proximal and distal
vertebrae are also marked in the same way. Lines are drawn from the apex to the centers
of the end vertebrae and form the Furguson angle, [Figure 6]. If the vertebral body is
wedged, convert the wedge into a rectangle and determine the center. It is sometimes

difficult to select the apex and end vertebrae, or to determine the exact center of the apex

14

vertebrae when it is wedged. Because of the considerable error in the measurement
caused by identifying the exact center, the Ferguson's method is not recommended by
some researchers, (Keim 1978, Kittieson 1970).

 

Hg5.Measudngspinflqnvcmagnhudeinsaginalphm(kyphoficmdlmdodcmgles)
by Cobb's method for scolioticspine. (McAlister 1975)

chastikoglou and Bergquist (1969) did consecutive measurements on X-ray films
of an unchanged scoliotic deformity of the spine by both Cobb's and Ferguson's methods.
They found that measurements performed by Cobb's method gave "consistently higher

values" than those performed by Ferguson's method, however, there was no appreciable

 

15

difference between measurements performed by the two methods. Robinson et al. (1983)
used both methods in measuring scoliotic curvature before treatment. They found a linear
correlation between the two measurements but reported that "the Cobb angle was 1.38
times larger than the Ferguson angle for a given curve". However, the Cobb method is the
preferred one because it is easier to use and more reproducible, and has been accepted as
the standard method.

 

 

Fig. 6. Measuring spinal curve magnitude in frontal plane (scoliotic angle).by Ferguson's
method. (McAlister 1975)

mar-1

16

There is a standard error of measurement of 2.2 to 3.0 degrees with the Cobb
method if the same end vertebrae are used for each measurement, (Beckrnan 1979, Jeffries
1980, Nordwall 1973, Wilson 1983). If several observers pick the end vertebral bodies
independently, the standard deviation of curve measurement is increased to 4.5 degrees

because of selection of different end vertebrae by various observers, (Oda, 1982).

White and Panjabi (1978) stated that Cobb‘s method could not give an accurate
picture of the type of curvature present as "two scoiotic curves which were markedly
different might have the same Cobb angle", [Figure 7a]. They suggested "a more precise
and descriptive quantification of spinal deformity by also measuring the radius of curvatm'e

at the apex of the curves", [Figure 7b].

Voutsinas and MacEwen (1984) measured the sagittal profiles of the spine by
using a new method called indices of kyphosis and lordosis and compared with the result
measured by standard Cobb's method made by one investigator. The indices of kyphosis
and lordosis were determined by (W/L x 100), with L representing the inside length of the
curve and W the maximum inside distance of the curve from the L line, [Figure 8]. This
study showed that Cobb’s method of measurement of kyphosis and lordosis matched well
with the indices in the normal spine but not in pathologic conditions. They found that "the
kyphosis and lordosis indices more accurately represent an are, based on its length and

width, and were easily reproducible".

Jeffries et al. (1980) used a computer measurement method to analyze scoliotic
angle. They first reconstructed the spine mathematically by a computer program and then
measured the scoliotic angle by both the computer method and manual Cobb's method.
With the computer method, the end points on the curve to be measured were chosen as

the inflection points (second derivative equal to zero) and the two tangential lines at the

l7

inflection points made up the scoliotic angle, [Figure 9]. A high degree of correlation
(0.968) between the computer and the Cobb's methods was found.

8)

 

b)

 

Fig.7.Radiusofcmvatnre. a)‘1VvoscolioticspinesmayhavethesameCobb'sanglebut
difi'erentshapes. 'Ihelattermaybedefinedbytheradiusofcurvatme. b)Mathematical
interpretation. (White and Panjabi 197 8)

18

 

 

Eg81hetwospinalcmvanues(aandb)repreecntedbyflfisschcmaficdmwingam
obviously quite different in magnitude. However, using Cobb's method to measure the
Moanitiesfihedegreesofemvaturearcidentical. 'Ihedifl'erencesinthecurvesaremore
aecmatelyreflectedwhcnthelengthofthecmves(L)andthcirrespectivewidths(Wand
W') are taken into consideration. (Voutsinas and MacEwen 1984)

A

 

Fig.9. a)Madrematicaldedvationofdreinflecdonpoints,CandG,anddleapexofdre
curve, pointE. b) Tangents to the curve at points C and G construct the scoliotic angle,

a. (Jemies et al. 1980)

19

Cobb's angle fails to describe the true severity of the curve because it reflects only the end
vertebral bodies and not changes within the curve itself as stated by White and Panjabi
(197 8), but the computer method showed the true difference between the two curves with
the same Cobb angle. Thus this computer method accurately identifies the actual slope of
a curve at the inflection points rather than the tilt of a single vertebral body. Their study
also showed that the computer method was more reproducible with a standard deviation

of only 1.3 degrees.

-' 5-3-1

Appelgren and Willner (1990) separated the standard Cobb angle into three angles, ill
called the end vertebral angles, in a S-shaped scoliotic spine, [Figure 10]. The end
vertebral angle were defined as the angle between each end vertebrae and the horizontal
plane. The three angles were measured and called A, B, C where the middle end vertebral
angle B is responsible for the improvement of the scoliosis in the brace and if end vertebral
angles A and B are not equal, the thoracic curve is asymmetric. HA is greater than B
[Figure 11a], the result of brace treatment was more successful than of the symmetric
[Figure 11b] curves; if B is greater than A [Figure 11c], the result of brace treatment was
not good. All theses could not be observed by using the Cobb method only.

2.A.2. Rotation Measurement

The determination of vertebral rotation is an important part of scoliosis evaluation,
particularly in cases being considered for fusion, (Cobb 1948, Moe 1958, Nash 1969).
Cobb (1948) described a standard technique for measurement of rotation based on "the
position of the tip of the spinous process in relation to the underlying vertebral body".
Values ranged from 0 to 4+; however, he did not correlate these grades with either the

degree of rotation or the clinically deformity. Moe (1958) stated that the spinous

20

A + a - Cbbb

b angle thoracic.anqle

 

 

 

Fig.10. Descn‘ption of the Cobb angle and the end vertebra angle. (Applegren and Willner

1990)
A
n ‘ 3
c

“metric Ara Symmetric Asymmetric B’A

 

Fig.1]. Symmetric and the two difl‘erent types of asymmetric thoracic ’curves in Soshaped
scolioses. (Applegren and Willner 1990)

21

processes were often difficult to visualize and suggested using the "pedicle shadows"

instead. but no actual measurement system was proposed.

Nash and Moe (1969) compared the rotational measurements based on either
spinous process or pedicle shadow displacement and found a method to determine rotation
based on pedicle displacement. At the same time, they correlated the approximate range
in degrees represented by each gradation of rotation. However, the apparent pedicle
offset image depends on a number of factors, including the axial rotation of the vertebrae,
the shape of the vertebral body, and the inherent symmetry of the body geometry of the
vertebrae itself, (Stokes 1986). Although this pedicle displacement method cannot
accurately and quantitatively determine the vertebral rotation, it has been long treated as a
standard.

Stokes et al. (1986) reported a radiographic method for measuring the axial
rotation of vertebrae in degrees by a simple mathematical formula based on the offset of
the pedicle image from the vertebral body center and "a depth estimate". It was found that
measln'ements of vertebral rotation could be made clinically from single-plane radiographs
with a standard deviation of 3.6 degrees based on a study of known rotations of a
radiographic phantom. ’Ihey mentioned that measurement from clinical films were

unlikely to be made more accurately than this because of inherent geometric constraint.

2.A.3. Three-dimensional Measurement

Simple single projectional film, such as radiographic film, or multiple single
projectional films are not able to describe the three-dimensional geometric natme of spinal
deformity, particularly with regard to rotation, even if kyphosis, lordosis or scoliosis are
measured from the sagittal film or frontal film, or other directional films and the rotation is

 

22

measured simultaneously. Three-dimensional deformity requires a true three-dimensional
descriptive method.

Brown et al. (1976) first analyzed the three—dimensional configuration of spinal
segments in vivo by bi-planar radiographic technique. They determined the coordinates of
anatomical points in space by an anteroposterior and a lateral film, and then an orthogonal
Cartesian coordinate system for each vertebrae based on its four anatomical points was
reconstructed, with one axis directed superiorly along the longitudinal line of the vertebrae
and with the center located at the vertebral mass center. Once each set of vertebral bodies
had been assigned to its individual "mass center" coordinate system, the location and
orientation of each vertebrae were uniquely described in terms of a translation vector and
three Eulerian angles which describe the spatial orientation of the individual "mass center"
coordinates with respect to the reference frame coordinate system. The three Eulerian
angles provide the important information of spinal deformity with one angle responsible
for kyphosis and lordosis, one angle for scoliosis and the remaining one for rotation of
each vertebrae. ‘Ihey also described the spinal relationship between adjacent vertebrae by
choosing the reference frame coordinate system as the inferior vertebral coordinate

system.

