Elastography Using Optical Coherence Tomography:
Development and Validation of a Novel Technique
by
Alexandra H. Chau
SB. Mechanical Engineering
Massachusetts Institute of Technology, 2002
Submitted to the Department of Mechanical Engineering in Partial Fulfillment of the
Requirements for the Degree of
Master of Science in Mechanical Engineering
at the
Massachusetts Institute of Technology
September 2004
2004 Massachusetts Institute of Technology
All rights reserved
Signature of Author:
Signature redacted
Department of Mechanical Engineering
August 6, 2004
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Certified by:
B
B
el
Brett Boumna
Associate Professor of Dermatology, Harvard Medical School
Member of the Faculty of the Harvard-MIT Division of Health Sciences and Technology
Thesis Supervisor
Certified by:
_Signature redacted
Mohammad Kaazempur-Mofrad
'Rincipal Research Scientist, Department of Mechanical Engineering
Thesis Reader
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Accepted by:
Ain Sonin
Professor of Mechanical Engineering
Chairman, Department Committee on Graduate Students
Ih
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2
Elastography Using Optical Coherence Tomography:
Development and Validation of a Novel Technique
by
Alexandra H. Chau
Submitted to the Department of Mechanical Engineering
on August 6, 2004 in partial fulfillment of the
requirements for the Degree of Master of Science in
Mechanical Engineering
ABSTRACT
Atherosclerosis is an inflammatory disease characterized by an accumulation of lipid and
fibrous tissue in the arterial wall. Postmortem studies have characterized rupture-prone
atherosclerotic plaques by the presence of a large lipid-rich core covered by a thin fibrous
cap. Studies employing finite element analysis (FEA) based on ex vivo plaque geometry
have found that most plaques rupture at sites of high circumferential stress, thus diagnosis
of plaque vulnerability may be enhanced by probing the mechanical behavior of
individual plaques. Elastography is a method of strain imaging in which an image
sequence of the artery undergoing deformation is acquired, pixel motion is estimated
between each frame, and the resulting velocity field is used to calculate strain. In this
thesis, optical coherence tomography (OCT), a high-resolution optical imaging modality,
is investigated as a basis for FEA and elastography of atherosclerotic plaques.
FEA was performed using plaque geometries derived from both histology and OCT
images of -the same plaque. Patterns of mechanical stress and strain distributions
computed from OCT-based models were compared with those from histology-based
models, the current gold standard for FEA. The results indicate that the vascular structure
and composition determined by OCT provides an adequate basis for investigating the
biomechanical factors relevant to atherosclerosis.
A new variational algorithm was developed for OCT elastography that improves upon the
conventional algorithm by incorporating strain smoothness and incompressibility
constraints into the estimation algorithm. In simulated OCT images, the variational
algorithm offers significant improvement in velocity and strain accuracy over the
conventional algorithm, particularly in the presence of image noise. Polyvinyl alcohol
(PVA) phantoms of homogeneous and heterogeneous elastic modulus distribution were
developed for further testing of the variational algorithm. Testing with the phantoms
indicated that motion- and strain-induced decorrelation between images presents a
practical challenge to the implementation of OCT elastography. Analysis of the
3
experimental results led to the identification of potential improvements to the
elastography algorithm that may increase accuracy. These improvements may include
relaxation of the strain smoothness constraint to incorporate strain discontinuities at
boundaries of elastic modulus in heterogeneous regions, and enforcement of geometry
compatibility to prevent the estimation of non-physical velocity fields.
Thesis Supervisor: Brett Bouma
Title: Associate Professor of Dermatology, Harvard Medical School
Member of the Faculty of the Harvard-MIT Division of Health Sciences and
Technology
4
Acknowledgements
I would like to thank my thesis supervisor, Brett Bouma, for sharing his guidance
and insight and for always being very supportive. At MIT, Mohammad KaazempurMofrad has lent his expertise and made invaluable contributions to this work. I have also
benefited from insightful discussions with both Gary Tearney at MGH and Roger Kamm
at MIT.
Ray Chan has taught me volumes over the past three years, and I would not have
been able to complete this work without him. The Wellman Optical Diagnostics Group is
full of kind, intelligent researchers who are always willing to help, and I am indebted to
Nicusor Iftimia, Milen Shishkov, Seemantini Nadkarni, and Briain MacNeill for all their
technical assistance. A portion of this work was done in conjunction with Chiraag Dharia,
an undergraduate summer student from the University of Iowa.
Finally, I appreciate the influence of friends and family in my life. I would like to
thank Baris Erkmen for his friendship, encouragement and help. My brother, Phu, has
motivated me throughout my education. Most importantly, I am very grateful to my
parents, Peter and Kim, who have consistently shown their love, support, and confidence
in me.
5
6
Table of Contents
1
IN TR OD UC TION .............................................................................................................................
15
1.1
M OTIVATION..............................................................................................................................
15
1.2
PATHOLOGY OF A THEROSCLEROSIS .........................................................................................
15
1.2.1
Initiation...............................................................................................................................
16
1.2.2
Progressionand rupture......................................................................................................
17
1.2.3
Plaquestability.....................................................................................................................
18
BIOMECHANICS OF PLAQUE RUPTURE........................................................................................
19
1.3
1.3.1
Finite element analysis........................................................................................................
19
1.3.2
Materialproperties..............................................................................................................
20
1.4
ELASTOGRAPHY AS A MEANS OF PROBING BIOMECHANICS......................................................
21
1.5
THESIS GOALS............................................................................................................................
22
2
OPTICAL COHERENCE TOMOGRAPHY.............................................................................
25
2.1
INTRODUCTION...........................................................................................................................
25
2.2
THEORY .....................................................................................................................................
25
2.2.1
Coherent interferometry.......................................................................................................
25
2.2.2
Low coherence interferometry..........................................................................................
27
2.2.3
Tissue scatteringand speckle ...........................................................................................
31
2.2.4
Detection and demodulation..............................................................................................
32
2.2.5
Sensitivity..............................................................................................................................
36
2.3
IMPLEMENTATION ......................................................................................................................
39
2.3.1
Source and interferometer....................................................................................................
39
2.3.2
Reference arm scanning .......................................................................................................
39
2.3.3
Planarimaging.....................................................................................................................
40
2.3.4
Catheter-basedimaging.......................................................................................................
41
2.3.5
Signal acquisition.................................................................................................................
43
2.4
IMAGING BASED CHALLENGES TO ELASTOGRAPHY...................................................................
44
2.5
SPECTRAL METHODS FOR OCT...............................................................................................
45
2.5.1
SD-O CT ................................................................................................................................
45
2 .5 .2
FD ....................................................................................................................................
47
SUMMARY ..................................................................................................................................
48
2.6
3
OCT AS A BASIS FOR FINITE ELEMENT MODELING.....................................................
3.1
INTRODUCTION...........................................................................................................................
51
51
7
3 .2
M ETH OD S................................................................................................................................... 5 3
3.2.1
OCT imaging........................................................................................................................ 53
3.2.2
Histologyprocessing and registration................................................................................ 53
3.2.3
Segmentation ........................................................................................................................ 54
3.2.4
Finite element analysis......................................................................................................... 54
3 .3
R ESU LTS .................................................................................................................................... 57
3.3.1
3.4
DISCUSSION ............................................................................................................................... 66
3 .5
S U MM A RY ..................................................................................................................................
4
68
ELASTO GRAPHY ........................................................................................................................... 69
4.1
INTRODUCTION ........................................................................................................................... 69
4.2
VELOCIMETRY ........................................................................................................................... 70
4.2.1
Conventional elastography................................................................................................... 70
4.2.2
A variationalframeworkforincorporationofprior knowledge ........................................... 73
4.2.3
Numericalsolution to variationalenergy minimization ....................................................... 74
4.2.4
Multi-resolutionapproach.................................................................................................... 78
4.3
STRAIN CALCULATION ............................................................................................................... 79
4.3.1
Small strains......................................................................................................................... 79
4.3.2
Large strains........................................................................................................................81
4 .4
1.
S U M M A R Y ..................................................................................................................................8 3
ELASTOGRAPHY VALIDATION IN SIMULATION ................................................................ 85
4.5
INTRODUCTION ........................................................................................................................... 85
4 .6
M ETH O DS ...................................................................................................................................8 5
4.6.1
Simulation of OCT images.................................................................................................... 85
4.6.2
Elastography......................................................................................................................... 91
4 .7
4.8
R ESU LTS .................................................................................................................................... 95
4.7.1
Parameter selection .............................................................................................................. 95
4.7.2
Strain resolution................................................................................................................. 110
4.7.3
Modulus sensitivity ............................................................................................................. 114
4.7.4
Strain sensitivity ................................................................................................................. 117
SUMMARY ................................................................................................................................ 120
ELASTOGRAPHY VALIDATION IN PHANTOMS ................................................................. 123
5
8
Sensitivity analysis................................................................................................................ 62
5.1
INTRODUCTION ......................................................................................................................... 123
5.2
POLYVINYL ALCOHOL (PVA) .................................................................................................. 123
5 .3
M E T H OD S ................................................................................................................................. 12 4
5.3.1
Phantom construction......................................................................................................... 124
5.3.2
Loading experiments........................................................................................................... 126
5.3.3
Numerical modeling ........................................................................................................... 128
5.3.4
Elastography....................................................................................................................... 129
RESULTS AND DiscussION ....................................................................................................... 129
5.4
5.4.1
P VA phantom parameters................................................................................................... 129
5.4.2
Planarphantomstretching................................................................................................. 131
5.4.3
Cylindricalphantom inflation............................................................................................. 136
DiscuSSION ............................................................................................................................. 149
5.5
5.6
6
5.5.1
D ecorrelation...................................................................................................................... 149
5.5.2
Motion artifacts .................................................................................................................. 150
5.5.3
Energyfunction................................................................................................................... 151
SUMMARY ................................................................................................................................ 152
SUM M ARY AND FUTURE W ORK ............................................................................................ 155
10
List of Figures
1-1: PATHOGENESIS OF ATHEROSCLEROSIS...................................................................................
16
FIGURE 1-2: ELASTOGRAPHY BLOCK DIAGRAM .............................................................................................
22
FIGURE 2-1: SCHEMATIC OF A MICHELSON INTERFEROMETER ..................................................................
26
FIGURE 2-2: SCATTERING EVENTS IN TISSUE .................................................................................................
32
FIGURE 2-3: ELECTRONIC DEMODULATION ................................................................................................
34
FIGURE 2-4: SCHEMATIC OF THE RAPID SCANNING OPTICAL DELAY LINE
40
FIGURE
(RSOD).....................................
FIGURE 2-5: SCHEMATIC OF THE XY SCANNER FOR PLANAR IMAGING.......................................................
41
FIGURE 2-6: SCHEMATIC OF THE ROTARY JUNCTION WITH ATTACHED CATHETER.....................................
42
FIGURE 2-7: CATHETER SCHEM ATIC ..............................................................................................................
42
FIGURE 2-8: DISTAL OPTICS OF THE CATHETER'S INNER CORE ..................................................................
43
FIGURE 2-9: SCHEMATIC OF SPECTRAL DOMAIN
OCT (SD-OCT) ..............................................................
FIGURE 2-10: SCHEMATIC OF OPTICAL FREQUENCY DOMAIN IMAGING
(OFDI)..........................................
46
47
FIGURE 3-1: FINITE ELEMENT MODELING PROCESS FOR A LIPID RICH CORONARY PLAQUE CROSS-SECTION... 56
FIGURE 3-2: LIPID RICH PLAQUE HISTOLOGY, STRESS, AND STRAIN AS A FUNCTION OF ANGLE..................
59
FIGURE
3-3:
CALCIFIED PLAQUE
IMAGES ................................................................
60
FIGURE
3-4:
CALCIFIED PLAQUE EFFECTIVE STRESS AND CYCLIC STRAIN .................................................
61
OCT AND HISTOLOGY
FIGURE 3-5: CALCIFIED PLAQUE HISTOLOGY, STRESS, AND STRAIN AS A FUNCTION OF ANGLE .....................
62
FIGURE
3-6:
SEGMENTATION SENSITIVITY FOR THE LIPID RICH PLAQUE .....................................................
64
FIGURE
3-7:
SEGMENTATION SENSITIVITY FOR CALCIFIED PLAQUE .............................................................
65
FIGURE 4-1: CONVENTIONAL VELOCIMETRY ...............................................................................................
71
FIGURE 4-2: A MULTI-RESOLUTION APPROACH TO ELASTOGRAPHY ...........................................................
78
FIGU RE 4-3: NORM AL STRA IN .......................................................................................................................
79
FIG U RE 4-4 : SH EAR STRA IN ...........................................................................................................................
80
FIGURE 4-5: L ARGE DEFORM ATION ...............................................................................................................
81
OCT IMAGES...............................................................................
87
FIGURE 5-2: FINITE ELEMENT MESH FOR SIMULATION................................................................................
88
FIGURE 5-1: GEOMETRY OF SIMULATED
FIGURE 5-3:
OCT POINT SPREAD
FIGURE 5-4: SIMULATED
FUNCTION ..............................................................................................
90
O CT IM AGE ...........................................................................................................
91
FIGURE 5-5: TRUE AXIAL DISPLACEMENT AND STRAIN FOR THE STIFF 0.5 MM DIAMETER INCLUSION ........... 93
FIGURE 5-6: TRUE AXIAL DISPLACEMENT AND STRAIN FOR THE COMPLIANT
0.5
MM DIAMETER INCLUSION. 94
FIGURE 5-7: SIMULATED IMAGES OF STIFF INCLUSION WITH NOISE ............................................................
96
FIGURE 5-8: MAXIMUM CORRELATION COEFFICIENTS FOR VARYING NOISE LEVELS AND CORRELATION
W IN DO W SIZE S .....................................................................................................................................
97
11
FIGURE 5-9: CONVENTIONAL ALGORITHM VELOCITY FOR VARYING NOISE LEVELS AND CORRELATION
W IN DO W SIZ E S .....................................................................................................................................
99
FIGURE 5-10: CONVENTIONAL ALGORITHM STRAIN FOR VARYING NOISE LEVELS AND CORRELATION WINDOW
S IZE S ..................................................................................................................................................
10 0
FIGURE 5-11: VARIATIONAL ALGORITHM VELOCITY FOR VARYING NOISE LEVELS AND CORRELATION
WIN D OW SIZE S ...................................................................................................................................
10 1
FIGURE 5-12: VARIATIONAL ALGORITHM STRAIN FOR VARYING NOISE LEVELS AND CORRELATION WINDOW
S IZ E S ..................................................................................................................................................
FIGURE
5-13:
10 2
CONVENTIONAL AND VARIATIONAL ALGORITHM VELOCITY RMS ERROR FOR VARYING NOISE
LEVELS AND CORRELATION W INDOW SIZES........................................................................................
103
FIGURE 5-14: CONVENTIONAL AND VARIATIONAL ALGORITHM STRAIN RMS ERROR FOR VARYING NOISE
LEVELS AND CORRELATION WINDOW SIZES........................................................................................
FIGURE
104
5-15: VARIATIONAL ALGORITHM VELOCITY FOR VARYING NOISE LEVELS AND WEIGHTING
PA R A M ETER V A LU ES ..........................................................................................................................