Scholten and Veldghuizen (1987) summarized the error in using the conventional
Cobb method to measure the magnitude of the deformity of a scoliotic spine. The system
error is made when projecting the three-dimensional spinal shape into a two—dimensional
image on a radiographic film. The experimental error is caused in measuring the Cobb
angle in a fiontal film. They used "a three—dimensional geometrical non-linear mechanical
model to represent the spine as a collection of 17 rigid bodies", each corresponding to a
thoracic or lumbar vertebrae, interconnected discs, the intervertebral joints and the

ligaments. Each vertebrae was defined by local coordinates in a way similar to the study

23

of Brown et al. (197 6). In order to compare the deformity of a scoliotic spine described
with two- and three-dimensional parameters, a space angle, a three-dimensional angle and
the Cobb angle were calculated, with the space angle defined as "the smallest angle
between the two local saggital planes of the end vertebrae of a curve", the three-
dimensional angle defined as "the difference in rotation about the local anterior axes
between the two end vertebrae of the curve "(Eulerian angles), and the Cobb angle as the

angle "between the projections of the local upward axes of the upper and lower end 2

 

A s

vertebrae of the curve". The defined three-dimensional angle was always smaller than the
defined space angle and the differences became larger by increasing the lateral deviation
and axial rotation to the convex side of the curve. For the Cobb angle, the projection
error of spatial angle in a plane in mild scoliosis (Cobb angle 20-30 degrees) was 1.5-2.0
degrees, the observation error for the recognition of the anatomical landmarks went up to
3-5 degrees, the total error of the Cobb method would be 4-7 degrees. They also found
that the Cobb angle was mainly influenced by the lateral deviation and less by the axial
rotation, so they suggested an accurate method to measure the axial rotation in addition to

Cobb's angle measurement.

DeSilva and Yang (1991) suggested a new method different from Cobb's to
precisely describe a scoliosis spine based on a differential geometry point of view. They
modeled the spinal column as a space curve which connecting the centroids of the
transverse vertebral sections. The mathematical curvature of the spine in the sagittal plane
and the frontal plane were measured by projecting the curve into the two planes. The term
"tortuosity' which describes the rotational asymmetry about the vertical axis, with axial
rotation of the vertebrae, of the plane of maximum curvature of the spine. They stated
that the curvature in the frontal plane was a more meaningful measure than Cobb's angle
and the tortuosity described precisely "the phenomenon observed by surgeons in which

there was asymmetric rotation of the vertebrae along the spine".

The three Eulerian angles describe each vertebrae's orientation and "influence the
appearance of the projections of the vertebrae", (Drerup and Hicrholjcr 1992).
Conversely, the angles should be able to be determined from the projections, if the space
shape of the vertebrae is known. Based on this idea, Drerup and Hicrholjcr (1992)
exploited the shape information as much as they could from only fiontal radiographs of
scoliotic patients under the assumption of "parallel projection and a circular cylinder
model of each vertebrae". The lateral tilt angle was equal to "the tilt of the long axis of
the elliptical projection of the end plates". The axial rotation angle was determined by
added pedicle reference points of the vertebral model. From a single frontal plane film,
they got the position parameters, lateral tilt angle and axial rotation, which were plotted as
functions of the longitudinal coordinate separately. In addition, the length of the scoliotic
curve and the location of the apex which are important parameters in describing scoliosis
were also determined. In order to do that and obtain more information, the measurement
points of the spinal column were fitted by a least-square harmonic approximation. They
mentioned that the major advantage of using harmonic functions was that "a direct
interpretation of the function parameters in term of clinical parameters" would be possible,
and the least-square method improved the reliability of the data by smoothing them. The
results were compared to conventional clinical results and a satisfactory agreement was

found.

2.B. External Measurements to Describe Spinal Deformity

It is not possible to know the real condition of the spine from an external deformity
measurement because of the muscles, fat, and ribs lying between the spine and the skin.
Further, there is a difference in the appearance of a deformity in a thin and a obese person

even if they have the same spinal condition. But, as the external surface deformity is

actually caused by internal spinal deformity, it may be possible to detect the internal

deformity by external sm'face parameters.

Shinoto et al. (1981) analyzed scoliosis and kyphosis deformity quantitatively by
the Moire method. They got several three-dimensional points on the back and'fitted those
points by a cubic function. When a peak existed on both the right and left sides of the
curve, the distance from the each peak to the centerlinc and the height difference between
the two peaks were calculated, [Figure 12]. The angle used to estimate the deformity of
scoliosis was formd by the line linking two peaks (double tangent line) and a horizontal
line and it was named "hump angle", [Figure 13]. Another parameter to describe scoliosis
was the distance fiom the centerline to the point midway between the two peaks named
"lateral deviation", [Figure 13]. In spite of some variation, there was a tendency for the
maximum value of "hump angle" to increase with the increase of the rotation of the
vertebrae measmed from X-ray film by the method of Nash and Moe (1969). They
concluded that the parameter "hump angle" could be said as "indicating the rotation of the
spine to some extent". A linear relationship was found between "lateral deviation" and
Cobb angle in degrees of the spine on X-ray film. The coefficient of conelation was 0.82
(p<0.01) and it seemed that "lateral deviation" was useful to estimate the Cobb angle of
the frontal shape of the spine. A linear relationship was also found between "hump angle"
and Cobb angle with the coefficient of correlation as 0.76 (p<0.01).

Willner (1981) compared the results between Moire and X-ray in 216 cases with
structural scoliosis. His study demonstrated that there was a coefficient relationship
between a difference of Moire fringe number on left and light side of the body, which
mean hump height, and Cobb angle on X—ray film. He also reported that there was a

higher coefficient relationship in the case of thoracolumbar or lumbar scoliosis than for

 
         

                 

@W

.‘O.....
- .O.-.......

 

 

 

0
Fig.12. Cross-sectional contour line from Moire fiinge surface topography.(Shinoto et
al. 1981)

 

Fig. 13. Hump angle a: indicating the rotation of the spine, CD is the line of double
tangent, lateral deviation is expressed by ( a - b). (Shinoto et al. 1981)

27

thoracic scoliosis.

Moreland et al. (1981) summarized 558 examinations of 322 scoliotic patients by
comparing the surface topography change to that of the Cobb angle and a good
correlation was found. "Distance pattern shapes" seen on the Moire topographies could
be identified that correlated with "the anatomic regions of the scoliosis". By analyzing the
components of the Moire fringe topogram, it is possible to identify clearly the region of

"spinal involvement".

Stokes et al. (1983) compared the back surface topography with three-dimensional
spine shape in scoliosis. This study showed again that the back surface deformity in
patients was a measure of idiopathic adolescent scoliosis and it was found that the back

surface deformity tends to be considerably smaller than the actual deformities of the spine.

Hierholzer and Drerup (1986) reported a shape analysis study of the symmetry line
of the back surface. The symmetry line [Figure 14] was construcwd in "a point-wise
manner" by analyzing the shape of the horizontal profile where "that point was searched to
divide the profile into a left and a right half of minimum symmetry" and the serials of
"symmetry points" of all horizontal profiles formed the symmetry line. The symmetry line
is actually a three—dimensional curve on the back surface and its lateral view projection can
provide the kyphosis and lordosis angles directly. The surface normal can be found along
this line, which can be used to measure the surface rotation angle. Scoliosis produces not
only a lateral deviation of the symmetry line, but also a rotation of the surface normal.
They stawd that "the symmetry line was in close relation to the line of the spinal processes
even though this statement still required to be verified quantitatively". They concluded
that back surface asymmetry could be described by the symmetry line, which is a
generalization of the medial sagittal profile. Interal deviation and surface rotation, which

were calculated from surface curvature, might be used for an estimation of the three-
dirnensional shape of the spinal midline. The midline can be constructed from the line of
the spinous processes and the vertebral rotation in the case of scoliosis according to

Turner-smith (1983).

 

Fig-14." Surface Normal along the symmetry Line. (Heir-helm:- and Drerup 1986)

Frymoyer (1985) emphasized the importance of external measurements.
"Curvatures which are at risk for rapid progression such as congenital scoliosis are best
evaluated by radiography, however, the major problem of idiopathic scoliosis is cosmetic
except in severe cases where cardiopulmonary function is compromised. As a result, the
external shape measurements and their progression and change may be a more meaningful

clinical measurement, particularly in the assessment of the rib hump deformity". He

29

concluded that "the most intriguing possibility is that external shape changes may be a
more important predictor of unfavorable cosmetic curve progression than the traditional

measures of Cobb angle and Nash-Moe index of rotation".

Part 3: Measurement Techniques of Spinal Curvature and
Mobility

3.A. Radiography Measurement Techniques E
,l

Plane radiography is the most used method for recording and quantifying spinal
curvature. To measure curvature in the sagittal plane (kyphosis and lordosis angles), a
lateral film is required and to measure curvature in frontal plane (scoliosis angles),
anteroposterior or posteroanterior film is required and then either Cobb's or Ferguson's
method is applied. However, if there are coupling conditions, sagittal plane curvature
coupled with frontal plane curvature or rotation of vertebrae coupled with lateral tilt, the
angles measured fiom those films will be erroneous. Movements in sagittal plane
generally occurs without significant lateral tilt or axial rotation, (Pearcy 1985). Portek et
al. (1983) showed that the plane radiographic technique and the three-dimensional
technique in measuring intervertebral movements in sagittal plane correlated well with
each other. Pearcy (1985) proved that lateral tilt is actually accompanied by axial rotation
and sagittal plane movements. As a result, sagittal, anteroposterior or posteroanterior

films can not correctly represent the true spinal curvature.

Measuring of axial rotation fiom single film is performed by Pedicle Shadow
Ofl‘set technique (Nash and Moe 1969). However, because rotation is coupled with
sagittal movements and lateral tilt, "this method will be liable to error unless careful
consideration of the other planes is made", (Pearcy 1986).

30

Since X-ray tubes emit energy in a radial fashion just like any other kinds of points
source, such as the Sun, to accurately represent the true shape by the shadows or images,
it is required that the object be coplanar with the X-ray film plane, or a larger image will
be produced than the object itself. As a result, a correct magnification factor should be
taken into account depending on the tube position. However, since all of the geometric
information is compressed into a single plane, the magnification actually varies with the
position of the part of the skeleton being measured. This will produce a systematic error
in single plane measurement. DeSmet et a1. (1982) found differences in radiographic

measurements between posteroanterior and anteroposterior films.

In order to document the change in the patient during treatment of spinal
deformity, an accurate wchnique of observation is essential. Simple single plane or
multiple single plane radiographs cannot meet this requirement from two respects. First, it
cannot describe the true three—dimensional geometric nature of the spinal deformity,
particularly with regard to rotation. Secondly, because X-ray tubes act as point sources
emitting energy in a radial fashion, any object not coplanar with the X-ray film plane
produces a nonlinear image larger than itself, as a result, single plane radiographs loses the
accuracy required to detect the magnitude of change in spinal geometry needed.

Stereo-radiography provides a three-dimensional measurement of spinal curvature
and requires a three-dimensional location of each landmark on the vertebral column.
Lysell (1969) used a single X—ray source and film plate in a fixed relation to each other
with the subject on a rotating table to produce a double exposure. More commonly, it is
used with two X-ray source positions and a single X-ray film plate. While the X-ray film
is changed between exposures. In each position, two oblique radiographs of subject are

obtained. Brown et al. (197 6) applied a biplanar technique with two X-ray source

 

31

positions and two film plates for describing the three-dimensional configuration of the
spinal segment from an anteroposterior and a lateral radiograph. DeSmet et al. (1976)
determined the three-dimensional location and rotation of each vertebrae using a biplanar
method but with posteroanterior and posterior oblique radiographs. They found that the
frontal and lateral radiographs were not satisfactory projections for identifying the
vertebral landmarks. '

Once pairs of radiographic films are obtained, the analysis accuracy depends on the
identifications of the locations of anatomical landmarks. This is the most inaccurate part
of these wchniques and several methods have been devised to reduce errors introduced by
manual identification of landmarks. Selvik et aL (197 6) placed metal marks on the
vertebrae prior to examination in their study of measurement of the movement of spine.
This method is only applicable for patients undergoing surgical procedures on their spines.
Generally, more landmarks than the three required for three-dimensional analysis are used.
Rab and Chao (1980) used nine landmarks on lumbar vertebrae and those landmarks have
been shown to be repeatedly identifiable. To avoid this redundancy in landmarks,
optimization procedures have been used by Stokes et al. (1980), considering the vertebrae
as rigid bodies. The landmarks on a vertebrae, if it is a rigid body, should have the same
spatial relationship to each other and their coordinates may be modified to satisfy this
relationship. Drerup et al. (1978) described a geometrical reconstruction of the X-ray film
position method to enable the manual positioning of the marks on the radiographs to be
performed more accurately. Once the two-dimensional coordinates of each landmarks on
each radiograph have been satisfactorily identified, they can be entered into a computer
either manually or with a digitizer. The data can be rapidly processed utilizing previously
obtained calibration data of the positions of the X-ray tubes relative to the film plates.
However, due to the complexity of the identification of landmarks and the data entry
procedures, a skilled operator is required.

32

Measurement of spinal mobility is very difficult using radiographic techniques.
Most of the work done by radiography provided only the ranges of spinal movement and
not the pattern of movement dynamically, (Portek et al. 1983, Burton 1986, Adams et al.
1986). Stereo-radiograph techniques have been the most accurate method in measuring
static spinal curvature and should be for dynamic intersegrnental vertebral motion.
Overall, stereo-radiograph techniques have the following shortcomings: a)
stereoradiographic techniques depend on the ability to identify landmarks on the vertebrae;
b) use more exposure than conventional single plane techniques; c) require a complex
marking and measuring procedure and a complicated theoretical development and d) are
not suitable for spinal mobility examinations clinically. As a result, the stereo-radiograph

techniques are mostly used in research laboratories.

Because patients undergoing long-term treatment for spinal deformity require
periodic radiographic evaluations, the risks of radiation have been of increasingly concern.
Regarding radiation risk to breast tissue, Nash et al. (1979) suggested a simple technique
of taking posteroanterior rather than anterioposterior radiographs for simple single plane
evaluation and the additional wchniques, such as "higher speed film, more aggressive
collimation, shielding, screens, and anthropomorphic filtering", can further reduce such
risks, thereby keeping the long-term diagnostic radiographic follow-up of spinal deformity
reasonably safe. For the increasing number of children undergoing periodic follow-up and
treatment, they suggested "the number of different directional radiographic fihn should be
used judiciously depending on the patients' condition". Their study has served as "a guide
and a remainder to the practitioner and radiologist" concerning the risk of radiation to
spinal deformity patients, (Nash et al. 1979). Ardan et al. (1980) used a low dose

radiographic technique in assessment of scoliosis in children. Radiation exposure can be

33

minimized by "using experienced technicians to minimize retakes, proper collimation and

proper monitoring of X-ray tube performance and filtration".

More work has been done (Fabrikant 1982, Gregg 1977, Gross et a1. 1983,
Wagner 1983, Webster 1981) to reduce radiation risk in spinal deformity detection and
treatment. However, risks exist for four conditions which have not been solved. The four
conditions are: a) long-term treatment; b) school screening or radiographic examination
and treatment of children; c) accurate quantification the three-dimensional magnitudes of
spinal deformity by stereoradiographic techniques; d) spinal mobility examination or spinal
deformity examination dynamically. The radiation risk, along with other disadvantages of
using radiographic techniques for spinal deformity, forces researchers to investigate
external or back surface noninvasive techniques to measure three-dimensional spinal

deformity, as they are safe, easy, and economic to use.

3.3. External Measurement

3.13.1. Optical Techniques in Documentation Spinal Deformity

Due to the increasing interest in noninvasive methods to measure spinal deformity,
many optical techniques have been developed in recent years. These methods include:
Moire Fringe Topography, Stereophotogrammetry, Rasterphotography, and

television/computer systems.

Moire fringe photography ( meaning "water silk" ) technique was first described by
Lord Rayleigh (1874) and involved "producing patterns on surfaces of objects by the
optical interference of two separate grating patterns". Takasaki (1970) applied this

method to the human body and was able to demonstrate accurate surface shape contours

by "simple and elegant means". The interest in Moire topography for spinal deformity
examination has increased in the past years,(Adair et a1. 1978, Ohtsuka et al. 1981, Willner
1979 and 1982).

Generally, there are two types of Moire topography: shadow and projection. In
shadow Moire topography, standard gratings are arranged and the shadow of the grating
is projected on the back of the human body using an illumination and a transformed
grating according to the shape formed. By overlapping the standard gratings and the
transformed gratings, Moire fringes can be generated which indicate the contour line of
the back. In projection Moire topography, a grating image is projected on the back to
form a transform grating image. The transformed grating image will then be formed on
the standard grating through lens so as to generate contour line Moire fringes between the
transformed grating and the standard grating. Both methods provide the necessary
information, but the shadow type requires more equipment and is less accurate than the
projection type. The projection Moire fringe topography is the preferred type, (Stokes et
al. 1984, Shinoto et al. 1981).

Moire topography is a noninvasive and noncontacting method for recording the
back shape and showing whether it is symmetrical. However, two particular problems are
associated with the use of Moire topography. First, the fringe contours of the Moire
topography are especially sensitive to the subject's posture. If the posture changes even
slightly, the appearance of the fringe will change markedly. Therefore, for comparative
studies, the subject must be rigidly constrained. Second, the contour itself does not
provide numbers which can be used for further analysis. Various methods (\Vrllner 1979,
Moreland et al. 1981) have been proposed to assist in the analysis of Moire topography
quantitatively, but the measurement and analysis procedures tend to add time and

corrrplication to the Moiré technique.

 

35

In order to find the clinical value of Moire topography in the management of spinal
deformity, Sahlstrand (1985) used it as an adjunct to clinical assessment and
roentgenographic examination in a consecutive series of 139 scoliosis observations. He
concluded that even considering the disadvantages of Moire topography measurement, it
was "a valuable diagnostic tool and represents a practical way of recording the status of
the back in children with scoliosis". When it was used to complete a clinical evaluation,
one could gain better diagnostic accuracy and reduce the need for X-ray. More recently,
Denton et al. (1992) used both radiography and "instant Moire photography" to determine
the degree of progression of spinal curvature. They found that in order to minimize
errors, it required the clinician to interpret all three phases of scoliosis analyses - "physical
examination, Moire photography and Cobb angles" and at best, the accuracy of Moire
analysis is considerably less than that of radiography. Therefore they concluded that it
should be intended only as an adjunctive technique to radiology.

Stereophotogrammetry has been widely used in analyzing object stn'face shape in
biology and mdicine. This method uses two cameras with known viewing angles to take
different views of the object surface. Three-dimensional information is determined by
optical triangulation from two separate two-dimensional pictures. This techniques takes

hours for a skill operator to calculate, measure and analysis.

Based on the stereophotogrammetric method, Frobin and Hierholger (1981)
suggested a rasterstereophotographic method for measurement of body shape to analyze
spinal deformity. This method is similar to conventional stereophotogrammetry except
that one of the two cameras is replaced by a projector with a raster diapositive. The
projector and the camera may be considered as a rastereophotographic system with the

raster diapositive and camera image forming the stereo image pair. Since the raster

36

diapositive is a priori known, only the camera image need be evaluated. Identification of
corresponding points in the image pair is carried uniquely by a determination of the row
and column numbers of the raster intersection. In application, this method has a
equivalent range to that of Moire topography. However, the accuracy of the

rasterstereographic system depends on the geometry of the objects to be measured.

Turner-Smith (1988a) described a television/computer three-dimensional surface

lfi—l

shape measurement system which was designed primarily for human back shape. In this

 

system, a projector and television camera were mounted together in a box which could a
rotate about a horizontal axis. The projector shone a horizontal plane of light, which was

viewed at an angle from below by the television camera, linked directly to a rrrinicomputer.