10 6
FIGURE 5-16: VARIATIONAL ALGORITHM VELOCITY RMS ERROR FOR VARYING NOISE LEVELS AND
W EIGHTIN G PARAM ETER VALUES.......................................................................................................
107
FIGURE 5-17: VARIATIONAL ALGORITHM STRAIN FOR VARYING NOISE LEVELS AND WEIGHTING PARAMETER
V AL U E S ..............................................................................................................................................
10 8
FIGURE 5-18: VARIATIONAL ALGORITHM STRAIN RMS ERROR FOR VARYING NOISE LEVELS AND WEIGHTING
PA RA M ETER V A LU ES..........................................................................................................................
10 9
FIGURE 5-19: SIMULATED IMAGES OF VARYING DIAMETER INCLUSIONS .....................................................
110
FIGURE 5-20: VELOCITY FOR STIFF INCLUSIONS OF VARYING DIAMETER ....................................................
I11
I11
FIGURE 5-21: AXIAL STRAIN FOR STIFF INCLUSIONS OF VARYING DIAMETER ..............................................
FIGURE 5-22: STRAIN
CNR FOR STIFF INCLUSIONS
OF VARYING DIAMETER ................................................
112
FIGURE 5-23: VELOCITY FOR COMPLIANT INCLUSIONS OF VARYING DIAMETER ..........................................
113
FIGURE 5-24: AXIAL STRAIN FOR COMPLIANT INCLUSIONS OF VARYING DIAMETER ....................................
113
FIGURE 5-25: STRAIN
CNR FOR COMPLIANT
FIGURE 5-26: VELOCITY FOR
0.5
FIGURE 5-27: AXIAL STRAIN FOR
FIGURE 5-28: STRAIN CNR FOR
FIGURE
5-29:
INCLUSIONS OF VARYING DIAMETER......................................
114
MM DIAMETER INCLUSIONS OF VARYING STIFFNESS ................................
115
0.5
MM DIAMETER INCLUSIONS OF VARYING STIFFNESS ..........................
116
MM DIAMETER INCLUSIONS OF VARYING STIFFNESS............................
117
0.5
VELOCITY AND AXIAL STRAIN FOR A COMPLIANT INCLUSION WITH VARYING APPLIED STRAIN
M A GN ITU DE .......................................................................................................................................
1 18
FIGURE 5-30: STRAIN CNR FOR A COMPLIANT INCLUSION WITH VARYING APPLIED STRAIN MAGNITUDE ... 119
FIGURE 6-1: PHOTOGRAPH OF HETEROGENEOUS CYLINDRICAL PHANTOM ..................................................
126
FIGURE
6-2:
SCHEMATIC OF PLANAR PHANTOM STRETCHING APPARATUS ..................................................
127
FIGURE
6-3:
SCHEMATIC OF CYLINDRICAL PHANTOM INFLATION APPARATUS ............................................
128
FIGURE 6-4: DEPENDENCE OF PVA STIFFNESS ON SOLUTION CONCENTRATION ..........................................
130
12
FIGURE 6-5:
FIGURE
6-6:
OCT IMAGES
OF PVA PHANTOMS COMPARED TO HUMAN AORTA ...........................................
DEPENDENCE OF PVA STIFFNESS ON NUMBER OF FREEZE-THAW CYCLES ...............................
130
131
132
FIGURE 6-7: PLANAR HOMOGENEOUS PHANTOM .........................................................................................
FIGURE
6-8:
CORRELATION COEFFICIENTS FOR STATIC HOMOGENEOUS PLANAR PHANTOM ........................
133
FIGURE
6-9:
CORRELATION COEFFICIENTS FOR THE STRETCHED HOMOGENEOUS PLANAR PHANTOM..........
134
FIGURE
6-10:
STRETCHED PLANAR HOMOGENEOUS PHANTOM VELOCITY FROM ELASTOGRAPHY ...............
135
FIGURE
6-11:
STRETCHED PLANAR HOMOGENEOUS PHANTOM STRAIN FROM ELASTOGRAPHY....................
136
OCT IMAGES
138
FIGURE 6-12:
OF HOMOGENOUS PHANTOM INFLATION...........................................................
FIGURE 6-13: UNDEFORMED FEA MESH OF HOMOGENOUS CYLINDRICAL PHANTOM ..................................
FIGURE
6-14: PHANTOM FEA MODEL
BOUNDARIES OVERLAID ONTO CORRESPONDING
OCT IMAGES
........
FIGURE 6-15: INFLATED HOMOGENEOUS PHANTOM PREDICTED VELOCITY FROM FEA ...............................
139
140
141
FIGURE
6-16: INFLATED
HOMOGENEOUS PHANTOM PREDICTED STRAIN FROM FEA ....................................
142
FIGURE
6-17: INFLATED
HOMOGENEOUS PHANTOM VELOCITY FROM ELASTOGRAPHY ................................
143
FIGURE
6-18: INFLATED
HOMOGENEOUS PHANTOM STRAIN FROM ELASTOGRAPHY ....................................
144
FIGURE
6-19: INFLATED
HETEROGENEOUS PHANTOM IMAGES.....................................................................
144
FIGURE
6-20: INFLATED
HETEROGENEOUS PHANTOM VELOCITY FROM FEA ...............................................
146
FIGURE
6-21: INFLATED
HETEROGENEOUS PHANTOM STRAIN FROM FEA ...................................................
147
FIGURE 6-22: INFLATED HETEROGENEOUS PHANTOM VELOCITY FROM ELASTOGRAPHY .............................
148
FIGURE 6-23: INFLATED HETEROGENEOUS PHANTOM STRAIN FROM ELASTOGRAPHY .................................
149
13
14
Chapter 1: Introduction
1.1 Motivation
Cardiovascular disease (CVD) is the leading cause of death in industrialized
nations. In the United States, there are an estimated 64 million people with CVD. CVD
was responsible for approximately 38.5% of all deaths in the U.S. in 2001, with 54% of
those deaths attributed to coronary heart disease. Both coronary heart disease and strokes
are often caused by atherosclerosis, an inflammatory disease of the arteries, which
directly or indirectly accounts for nearly three-fourths of all deaths from CVD [1].
1.2 Pathology of Atherosclerosis
Atherosclerosis is characterized by an accumulation of lipid and fibrous tissue in
large and medium-sized arteries. In all stages of the disease: initiation, progression, and
rupture of plaques, chronic inflammation is present, resulting from the interaction
between modified lipoproteins, monocyte-derived macrophages, T cells, and the normal
cellular elements of the arterial wall. The inflammatory process can eventually lead to the
formation of lesions, or plaques, which may occlude the lumen. Advanced lesions may
lead to plaque rupture and thrombosis, and may ultimately result in myocardial infarction
and stroke (Figure 1-1).
15
A
AN
Endoehad
per-meb"Y
LetAacyt
mraow~
Endothefa
adhesion
aucct
Loukocyl*
a,
adhosico
C
Macrophaga
Smooth-muscle
migraton
Foamcg
formatdion
TCafl
activation
Opragalio of
phateles
arwf try
of Iatkoctn
D
eccumufaton
Formation of
"ncroc oro
Fibrouscap formation
Plaque
rip-
Thinnng of fibrous cap
Hemorrhage from plaque
m *rwe
**
Figure 1-1: Pathogenesis of Atherosclerosis. A) Endothelial dysfunction, B) Fatty-streak formation,
C) Advanced lesion formation, D) Unstable plaque rupture. (Taken from [21.)
1.2.1 Initiation
The first step in atherosclerosis is the development of a fatty streak. The
endothelium, the innermost layer of the artery that contacts the blood, is normally
resistant to adhesion by leukocytes. Under certain circumstances, endothelial dysfunction
occurs, characterized by increased endothelial permeability, platelet aggregation,
leukocyte adhesion, and generation of cytokines. Leukocytes then begin to attach to the
endothelial cells, a mechanism in which vascular cell adhesion molecule-1 (VCAM-1)
plays a significant role. Many of the risk factors for atherosclerosis are also risk factors
for endothelial dysfunction: hypercholesterolemia, hypertension, smoking, diabetes, and
positive family history of premature atherosclerosis [2, 3]. Atherosclerotic lesions
develop at certain sites of the arterial tree, including branches, bifurcations, and
curvatures - all regions that experience altered blood flow patterns, including shear stress
and increased turbulence [2]. Some regions of the arterial tree seem to be protected from
16
atherosclerosis - usually regions with orderly blood flow, resulting in laminar shear
stress. This
may be
explained by the existence of genes with potentially
"atheroprotective" properties that contain shear-stress response elements within their
promoter regions, such as superoxide dimutase, which limits VCAM-l expression and
other inflammatory pathways [4].
Once leukocytes have adhered to the endothelial wall, they enter the intima through
endothelial cell tight junctions. This process is enhanced by the presence of
chemoattractant cytokines that recruit leukocytes into the intima, such as monocyte
chemoattractant protein-1 (MCP-1). Inside the intima, these monocytes acquire
morphological characteristics of macrophages. They increase expression of scavenger
receptors for modified lipoproteins, enabling them to internalize these modified
lipoproteins. Lipoproteins accumulate in the macrophage as droplets in the cytoplasm,
giving rise to a complex known as a foam cell, for its foamy appearance under a
microscope. Development and accumulation of macrophages is enhanced by macrophage
colony-stimulating factor (M-CSF) [5].
1.2.2 Progression and rupture
It was previously believed that the development of a fatty streak into an
atherosclerotic lesion was a continuous process. Macrophages in the fatty streak were
thought to release growth factors that stimulated smooth muscle cell (SMC) proliferation.
As SMCs accumulated in the plaque, they synthesized an abundant extracellular matrix,
leading to the development of the fibrous cap [6].
However, clinical observations have shown that atherosclerotic lesion stenoses
develop discontinuously [7], caused by physical disruption of plaque, followed by
thrombosis, leading to the sudden expansion of the lesion. Three types of physical plaque
disruption have been identified: superficial erosion of endothelial cells, microhemorrhaging within the plaque, and fracture of the plaque cap [8].
Superficial erosion of endothelial cells can form the nidus of a platelet thrombus
by uncovering subendothelial collagen and factors promoting platelet adhesion and
activation. Superficial erosion is common and usually asymptotic [5].
17
Disruption of microvessels in the atheromatous lesion may also contribute to
plaque progression. Lesions may develop microvascular channels as a result of
neoangiogenesis. These microvessels are fragile and prone to micro-hemorrhage, which
leads to thrombosis in situ. This, in turn, leads to thrombin generation, which can
stimulate SMC migration and proliferation. Activated platelets also release growth
factors that stimulate more collagen synthesis by SMCs, potentially leading to a growth
spurt of the plaque [5].
The most common mechanism of plaque disruption is fracture of the fibrous cap.
Activated macrophages in the atheroma can produce proteolytic enzymes that degrade the
collagen in the fibrous cap covering the plaque. The cap then becomes thin, weak, and
susceptible to rupture. Fissure of the cap allows thrombogenic materials in the lesion's
core to come into contact with coagulation factors in the blood, resulting in thrombus
formation [9]. Three-quarters of acute myocardial infarctions are caused by a ruptured
fibrous cap, yet rupture may be clinically silent. In these cases, the thrombus is not
completely occlusive, and in the resulting healing process, the plaque may evolve from a
fatty lesion to a more fibrous and stenotic plaque.
1.2.3 Plaque stability
Histopathologic examinations have shown that there is substantial variation in the
size of lipid pools and the thickness of fibrous caps. Different combinations of cap
thickness and core size may occur, with different clinical outcomes. The most clinically
stable plaques are fibrous plaques, with dense fibrous tissue and small amounts of lipid.
Stable plaques feature an abundance of SMCs embedded in a dense collagen matrix,
giving the plaque a thick fibrous cap [10]. These lesions can remain clinically silent or in
the long term can lead to stable angina pectoris. Vulnerable plaques are histologically
characterized as having large lipid pools and thin fibrous caps. Postmortem studies have
identified one type of vulnerable plaque, the thin-cap fibroatheroma (TCFA), as the
culprit lesion in approximately 80% of sudden cardiac deaths [11-13]. The TCFA is
typically a minimally occlusive plaque characterized histologically by the following
features: 1) a thin fibrous cap, 2) a large necrotic core, and 3) the presence of activated
macrophages at the shoulder region of the plaque [14, 15]. Macrophage activity can lead
18
to degradation of the structural integrity of the fibrous cap, making the plaque most likely
to rupture at the shoulder region. These plaques are considered to be "rupture prone"
because they are often found underlying coronary thrombosis at autopsy [2, 16].
There are multiple paths to acute coronary events [15, 17, 18]. One is where the
plaque becomes progressively occlusive,
initiating ishchemic processes
in the
myocardium. Alternatively, the plaque may be only mildly or moderately stenotic but
may still fissure. The resulting thrombus may repair, leaving increased stenosis, or may
be totally and fatally occlusive. Seventy percent of high grade stenoses have had a prior
disruption that healed [19], indicating that a large number of plaque ruptures are
clinically silent. Studies have noted that plaque fissure is often a critical aspect of acute
cardiac events [19, 20], suggesting the importance of studying the biomechanics of
plaque rupture.
1.3 Biomechanics of Plaque Rupture
1.3.1 Finite element analysis
Investigation of the biomechanics of atherosclerotic lesions has relied on finite
element analysis (FEA) for computing stress and strain in idealized vascular crosssections [21, 22] or histology derived realistic geometry sections [23-25]. A 1989 study
comparing FEA and histology found a connection between mechanical loading of the
vessel and plaque rupture [22]. More specifically, FEA suggests that plaque rupture is
highly dependent on circumferential tensile stress in the fibrous cap, and that most
plaques rupture at sites of high circumferential stress [16, 22, 24, 26, 27], typically in the
shoulder region of the cap [22, 24, 26]. Conversely, a stiff region in a plaque lesion tends
to mask the tissue behind it, thereby reducing stress and strain.
Using an IVUS-based model for rupture prediction [28], a 1993 study imaged and
modeled 16 in vitro coronary arteries. An angioplasty balloon was inserted into each and
inflated until rupture. Histological comparison with FEA showed that 17 of 18 fractures
occurred in segments with high stress, and 82% occurred at the location of high stress.
No correlation was found between pressure and peak stress, but this was attributed to the
low resolution of IVUS.
19
Stress distribution depends on not only local plaque geometry, but also the
curvature and flexure of the artery [29]. As it beats, the heart can reversibly increase in
size due to volume loading of the circulation, lengthening and curving the coronary
arteries and increasing mean arterial pressure. It is even possible for an in vivo artery to
be axially stretched and also increase in cross-sectional area. A shortcoming of current
cardiovascular FEA is that most studies do not account for these effects.
1.3.2 Material properties
The reliability and accuracy of prior FEA studies of coronary plaque
biomechanics is suspect for two primary reasons. First, plaque tissues are characterized
using uniaxial tension tests, while in vivo, the tissues are subjected to biaxial stress.
Additionally, elastic modulus values for plaque tissue components have not been reliably
characterized and estimates in the literature vary widely.
Plaque modulus values are difficult to measure accurately for many reasons.
Tissue must be harvested and analyzed immediately after death to avoid the confounding
effect of autolysis. In most cases, autopsy results in a delay of 24-48 hours. Because of
the coronaries' small size, is it quite difficult to obtain a uniform piece of tissue suitable
for mechanical testing. Typically, testing requires samples that are at least 0.5 mm wide
and 3 mm or more long. To avoid dehydration, the tissue needs to be tested while
immersed, which means the setup needs to be transparent in order to see fiduciary
markings. Another confounding factor is that uniaxial test data has shown that the tissue
does not stretch smoothly and finally tear, rather there is a series of localized internal
fractures before the final failure [30, 31].
The choice of material model may also impact the accuracy of plaque FEA
studies. Soft tissue is generally considered to be incompressible [29], and tension tests
show that tissue behaves nonlinearly, i.e., the incremental modulus increases with
elongation. The modulus is determined by the presence of the elastin, collagen, and
smooth muscle cells [32]. Rubber elasticity is a good model for elastin, but it is not as
suitable for collagen. Additionally, because of the orientation of the molecular
components, most tissues behave anisotropically, which is usually not taken into account.
An alternative approach is to model the extracellular matrix as a soft matrix with
20
embedded reinforcing fibers, like a composite material, to account for anisotropy [33].
Fortunately, if the biological tissue is studied on an appropriate size scale with a quick
loading, it can be approximated as linear, elastic, and isotropic, greatly simplifying
analysis.
1.4 Elastography as a Means of Probing Biomechanics
The mechanical properties of a biological tissue depend on the distribution and
structural organization of its molecular building blocks; thus pathological tissues have
different mechanical properties from their normal tissue counterparts. For this reason,
palpation has long been used as a clinical method for detecting pathological tissues, such
as breast lesions. When a physician applies a mechanical load, using manual compression
to the breast, she feels a lesion as a hard lump because the stiffness of the lesion is
typically greater than that of the surrounding normal tissue [34].
Being able to detect the stiffness of a tissue can theoretically enable detection of
pathological tissues even when normal and pathological tissues cannot be distinguished
visually, either by eye or through traditional imaging modalities. Tissue stiffness, which
depends on both the elastic modulus and the geometry of the tissue, determines how
much strain the tissue will undergo when a load is applied. Thus, a method of strain
imaging would allow indirect observation of the elastic modulus distribution, providing a
basis for diagnosis of pathological tissues [34].
Elastography is the process of estimating strain maps, or elastograms, from
images of a tissue undergoing deformation [34], as shown in Figure 1-2. The first step is
to image the response of the tissue to an applied load [35, 36]. The applied load can be
considered static if the data acquisition time is small compared to loading time. Next,
image processing techniques for motion and strain estimation are used to produce
elastograms. The elastogram is an image of the strain present in the tissue, and it serves
as a way to visualize the tissue stiffness. In conventional elastography, strains are small
(<1%) and the constitutive equation is linear. In reality, tissue incompressibility results in
a 3D strain field, but thus far, elastographic strains have mainly been calculated in the
axial direction.
21
Image
acquisition of
tissue
undergoing
deformation
Motion
estimation
Strain
estimation
c>
Elastogram
Figure 1-2: Elastography block diagram.
Applications of elastography to date include in vivo imaging of breast tissue for
detection of cancerous lesions and for strain based classification of tissue composition
[34]. In typical applications, imaging is performed with ultrasound or MRI while the
breast tissue is externally loaded. Elastography has also been applied to atherosclerotic
plaque imaging in vivo with intravascular ultrasound, using intrinsic blood pressure as the
loading condition. An elastogram of an atherosclerotic plaque can potentially enable
identification of different plaque components and high strain regions [37]. Elastography
can be further used in conjunction with an iterative finite element analysis scheme to
estimate elastic modulus and stress distribution [38]. As areas of stress concentration are
predisposed to rupture, this method would be useful in diagnosing vulnerable plaques.
1.5 Thesis Goals
The objective of this thesis is to investigate the use of a relatively new imaging
modality, optical coherence tomography (OCT), as a means for characterizing the
biomechanics of atherosclerosis. OCT is the optical analog of ultrasound imaging but it
provides significantly higher spatial resolution (10x - 100x) and tissue contrast, enabling
a more precise characterization of tissue microstructure. OCT has previously been
demonstrated as a highly sensitive intravascular imaging modality in vivo, capable of
characterizing the structure and composition of normal vessels and atherosclerotic
plaques [39, 40]. Thus, OCT has potential as a basis for FEA of in vivo arterial geometry,
as well as a high resolution basis for elastography.
The principles of OCT imaging are described in Chapter 2, while Chapter 3
demonstrates the first use of OCT-based FEA of atherosclerotic plaques and shows its
validity by comparison with histology-based FEA.
22
The remaining chapters focus on the use of OCT for elastography. Chapter 4
reviews conventional methods used in ultrasound elastography and describes a novel
robust velocimetry algorithm appropriate for OCT. Validation of the new algorithm using
simulated OCT images is presented in Chapter 5 and using customized phantoms of
known mechanical properties in Chapter 6. Chapter 7 provides a summary and
conclusion for this thesis research.
23
24
Chapter 2: Optical Coherence
Tomography
2.1 Introduction
Optical coherence tomography (OCT) is a highly sensitive, non-contact
interferometric imaging modality capable of providing cross-sectional images of the
internal microstructure of living tissue at high resolution (3-10 pIm axially x 25 ptm
laterally), with a penetration depth of 2-3 mm. OCT has been applied to retinal,
cardiovascular, and gastrointestinal imaging, and is particularly attractive because it is
relatively non-invasive and has a resolution approaching that of histological sectioning.
In this chapter we first present a theoretical description of OCT, following the
derivations of Hee [41] and Schmitt [42]. Next is a description of the implementation of
the OCT system used for subsequent experiments. We then discuss OCT imaging in the
context of elastography, as well as some of the challenges faced in OCT-based
elastography. Finally we preview spectral OCT techniques, currently in development,
that may help solve these challenges.
2.2 Theory
2.2.1 Coherent interferometry
Optical coherence tomography is essentially an optical interferometric ranging
system. A common and simple implementation utilizes a Michelson interferometer
.
(Figure 2-1). The light source is split into a reference field, ER, and a sample field, Es
The reference field reflects off the reference mirror while the sample field scatters off the
sample. The reference beam and the reflected sample beam, Es', recombine on the surface
of the photodetector. A compensator may be placed in the sample arm of the
interferometer to correct for the pathlength difference arising from the fact that the
25
reference field travels through the beam splitter three times, while the sample field travels
through the beam splitter only once.
Reference Mirror
I
_
A
Sample
ER (t)
'R
Broadband
Light Source
Es(t)
IS
HHH
E0 (t)
Detector
Figure 2-1: Schematic of a Michelson interferometer. Broadband light is emitted from the source,
where it is split into the reference and sample beams. After reflecting off the reference mirror and
sample, the beams recombine at the detector. The reference mirror is translated to enable scanning
of the reference beam pathlength, thus probing different depths in the sample.
To analyze the output of the interferometer at the detector, first consider the case
of a perfectly reflecting mirror in the sample arm. For monochromatic light, the reference
and sample fields can be expressed as phasors:
ER= ARexp
[-j (2,3R1R
-
wt)]
(2.1)
and
,
Es = As exp -j (20s/s - wt)
(2.2)
where A is the amplitude, j is the propagation constant, I is the distance from the beam
splitter to the reflector, and w is the optical frequency of the light source. The intensity
of the field impinging on the detector is
26
ID
(Er + E)
+ Is)+ Re EE) ,
= 0.5(R
(2.3)
where IR and I's are the mean intensities of the reference and reflected sample fields,
respectively. The interferometric term can also be expressed as
Re{E*Es} = ARAs cos (20RlR
(2.4)
-20sl).
In free space, the propagation constants in the reference and sample arms are equal,
yielding
O=
OS
=
27r
A
(2.5)
and thus
(2.6)
Re{E*Es= ARAs cos 27rj ,
where A 1 =
R
- ls is the pathlength difference between the reference and sample
beams. The optical intensity impinging on the detector and the resulting photocurrent
varies sinusoidally with a phase determined by the pathlength mismatch. Since all the
interference information is contained in this term, the following analysis will neglect the
DC components corresponding to the sample and reference mean intensities.
2.2.2 Low coherence interferometry
Considering the case of a low coherence light source consisting of a finite
bandwidth of frequencies, the reference and sample fields can be redefined as
ER
(W) =
AR (w)exp[-(23R
-0
W) R
)
(2.7)
and
Es (w) = As (w)exp [-j(20s (w) Is
-
wt)].
(2.8)
27
The intensity of the interference signal at the detector can be found by summing the
interference due to each monochromatic wave component
ID
oc Ref -_
c E,(w)Es (w--
dw
{f
,} (2.9)
S (w) exp [-jA# (w)]
TOO
271
=
Re
AR
(w)As (w)
where
S(w)
(2.10)
and
AO (w)=-
2
0R
(w)
1R-
20s (w) Is.
(2.11)
In the case of spectrally uniform reflectors in both fields, S (w) is equivalent to the power
spectrum of the light source.
Now consider that the reference and sample arms are in nondispersive media.
.
Assume the spectrum of the light source is S (W - w 0 ) with center frequency w,
Assuming identical propagation constants, Taylor expansion around the center frequency
yields
()
= Os (w) = 0 (O ) +
(
(2.12)
)(P - LO
)
OR
Substitution into (2.11) gives
A&(w) = 3(w 0 )(2A 1) +
(2 (w 0)(w - w1)(2A I)
(2.13)
The integral in (2.9) becomes
ID O
Re exp[--JoWATrp
S (W - wO)exp[--j(Lo - Wo) A
d(L)
),(2.14)
where Ar, is the phase delay mismatch defined as
ATr
(w0 ) (2A1) _2A1l
WO
Vp
and A7, is the group delay mismatch defined as
28
(2.15)
(wo)(2A1)
AT9 =
2A.
V
=
(2.16)
Thus the center frequency phase velocity is
P
(2.17)
W
and the group velocity is
1
V =(
(2.18)
.
In this case of perfect reflectors in both arms of the interferometer, the intensity hitting
the photodetector is the autocorrelation of the source. Equation (2.14) illustrates that the
autocorrelation is simply the inverse Fourier transform of the power spectral density, as
the Wiener-Khinchin theorem states. Thus the sensitivity of the interferometer to path
length difference depends on the shape and width of the spectrum of the light source.
From equation (2.14) it is evident that the interferometric term consists of a carrier and an
envelope. The carrier oscillates with increasing path length mismatch 2A l at a spatial
frequency of 0 (wo). The envelope is the inverse Fourier transform of the source
spectrum, and it determines the axial point spread function of the interferometer.
Now consider a light source with a Gaussian power spectrum
I2
S (W- wo) =
2
a
eP ( -
2
(2.19)
2U2
which has been normalized to unit power
= 1,
S (W)
(2.20)
is the standard deviation power spectral
where wo is the center frequency and 2o-
bandwidth of the source. Substitution into (2.14) gives
Ioc Re
exp _ 2
20-
I
exp
joATP i
(2.21)
29
Thus the interferometric term has a Gaussian envelope with a standard deviation temporal
width 2o, (seconds) that is inversely proportional to the power spectral bandwidth
2
2u = -.
(2.22)
0-
Thus the envelope falls off quickly as the group delay mismatch Ar, increases and is
modulated by interference fringes that oscillate with increasing phase delay mismatch
AT,. Equation (2.21) thus shows the axial resolving properties of the OCT system.
Interference fringes are formed at the detector only when the sample and reference are
matched sufficiently so that the group delay mismatch falls within the Gaussian envelope.
In free space, with the phase velocity and the group velocity both equal to the
speed of light, c, the axial resolution of the OCT system is given by
AlFWHM
21n2 A]
~ O.44
(2.23)
A2
AA
(2.24)
where AA is the full-width at half-maximum (FWHM) of the source spectrum. Thus a
wider bandwidth source provides higher resolution images.
In two-dimensional imaging, a cross-sectional image is built up from multiple
axial scans, so the resolving properties of the optics used for beam scanning and focusing
onto the sample must also be considered. As reported by Kempe and Rudolph [43], the
transverse point-spread function is not affected by the temporal coherence of the light
source for coherence times greater than approximately 20 fs. Thus when the bandwidth of
the source is not too broad, the axial and transverse resolutions can be treated
independently. The transverse resolution is determined by the size of the spot where the
laser has been focused onto the tissue. Using a larger NA lens creates a smaller spot size,
but the Rayleigh range (the depth through which the laser remains focused) decreases as
well. Using an objective with a smaller NA increases the ranging depth while increasing
the spot size.
30
2.2.3 Tissue scattering and speckle
The above analysis is assumed to be true when the sample is treated as an ideal
mirror; i.e. reflection off the sample only introduces a time delay into the sample beam
while the amplitude and coherence remain unchanged. While this is generally unrealistic
for a real sample, treating the sample as a series of flat reflectors can provide a
satisfactory description of OCT imaging, in which the interference signal is expressed as
a convolution
D
(lR
R(1) 9 exp
2
WsA (2.25)
where R (Is,) is the fraction of power reflected from a layer at depth Is, in the sample,
and the pathlength mismatch of the sample beam relative to the reference beam is
Al'-i
a= R -i S'
Unfortunately, biological tissues rarely conform to this model well.
Tissues
consist of a collagen and elastin matrix filled with cells, blood vessels, nerves, and other
structures. These components range in size from 100 nm to several mm. The refractive
index in tissue is inhomogeneous, so light that is focused into the tissue scatters at various
angles [44, 45]. Figure 2-2 shows the scattering events that may happen in the tissue.
The effect on the temporal and spatial coherence of the sample beam due to all these
scattering events is under continued investigation. The analysis provided here uses a
single-backscatter model that accounts for two of the possible scattering reactions: total
loss of coherence and no loss at all.
31
Reference
0
0
Source0
(iii a)
Detetor(i)
Detecto)
(iii b)
inge(baksctte
Single backscatter
Extinction by absorption or
wide-angle scatter
(iii a) Large-scale index variations
(iii b) Multiple forward scatter
(ii)
Figure 2-2: Scattering events in tissue. (Taken from [44].)
Studies of the effect of other scattering events have shown that low-angle multiple
forward scattered light degrades both resolution and contrast, but wide-angle scattered
light only degrades contrast with minimal effect on resolution [46, 47]. Additionally, the
maximum depth that can be probed by OCT is limited by the single-scattering coefficient
and the mean scattering angle of the tissue. Lastly, the light source's coherence time
determines the width of the axial point-spread function and the size scale of the speckle
generated by multiple scattering.
2.2.4 Detection and demodulation
Light impinging on the detector creates a photocurrent.
A transimpedance
amplifier converts the current i into a voltage v = iR with a gain of R.
Equation (2.21) illustrates that the interferometric signal consists of a carrier
signal modulated by the axial point spread function envelope. Scanning of the reference
mirror maps the point spread function from a function of A l to a function time by
32
A l = -v~t
(2.26)
where time t = 0 occurs when the path lengths are matched and increasing t indicates
that the reference path length is increasing.