Three-dimensional coordinates of points on the line of light formed by the plane as it fell

on an object could be calculated with a knowledge of the geometry of the system. This

system has been used by Turner-Smith et al. (1988b) to provide data for the assessment of

scoliosis. They found the correlation of lateral asymmetry from the surface shape analysis

with Cobb angle from X-ray measmement in 119 patients were in the range from r=0.77

to r=0.94. Comparing with structured light methods such as Moire topography and

rasterstereography, this television/computer system is more suitable for clinical use,

eliminating the time required for chemical film processing and digitizing the image.

3.13.2. Other Techniques in Assessment of Spinal Deformity

In an order to measure spinal deformity with simple, easy and noninvasive tools,
many new techniques have been developed and some have been used clinically. Among
all the new techniques, inclinometer, goniometer, flexicurves, spondylometer, and pins

in spinous processes are used most, (Pearcy 1986).

37

Asmussen and Heeboll-Nielsen (1959) described an inclinometric method of
measuring the spine which allowed them to define numerically spinal posture and
movement. However, they chose as points of reference the lowest lumbar lordosis and
highest dorsal kyphosis, which have little anatomical functional significance. Lobel (1967)
designed a special inclinometer that consisted of a dial divided into degrees and fixed to
two plastic buttons. When the two buttons were held against the spine, the weighted
needle remains vertical and indicated the angle of spinal incline. Adams et al. (1986) used

an electronic inclinometer technique to measure lumbar curvature. These electronic

['1‘ 4V ':

inclinometers were small metal cylinders with a flat base which could be attached to the
skin. Each contained a miniattn'e pendulum as part of an optoelectric circuit that had a
voltage output proportional to the angle between the flat base and the vertical. For
measuring lumbar curvature, the inclinometers were attached to the skin overlying sacrum
and L1 spinous processes and the summation of the output angles at the two landmarks
made up the lumbar curvature. This technique requires frequent recalibration of the
instruments and care to avoid skin wrinkling. The electronic inclinometer technique is

suitable for research applications but is not good for routine clinical use.

Similar to electronic inclinometer, Paquet et al. (1990) developed a new
electrogoniometer for the measurement of sagittal dorsolumbar mobility. The
electrogoniomter consisted of a standard potentiometer fixed to a plate at the sacral level
and connected by a flexible slat to the other plate at the thoracic level. This
electromechanical device provided the amplitude, velocity, and acceleration of movements
by means of a computer interface. A high correlation between this device and the
inclinometer technique has been found. This new electrogoniometer can provide
continuous measurements of sagittal dorsolumbar motions with an accuracy comparable to

that of the inclinometer method. However, this device requires special consideration of

38

the antlrropomorphic features of patients and the selection of a flexible slat of appropriate
length for the spinal segment to.

Burton (1986) used a draftsman's flexible curve (flexicurves or flexi-rule) to
measure regional lumbar sagittal mobility. This flexicurve was capable of bending in one
plane only and maintaining an adopted shape that could be transferred to paper. The
measuring technique was performed by three steps: identification of bony landmarks,
moulding of flexicurve to the spine, and measurement of the angle of the transferred curve
on paper. This method provided an accuracy comparable to other methods, but it is
cumbersome and laborious. All three methods, inclinometers, goniometers and
flexicurves, are able to separate back movement from hip motion, and thus have a clinical

advantage, (Pearcy 1986).

A spondylometer was developed by Dunham (1949) for the measurement of spinal
movement in patients with ankylosing spondylitis. This device has a long linkage attached
to a protractor pointer. The protractor is held against the base of the spine and the end of
linkage is held against a point higher up the back. Movement of the trunk results in a
displacement of the linkage and an angular reading on the protractor. Reynolds (197 5)
applied this device to measure spinal anterior flexion and extension from the erect
position. A spondylometer was modified by fixing a protractor over the hinge joining the
two arms of the instrument and was used to measure the change in the angle between the
arms during flexion and extension. After a comparison of this techniques to goniometers,
he found difficulties with technical feasibility and reproducibility of some of the

measurements using the spondylometer.

In order to get more accurate information about spinal curvature and mobility, the

complexity and inaccessibility of the spine has led to invasive techniques on groups of

39

volunteers using pins in the spinous processes, (Gregerson and Lucas 1967 , Lumsden and
Morris 1968). The insertion of Steinnman pins into spinous processes of volunteers has
been used to examine axial rotation of intervertebral joints, this method is obviously of
limited applicability.

40

Mathematical Development

Part 1: Curve Representation and Curve Fitting
LA. How to Represent a Space Curve

Generally, there are three ways to represent a space curve: implicit, explicit and in

parametric form, (Eisenhart 1960).

In three-dimensional space, a single equation usually represents a surface, and at
least two equations are needed to specify a curve. Thus the curve appears as the
intersection of the two surfaces represented by the two equations in Cartesian coordinates
(x, y, 2) as:

(1-1) F(x, y, z) = O , G(x, y, z) = O;
The curve represented by (1-1) is expressed in implicit form.

If the implicit equations (1-1) is solved for two of the variables in terms of the

third, say fory and zin termsofx, theresultcan be written in the form:
<1-2) y=y<x). z=z<x>; A
The curve represented by (1-2) is expessed in explicit form.

The Cartesian coordinates (x, y, z) of the point on the curve can be expressed as
real-valued functions (f1 , f2, f3) of a parameter u:

(1-3) x= f,(u). y = AW). 2 = f,(u);
The curve represented by (1-3) is called in parametric form.

41

Alternatively, in vector notation the curve can be specified by a vector-valued
function:
(no 7=Rmx
where: F=£+yj+zlt3
Rm) = f;(u)i + f,(u)i+ f,(u)12;

3,3,]? are coordinate unit vectors corresponding to x, y, z coordinates;

A parametric representation of a curve specifies not only the curve but also the
particular manner in which the curve is described, (Willmore 1959). This is easily seen if
parameter u is interpreted as time and the curve is considered as the locus of a moving
point. In differential geometry, a parametric representation is not only a convenient way
of description but it is also a useful tool for further study of the properties of the curve,
(W illmore 1959). Theoretically and practically, the independent variable in curve
equations would be chosen as mm in parametric form than one of the coordinates.

Let us consider a point P(x,y,z) and a neighboring point Q(x+Ax, y+Ay,z+Az) on
a curve represented in parametric form, [Figure 15]. Draw the line PO and let the
segment of this line between P and Q be denoted by Ac, while the arc of the curve
between P and Q is denoted by As, then
(1-5) At:2 = A7::1 -t-Ay2 +Az’;
As Q approachs P, we obtain
(1-6) Ac 5 At ;
(1-7) ds’ = dx’ + dy’ +dz’;

- £2 Q: 912:.
r.e. (d9) ”(19 +(ds) 1.

i.e. (x’)2 + (y’)2 «l-(z’)z =1;

42

, dx
wherex =— ,y
ds

 

Fig. 15. The points on a space curve A, as Q approaches to P,
the length ofline segment PQ approaches to that ofcurve
segment PQ.

 

If we let the arc length 3 be time, J(x’)2 «l-(y’)2 + (2’)2 will be the "speed" of a

point travelling along the curve. If we choose arc length as the independent variable, the

curve will always have unit speed, i.e. \Rx’)2 + (y’)2 + (z’)2 = 1. This property simplifies

 

the calculations and makes it easy to understand and track the mathematical inductions.

For the spine, if we choose the independent variable as one of the coordinates
(x,y,z). For example:
(1-8) x = x(z) , y = y(z) , z = z;
where: positive x is anatomical anterior ,
positive y is anatomical left side ,
positive 2 is anatomical up;
We can successfully locate a point on the curve by the 2 value in the standing erect
condition, but we will have trouble in large flexion or extension conditions. It may be
found that different points have the same 2 value. On the other hand, if the arc length is

chosen as the independent variable, Starting at sacrum and increasing as we move a point

43

along the curve directed to the T1 vertebrae. Any point can be located by their different

arc length on the curve no matter what condition is chosen.

In our spinal curvature study, the spinal curve is represented in parametric form
with arc length as the parameter:
(1-9) X = fi(S) . y = M8) . z = f3(S):

where: s is the arc length starting at sacrum and increasing along the curve to the

T1 vertebrae.

LB. Curve Fitting

"Curve fitting is the process of fitting particular classes of functions to discrete
data in an exact or approximate manner", (Lindfield and Penny 1989). If the discrete data
is accurate, exact curve fitting methods will produce one or several functions to join those
discrete data into a continuous curve without losing the accruacy. In the case that the
known data is inaccurate, it is important that the fitting function follows the trend of the
data, rather than passing through each data point. This procedure is called approximate
curve fitting. Among approximate curve fitting methods, the least squares criterion
applied to polynomial or trigonometric functions is most frequently used. Other methods

such as the minimax criterion may also be employed for some special cases.

In this spinal curvature study, discrete data points are collected with small error
and an exact fitting method is the better choice. Usually, there are three methods for exact
curve fitting: collocation, osculation and piece-wise curve fitting, (Lindfield and Penny
1989).

Collocation fitting means that "a function is chosen and its coefficients adjusted so
that it passes through all the data points", (Lindfield and Penny 1989). In the collocation
method, a system of equations is solved to determine the coefficients and ill-conditioning
will cause trouble. Furthermore, the chosen function has its own properties and might not

match the curve to be fitted. This can be seen in Figure 16.

 

 

 

- ll
l2” \
\
t
to ['4-
‘II
3..” «roentgen-tial
3‘”
'.' 4* .Q.
’0‘ ’ \ / . .“
+ ‘s
s
\
32 s
~ +‘ W
>0 : e c e - c ' : c I: c ’ :
0.2 0.4 0.6 V0.0 4.0
A '
K

- Fig. 16. thlc spline and 6th degree polynomial passing through fitted the data points.
(Lindfield and Penny 1989)

The osculation fitting method is defined such that "a function is chosen and its
coefficients adjusted so that it both passes through all the data points and has given values
for some of its derivatives", (Lindfield and Penny 1989). This method has advantages

when one or more values of the derivatives are known at specific points. Obviously, it is

not useful in our spinal curve fitting.

45

Piece-wise curve fitting means that "a function is chosen and its coefficients are
adjusted so that forms of the function fit subgroups of the data", (Lindfield and Penny
1989). Thus the data is represented by a series of different forms of the function in a
piece-wise manner rather than by a single function. The most used piece-wise method is
the mhigsnline. The term "spline" is derived from the name of a device traditionally used
by a draftrnan to join points in drawing a smooth curve. It consists of a steel strip, held in
position by weights. Since a steel strip is subject to the laws of elastic deflexion, its shape,
between adjacent weights, is a cubic polynomial function. The polynomial functions are
different in each interval but at each fixed point there is no discontinuity in slope or
curvature. The cubic spline is mathematically generated to mimic this behaviour, (de Boor
1979). Between points (shxi) and (s, +in +1 ), the function of cubic spline can be expressed
as:

(1-10) x = a, +b,.(s-.s',.)+c,(.r'-s,)2 +d,(s-s,.)3;

Let us derive a method to find the coefficients a, ,b.., c, , d, as follows.

If (s,.x,) and (smxm) make up the interval, they should satisfy the function:
(1-11) x, = 0,;

(1-12) xM = a, +bih, -t-c,.h,2 +d,.h,3 ;

where h, = SM - 3,-
Since we require the slope and the cru'vature at the end of each interval match that

of its neighbor, we have

(1-13) ats=s,. , c, =52}-

(1-14) at s=s,.,,, tie-(515.12

d’x
where I; =73? , or the curvature at s,

substituting (1-11), (1-13) and (1-14) into (1-12) leads to

(1-15) b, .-. (huh-15L hungry.) ;

Now, we have the expressions for the coefficients in terms of 1;. , 1;.“ which can be
determined as following.
(1—16) ats=s, x,’=b,;
Considering the previous interval, the subscript i in (1-10) is replaced by (i-l), then
(1-17) x; = bH + 2c,_,h,_l + ad,_,h’.-.r ;

where ItH = x, - Jt,,_1
The requirement that the slopes of the two joining cubics at the point (s,,x,) be identical,

leads to equating (1-16) and (1-17)

(1-18) hi-l'i-I +2(h‘ +hl-l )n + hm“.1 = 6[(xi+lh: xi) __ (xi ;x‘-L)] ;
r-l

 

where i = 2 , 3, ...,(n-1)
By (1-18), a set of (n-2) equations is formed with unknowns r, . Two more equations are
needed in order to solve it. Assumptions are made at the extreme points, i = 1 and i = n ,

that the curvature at these extreme points be zero. It is then only necessary to solve (n-2)
equations with (n-2) unknowns. An alternative assumption is to make I} a linear

extrapolation from r2 and r3 , and r. a linear extrapolation from r,,_2 and r“. The latter
assumption gives:

(1-19) hurt “(hr +h2)’2 +hr’3 =0 ;

(1-20) It,,_1r,,_2 - (hm,2 + Ir,,_1)r;,_l + hHr. = 0 ;

(1-18), (1-19) and (1-20) make up n equations with n unknowns. The number of
equations may be large, but the matrix of coefficients is diagonally dominant and banded,
and an iterative method can be used since convergence is guaranteed for diagonally

dominant systems, (de Boor 197 8).

47

LC. The Use of Quintic Spline in Spinal Curve Fitting

Let's differentiate the cubic spline functions (1-10) twice, yielding
(1-21) x31: 2c, +6d,h, ;
Obviously, second derivatives of the cubic splines are linear functions. In differential
geometry, we need the derivatives to evaluate curvature, principle normal and binorrnal
vectors. These linear relationships give artificial information for the actual smooth
changes of the second derivatives of the human spine curve. As a result, we need an exact
curve fitting function with first and second order derivaive continuous and smooth. This
requirement forces us to raise the order of the polynomial spline functions. "Usually, there
is an essential difference between splines of even and odd degree, such as, polynomial
splines of even degree interpolating to a prescribed function at mesh points need not exist
and yield the expected extension of cubic spline properties must be modified for splines of
even degree", ( Schumacher 1969). It has been found in practice that there is an
advantage in curve fitting using quintic spline instead of the simple cubic spline for
smoothing first and second order derivatives, (Schumacher 1969 ). Quintic splines have
been used also in cam design and kinematic design of a cam system, (MacCarthy at al.
1985 and MacCarthy 1988).

The quintic spline function is represented in the form
(1-22) x(s) = a, + bih + cih2 + dih3 + e,h‘ + fih’ ;

with h=s-s, for s, Ss $55,,

The evaluation of the five coefficients in (1-22) is quite similar to that of cubic
spline coefficients in (1-10). The following assumptions have been made:
i) x(s) and its derivatives x’(s),x"(s),x”'(s),x” (s) are continuous in s e [31.53,]
ii) x(s,) = 1:,

iii) x"'(s,)=x'"(s,)=x”(s,)=x”(s,)=o;

48

which makes the quintic spline to be natural spline.

The resultant system of equations of 7;. (I; = x” (3,9) is also diagonal dominant and

thus unique solutions to this system of equations are guranteed.

Part 2: Differential Geometrical Analysis
2.A. Tangent

A space curve may be specified by a vector-valued function in parametric form:
(2-1) 7=§(s) ; sl SsSs_;

where s is the arc length and i" is the position vector introduced in equation (1-4).

The positive direction along the curve at any point is taken as that corresponding
to algebraic increase of 8. If ms) is twice differentiable, its derivatives with respect to s
will be denoted by 7’, F" . Let P, Q be the points on the curve whose position vectors are
i" , 7+6? corresponding to the values s , s+53; then 8? is the vector FQ, [Figure 17].

The quotient 5%“. is a vector in the same direction as 8?; and firm-6: becomes tangent
b-vo 53

e e e 0 y e o e o c
at P. Moreover, as shown in prevrous section, 11m 3: rs a unrt vector in the posrtrve
amo

tangent direction at P. We denote this by f and call it the unit tangent at P. Thus

.. . 8? d7
(2 2) t lam 5

Ifx, y, z are the Cartesian coordinates with unit vectors (5,},12):
(2-3) 7=xi+yi+z£

and f=f’=x’i+y’}+z’lt

 

49

A
Fig. l7.Interpretarionoftangentvector:whenQ—I-p, abs-:- 1".

a-r-———-7
~0- - -
L

2.B. Principle Normal. Curvature. Radius of Curvature

The curvattn'e of a curve at any point is generally defined by the change of the
direction of tangent with respect to the changing of arc length, (Willmore 1959, Bisenhart
1960).

Suppose 86 is the angle between the tangents at P and Q (Figure 18), 5% is the

average curvatln'e of the arc PO; and its limiting value as or tends to zero is the curvature
at the point P. We shall denote it by k. Thus

. 60 d0
( ) k 1111183 I 9

Let? bethetangentatPandf-tof atQ(Flglue18),ifthevectorsB-l£ and 877 are
respectively equal to these, then 8? is the vector 5'13 and 66 the angle EBF. The quotient

50

6%: is a ““0? Parallel to 8? , thus as 83 tends to zero 5%5 becomes % and is

perpendicular to f at P. Moreover, since if: and Bi" are unit length, then

. at d; . se._ .
‘2") lyre-Edgar“

where ii is a unit vector perpendicular to f , and in the plane of the tangent at P and a
consecutive point. This plane, containing two consecutive tangents and therefore three
consecutive points at P is called the osculating plane at P. ii is called the principle normal
at P and always points to the concave side of the curve. From equation (2-5),

2..
where 7” =%;;

and the curvature .
(2-7) It = M = Jx'“ + y"2 + 2'”

 

The osculating circle lies in the osculating plane and is tangent to the curve at P.

Its radius p is called the radius of curvature, and its center C, the center of

curvature.(Figure 19) Thus

The center of curvature C lies on the principle normal and the vector PC is equal to pfi or
ii lk.

2.C. Binormal. Torsion. Frenet Frame

The binormal 5 is defined as the unit normal vector to the osculating plane and
forms a right-handed frame with f and ii in the order as f , fi , 5. We have:

51

 

,Fig. 18. Interpretafionofpdndplenormalvectornandcurvmek

 

A
Fig; 19. Radius oromtte.

U‘)
ll
3‘)
O
N)
ll
0

(2-10) f-fi =3-
and Fxri=b,fix5=? ,bx? =55;

If we differentiate the relation b 0? = O, we get:

Wang! ‘

52

i" o 5 + f 0 5’ = 0
i.e. its o8 + i‘ o8’= o
i.e. f 05’: o
i.e. 5’perpendicular to f , and 5’also perpendicular to 5, this means that 5’ should
parallel to ii . Let's write this relation in a formulae:
(2-11) at; = -1:ii

In equation (2—5), the scalar k measures the change of f with respect to the
change of arc length, and here 1: measures the change of 5 with respect to the change of
arc length. 1: is called the torsion of the curve at the point P measuring the rotation of the
osculating plane. It is interesting to note that a curve lies in a single plane if t = 0. If we
model a space curve as a circular helix, 1: is positive when it is right-handed and negative
for a left-handed helix. When 1 is positve, the curve is also called "sinistrosum" and
"dextrosum" if r is negative, (Eisenhart 1969).