From equation (2.15), the electric signal has a carrier frequency of
v
D
(2.27)
VP
where the tilde denotes the electronic frequency counterpart of the corresponding optical
frequency. Assuming free space propagation,
=v
fD
A0
(2.28)
This can also be interpreted as the Doppler shift caused by the moving reference mirror.
With a broadband source, the frequency range becomes
- 2v
Af =
Af
2v
(2.29)
A2
Once the interferometric signal is detected, it is demodulated to obtain the
envelope, thus extracting the depth-resolved reflectivity of the sample. Demodulation can
be performed either electronically, with subsequent acquisition of the demodulated
electrical signal, or digitally, where the interference signal is acquired first and
demodulation occurs in software.
One method of electronic demodulation is through envelope detection, which
entails rectification of the signal followed by low-pass filtering (Figure 2-3). Assuming a
carrier frequency of D with an envelope A (t), the electrical current signal coming from
the photodetector can be represented as
x(t) = A(t)cos(at).
(2.30)
33
First the signal is rectified, using half-wave rectification. When the voltage across the
capacitor is positive, the diode allows current to pass, charging the capacitor. When the
voltage drops, the diode circuit becomes essentially open and the capacitor loses charge
through the resistor. The output signal from this circuit tracks the envelope with ripples,
which can be filtered with a low pass filter.
R
C
VR t
1
0
-1
I
I
I
1 .5r
1
(t)RM
t
-V
0.5
.A
0
1L
3
4
I
I
5
-
6
Time [s]
- -
7
8
9
Figure 2-3: Electronic demodulation. A half-wave rectification circuit (top) is used to demodulate a
cosine carrier signal subjected to amplitude modulation (center). The output of the circuit (bottom)
tracks the envelope shape with ripples that can be removed with a low pass filter.
Software demodulation may be performed using the Hilbert transform. The
Hilbert transform is used to convert a real signal into an analytic signal, i.e. a signal that
has no negative frequency components. A real signal A cos (wt +
34
#)
can be converted to
a positive-frequency complete sinusoid by generating a phase-quadrature component
A sin (wt +
#)
to serve as the imaginary part:
A exp [i (wt + 0)] = A cos (wt + #) + JA sin (wt + #).
This is accomplished using the Hilbert transform y (t)
=
(2.31)
7 {x}, which simply
introduces a phase shift of - 7/2 at positive frequencies and r/2 at negative frequencies.
Combining the original real signal with the Hilbert transform yields the complex analytic
signal z (t) = x (t) + j-4 {x}. To see how this applies to demodulation, consider the
interferometric signal
x(t) = A(t)cos(Dt)
= A
(2.32)
7r/2 phase shifts, the Hilbert transform is
=
A(t)
2
exp
jLat
j + exp -jLt + j-
-
y (t =
'
Applying the
2
(exp[t] + exp [-jLt]).
2 (-j exp [jDt] + jexp [-jct])
2
= A(t) sin (it).
(2.33)
The analytic signal is then
z (t) = A (t) cos (0t) + A (t) j sin (st)
(2.34)
SA (t)exp[ILt].
The envelope is now just the magnitude of the analytic signal z (t):
A(t)
=
(t).
(2.35)
A main feature of each of these methods of demodulation is that neither requires
exact knowledge of the carrier frequency. This is particularly advantageous because the
carrier frequency in the OCT signal may not be constant due to nonlinear motion of the
reference mirror.
35
2.2.5 Sensitivity
The sensitivity, or signal-to-noise ratio (SNR), of the interferometric signal is a
measure of the minimum detectable reflectivity of the OCT system. The SNR is defined
as
SNR= signal power
(
1.-
'
noise variance
From equation (2.3), the photocurrent at the detector is
hil
PR + PS +
77I
ReE*Es
RZd
+
(2.37)
,
=
where q is the quantum efficiency of the detector, e is the charge of an electron, hV is
the photon energy,
i1
is the impedance of free space, idak is the dark current of the
photodetector, andPR and P are the reference and sample field powers, defined as
P = A|/2ro. The peak current occurs when the third term, the correlation term, is
maximized and is equivalent to
(2.38)
V
.
=S
The signal power is then
Pn
= (iR
Plignal
hV
_1_12
(2.39)
()2
P
Noise sources are modeled as wide-sense stationary (WSS) stochastic processes.
A WSS process p (t) has a constant mean
E [p(t)] = (p (t)) = m,
(2.40)
and a statistical autocorrelation
R, (t1 , t2 ) = E (p (t) - m,)(p (t2 ) - m,
36
=
(p (ti) p (t ))
2
-n2
(2.41)
that is a function of -r= t2- t1 only. By the Wiener-Khinchin theorem, a WSS process
has a power spectral density
S, (w)
=
f
R, (r) exp [-jwt]dr .
(2.42)
Now the signal to noise ratio can be written as
Psignal
SN=var [n (t)]
P gnal
Pignal
Rn (0)
f'0Sn (Lo dt,17
2.3
where S, (w) is the power spectral density of the noise in the signal.
Noise in the OCT signal comes from three sources: thermal noise, relative
intensity noise, and shot noise. Thermal noise is generated as heat dissipated by the
feedback resistor of the transimpedance amplifer. Relative intensity noise (RIN)
describes any noise source with a power spectral density that scales with mean
photocurrent power. Sources include power fluctuations from the source and mechanical
motion of optical mounts. By selecting the reference arm power and the electronic
components appropriately, thermal noise and RIN can be almost eliminated so that shot
noise is the dominant noise source.
Shot noise is due to fluctuations in current at the photodetector caused by the
quantized nature of light and charge. The photodetector emits charge at a mean rate
governed by the impinging optical power, but the time between specific emissions is
random. Shot noise is a white noise process with mean (i) and spectral density
S, ()
= e (i).
(2.44)
Thus shot noise power is proportional to electron charge and the square root of the
photocurrent power (i)2
In the shot noise limit, the noise spectral density is given by
S,, (w) = e (i) R 2
(2.45)
because the variance scales as R 2 when the current goes through the transimpedance
amplifier. Thus the noise density is given by
37
Noise = rje2e-PR2BW,
hv
(2.46)
Signal
o
Noise
SNR
r,
P
s
.
where B W is the bandwidth of the detection electronics. The SNR is then given by
hiu 2BW,
(2.47)
Rearranging (2.47) shows the relationship between SNR, resolution, sample power, and
speed of imaging is
SNR x BW
= constant.
(2.48)
The bandwidth B W is essentially equal to the electronic bandwidth Af used for
detection of the interferometric signal. Electronic bandwidth is linearly related to the
reference mirror scanning velocity v, and the light source's wavelength bandwidth AA,
which yields
BW - Af oc v, AA.
(2.49)
Since the axial resolution is inversely proportional to the spectral bandwidth of the
source,
BW - Afc a V.
Al
(2.50)
Thus
SNR oc
A
V8
(2.51)
which can be alternatively rewritten
SNR c
P
xresolution
smp"e
speed
(2.52)
Thus there is an inherent tradeoff between image quality, resolution, power, and imaging
speed.
38
2.3 Implementation
2.3.1 Source and interferometer
In addition to a short temporal coherence length, a suitable light source for OCT
imaging requires both emission in the near infrared and high irradiance. Emission in the
near infrared is necessary because the light in this spectral range adequately penetrates
into tissue. Attenuation of the OCT signal with depth in tissue is determined by both
scattering and absorption. The blue and ultraviolet spectral ranges are not appropriate
because of the short mean scattering length in tissue at these wavelengths, which would
limit OCT imaging to only superficial layers on the order of a hundred microns thick. At
longer wavelengths, scattering is reduced and absorption, primarily that of water,
becomes dominant. The greatest light penetration depths have been achieved using
sources with emission wavelengths of approximately 1300 and 1600 nm [48, 49]. Note
that the optimal source for an imaging application is not entirely determined by
penetration depth - backscatter contrast is also somewhat dependent on wavelength and
affects contrast between tissue components in OCT images.
High irradiance is necessary to achieve a wide dynamic range and high detection
sensitivity to image weakly scattering structures within the tissue. From equation (2.25),
it is evident that the interference signal is proportional to the square root of the power
reflected from the sample. A dynamic range of 90dB can be attained with a source power
of a few hundred microwatts.
The OCT system used for these experiments utilizes a source with a 1310 nm
center wavelength and 100 nm spectral bandwidth. This wavelength was chosen because
it provides adequate penetration and contrast in cardiac tissue [49, 50] while also
affording readily available, high-performance optical components originally developed
for the telecommunications industry.
2.3.2 Reference arm scanning
An important consideration in designing an OCT system is how to rapidly scan
the reference arm pathlength. A rapid scanning optical delay line (RSOD) provides a
means to probe depths of up to 3 mm at a repetition rate of 2 kHz (Figure 2-4). The
39
RSOD allows independent control of the group and phase delay, advantageous because
the phase delay can be chosen to select an appropriate interferometric fringe
frequency,
while the group delay can be adjusted to provide adequate ranging depth and scanning
speed. Group delay arises from the rotation of the scanning mirror, while phase delay is
determined by the offset between the mirror's axis of rotation and the focus of the lens.
The RSOD design is also advantageous because it allows control of group velocity
dispersion through adjustment of the separation between the lens and diffraction grating.
This capability is particularly important for applications requiring a fiber optic catheter
probe.
GRATING
d
L
AXIS OF
ROTATION
SCANNING
MIRROR
Figure 2-4: Schematic of the rapid scanning optical delay line (RSOD). Light from the system is
incident on the diffraction grating. Diffracted light travels through the lens to
the scanning mirror,
where it is reflected back to the system. (Taken from [511.)
2.3.3 Planar imaging
Imaging of flat samples is performed using the XY scanner shown in Figure 2-5.
The sample arm light travels through an optical fiber to the scanning head where it is
collimated by a lens. The collimated light then reflects off a mirror mounted on a
galvanometer to a low NA objective, through which it is focused onto the sample. As the
galvanometer changes the angle of the mirror, the beam focus is scanned across the
sample. In this way, the sample beam is swept across the sample to provide 2D crosssectional images. The speed of the galvanometer is set for imaging at 4 frames per second
with 500 A-lines per frame.
40
Galvanometer
Fiber optic input
Collimator
-Objective
Sample
tissue
Figure 2-5: Schematic of the XY scanner for planar imaging. Light from the system is collimated and
reflected off a galvanometer-mounted mirror, where it is focused by a low NA objective onto the
sample. Rotation of the galvanometer scans the beam across the sample's surface.
2.3.4 Catheter-based imaging
Imaging of cylindrical samples with a narrow lumen diameter is made possible
through a custom built catheter connected to an optical rotary junction [52]. Light from
the sample arm of the interferometer first travels into the rotary junction, then into the
catheter. The rotary junction (Figure 2-7) allows an optical coupling between the OCT
interferometer and the rotating catheter. Coupling is achieved through free space with a
pair of lenses forming a telecentric telescope. One lens is fixed to the non-rotating optical
fiber and acts as a collimating lens, while the second lens is attached to a short length of
optical fiber that is connectorized at the distal end. The lens/fiber combination is mounted
in a cylindrical steel tube held by bearings. The tube is aligned so that the rotation axis of
the bearings is coaxial with the optical beam from the first, stationary lens. A computer
controlled motor is connected to the bearing-mounted tube to provide continuous,
41
velocity controlled rotation. The distal connector of the rotating fiber allows rapid
interchange of catheters.
FIXED END ROTATING END
-
TRANSPARENT
SHEATH
WITH MNORAL
GUIDEWIRE
12cm
Figure 2-6: Schematic of the rotary junction with attached catheter. The rotational coupler allows an
optical connection between the fixed fiber input and the rotation catheter core.
The catheter itself (Figure 2-7) consists of a flexible, rotating inner sleeve that fits
loosely inside a stationary outer sheath. The catheter must be flexible to allow it to bend
as it navigates the lumen of the sample (e.g. snaking through the coronary tree). The outer
sheath is stationary and the exterior is smooth, providing for safe passage through the
sample. The sleeve itself consists of an optical fiber enclosed in a torque carrying coil.
Torque applied at the proximal end of the inner sleeve is transmitted through the length
of the sleeve to rotate the distal optics. The proximal end of the catheter is connected to
the rotary junction.
Pullback
i
I
.
Optical
beam
I
Inner core
Transparent sheath
Distal optics
Figure 2-7: Catheter schematic. The inner core rotates within the stationary transparent sheath.
Pullback of the core may be performed simultaneously to achieve helical scans of a cylindrical
sample.
The optics at the distal end of the catheter (Figure 2-8) focus the beam onto the
sample at the proper angle [53]. The fiber is attached to a gradient-index (GRIN) lens.
42
The pitch of the lens is chosen to yield the required Gaussian beam parameters. The
confocal parameter and the spot size are chosen so that the spot size of the beam is
comparable to the axial OCT resolution while maintaining an appropriate working
distance (-3 mm). A right angle prism is attached to the distal surface of the GRIN lens
to redirect the beam perpendicular to the axis of the catheter. UV-curing optical epoxy is
used to secure the optics.
Fiber
Epxoxy GRIN Lew-;
Pwn
Figure 2-8: Distal optics of the catheter's inner core. A GRIN lens and prism are used to focus and
direct the sample beam onto the surface of the sample.
Images are acquired by rotating the catheter within the sample while scanning the
reference arm of the interferometer as usual. As the catheter spins, the beam is focused
onto the surface of the tissue and scanned circumferentially. In this system, the motor
speed is chosen to acquire 4 frames per second with 500 A-lines per frame.
2.3.5 Signal acquisition
For the experiments to be presented, either the electronically demodulated signal
data is collected at a sampling rate of 500 kHz, or the interferometric fringe signal is
collected at 5 MHz. All signal acquisition is performed using a National Instruments
DAQ board (Model #6110).
43
2.4 Imaging-based Challenges to Elastography
OCT elastography focuses on tracking each image's speckle pattern from frame to
frame. It is based on a frozen speckle model, which assumes that the sample deformation
is sufficiently small and the imaging rate is sufficiently fast that the speckle pattern over a
small spatial region remains unchanged from frame to frame except for a translation.
Thus elastography essentially tracks the displacement of small stationary patches in the
image. Decorrelation of the speckle patterns between imaging frames presents a
challenge to elastography.
Speckle decorrelation can be caused by Brownian motion of reflecting particles
in the sample. In a viscoelastic medium, suspended particles undergo Brownian motion,
which is directly related to the viscoelastic properties of the medium [54]. Thus even
under static conditions, the underlying reflector distribution may change, resulting in
changes in the OCT speckle pattern. Biological tissues behave viscoelastically, so it is
important that the imaging rate be fast enough that Brownian motion does not induce
large inter-frame displacement.
A greater source of speckle decorrelation is strain induced movement of the
reflecting particles. Very large inter-frame strains violate the frozen speckle model, so
image acquisition rates must therefore be chosen appropriately for a given strain rate.
Additionally, recall from equation (2.25) that the signal at the detector is the convolution
of the reflectance in the sample with the axial PSF of the OCT system. Thus the
relationship between the movement of reflecting particles and the movement of the
speckle pattern is not necessarily direct, leading to potentially more inaccuracies in the
displacement estimates.