The orthonormal system formed by f , ii , 5 is known as the Frenet frame at each
point on the curve. This frame varies its orientation as s traces out the curve. The vectors
5 and ii makeup theosculatingplane, fl and 5 make upthe normalplaneand 5 and f
the so-called rectifying plane. The orientation of each plane can be defined by its normal
vector, for example, the orientation of the oscualting plane can be found from its normal

vector 5 .

When we move the Frenet frame along the curve, it is similar to moving a
trihedron with its principle axes determined by ?, 23,5 . As the are changes, the orientation
of the trihedron changes with its principle axes keeping the alignment with ?, 3,5.

__ '—-'V w

53

Part 3: Kinematic Analysis
3.A. The Reference Coordinates

In order to describe the orientation of the spinal column kinematically, a referrence
coordinate system is needed. Usually there are three choices of reference to describe the
kinematics of spine (White 1979, Stokes 1984): lab or room coordinates, central
coordinates and spinal coordinates, [Figure 20]. Lab or room coordinate system [Figure
20a] is the one fixed in space and independant of the subject. The central coordinate
system [Figure 20b] is one with origin at the sacrum and the axes aligned with the
principle axes of the pelvis. The spinal coordinate system [Figure 20c] also originates at
sacrum but with the z direction axis aligned with the line connecting the ends of the spinal
curve. In this spinal curvature study, all the raw position data relative to lab coordinates
has been transform into the data relative to the central or pelvis coordinates. The
detailed steps are shown as follows:

To get the coordinate system of the pelvis or the reference, the following vector
analysis is required, see Figure 21. Left and right ASIS form a vector, called fiL:
(3-1) EL = Paett asis) - P(lightasis);
where i5 (x) indicates the position vector of target x.

Sacrum and right asis form another vector sit:

(3-2) sir = fittight asis) - P(sacrutn);

The third vector (7? is formed by:

(3-3) (71’ = sir x it. ;

If EL, (71’ are chosen as the two coordinate directions Y, , Z, of the pelvis, the unit
vectors will be:

<3-4) é, =iL/IELI;

posterior

‘

 

 

posterior

 

 

 

 

1c] spinal coordinate system
[b] centl'olcoordlnate system

Fig. 20. The possible coordinate systems in describing three-dimensional spinal
curvature and mobility.

55

(3-5) 2, =t7P/II7PI;
ThethirdcoordinateX,isfoundby:

(3-6) é,=‘é,xé,;

 

(Pelvis Model: XP,YP,ZP form
the pelvis coordinate system)

 

Y

X
(Lab eoordinatesystem: Oistheorigin)

Fig. 21. The construction of pelvis coordinate system.

Thus, X ,, Y, and Z , construct a right-handed Cartesian coordinate system and
point to the anatomical directions:

X ,z to anatomical anterior direction;

Y,: to anatomical left side;

Z ,: to anatomical superior.

The sacrum has been chosen as the center of the reference coordinate system, thus
a transformation of target position vectors is needed to move all the raw target positions
from the lab coordinates to the pelvis system. Two steps accomplish the move:
Step 1: Translation. '
(3-7) 13,00 = 130‘) - P(sacmm);

where: is, (x) stands for all the targets new position vector after translation.

56

Step 2: Rotations.
(3-8) 1520‘) = [A]- 13,00;
where: A

3.0:) 3.0) 3.(Z)
(3-9) [A]= 3,00 3,0) 3,(Z)
3.,(1) 3&2) 3’,(Z)

and e, (1:) indicates the x component of unit vector 6,.
3.8. Kinematic Analysis of Rigid Body Motion

Kinematically, there are six degrees of frwdom for a rigid body, three being
translational and three rotational. The three translations can be decided by the change of a
position vector. In Largrangian mechanics, the three degrees of freedom of rotation are
three independent generalized coordinates, which are most common parametrized by Euler
angles, (Greenwood 1989).

Euler's angles are the traditional way to analyze three-dimensional rigid body
motion. However, it has been found that, for clinical usage, this description of human
motion is not easily understood by clinicians.- Grood and Suntay (1983) provided a way to
describe three-dimensional knee joint motion in a manner which "faciliates the
communication between biomechanician and physician". "Considering the angular
position and the corresponding rotational motion between two arbitrary bodies, three
spatial axes about which the corresponding rotational motions occur can be specified in
order to decide the angular position in three dimensions". A "nonorthogonal joint

coordinate system" is formed with unit base vectors of this coordinate system denoted as

3,, 32 , 6,. Two of the axes are called body fixed axes since they are embedded seperately

57

in the two bodies whose relative motion is to be described. Their direction is specified by
unitbasevector é, inbodyA,andé‘3 inbodyB. Thethirdaxis é, isthecommon

perpendicular to the body fixed axes, its orientation is given by

6: x5“

'33 xé,|

(3-10) 2, =

Because of the different nature of 32 from body fixed axes 2, and 23, 22 is called

the floating axis. This floating axis is equivalent to Euler's "line of nodes".

 

In this spinal curvature study, a "joint coordinate system" has been used in the
description of several types of movements: the orientation of thoracic cage relative to
pelvis, the gross motion of the cervical spine, the gross motion of the whole thoracic
spine, the gross motion of the whole lumbar spine, the orientation of each point on the
spinal curve relative to pelvis and other adjunct analysis. In each of these analyses the
relative movement between two rigid bodies were calculated and a joint coordinate system

was formed as follows:

In Figure 22, xyz is the coordinate system of rigid body A and XYZ is the
coordinate system of rigid body B with:

)2 , 52: positive to the anterior direction;

17, y: positive to the left side;

2 , 2 : positive to the superior direction.

)2 is chosen from A as one of the two fixed axes and noted as 2,, i is chosen from

B as the other fixed axis and noted as 2,. The floating axis 2, is found from a cross

product:

58

 

N) N)

(3-11) 5, =f

N’ N)

 

 

In this "joint" coordinate system, E2 is the axis about which flexion-extension
occurs (we call é, as the flexion/extension axis), 23 is the rotational axis and E, is
responsible for side bending. Let's define:

q): flexion-extension angle ( flexion + );

0: rotaional angle (to right + );

(p: lateral side bending angle (to right + ).

Then:
(3-12) 22,-2 =cos (90+(p) =9 o=- asin ( 22.2);
(3-13) 62052 = cos (90 + 0) => 0 = - asin ( 2202);

(3-14) é,-é,=eos(9o-¢) = o = asin(é,o 2,);

21
x floating

e2

 
 

“’2‘

Joint Coordinate system

Fig.22 : Joint Coordinate System Formation

59

33.1) To calculate the movement of thoracic cage relative to pelvis. Three targets have
been placed on the surface points over bony landmarks of the thoracic cage at sternal
notch (D), xyphoid (E) and T1 spinal process (F), [Figure 23]. The position vectors of
these points were used to form the thoracic coordinate system. Then:

(3-15) 55 = P(D) - Pas)

(3-16) ii) = fiat) - 15(1))

The vectors 58 and F-b form the sagittal plane of the thoracic cage, and the normal
vector points to the side:

(3-17) im =( F'bx 15's )/I F'bx 6'15 |

and (3-18) 2m = Eli/l 55 I

therefore (3-19) 22m = fa“) x 2m

)2 (T), 5(1) and 2 (T) are the unit vectors of the coordinate system of the thoracic
cage. To calculate the orientation, this system has been treated as that of rigid body A and
the reference system (the pelvis ) has been treated as that of rigid body B, a "joint
coordinate system" has been formed as mentioned before.

0: Siemal notch '
E: hold
F: process

 

Fig.23 : Thoracic Cage Motion Calculation

3.B.2) To calculate the gross motion of the cervical spine[Figure 24], the gross motion of
file cervical spine actually can be expressed by the relative orientation of head to thoracic
cage, therefore head and thoracic cage were modeled as the rigid bodies A and B. Three
targets were put on the head locating at the right and left temple and the forehead, such
that the targets formed a anatomical transverse plane.

(3-20) far) = M'NII hfivl

(3-21) 201) =(0'1'v x Afiv )Ilo'z'v x Mill

and (3-22) fiat) = for) x 2m)

12(H), 501) and 2(H) are the unitvectorsofthe coordinate system ofhead and
if ('1'), 17(1) and 2 (T) are those of the thoracic cage. A "joint coordinate system" has
been formed between the two bodies and the gross motion of the cervical spine was
calculated, (Soutas—Little and Cao 1993).

w . I
Head .‘ _
“M“ L Xi"! whearentpte

Yl'l] N: Right Temple

: d
Trunk 0 Forehea

Model XII]

Figure 25 Cervical Spine Motion Measurement Targetting

3.B.3) To calculate the gross motion of the thoracic spine, the rigid body A was chosen as
the thoracic cage formed by T1, sternal notch, and xyphoid process and B as the lower
thoracic body formed by T10, T12, and xyphoid process. Targets were putted on these
bony locations. [Figure 25]

61

(3-23) 2031') = 11'1/l1r'1'1 l

(3-24) fair) =(1i'E x 171 )/| 175 x 1'11 |

and (3-25) 12031) = fair) x 2031‘)

where: 12 (BT), fair) and 2 (BT) are the unit vectors of lower thoracic rigid body B. A
"joint coordinate system" has been constructed between the thoracic cage and lower
thoracic body and the gross motion of the thoracic spine has been calculated.