OCT's limited depth range and depth penetration present additional challenges for
elastography. In the case of inflation of cylindrical samples imaged through a catheter,
applied strains may be large enough to cause the sample to move out of the image's field
of view, making motion tracking impossible. This is particularly problematic for elastic
coronary blood vessels, which undergo large strains during the cardiac cycle. Increasing
the depth range of OCT is possible but compromises detection sensitivity. The image
penetration depth within tissue is limited by scattering and absorption of light. While the
44
penetration depth of OCT in normal coronary arteries exceeds the typical wall thickness,
in severely atherosclerotic vessels, both attenuation and wall thickness are increased. In
these cases, OCT is unable to visualize the outer portion of the vessel wall and accurate
elastography may be challenged.
Finally, motion artifacts can give rise to significant challenges for elastography.
In catheter-based imaging, motion artifacts can arise from nonuniform rotation of the
rotary junction motor, movement of the catheter inside the vessel, and movement of the
imaging core inside the catheter. The scanning beam in planar imaging has fewer degrees
of freedom than the catheter, so the image sequence is relatively free of motion artifacts.
2.5 Spectral Methods for OCT
New developments in OCT technology hold the potential to alleviate imaging
based challenges to elastography, mainly by increasing imaging speed. Spectral OCT
methods increase the signal-to-noise ratio of the imaging system, and the resulting gain
can be used to increase the imaging acquisition rate. Instead of scanning the reference
arm and collecting the fringe pattern corresponding to each sample depth sequentially in
time, it is possible to interrogate the entire depth profile all at once. Keeping the reference
arm pathlength constant, the spectral density at the detection arm is acquired. The Fourier
transform of this signal gives the same depth profile information as that of the envelope
in time domain OCT. There are two methods of spectral OCT. In spectral domain OCT
(SD-OCT) the spectral density at the detection arm of the interferometer is measured
using a spectrometer. In optical frequency domain imaging (OFDI), a monochromatic,
wavelength-swept source is used with a single photodetector to acquire the detector arm's
spectral density.
2.5.1 SD-OCT
In SD-OCT (Figure 2-9), light from the interferometer is directed to a grating,
which separates the light into its frequency components. The separated light is then
focused onto a detector array, where the signal for each frequency component is acquired.
45
Reference arm
Broadband
Source
CMirror
Sample armPrb
Grating
Detector
Sample
I
Array
tissue
Figure 2-9: Schematic of spectral domain OCT (SD-OCT). A broadband light source is split into a
stationary reference arm and a sample arm. The combined light reflected from each arm is dispersed
by a grating, and the resulting spectrum is imaged onto a detector array.
To establish the relationship between SNR in SD-OCT and time domain OCT,
consider the case of M detectors. The shot noise per detector is proportional to the
reference arm power times the detector bandwidth. Total noise is determined by the sum
of the shot noise for each detector. The reference power at detector i is
i,ref
=
1
I
M
(2.53)
ref
Thus the total shot noise for the detector array is
SD2n 2
NoisesD = 2
m
E,i=
Pe BW
21e2 (Pref
E, (MF=M
BWJ
M
(2.54)
= 7e
P,f B W
E, M
I
= NoiseTD
M
Since the signal in SD-OCT remains the same, the SNR is M times greater.
The SNR increase is a result of reduced shot noise. Recall that shot noise is white
and proportional to the optical power and that the best theoretical SNR is achieved when
the detection is shot-noise limited. By replacing the single element detector in time
46
domain OCT with a spectroscopic multi-element detector, shot noise can be reduced
significantly. Each element of the detector array reads only a small portion of the spectral
width of the signal. The optical power on each element is inversely proportional to the
number of detector elements, so the shot noise density per element is reduced. Since the
signal is not affected, the SNR increases by the number of elements in the detector array.
2.5.2 OFDI
OFDI uses a wavelength-tunable light source and a single photodetector (Figure
2-10). As the source is rapidly swept through multiple wavelengths, the interference
information for each wavelength is simultaneously acquired. Both SD-OCT and OFDI
offer the benefit of high speed imaging, but OFDI offers a few more advantages for
imaging at 1300 nm. OFDI has greater ranging depth (>4 mm) than SD-OCT (2 mm).
Dual-balanced detection and polarization diversity are more easily and cheaply
implemented in OFDI. Finally, OFDI is less sensitive to phase instabilities caused by
catheter motion. OFDI is particularly promising because it is possible to increase imaging
speed and depth penetration simultaneously. Since increasing imaging speed can
significantly reduce motion artifacts, OFDI is attractive for elastography.
Referenc a arm
Swept
source
Mirror
Coupler
Photodetector
Probe
Sample arm
Sample
tissue
Processor
Figure 2-10: Schematic of optical frequency domain imaging (OFDI). The reference and sample arm
path lengths are held constant, while the wavelength of the monochromatic source is swept and
interference is detected by the photodetector.
The possible depth range in OFDI is given by
47
Az =
where 6A = AA/N,
N(2.55)
4n6A
is the sampling wavelength interval, and N, is the number of
samples within FWHM range of the spectrum AA [55]. The sampling interval is chosen
to be smaller than the instantaneous linewidth of the source because otherwise, the
amplitude of the coherence function decays with z, which causes a decrease in
sensitivity as the depth increases. The narrow linewidth of a swept laser typically
provides superior spectral resolution than the grating-based spectrometer used in SDOCT, leading to higher sensitivity as a function of depth and larger usable depth range. A
large difference in optical pathlength will cause a high frequency in the spectrum and,
according to the Nyquist theorem, the sampling rate must be at least twice this frequency.
For z =
zmax
the period of the cosine fringes is Ak
=
w/nzmm
. Thus the spectrometer
needs to be able to measure a period of at least Ak/2. Similar to the noise analysis for
SD-OCT, OFDI's signal to noise ratio is also M times greater than that of TD-OCT,
where M is the number of samples within the FWHM of the source spectrum.
OFDI has been demonstrated using a novel swept laser developed by Yun et al.
with a tuning rate of 15.7 kHz [56]. The laser generates cw polarized light with a center
frequency of 1320 nm and an average output power of 6 mW, with an incident power to
the sample of 3 mW. The FWHM of the spectral source is 63.5 nm, corresponding to
axial resolution of 12.1 ptm. The narrow linewidth of the laser (0.06 nm) gives a possible
6.4 mm ranging depth; however the limited sampling rate of the current hardware
implementation allows only a 3.8 mm depth range. The OFDI system acquires 5 mm
wide images at 30 frames/second (520 A-lines per frame), with a sensitivity of 110 dB.
2.6 Summary
OCT is an interferometric ranging system capable of high speed imaging at a high
resolution of less than 10 ptm. This high resolution makes OCT an ideal candidate for
elastographic strain imaging, though there may be challenges. The frozen speckle
assumption, vital to elastography, can be violated by Brownian motion within the sample,
48
high inter-frame strain, and motion artifacts of the imaging system. New spectral methods
for OCT in development may help alleviate these problems by providing faster imaging
rates and thus reducing the magnitude of motion between frames due to all three effects.
49
50
Chapter 3: OCT as a Basis for Finite
Element Modeling
3.1 Introduction
Investigation of the biomechanics of atherosclerosis has relied upon finite element
analysis for computing stress and strain distributions in idealized vascular cross-sections
[21, 22] or more realistic morphology derived from histology of autopsy specimens [23,
25, 57]. While these studies have provided important insight into disease progression and
acute events, several pressing questions remain that cannot be addressed with a
retrospective, ex vivo methodology. In addition, a non-destructive means for determining
vascular structure and composition would facilitate the investigation of biomechanical
response to new interventional strategies.
The current gold standard for determining plaque geometry and composition is
histology. Although histology provides excellent resolution and, through the application
of tissue-specific stains, can provide clear delineation of tissue composition, histology
processing is known to cause geometry artifacts. During processing, non-uniform tissue
shrinkage can give rise to distortion and warping of vascular geometry. Additionally,
since histology processing requires manipulation of tissue sections as thin as 5-10 gm,
further warping, folding, and tearing of the tissue section is common. These artifacts are
difficult to characterize and limit the accuracy of geometric models for biomechanical
analyses.
A more attractive approach would be to base model construction on a noninvasive imaging modality. Although angiography is routinely used for detecting
coronary stenoses and for directing intravascular intervention, it is limited to the
visualization of the vascular lumen and does not provide information regarding the
structure of the vessel wall. Magnetic resonance (MR) and computed tomography (CT)
imaging have been increasingly used for evaluating vascular structure but lack sufficient
resolution (~ 500 x 1000 pm for in vivo MR [58], ~ 600 x 600 pm for CT [59]) for
construction of detailed finite element models which incorporate local variations in
51
plaque composition. Intravascular ultrasound (IVUS) is used routinely for assessing
vascular structure and stent deployment but is limited to a resolution of -100 x 300 pm
and provides limited contrast between typical plaque tissue components. An IVUS-based
model was developed to predict rupture during in vitro balloon angioplasty [28]. IVUS
images of 16 coronary specimens were digitized and outlined to create the structural
models. An angioplasty balloon was inserted in each specimen and inflated until rupture
occurred. Histologic confirmation of rupture location showed that 17 of 18 fractures
occurred in segments having high stress and 82% of these fractures occurred at locations
of high stress. No correlation was found, however, between predicted peak stress and
balloon pressure at fracture. The lack of correlation was attributed to the low resolution
of IVUS and its inability to detect local variations in material composition.
Optical coherence tomography (OCT) is a recently developed optical analog of
IVUS that provides high-resolution (10 x 25 pm) cross-sectional images of human tissue
[60, 61]. Studies conducted ex vivo have demonstrated that this new imaging modality is
capable of accurately characterizing the structure and composition of normal vessels and
atherosclerotic plaque [40, 62, 63]. OCT catheters, suitable for intravascular application,
have recently been developed and applied for imaging coronary arteries in patients. In
vivo OCT imaging is facilitated by an 8 to 10 cc saline flush to allow clear visualization
of the vessel wall [39]. Potential applications of OCT-based finite element modeling
include patient-specific modeling and diagnosis and studies of disease progression and
treatment efficacy. With its ability to provide high-resolution images of plaque structure
in vivo, OCT is ideally suited for investigating plaque biomechanics using finite element
modeling. However, the major drawback of OCT as a basis for modeling is the limited
depth penetration inherent to the modality, which is often not sufficient to visualize
through the entire thickness of the vessel wall. Thus, ambiguities in external structure
identification can result, which can compromise the accuracy of the finite element
analysis.
In this chapter, we determine the geometry of excised human coronary vessels
using both OCT imaging and the conventional histology method. Using these geometries,
we construct finite element models and then compare stress and strain distributions in
order to assess the feasibility of OCT as a basis for finite element analysis. Finally, we
52
investigate the effect of OCT's limited depth penetration and subsequent outer contour
ambiguity by examining the sensitivity of stress and strain distributions to perturbations
in outer contour geometry.
3.2 Methods
3.2.1 OCT imaging
Excised coronary arteries were collected from autopsies and stored in PBS at 4'C
until imaging occurred, within 72 hours. The OCT catheter had a diameter of 1.0 mm
and, through rotation and longitudinal displacement (pullback) of the internal
components, provided cross-sectional images of the entire length of the vessel segments.
The catheter pullback rate was 0.5 mm/s, and the frame rate of the OCT system was 4 Hz
(500 angular pixels x 250 radial pixels). The axial resolution was 10 pm and the
transverse resolution was 25 pm. A visible light beam coincident with the infrared
imaging beam was used to determine the longitudinal location of the imaging site. Ink
marks were placed at imaging sites of interest for longitudinal registration with histology.
In addition, a narrow diameter thread was placed between the catheter sheath and the
lumen. An ink mark was applied to the vessel to denote the thread location in histology.
Visualization of the thread on OCT and the ink mark in histology permitted rotational
orientation of the two data sets. An example OCT image of a lipid rich plaque is shown in
Figure 3-1, A.
3.2.2 Histology processing and registration
After imaging, the vessels were fixed in formalin and cut into segments for
histology processing. Arteries with significant calcium content were decalcified as
necessary. The vessels were embedded in paraffin and sliced into 4 pm sections for
staining with hematoxylin and eosin (H&E) or Movat's pentachrome. Histology slides
were registered first with the ink marks made during imaging. Fine registration was
accomplished using morphologic landmarks in the vessel wall such as calcium crystals
and nodules, eccentric stenoses and lipid rich cores (Figure 3-1, A and B).
53
3.2.3 Segmentation
Histology segmentation (Figure 3-1,D) was performed by a vascular pathologist.
OCT segmentation was performed by expert OCT readers who were blinded to the results
from histology (Figure 3-1,C). Criteria for characterizing tissue types by OCT have been
established previously [40]. Fibrous plaque is identified by homogeneous signal rich
regions. Lipid rich plaques are characterized by signal poor regions with diffuse borders,
and calcific plaques exhibit signal poor regions with distinct borders. As the infrared
OCT beam propagates through the lumen and into the tissue, it experiences attenuation
through scattering and absorption. In heavily diseased arterial segments where the vessel
wall thickness has significantly increased, attenuation can limit the ability of OCT to
characterize deep structures near the adventitial surface of the vessels. In these cases,
accurate segmentation of the full arterial cross-section can be challenging. Three OCT
readers independently determined segmentations and the discrepancy between the results
was used as a basis for assessing the impact of attenuation on mechanical modeling.
3.2.4 Finite element analysis
Vessel contours from segmentation were imported into ADINA (Watertown,
MA), a commercial finite element software package. A structured finite element mesh
was created using 2D plane strain elements. The element edge length was approximately
30 pm, which, based on grid convergence studies, was sufficiently small. Each
segmented region was assigned uniform isotropic material properties. All regions were
given rubberlike Mooney-Rivlin material properties [64, 65], as defined by the strain
energy density function
W = D (eD2(13) _ 1),
(3.1)
where W is the strain energy density, D, and D2 are material constants, and I, is the first
invariant of the Cauchy-Green deformation tensor. The product D1 *D 2 is proportional to
the elastic modulus of the material, while D 2 is related to the strain-stiffening behavior of
the material. The actual values used were taken from previous literature [66] (Table 3-1)
in which values for fibrous plaque and calcification were calculated from previously
54
published uniaxial test data [67], lipid properties were adapted from previously published
lipid data [68], and arterial wall properties were taken directly from a previous study [69].
Table 3-1: Material properties for atherosclerotic plaques
Material
Fibrous plaque
Arterial Wall
Lipid
Calcium
Di [Pa]
5105.3
2644.7
50
18,804.5
D2
13
8.365
5
20
The boundary conditions consisted of a fixed node on the right side of the outer
boundary of the vessel and a point 180 degrees away that is free to move in one direction
only (, E and F). A pressure load was applied to the lumen of the vessel in 24 steps over
the range of 0 to 16 kPa (0 to 120 mmHg).
55
B
Excised
coronary
OCT
imaging
D
OCT IMAGING
vs
HISTOLOGY
LIPID
FIBROUS
PLAQUE
ARTERIAL
WALL
EZ
F
Y
Y
FINITE ELEMENT
MODELING
Z
P1 free fixed
P2 fixed fixed
MAXIMUM
EFFECTIVE
STRESS
[kPal
A 343.344
MINIMUM
)K 0.001532
11
115
13
H
MAXIMUM
A 984.966
MINIMUM
)K 0.002731
MECHANICAL
ANALYSIS
CYCLIC
STRAIN
..........