3.B.4) To calculate the gross motion of the lumbar spine, similar to the procedures of
3.B.2), the upper body was chosen as the lower body of the thoracic spine and the lower
body was chosen as the pelvis. Therefore, 12 (BT), 17(BT) and 2 (BT) are the unit vectors
oftherigidbodyAand 12, Y and 2 arethe unitvectors ofrigid body B. A "joint
coordinate system" has been constructed between the two bodies and the gross motion of
the lumbar spine has been calculated.

D: Sterne! Notch
E: Xyphoid Process
F: 1'1

H: T10

I: 1'12

 

 

Figure 25.: Thoracic Spine Movement Heasurement Targetting

62

3.B.5) To calcualate the orientation of each point on the spinal curve, the spinal curve can
be thought as the path of the movement of a trihedron. The orientation of this trihedron
on the spinal curve has been related to the anatomical orientation of the trunk. As shown
in Figure 26 , one more target has been placed on each of the spinal processes than was
targetted originally. The upper targets and the base targets were fitted by quintic splines
seperately, and two curves were constructed, [Figure 26a]. Between two adjacent targets
on the same curve, ten small segments were made with equal length as shown by Figure
26b.

base curve

  

 

 

 

linkage

(a) (b)
Fig. 26. The coordinate system (aAp, is , ’t‘) of the trihedron construction based on points

on the base and up curves.

Pointsrandnareadjacent on thebasecurveandmis the point on the upper curve with
thesamecountfromDasthatofnfromC. iisthetangentialunitvectorotthebase
curveatn, ii anth areformedbym,n andr. Aunitvectorli‘t is formedasfollows:
(3-26) fs=t7ixv31|5ixt7il;

where it points anatomical lateral side;

The a?) unit vector is calculated by a cross product:

(3-27) 212:1? xi;

A
where ap points anatomical anterior;

 

63

A A
The three unit vectors a}, Is , and 1 construct a right-handed coordinate system of the
trihedron at each point on the spinal curve. The trihedron is treated as rigid body A and
the pelvis as B, and a "joint coordinate system" is constructed between the two bodies and

the orientation of each point on the spinal curve can be calculated.

 

Experimental Methods
1. Apparatus and Working Space Calibration

The spinal curvature study obtained position and motion data using a motion
analysis system made by Motion Analysis Cooperation. Four video cameras with 60 Hz
sampling frequency were used to take the images made up by pixels of reflective targets in
a calibrated space. The three-dimensional centroid location of each target was determined
based on at least two different camera images with more than 3 pixels of each by using the
Expertvision Three-dimensional (EV3D) digitizing program. Each target's location was
expressed in the lab coordinate system with its position vector:

i5 = xi + y} + z];

wherei, }, it arethebaseunitvectorsofthelabcoordinates.

The calibrating of the working space was performd in the following procedures:
1) Choose the best space volume for the study. The smaller the space volume , the more
accurate the position vectors will be. For this study, the volume was chosen as 1x1x0.8
m, where the last number indicates the height of the volume, [Figure 27].
2) Sixteen control points (targets), four on each standing structure, were place on the
boundaries of the chosen volume. The positions of those points were decided for
calibration, as shown in Figure 27.
3) Position the cameras such that each could see all the points and had the closest view.
5) Calibrate each control point position to match up the designed coordinates.
6) Run data acquization and calibration programs to get calibration results.

The residual number was reported for each camera and the closer the residuals for

the cameras, the more accurate the calibration. For all the tests in this study, the residual

—- I

65

values were within 0.13 apart and less than 0.35, which fitted the system requirements of
residual values of less than 2.0.

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Figure 27 Calibration space and control points.

2. Experimental Preparation

In this study, the targets located on the spinal processes were target linkages. Two
small targets were rigidly connecmd by a rigid light bar on the ends, see Figure 28a. To
calibrate the linkage for our purpose, it was mounted on a level table surface and adjusted
to make the bar perpendicular to the table, see Figure 28b. To calibrate the linkages to be
used as a system, they were mounted on the table again and the bars were adjusted such
that they were parallel to each other and perpendicular to the table, see Figure 28c.

The subject used for the test was a young male with normal spinal condition. The
purpose of the research was explained and a signed informd consent obtained, (IRB #89-
559)

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67

Two targets were mounted at the left and right ASIS and another two at the sternal notch
and the xyphoid. The linkages were mounted on the spinal processes such that the bars
were perpendicular locally to the skin surface. The bony process levels chosen were S 1 ,
L5, L3, L1, T11, T9, T7, T5, T3, and T1.

3. Testing Procedures and Data Analysis

During testing, the subject was located in the calibrated space. Four static
conditions were chosen for spinal curvature analysis: standing erect, standing with
maximum right side bending and standing with forward flexion, and standing with right
rotation. Four seconds of data were collected for each condition or each trial. Three
dynamic conditions were chosen for spinal kinematics analysis: bending, flexing, and axial
rotation. After tracking the target images, the position of each target was obtained in the
lab coordinates. All the data was smoothed once by EV3D programs. The position data
of each target of each frame was then combined in a position file. i

The data analysis was performed using a SUN4/260C workstation. The program,
creawd by this author, for spinal curvature analysis accomplishes these steps of analysis:
a) transforming all the position data fiom the lab coordinate system into the pelvis
coordinate system;

b) curve fitting by quintic splines for the targets on the spinal processes and upper;
c) differential geometrical analysis for the base spinal curve;

(1) calculating the orientation of each point on the spinal curve relative to pelvis.

Another program, created by this author, for spinal kinematics analysis, performs
the followings analysis:

a) calculation of the movement of the thoracic cage relative to pelvis;

 

b) calculation of the gross motion of the thoracic spine;
c) calculation of the gross motion of the lumbar spine;

69

Results and Discussion

Spinal Kinematics Analysis
1. The Movement of Thoracic Cage Relative to Pelvis

Because of the su'uctln'e of the thoracic cage, it can be easily modeled as a rigid
body. Also because the thoracic spine is less mobile than lumbar spine, it is possible to use
the movement of thoracic cage relative to pelvis to approximate the gross motion of the
lumbar spine. Figure 29 shows the time history of the three-dimensional angles of the

subject performing three different voluntary movements.

Figure 29a is three directional angle time historys for voluntary flexion/extension
movement. The flexion(+)/extension(-) angle started at -10 degrees, which indicates the
thoracic cage tilted 10 degrees backward for the starting posture. The subject forward
flexed first to about 55 degrees, then backward extension to about -35 degrees. Bending

and rotating angles remained approximately unchanged throughout the movement.

Figlue 29b is of voluntary side-to-side bending movement. The side-to-side
bending angle started at almost 0 degree, bending to left first to about -45 degrees, then to
right to about 45 degrees. It shows the symmetry for the subject in side-to-side bending
movement. The remaining two angles, flexing and rotating, remained nearly unchanged

throughout the movement.

Figure 29c is of voluntary axial rotation movement. The rotation angle started
close to 0 degree, rotating to right first to about 40 degrees, then to left to about -40
degrees. It shows the symmetry for the subject in axial rotating movement. The flexing

70

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71

angle remained unchanged throughout the movement. However, the bending angle varied
with the rotating angle in a way that always went to the opposite direction. This is the
well known spinal movement coupling. The ratio of the two angles, primary movement
and coupled movement (here is rotating and bending), is called the coupling ratio. Here
the coupling ratio is about (4 degrees rotating) : (-1 degrees bending). The minus sign

indicates the coupled movement goes to the opposite direction of the primary movement.

2. The Gross Motion of the Thoracic Spine and the Gross Motion of the Lumbar
Spine

Even though the thoracic spine movement is restricted by libs, the libs themselves
are deformable and the lower part of thoracic spine is free of these restriction. Therefore
the movement of thoracic cage combines the movements from both lumbar and thoracic
spine, with the thoracic spine contributing less. It is possible to separate these two parts
by different rigid body modeling. Figure 30 and 31 show the time history of the three
angles for the lumbar spine and thoracic spine druing the three voluntary movements.

Figrne 30a is the gross motion of the lumbar spine during voluntary
flexion/extension movement. The flexing/extension angle started at -10 degrees, which
meant the initial posture for lumbar spine was -10 degrees lordotic, forward flexing first to
about 40 degrees, then backward extending to about -30 degrees. These numbers show a
large mobility of the lumbar spine for the subject. The other two angles, bending and

rotating, were nearly unchanged throughout the movement.

Figure 30b is the gross motion of the lumbar spine during side-to-side bending
movement. The side-to-side bending angle started at about 0 degree, bending to left first
to about -25 degrees,

Figure 30: Thoracrcsprne motion: drreedifl’erentconditions

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73

then to right to about 25 degrees. It shows the symmetry of the subject performing side-

to-to bending. The other two angles, flexing and rotating, remained almost unchanged.

Figure 30c is the gross motion of the lumbar spine during axial rotating movement.
The rotating angle started at 0 degrees, rotating to right first to about 20 degrees, then to
left to about -20 degrees. The flexing angle remained unchanged, but the bending angle
showed a slight variation. When rotating to one side, the lumbar spine bent to the same
direction. This is the coupling characteristics of lumbar spine which has been discussed by
many researchers (White and Panjabi 1979). The coupling ratio here is about (6 degrees
rotation) : (1 degrees bending), which also matches with other researcher's, (Pearcy 1985).

Figure 31a is the gross motion of the thoracic spine as a whole during
flexing/extension movement. The flexing/extension angle started at 0 degrees, forward
flexing first to about 18 degrees, then backward extension to about -8 degrees.
Comparing this result with that of lumbar spine, the thoracic spine showed a more
restricted mobility in the sagittal plane. The other two angles, bending and rotating,
remained almost unchanged throughout the movement.