1%*
I
2.760
1.800
0.840
MAXIMUM
A7.486
MINIMUM
)K -2.609
Figure 3-1: Finite element modeling process for a lipid rich coronary plaque cross-section. The OCT
(A) and histology (B) images are segmented (C,D) into lipid rich, fibrous plaque, and arterial wall
regions. Each segmented image is used to create an undeformed finite element mesh (OCT mesh (E),
with closeup (F)). Application of an internal pressure load results in stress at systolic pressure (G,H)
and cyclic strain (I,J) distributions. Results for the OCT-based model are shown in the left column;
results for the histology-based model are shown in the right column.
56
3.3 Results
The results of two example cases are presented. The first case is that of a lipid
rich plaque (Figure 3-1). Note that even though the OCT and histology images exhibit a
close correspondence, they do not have identical geometry. In the OCT image, the lumen
boundary is smooth with no jagged edges, but in the histology geometry there are many
sharp edges in the lumen, particularly between the 5 and 6 o'clock position and the 10 and
11 o'clock position. Not coincidentally, this is also where the thin tissue section has
folded upon itself during histology processing, as evidenced by the dark radial bands
through the vessel wall at those locations (and to a lesser extent at the 3 and 9 o'clock
positions). The folding may also explain why the artery appears to be more ellipsoid in
histology than in OCT.
The finite element meshes were inflated to systolic pressure and effective stress
was examined (Figure 3-1, G and H). The two stress distributions are qualitatively
similar, with corresponding regions of high and low stresses. As expected, the lipid pool
exhibits low stresses because it is a very compliant material and thus cannot carry much
load. Being the stiffest material, the fibrous plaque carries most of the pressure load and
therefore has higher stresses than the arterial wall and the lipid. High stresses are found in
the fibrous cap covering the lipid pool, because the thin cap has to carry the share of the
load that the lipid cannot support. High stresses are also found near the 5 o'clock position
due to the curvature of the region. In both cases, this is the most kinked region of the
lumen, naturally leading to focal stress concentrations. Although the maximum stress in
the histology-based model is higher than in the OCT case (985 vs. 343 kPa), its location
corresponds to a portion of the lumen that was folded in histology. This geometry artifact
causes an erroneously high focal stress concentration in the histology-based model. If
these artifactual maximum stress magnitudes are not considered, the stress distributions
are similar between the OCT and histology models, a fact which is more easily
appreciated by examining the stress distribution on a sector by sector basis. Each model
was divided into fifty angular sectors centered at the centroid of the lumen. The mean
stress, as well as the 25th and 75th percentiles of stress were calculated for each sector and
57
plotted, along with the "unwrapped" geometry, as a function of angle (Figure 3-2, B).
The mean stress, as well as the spread, are indeed similar throughout the cross section.
Cyclic strain, the difference in maximum principal strain between systolic and
diastolic pressures (120 and 80 mm Hg), was also examined (Figure 3-1, I and J). Again
the overall strain distribution is qualitatively similar for both models with higher strains
occurring in the lipid pool because it is compliant and displaces easily. Regions of the
lipid pool border with the greatest curvature are expected to correspond to locations of
highest strain. In this example, the histology-based model depicts a lipid pool having a
border with significantly higher curvature than in the OCT model. The strain at this
location was found to be 100% higher in the histology case compared with the OCT case.
Despite these differences, the overall strain distributions are similar (Figure 3-2,C).
58
A
200
I
Histology
OCT
150 --
I
I
-x-
CL-e
U)
CD
50
-
(D
0
0
-- Histology
- - OCT
-
-3
-2
-1
0
0 [rad]
1
2
3
Figure 3-2: Lipid rich plaque histology, stress, and strain as a function of angle. The "unwrapped"
histology image (A), mean effective stress (B), and mean cyclic strain (C). Mean values for histology
are plotted with x's, OCT values with solid circles. Errorbars indicate the 2 5th and 7 5th pecntil
values.
The second example is that of a calcified plaque (Figure 3-3). The calcified region
appears in the third quadrant. In the histology image, there is a small fold in the tissue at
the 2 and 5 o'clock positions, and the tissue has undergone some tearing and separation
of layers.
59
CALCIFIED
REGION
FIBROUS
PLAQUE
ARTERIAL
WALL
Figure 3-3: Calcified plaque OCT and histology images. An OCT image (A) and its corresponding
histology section (B) are segmented into regions of calcium, fibrous plaque, and arterial wall (C and
D).
Typically, portions of the vessel cross-section with greater thickness should
exhibit lower peak stress due to a distribution of the applied pressure load over a greater
area. Correspondingly, high stresses in the calcified plaque model (Figure 3-4, A and B)
are observed at the regions where the model vessel wall is thinnest overall - from 12 to 5
o'clock. Within the thin walled region, there are focal stress concentrations at the sharp
corners of the mesh at 12, 1, and 5 o'clock in the histology model, and at 12 and 5 o'clock
in the OCT model. Another curvature based stress concentration occurs at 10 o'clock.
While the maximum stress in the histology-based model is nearly 300% greater than that
of OCT-based model, the overall similarity of the stress distribution as a function of
angle (Figure 3-5, B) suggests that these maxima are merely outliers.
60
Elevated cyclic strain levels (Figure 3-4, E and F) occur again in the thinnest part
of the artery. Because it is so stiff, the calcified region exhibits very little strain.
Additionally, the part of the artery that lies beyond the calcified region is "shielded" from
the load by the calcium nodule, and thus also sees little strain. Strain distributions are
similar between the two models (Figure 3-5,C).
(
A
EFFECTIVE
STRESS
[kPa]
I
153
100
46
CYCLIC
STRAIN
[%]
3.067
2.000
0.933
Figure 3-4: Calcified plaque effective stress and cyclic strain. The left column shows results for the
OCT-based finite element model, while the right column shows results for the histology-based model.
Effective stress is depicted in the first row, cyclic strain in the second.
61
50
-x-
0- 40o-
Histology
OCT
300( 20 0
0
= 10 0
w
n
Histology
OCT
-
3 ---
0
-3
-2
-1
0
0 [rad]
1
2
3
Figure 3-5: Calcified plaque histology, stress, and strain as a function of angle. The "unwrapped"
histology image (A), mean effective stress (B), and mean cyclic strain (C). Error bars indicate the 2 5 'h
and 7 5 th percentile.
3.3.1 Sensitivity analysis
The finite depth of penetration of OCT imaging poses a potential limitation for
accurate modeling of vessel biomechanical properties. As the infrared OCT beam
propagates into the tissue from the lumen, it experiences attenuation. Although the depth
of penetration is sufficient to visualize the entire cross-section of normal human
coronaries, it is frequently difficult to identify the adventitial structure in heavily diseased
vessels. To investigate the potential impact of ambiguous segmentation on the modeling
62
of biomechanical properties, a segmentation sensitivity analysis was performed. Each
OCT image was given to two trained, non-pathologist OCT readers, and the resulting
segmentation lines were used to create new finite element meshes.
For the lipid rich case, the alternate OCT readers found the plaque boundary to
have a slightly different shape, and they were unable to determine an outer boundary for
the arterial wall. Thus one new model was created using the alternate plaque boundary,
keeping all other boundaries the same (Figure 3-6, A). A second model was created
where the outer arterial wall boundary was taken to be a constant offset version of the
plaque boundary (Figure 3-6 C).
The stress fields (Figure 3-6, D, E, and F) are remarkably similar in distribution
and magnitude. The maximum stress of the alternate plaque model is 1% less than the
original OCT-based model, while the maximum stress of the constant offset outer
boundary model is only 2% higher than that of the original OCT-based model. The strain
fields (Figure 3-6, G, H, and I) are similar as well, with the maximum strain differing by
+0.5% and -1% for the alternate plaque and outer boundary models, respectively. One
difference in the strain distribution can be seen within the lipid pool. The alternate plaque
boundary model does not have a region of elevated strain in the center of the lipid pool as
the other two models do. This reduced strain is expected, as the increased wall thickness
results in greater stiffness overall.
63
ALTERNATE
PLAQUE
D
ALTERNATE
OUTER
BOUNDARY
ORIGINAL
MODEL
E
MAXIMUM
A 339.779
MINIMUM
)K 0.000733
F
MAXIMUM
A 343.344
MINIMUM
* 0.001532
EFFECTIVE
STRESS
[kPa]
MAXIMUM
A 350.170
MINIMUM
* 0.003844
115
75
35
Ii
H
CYCLIC
STRAIN
MAXIMUM
A 4.118
MINIMUM
* -1.666
MAXIMUM
A 4.076
MINIMUM
* -1.598
[0]
2.760
1.800
0.840
Figure 3-6: Segmentation sensitivity for the lipid rich plaque. The left most column depicts an
alternate plaque boundary model (blue line in the segmented OCT image (A)), the center column
depicts the original model (B) , and the right column depicts a constant offset outer boundary model
(black line in the segmented OCT image (C)). The second row depicts effective stress while the third
row depicts cyclic strain.
The same sensitivity analysis was performed for the calcified plaque. In this case,
the alternate OCT readers found the calcified region to be smaller than the pathologist
had indicated. They also indicated the fibrous plaque region as smaller, and they were
unable to completely determine the outer arterial wall boundary. A new model was
64
constructed using alternate calcium nodule and fibrous plaque boundaries as drawn by the
OCT readers (Figure 3-7, B). The alternate arterial wall boundary was chosen to be a
constant offset of the fibrous plaque boundary, with the offset distance approximated
from the wall thickness measured in the first quadrant of the image.
ORIGINAL
MODEL
ALTERNATE
SEGMENTATION
I.
(
EFFECTIVE
STRESS
[kPa]
D
MAXIMUM
600.812
MINIMUM
)K0.02055
153
100
46
CYCLIC
STRAIN
[%]
3.067
2.000
0.933
Figure 3-7: Segmentation sensitivity for calcified plaque. The left column depicts the original model's
segmentation (A), effective stress (C), and cyclic strain (E) results. In the right column, the alternate
segmentation is shown in red (B), and the stress (D) and strain (F) fields are displayed.
65
Despite the significant change in model geometry, the overall stress distribution
(Figure 3-7, B and C) does not change drastically in regions of interest. Regions of
elevated and low stress still correspond and the locations of stress concentrations remain
the same. The maximum stress differs by 10% and the maximum strain differs by 9%.
The cyclic strain distributions (Figure 3-7, E and F) also exhibit modest differences.
These results are not surprising despite the different shape of the calcified region and the
decreased thickness of the arterial wall beyond. The calcified region is very stiff and thus
has a "shielding" effect on the material around and behind it.
3.4 Discussion
In this study we have demonstrated the first use of OCT for finite element
analysis. OCT-based modeling and the accepted histology-based modeling methods
provide similar stress and strain distributions, but can yield disparate stress and strain
magnitudes. These higher focal stress concentrations can arise from artificial sharp edges
in the histology images that arise from nonuniform warping and folding of the tissue
during histology processing.
Despite ambiguities in the segmentation of adventitial structure due to OCT signal
attenuation, the resulting discrepancy in predicted cyclic strain was found to be modest
for the vessel sections examined in this study. Much of the stress variation and the
maximum values of stress and strain tend to occur near the inner lumen of the vessel and
are therefore only slightly influenced by changes in outer geometry. This is most striking
in the case of calcified plaque. Since calcium has a high stiffness, it has a stabilizing
effect on the surrounding tissue [66] and acts to shield the adventitial regions from much
of the mechanical load. It is also important to note that most acute myocardial infarctions
are the result of an intimal disruption of a thin-capped, lipid rich plaque. Assessing the
strain distribution near the lumen is therefore the most relevant goal for identifying and
investigating vulnerable plaque.
The validity of finite element analysis in general is limited by the accuracy of the
specific material model. Biological tissues are anisotropic and exhibit strain-stiffening
behavior.
66
Strain stiffening was incorporated into the analysis by using the Mooney-
Rivlin material model, but anisotropy was not accounted for in this study and should be
included in future similar studies. Another limitation of the modeling applied in this study
is that each tissue component was assigned a single modulus value. Although a more
accurate model would address the natural heterogeneity of biological tissues, the
measurement of modulus values for specific tissue types is challenging on even a
macroscopic level and there is significant variation in the values assigned to vascular
tissue in the literature. A recent study, however, has shown that stress fields are
remarkably robust to variations in elastic modulus [70].
Another limitation of this study is that the potential effect of residual stress was
not considered. When an excised arterial segment is cut, it springs open, suggesting that
in the uncut configuration, the artery is not in a stress-free state. Due to the lack of an
accurate model to quantify the residual stress in an artery [71], it is difficult to assess the
impact of residual stress on resulting stress and strain fields. Nevertheless, studies have
shown that the inclusion of residual stress tends to decrease the absolute magnitude of the
resulting stresses and make the stress and strain distributions more uniform [72, 73]. A
recent study found that the cyclic strain distribution remains relatively unchanged by the
inclusion of residual stress [74]. In this study, the omission of residual stress should not
affect the comparison of OCT and histology-based finite element modeling.
Similar to most previous studies of vascular biomechanics, these results are based
on two-dimensional analyses using the plane strain assumption. This assumption is valid
if the vessel is either constrained longitudinally or if the longitudinal dimension is
sufficiently large so that longitudinal strains are negligible. This may not be the case in
vivo, as some segments of coronary vessels can undergo extension and high curvature
during the cardiac cycle. Longitudinal variations in plaque geometry might also
significantly alter stress and strain fields. OCT-based finite element analysis is
particularly attractive since, through longitudinal pullback of the catheter during imaging,
3D images can be readily obtained. A similar 3D reconstruction based on histology slides
would require a large number of histology slides, and would thus be prohibitively
expensive and time consuming.
The purpose of this study was to demonstrate the utility of using OCT images for
finite element analysis of atherosclerotic plaques. This study can be extended by imaging
67
the excised vessels undergoing inflation. The OCT- and histology-based finite element
analyses can then be evaluated by comparing how well each analysis predicts the final
arterial shape at a given pressure.
Furthermore, finite element modeling of vessels
inflated either in vivo or through an ex vivo experimental setup can be used to estimate
patient-specific elastic modulus if the mechanical load and resulting deformation is
known. Elastography is a method of strain imaging where sequential images of a tissue
being deformed are used to estimate the strains in the tissue [35]. Thus an iterative
reconstruction method can be used with OCT elastography [75] to estimate the elastic
modulus of real tissue [38, 76].
3.5 Summary
Finite element analysis of coronary vasculature is a useful method to gain
understanding of the biomechanical factors relevant to atherosclerosis. In this chapter we
have presented the first use of OCT, an intravascular optical imaging modality, as a basis
for finite element analysis. Comparison with the traditional histology-based method
shows that OCT-based models exhibit similar stress and strain distributions. The results
of a segmentation sensitivity analysis show that the stress and strain predictions are not
significantly affected by segmentation ambiguities associated with OCT signal
attenuation. Since OCT can be performed in vivo and at multiple time points, our results
suggest that OCT-based finite element analysis may be a powerful tool for investigating
coronary atherosclerosis, detecting vulnerable plaque and monitoring response to therapy
in living subjects.
68
Chapter 4: Elastography
4.1 Introduction
Elastography is a method of probing the biomechanical response of a tissue
through analysis of an image sequence of the tissue undergoing deformation.
Elastography consists of two steps: motion tracking, or velocimetry, to estimate the
interframe displacement, and strain calculation. The resulting strain image is called an
elastogram, and it provides information about the mechanical response of the tissue to the
given load under given boundary conditions.
First described by Ophir et al. in 1991, elastography was developed as a
noninvasive method of breast cancer detection [35]. In the breast, cancerous cells are
often indistinguishable from normal fatty tissue in MR and ultrasound images. An
elastogram would enable detection of the cancerous lesion based on its elevated
mechanical stiffness.
Elastography has been applied to coronary blood vessels, using IVUS as the
imaging modality [37]. In IVUS elastography, one-dimensional A-lines from a single
short-axis cross-section are acquired as the artery pulsates over the cardiac cycle.
Consecutive A-lines in time, corresponding to the same circumferential vessel location,
are selected starting from a cardiac phase with minimal catheter motion artifacts. Arterial
tissue displacements as a function of the luminal pressure change are estimated with
cross-correlation analysis, and the corresponding one-dimensional strains are computed
from the tissue velocity gradient. In current practice, IVUS elastography is capable of
acquiring approximately 500 angles per revolution, with an imaging depth of ~7.5 mm,
and a spatial resolution of 200 pm for axial strains (oriented along each A-line). The
spatial resolution of IVUS elastography is a significant limitation since vulnerable
atherosclerotic plaques have structural components (e.g. fibrous caps) on the order of 50200 tm in dimension.
OCT imaging has an order of magnitude higher spatial resolution and
significantly enhanced soft tissue contrast relative to IVUS [39], at the cost of reduced
69
imaging depths. Vascular OCT elastography has the potential to provide high-resolution
characterization of strains in tissue lying within 1-1.5 mm of the lumen interface, which
is the region most susceptible to plaque disruption. Successful development of
intravascular OCT elastography requires robust arterial tissue velocimetry. High accuracy
in the velocity estimation is critical for subsequent strain and elastic modulus estimates.
The primary challenges to implementing OCT elastography are incomplete imaging of
the adventitial portions of significantly diseased vessels due to limited depth of
penetration and rapid noise- and strain-induced decorrelation of intensity patterns
between consecutive image frames due to the short optical wavelength of OCT.
Unfortunately, however, OCT elastography is challenging because the short optical
wavelengths used result in rapid noise- and strain-induced decorrelation of intensity
patterns between consecutive image frames.
This chapter describes the methods used in conventional IVUS elastography, as
the basis for a novel method of robust OCT elastography.
4.2 Velocimetry
4.2.1 Conventional elastography
Velocimetry in conventional elastography uses a block-matching approach based
on a single pair of A-lines. The first A-line is divided into overlapping reference blocks,
while the second A-line, consecutive in time, is divided into search blocks. For each
reference block, the similarity with each search block is calculated, resulting in a ID
correlation function. The relative offset distance of the search block yielding the
maximum similarity gives the displacement of the reference block (Figure 4-1). Since
displacement can equivalently be expressed as velocity (with units of displacement per
frame) both terms will be used interchangeably.
70
2
-
---
---
-
original
--
-.
0
50
100
n
delayed
150
200
30-
~0 .5 -
-
------
-
0
005
100
50
0
Lag
50
100
Figure 4-1: Conventional velocimetry. Conventional velocimetry compares a "reference" A-line with
a consecutive "search" A-line (top). A reference block, indicated by solid blue circles, is defined
within the reference A-line. Correlation coefficients are calculated between the reference block and
identically sized search blocks in the search A-line as a function of lag, the displacement between the
reference and search block positions (bottom). Here, the lag corresponding to the maximum
correlation coefficient is 5, corresponding to the delay between the reference and search A-lines.
There are a number of similarity measures available for the block-matching
technique,
including correlation
(equivalent to covariance),
correlation
coefficient
(equivalent to normalized covariance), sum of squared differences (SSD),
sum of
absolute differences (SAD), and others. Viola and Walker have found that in ultrasound
elastography, certain similarity measures give more accurate velocity estimates than
others, with the most accurate being correlation coefficient and SSD [77]. In this work,
correlation coefficients because they can be quickly computed using Fast Fourier
Transforms.
While conventional elastography has traditionally been performed only in the
axial direction, it is possible to take advantage of OCT's high transverse resolution by
extending the block-matching approach to two dimensions. For an M x N sub-block and
a reference position (x, y), the correlation coefficient, p, is calculated as
71
M/2
N/2
(
(1
PX, (U7 v)
7
P
(
s
S
IM/2
-M2 -N/2
2
f
PR 2
(-R
f
dd2
dx dy
N/2
-M2 -N/2
X1
M12
N12
(2-U,
ff(IS
- V) - /ptS)2 dx'dy'
-M12 -N12
where i
x'- x, # =
y-
IR
and 1 are the reference and search block intensities,
and u and v are the horizontal and vertical displacements of the search block relative to
the reference block. The velocity estimate can then be expressed as
{i (x, y)
v (x, y)j =arg max p. (U, v).
[u
(4.2)
vJ
This block-matching approach to velocimetry hinges upon a "frozen speckle"
assumption, in which strain between the reference and search A-lines is assumed to be
small enough that, over the length of the search block, the two speckle patterns are nearly
identical, except for a simple translation. For real tissues containing ensembles of
scatterers undergoing non-rigid deformation, the coherent impulse response from each
scatterer produces interference patterns in the backscattered signal which may not simply
translate between sequential images in time. This is particularly true for OCT images,
where displacements may easily be the same order of magnitude as the PSF. The
resulting speckle pattern motion may not directly reflect the motion of the underlying
tissue, causing rapid speckle decorrelation between image pairs. Under realistic
circumstances, imaging noise and decorrelation not only reduce the correlation value at
the true displacement within the correlation surface, but also introduce jitter which shifts
the location of correlation peaks in addition to multiple local maxima or false peaks
whose values can exceed the correlation at the true displacement. These false peaks lead
to an inaccurate velocity field using correlation alone, necessitating the development of
more robust velocimetry methods.
72
4.2.2 A variational framework for incorporation of prior knowledge
Strategies for improving velocity estimation include image sequence blurring for
noise suppression, the use of larger correlation windows, and smoothing of velocity fields
after estimation by correlation maximization. Based on observations, these strategies may
lead to limited improvements in velocity and strain estimates while compromising the
spatial resolution advantage of OCT for elastography. Image sequence blurring can
remove not only noise, but also fine image features that may be useful in motion tracking.
Large correlation windows reduce the ability to track fine changes in the velocity field
and may also lead to a violation of the frozen speckle model. Filtering of velocities or
strains either with median-filters or other smoothing kernels operates on the
measurements after they have already been made. These filtering techniques therefore
cannot make use of information present within the underlying correlation function to
improve velocity and strain estimates. A better approach to estimation would allow for
data-driven velocity filtering during the correlation maximization process itself. One such
scheme is the variational method described below.
The velocity estimation problem is posed as a variational energy minimization in
order to exploit velocity information present within the correlation functions while
adding robustness to estimation by incorporating prior knowledge about velocity fields in
the pulsating arterial wall. In this approach, image smoothing is avoided to preserve all
available information from the full-resolution data. The overall of variational energy
functional is
E (v (x, y)) = aED (v(x, y)) + bEs (v (x, y)) + cE, (v (x, y)).
This energy depends on the unknown velocity field v
=
[u (x, y)
(4.3)
v (x, y) and is a
weighted combination of three terms: a data fidelity term, ED(v), a strain field
smoothness term, Es(v), and an incompressibility term, E(v). The functional forms for
each of these terms are:
ED (V (X, Y))
=
ffpX1
(v (x, y)) dxdy
(4.4)
73
f
Es (v (x, y))
E, (v (x, y))
V2v (X, y)1 2 dxdy
(4.5)
ff V -v (x, y)| dxdy.
(4.6)
Minimizing the data fidelity term in the absence of the strain smoothness and
tissue incompressibility terms is equivalent to the correlation function maximization of
equation (4.2) and results in velocity estimates that are identical to those from
conventional velocimetry. The strain smoothness and tissue incompressibility terms
constrain velocity estimation to penalize deviations from prior knowledge about tissue
biomechanics. The strain smoothness term forces the second derivative of the arterial
velocity field to vary smoothly over the wall whereas the incompressibility model
couples the behavior of the u and v velocity fields so that points inside the wall do not
deviate far from incompressibility. By using this energy functional, information in
correlation functions from neighboring reference locations is effectively combined to
confer robustness to decorrelation, false peaks, and poorly defined regions of elevated
correlation coefficient values. The desired velocity field estimate will minimize the
overall variational energy:
ir (x, y) =
arg min
V(X'Y)+~(X'Y)
{aED
(v (x, y)) + bEs (v (x, y)) + cE, (v (x, y))}
(4.7)
v(X'Y)]
4.2.3 Numerical solution to variational energy minimization
In order to obtain a numerical solution to the energy minimization problem, the
image and velocity fields are discretized by sampling the continuous fields from a set of
regularly-spaced points defined on a rectilinear grid. For example, the discretized
reference image, iR , is expressed as
IR (Ax,4y)
74
IR(2A x, Ay)..
where Ax and Ay specify the spacing of the rectilinear grid. The continuous expression
in equation (4.7) can then be discretized to obtain:
V = argmin{aED (V) + bEs (V) + cE, (V)},
V=[
(4.9)
;,
where the discrete velocity components in the column and row directions are represented
respectively as the lexicographically-ordered column vectors
fj[ikjI v=
(4.10)
V[ikjk]
Here k is the lexicographical index of the k"h reference location of interest,
[i,
jk]
are
the row and column coordinates of this location within the reference image matrix
IR
[i, i], and U [i, J] and V [i, J] are the discretized velocity fields. The discrete data
fidelity term is:
ED (V)
-
Pk
[V [k] , i[k]]
(4.11)
k
where the correlation coefficient function in the discrete domain is
M/2
N/2
Z Z
[n - V[k],n -
iiis
m=-M/2 n=-N/2
Pk [V [k], i [k]] =
M/2
[k]]
N/2
Tr (IT
[sn],
" YR)(R *" AR
(4.12)
m=-M/2 n=-N/2
1
x
FM/2
N/2
n=ENTr
M=-M12
[k],I -/2[k]] -[s (is [ -i[k], f -i [k]]
s
n=-N/2
for reference image
lR [i, j]
and search image is [i, J]. The discretized strain smoothness
and incompressibility terms are respectively
Es (V)=
T DD+
T
D2Dn+
22
i
T
DD
r2+
+i
2c
(4.13)
75
and
E (V)
= diTDT
Drf + VjD TDfn
+iTT DjDii +
ni TDDT D
+
TDT1c D ii
f
V + _iTDT D
(4.14)
ri
where D 2r and D 2 c are second-order row- and column-difference matrices which operate
on velocities from neighboring locations in column vectors i and
ii.
Matrices Dir and
DIC are the corresponding first-order row- and column-difference operators. In the case
of the lexicographically-ordered velocity vectors generated from 2D velocity fields
defined on an M x N rectangular domain, the first-order row-difference operator Dir
and first-order column-difference operator DIC are defined as follows:
D 1 (M-1)N
Dir=
(4.15)
.
Dl(M-1)N
with
-1
1
-1
1
(4.16)
Dl(M-1)N
-1
1
and
-I
IM
MI
-I,
D
-C
M
I
(4.17)
.
-I, I
M
where
Dl(M-I)N
IM
is an (M - 1) x N first-order difference matrix and IM is an M x M
identity matrix.
To minimize equation (4.9), its first variation is derived to obtain the Euler
equations
76
+
a
+ cfI T DTDcifi = 0
b (DD2 r + DID 2 c) + c (DTDir +DDTic)
(4.18)
cVTDD
1rD 1C V=0
+ b (D D 2 r + D D 2C) + c (DTDir + DTDjc) Zi
+
a
&ir
where the first variations of the data fidelity terms are defined as
k i[]
a8
(4.19)
afi[k]
DE1
and
_
D
[ [k If[k]]
Op
AD
(4.20)
&V [k]
The Euler equations in equation (4.18) can be solved iteratively by forming the evolution
a
a
DED
[i
t
Of
-itl]-______
+ cDTDi1 + Au =
'
1r 1
OED ' 1--1 1
'
V1r
(fi t
-
equations,
(v
-
(4.21)
cD Dii-1 + AV =
where A = [b (DT D2 r + DLD 2 C)+ c(DTDi + D TDj
)1and
T
1
is the time-step taken at
each iteration. Rearranging to solve for the updated velocity estimate at time t yields the
matrix-vector equations
i t = (TA + I)'
KIt"
Ta
-a
Vt =
(TA + I)-'
Ht
- Ta
DED
Li"
H'
t -1 Iftlt-11
[ rtl
_'
AE
ED
A
'
TDT
-rDi
_ TCDT
.]
Dlft
-CD DD1ri t-1
(4.22)
-]
77
At steady-state, the time derivatives disappear and the resulting velocity estimates satisfy
equation (4.21). The final velocity estimates are obtained by assigning an initial guess
and iterating over equation (4.22) until the maximum change in the velocity field
magnitude is less than 0.01%.
4.2.4 Multi-resolution approach
The solution of equation (4.22) will always converge to a local minimum in the
variational energy function. The unknown velocity field must therefore be initialized
close to the global minimum in order to ensure good global convergence properties. In
order to achieve this, a multi-resolution approach to elastography is used (Figure 4-2).
Pixels in the full resolution reference and search images are averaged together in 5x5
blocks to obtain a low-resolution image sequence from which an initial low-resolution
estimate of the velocity field is obtained by correlation maximization in equations (4.1)
and (4.2). This estimate is used to initialize the variational method applied in the lowresolution domain. The robust low-resolution velocity estimates are then mapped into the
high-resolution domain and are used to define the high-resolution search region for
computing the full-resolution correlation functions at each reference position of interest.
The low-resolution estimates from the variational method also serve as a good initial
guess for iterative estimation of velocity fields from the full-resolution correlation
functions. The resulting full-resolution velocity estimates are then used for strain
calculations.
Input full
reout OCT
ssequence
Low-resolution
equence
Full-resolution
Fu-resolution
>
Low-resolution
correlation
Lwrslto
Low-resolution
method
Full-resolution
variational
method
Output full
resolution
velocities
Figure 4-2: A multi-resolution approach to elastography. Each original image is averaged in 5x5
blocks to create a pair of low-resolution images. Velocity estimates from the low-resolution image
pair are used as an initial guess for high-resolution velocimetry.
78
4.3 Strain Calculation
Strain, a measure of deformation, is defined as the motion of points in the sample
relative to the motion of other points in the sample. Strain depends not only on the
stiffness of the material and the load applied to it, but also on the geometry and boundary
conditions of the particular sample. Typical engineering materials undergo small strains
of less than 5%, but biological tissues may be subjected to much higher strains.
4.3.1 Small strains
The normal strain component is defined as the fractional change in the original
length of a line. Consider the body in Figure 4-3.