Figure 31b is the gross motion of the thoracic spine side-to-side bending
movement. The bending angle started at 0 degree, bending to left first to about -20
degrees, then to right to about 20 degrees. These numbers showed a large mobility of
thoracic spine in the side-to-side bending movement, and also a symmetry in the two
directions. The other angles, flexing and rotating remained predominantly unchanged.

Figure 31c is the gross motion of the thoracic spine axial rotating movement. The
rotating angle started at almost 0 degree, rotating to right first to about 18 degrees, then
to left to about ~18 degrees. The flexing angle again remained mostly unchanged. There

74

Figure 3 1: Lumbar spine motion: tine: difi‘erent conditions

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75

was a strong coupling between rotating and bending for this movement, the coupling ratio

was (2 degrees rotating) : (-1 degree bending).

Spinal Curvature Analysis
1. The Projection Figures of the Spinal Curve

As mentioned previously, the targets on and above the spinal processes have been
fitwd by quintic spline functions to get a base curve and upper crn've. The base curve was
the actual spinal cm which joined the spinal processes. It is possible to view this curve
from different view points. In this study, this spinal curve has been projected onto three
orthogonal planes: sagittal, frontal and transverse which were formed by pelvis
coordinates X & Z, Y 8: Z and X & Y respectively. Figure 32 to 35 are the projections
for each of the three trials of the primary movement conditions.

Figure 32 is the projections of the curve of standing erect condition. Based on the
sagittal plane view, the thoracic kyphosis and lumbar lordosis can be identified. Since the
current measurement technique is a noninvasive, external measurement method, the
thoracic kyphosis and lumbar lordosis can not be compared directly with either
radiography or other noninvasive methods. For better use of these projections and other
methods in this study, a normative standard data base should be set up. Based on the
frontal plane view, it is possible to identify the testing subject's scoliotic condition. The
transverse view provided us a clear impression of the excursions in side-to-side and
anterior/posterior directions of the spinal curve. Also, the area occupied by the spinal
curve in this projection can serve as another spinal deformity criterion, the larger the area,

the more deformity of the whole spinal column.

76

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The reason for plotting the three projections is that different possible spinal
deformities can be represented. Figure 33 is the projections of the curve for rotating to
light condition. Comparing these projection with those of standing condition, the sagittal
plane and frontal plane views are very similar to those of standing condition, but they
shows a little backward tilt of thoracic spine in the sagittal plane projection. The
transverse view provided us clear information that a rotating of upper trunk occured by
comparing it with that projection of standing condition. Figure 34 is the projections for
bending to right. The sagittal plane view remains similar to that of standing condition, but
the frontal and transverse views both show a large excursion of the spinal curve to the
right side. Figlne 35 is the projections for flexing forward. The frontal plane view
remains similar to that of standing condition, but the sagittal and transverse views both
show large excursions of the spinal curve to the anterior direction.

~

2. The Orientation of Each Point on the Spinal Curve Relative to Pelvis

The spinal curve can be treated as the path of moving trihedron whose principle
axes align with the anatomical axes of the trunk, therefore each point on the spinal curve
has a local orientational coordinates. A "joint coordinate system" has been formed
between the coordinates of each point and the pelvis, and the orientation of the trunk was
calculated everywhere on the curve relative with pelvis. To test the reproducibility of the
experiment, all the trials of each condition have been plotted together in one plot for the
angles, see Figure (36) to (39). To test the functions of different primary movements, one
average trial of each condition has been chosen and plotted in Figure (40).

Throughout all the uials, there was good reproducibility for each condition. For
the standing condition the thoracic kyphotic angle was 35 degrees and lumbar lordotic
angle was only 15 degrees. This result is different from most of the X-ray results.

 

78

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However, since the spinal vertebral surface measurement, and the space angle
measurement are not based upon the traditional two—dimensional Cobb projectional
measurement based on the end plates of proximal and distal vertebrae, the difference in
results is understandable. Spinal side tilting angle was within -5 to 10 degrees normal
range. Some differences might be produced by targeting error or tracking error. The
rotational angle is not the actual spinal vertebrae torsion angle, but the orientation of the
trunk relative to pelvis. The variation in results was obviously created by alignment of the
bars connecting the two targets. Therefore they cannot be used as the value in quantifying
spinal rotation. However, it is useful in judging the mobility of spinal rotation.

Comparing the results of bending, rotating and flexing with that of standing by
Figure 40, it was obvious which were the primary movements shown by the plots. It is
important to note here that the motion of the higher level of the spine is the accumulation
of the lower region. Primary rotation of the trunk was coupled with side bending and the
primary bending of the trunk was coupled with rotation.

3. Differential Geometrical Analysis of the Spinal Curve

Space curvature and torsion are the two parameters indicating the condition of a
space curve from a differential geometrical point of view. The space curvature plots of
the three trials of each condition were shown by Figure 41 to 44. Since space curvature
indicates the curvature in every direction, it is a compound value of all the directions in
space. As a result, it was not expected that this parameter would have the same
reproducibility as that of the kinematic angles. However, since the human spinal shape is
predominantly an "S" style, the space curvature can be modeled in a "non-zero, zero, non-

zero" format. The non-zero indicate the lumbar lordosis curve and thoracic kyphosis

87

Figure 41:

CURVATURE ANALYSIS (DIFFERENTIAL GEOM EI'RY) OF SPINAL CURVE
THREE TRIALS OF FLEXION FORWARD CONDITION

 

 

 

 

 

Figure 42:

CURVATURE ANALYSIS (DIFFERENTIAL GEOMEI'RY) OF SPINAL CURVE
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Figure 43:

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89

curve and the zero indicates a transitional point between the two curves. Throughout all
the conditions, the peak values of the thoracic curve were unchanged which indicated the
rigidity of thoracic spine and the large variation of lumbar curvattn'showslarge mobility of
lumbar spine.

Since bending added a comparable curvature in the frontal plane with that in the
sagittal plane, [Figure 44], the space curvature for this condition showed a consistently
curvature increase of lumbar curve and consistent upward shift of transitional point. This
means that frontal plane deviation or deformity can be qualitatively understood by the
space ctn'vature plot. The space curvature, as a differential geometrical parameter, is

useful in judging scoliosis condition.

By comparing the curvature plots of standing with rotating, [Figure 42], it was
found that there was an increase of lumbar curvature and a slightly decrease of thoracic
curvature. This matched up well with the result from kinematic point of view. Even
though there was a decrease of lumbar curvature of flexing from standing, a similar trend

existed motion in sagittal plane.

The torsion of a space curve measures the twisting of the osculating plane. The
results were very similar to that of the curvature, see Figure 45 and 48. The
reproducibility throughout the trials of the conditions were not good due to the sensitivity
of space curvature. However, it was found that the torsion values of the upper thoracic
and lower lumbar region were always close to zero which meant that the curves at those
regions were very close to planar curves. On the other hand, large variations existed in
the thoracolumbar region which meant that there was not only large three-dimensional
mobility but also curve directional changes.

 

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Overall, the difl‘erential geometrical information was not as reproducible as that of
kinematic results and this was due to the combination of three-dimensional information
into one value. Both the curvature and torsion provide us a more qualitative

understanding of the spinal me than a quantitative one.
4. Coupling Analysis of Lumbar and Thoracic Gross Movement by Cross Plot

Figure 49 is the cross plot of rotating and bending angle of lumbar spine during the
voluntary rotating movement. The resultant straight line proved that there was a coupling
happened during that movement. The positive slope means the coupled movement went
to the same side of primary movement. The straight line tilts more toward primary
movement axis, which means the coupling is weak. All these results agreed very well with
other researches, (White and Panjabi 1978, Pearcy 1985).

Figure 50 is the cross plot of rotating and bending angle of thoracic spine during
the voluntary rotating movement. The resultant straight line proved that there was a
coupling happened during that movement. The negative slope means that coupled
movement went to the opposite side of primary movement. The straight line makes up
almost an equal angle with both axes, which means the coupling is strong. Even there is
not much work have been done previously on testing the gross motion of thoracic spine in
vivo, the results developed here matched very well with the anatomical information,
(White and Panjabi 1978).

 

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94

Conclusion

Spinal curvattne and kinematics are important in spinal deformity analysis and
ergonomic design. Most of the studies done in the past are either based on radiology
projectional pictures or either too complicated or use grossly invasive techniques. An easy
and reasonable accurate noninvasive measurement method would benifit current programs
such as school scoliosis screening, long term spinal deformity treatment, and mobilty
measurements. In this study, not only the static spinal curvature but also the dynamical
movement have been quantified by a video camera system. and the data was easily

processed by a computer.

Kinematic analysis has been used to quantify the gross motion of whole thoracic
and lumbar spines, and the movement of the thoracic cage. The thoracic spine is
anatomically retricted by the ribs. The gross motion results showed that the thoracic spine
had a restricwd mobility for flexing/extending, but as large a mobility as that of lumbar
spine for side-to-side bending and axial rotating movements. Coupling has been found in
both thoracic and lumbar regions for axial rotating movements, but not for side-to—side
bending movements. The direction of coupling and approximate coupling ratio matched
well with other studies. The movements of thoracic cage combined the motion from both
thoracic and lumbar spine, and it is dangerous to use the movement of thoracic cage to

approximate lumbar spine mobility.

The relative orientation of each point on the spinal curve with the pelvis has also
been calculated. Good reporduciblity was found among all the trials for each condition.
The primary motions were easily recognized by the plots, and the accompanying motions
could also be seen.

 

 

95

Differential geometrical analysis was based on a space curve fitted by quintic spline
functions, and the curve was smooth in the second derivative. Space curvature and

torsion are two parameters that provide the information for the curve condition.

96

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