X2
E
0/F-------------------------C
0
C
Figure 4-3: Normal strain. The original geometry is shown as a solid
configuration is shown as a dotted line.
line, while the deformed
The normal strain at point 0 along the x-axis is
o'C'i-OC
=
OC
,(4.23)
and similarly, the normal strain along the y-axis is
OF=
(4.24)
Normal strains are positive when the body elongates and negative when the body
contracts.
79
Shear strain is defined as the tangent of the change in angle between two
originally perpendicular axes (Figure 4-4). Shear strain is positive when the first and third
quadrants become smaller and negative when they become larger.
X2
AL
E
E
,of
0
C
-10
xi
Figure 4-4: Shear strain. The original geometry is shown as a solid line, while the deformed
configuration is shown as a dotted line.
For small shear strains (less than 0.01), it is sufficient to define shear strain in
terms of the change in angle itself (in radians) instead of the tangent of the angle change.
Thus in the figure, shear strain at point 0 is
= Z COE - Z C'O'E'
-
2
- Z C'O'E'.
(4.25)
In the continuum limit, equations (4.23), (4.24), and (4.25) become
Ex
E
Y
lim
AX--4O
Ax
i
ylim
AY-0
Ay(4.27)
+(ov/ay)AyV--Ay
ax,
AY
,
(4.26)
ay
and
7, =
lim
AJ-0 2
AY-0
80
7F
2
(Ov/Ox)A x
Ax
(Ou/Oy)Ay
Ay
=-
Ov
Ox
(4.28)
In indicial notation, these can be expressed
I -9 +
,
2 Ox3
Oxi
(4.29)
.
where Y, = 2Ei
4.3.2 Large strains
Since biological tissues are not limited to strains under 5%, an alternate large
strain definition is needed to provide a more general analysis of deformation. Consider
the large deformation in Figure 4-5.
X2
XX
Figure 4-5: Large deformation. The original geometry is shown as a solid line, while the deformed
configuration is shown as a dotted line.
The relationship between the initial and final positions is
x = X + u,
(4.30)
where X is a point in the reference configuration, x is the same point in the deformed
configuration, and u is the displacement between the two points. Considering now an
infinitesimal piece of the material, the deformation gradient tensor F can be defined to
map from the initial configuration to the deformed configuration:
81
dx = FdX
(4.31)
or
Ox1
dx I
dx2
dx 3
Ox 2
1X
Ox1
(Ox 1
OX 2
OX3
dX
Ox 2
Ox 2
dX
X2
Ox3
Ox3
XI
aX2
.
(4.32)
3X
Ox3 dX 3
X3
The deformation gradient tensor can be concisely expressed using indicial notation
9xi
Ox.
(4.33)
OX*
Taking the derivative of equation (4.30),
OX
Ox
x
Ou
O9X
au
(4.34)
OX
The deformation gradient tensor can be rewritten in indicial notation as
FO
=,+a,
(4.35)
axi
where the first term on the right hand side is the Kronecker delta function. However, the
deformation gradient cannot be used as a robust measure of large strain because it
includes rigid body rotations. Instead, a strain measure is sought that is independent of
rigid body rotation, such as the Green-Lagrange strain tensor,
E
-
2
(F TF - I).
Each strain component can be expressed in indicial notation as
82
(4.36)
-
+ "OUk IF& + aOlk
I6ki
6i
,
' 2 I6ki6+096bak
z + 09 x j
6k au~ U+aUkOaUk _ 6
22
ax
x19x(4.37)
39
1
6
-- 6
22
+ U
Ou
ll+
2u
xxi
3
_
k
a+l aOO
x axi2
x
x
3
2
9xi
ax
231
This strain definition is valid for both large and small displacements and strains.
Applying equation (4.37) to a small strain case, the final term becomes a negligible
higher order term, leaving
E
+ O,
2 axi
(4.38)
ax
which is identical to the small strain definition of equation (4.29).
4.4 Summary
In this chapter, we have described the conventional block-matching approach to
velocimetry and a new robust velocimetry algorithm that builds upon block-matching by
incorporating the additional constraints of strain smoothness and tissue incompressibility.
We have also described suitable small and large strain definitions for calculating strain in
biological tissue.
83
84
Chapter 5: Elastography Validation in
Simulation
5.1 Introduction
Simulated image data provides an ideal framework for initial validation of the
variational motion tracking algorithm because it provides a means of testing algorithm
performance under ideal conditions. In simulation, the basic characteristics of OCT
imaging are retained while confounding real-world effects, such as imaging noise and
motion artifacts, are not. Validation in simulation is advantageous over validation in
phantoms because simulations can be performed with significant variations in geometry,
boundary, and loading conditions that would be difficult and time-consuming to
reproduce in phantoms.
In this chapter, simulated OCT images are used to assess the accuracy of the
algorithm, as well as its sensitivity to stiffness, geometry, and applied strain. The
simulation consists of a two-dimensional rectangular block with an embedded circular
inclusion subjected to a displacement load. To measure the strain resolution, the inclusion
diameter is reduced until the inclusion can no longer be distinguished from the
surrounding material in the elastogram. Strain sensitivity is measured by varying the ratio
of the inclusion stiffness to the background stiffness. Different strain levels are also
applied to determine the maximum strain detectable by the algorithm.
5.2 Methods
5.2.1 Simulation of OCT images
OCT imaging was modeled as the convolution of discrete scatterers in the sample
with a coherent point spread function (PSF) derived from measurements of a functional
OCT system [42, 78]. First, an FEA model of the simulated geometry, boundary
conditions, and load was created. Next, a "tissue field" was generated to represent the
reflectivity of scatterers in the simulated tissue. Multiple frames of tissue motion were
85
simulated by warping the tissue field according to displacement results from the FEA
model. Convolution of each frame with the PSF simulated the photodetector output of the
OCT system, which was then demodulated, resulting in the final image sequence.
Finite element analysis
The FEA model consisted of a two-dimensional rectangular block with an
embedded circular inclusion (Figure 5-1). The block's elastic modulus value was 1, while
the inclusion modulus value was varied for each simulation. The diameter of the
inclusion was also varied, but in all simulations, the top edge of the inclusion was located
1.5 mm from the top edge of the block. Note that the simulated image encompasses only
a portion of the entire FEA model's area; this is necessary to ensure that the image
simulates an inclusion in an "infinite" medium.
The center point of the bottom edge of the block was fixed in all dimensions,
while points elsewhere on the bottom edge were free to move horizontally. A downward
displacement load was applied to the top surface of the block. Over five time steps, the
block was compressed by 0.15 mm for a total of 4.3% gross strain. Thus each time step
induced ~ 0.86% strain relative to the initial geometry.
86
1.50 mm
0.50 mm
3.25 mm
3.50 mm
3.25 mm
Figure 5-1: Geometry of simulated OCT images. Two-dimensional finite element models of a
rectangular block with a circular inclusion of different elastic modulus values were created. In each
model, the block dimensions and the distance between the top of the inclusion and the top of the
block remained constant. The diameter and modulus of the inclusion was varied in each simulation.
A displacement load was applied to the top surface of the block, while the bottom surface was
constrained in the center and free to move horizontally elsewhere. The blue dotted line encloses the
image simulation region.
Both the block and the inclusion were modeled as nearly incompressible
(v = 0.495) linear elastic materials. For each simulation, the stiffness in the inclusion
was chosen to represent either lipid or calcium in a background matrix of fibrous plaque.
According to available literature, the stiffness ratio of lipid to fibrous plaque is
approximately 0.0001, while the stiffness ratio of calcium to fibrous plaque is
approximately 5 [66]. For the remainder of this chapter, the term "compliant simulation"
will refer to simulations with an inclusion to background stiffness ratio of 0.0001, while
"stiff simulation" will refer to simulations with a stiffness ratio of 5.
FEA was performed using ADINA 8.0 (Watertown, MA). The geometry was
meshed using 9-node, quadrilateral, 2D, plane strain elements. The mesh density, defined
by the edge length of each element, was 0.025 mm in and around the inclusion and 0.1
mm in the background region, resulting in approximately 3200 elements and 13,000
nodes for each model (Figure 5-2). Solution of each FEA model was performed by the
87
ADINA solver with run time on the order of seconds, and the resulting nodal
displacements were saved in a text file.
I
Figure 5-2: Finite element mesh for simulation. The background region is shown in blue and the
inclusion is shown in red.
Tissue scatterer field
The tissue field is a matrix representing the reflectivity of the simulated material.
Because the PSF is defined only in the axial direction, the lateral resolution of the tissue
field need not be finer than the lateral resolution of an OCT image and was set at 25 pItm.
The axial pixel size of the tissue field was arbitrarily set at 1 pm. The tissue field was
generated from a uniform probability distribution, with different variances used for the
background block and the inclusion to give different scattering properties. For the block
material, the variance was 50, and for the inclusion, the variance was 2, resulting in a 5:1
mean reflectivity ratio. These values were arbitrarily chosen to provide high contrast in
the final demodulated image and have minimal effect on the velocimetry results, due to
the use of correlation coefficients as the similarity measure. Since the correlation
coefficient is calculated by subtracting the mean from each image, only the intensity
distribution is important, not the actual intensity magnitude.
88
Simulation of tissue movement
To simulate tissue movement, the tissue field was warped according to the
displacements from FEA. Before warping, the tissue field was upsampled axially by a
factor of 10 and laterally by a factor of 2 to provide for smoother, more accurate warping
results. The high resolution tissue field was then warped according to
G'(x, y) = G(x - u(x, y), y - v (x, y))
(5.1)
where G is the original image, G' is the warped image, x and y are the image
coordinates, and u (x, y) and v (x, y) are the horizontal and vertical displacement fields.
Starting from the initial tissue field, a warped tissue field was created for each time step
of the FEA model, resulting in a six frame sequence of the tissue field undergoing axial
compression.
Following warping, each tissue field was laterally downsampled back to the
original pixel size of 25 tm. Axial downsampling was not performed because the
increased resolution provides for more accurate results in the convolution step. This
tissue field sequence, with a pixel size of 0.1 tm by 25 tm, was used for convolution
with the PSF.
Convolution with the point spread function
The axial PSF for the OCT system was measured by imaging a perfectly
reflecting surface and observing the acquired fringe pattern. The simulated PSF was
modeled as a sinusoid modulated by a Gaussian, with the Gaussian variance chosen to
match the measured PSF (Figure 5-3).
89
1
0.8 -
Measured
Simulated
-
0.6
-..
~0.2CD
0
.
..
.
.
.
.
...
0.2
Z -0.4
-
-
-0 .6
-0 .8
-1
-30
---.
---....-.-..
-
-
-
-20
--..
-10
0
Delay [mm]
10
20
30
Figure 5-3: OCT point spread function. The PSF of the OCT system was measured by imaging a
mirror, while the simulated PSF was generated by fitting a Gaussian-modulated sinusoid to the
measured points.
In this simulation, all light impinging on the sample was assumed to be perfectly
reflected. Light attenuation was modeled by multiplying each tissue frame by an axially
decaying exponential. Each tissue field frame was convolved with the PSF to yield a
sequence of simulated fringes, and each simulated frame was then axially downsampled
to the original pixel size of 1
tm by 25 tm. The Hilbert transform was used to
demodulate each frame (Figure 5-4).
90
I
I
I
Figure 5-4: Simulated OCT image. A simulated image of an embedded circular inclusion of 0.5 mm
diameter.
5.2.2 Elastography
Velocity and strain fields were calculated from the simulated image pairs and
compared with the true velocity and strain. Noise was added to the images to compare the
performance of the conventional and variational algorithms in response to image noise.
Noisy images were also used to investigate the effect of algorithm parameters: the
correlation window size for both algorithms and the weighting parameters, a, b, and c,
in equation (4.7) of the variational algorithm. Using selected parameters, simulated
images were used to probe strain resolution, modulus sensitivity, and strain sensitivity.
True velocity and strain
"True" velocities were taken directly from the output displacements of the FEA
model. For consistency with elastography derived velocities, the true velocities were
converted into units of pixels by scaling by the reciprocal of the image pixel size. Note
that positive v velocity indicates downward motion, while positive u velocity
corresponds to motion towards the right. Because strains are small (<5%), the small strain
definition was appropriately used to calculate linear axial strain by simply taking the first
derivative of the axial velocity in the axial direction. Under the applied loading
conditions, the magnitude of the vertical displacements is much larger than that of the
horizontal displacements, so only vertical displacements and strains were used for
assessing algorithm performance.
91
The displacement and strain fields of the stiff 0.5 mm diameter inclusion,
subjected to an applied gross strain of -0.86%, are shown in Figure 5-5. Horizontal
displacements are subpixel in magnitude and symmetric about the vertical center line.
Throughout the image, vertical downward motion is observed, with the highest
magnitude at the top of the image, closest to the application site of the displacement load.
The smallest displacements occur at the bottom of the image, closest to the vertically
constrained bottom edge. In the strain field, far away from the inclusion, nearly axially
uniform strain of -0.86% is observed, as would be expected if no inclusion were present.
The stiff inclusion experiences very low strain, but its presence creates a distinctive "X"
pattern of reduced strain in the background material.
92
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
26
24
22
20
18
16
14
12
-0.4
-0.6
-0.8
-1
-1.2
Figure 5-5: True axial displacement and strain for the stiff 0.5 mm diameter inclusion. The
horizontal (A) and vertical (B) velocity fields are displayed in image pixel units (1 pm axially and 25
pm laterally). Positive displacement indicates downward motion. Axial strain (C) is displayed as
percent elongation.
Displacement and strain fields for the compliant 0.5 mm diameter inclusion are
shown in Figure 5-6. Horizontal displacements are symmetric about the vertical center
line. The same vertical displacement gradient seen in the stiff inclusion is present, except
that the inclusion and the material above and below it experience elevated downward
93
motion. The axial strain field exhibits axial strains of approximately -0.86% far away
from the inclusion, with very high strains within the inclusion itself.
0.6
0.4
10.2
0
-0.2
-0.4
-0.6
-
I4
25
1L 120
15
I
C
I
I
10
0
-0.5
-1
-1.5
-2
i
f
Figure 5-6: True axial displacement and strain for the compliant 0.5 mm diameter inclusion. The
horizontal (A) and vertical (B) velocity fields are displayed in image pixel units (1 gm axially and 25
gm laterally). Positive displacement indicates downward motion. Axial strain (C) is displayed as
percent elongation.
94
Performance metrics
The accuracy of each elastography algorithm was assessed by calculating the
root-mean-square (RMS) errors for the velocity and strain as
2
iN
RMS, =
(
N
-6)(5.2)
and
RMS, =
-Z(&
(5.3)
-)2
respectively, where N is the total number of image pixels, vi and i9 are the estimated
and Zi are the estimated and real axial strains at
and real axial velocities at pixel i, and E&
pixel i.
To quantitatively assess whether differing strain in an inclusion is distinguishable
from the background strain, the strain contrast-to-noise ratio (CNR) was calculated as
mean (Einclusion)
mean
(Ebackground
(5.4)
std dev (Ebackground
)
N
-
where
Einclusion
is the axial strain in the inclusion and
Ebackground
is the axial strain in the
background region. If the CNR is greater than 1, the difference in strain between the
inclusion and the background is greater than the strain noise level in the background, and
it is reasonable to expect that the inclusion is detectable.
5.3 Results
5.3.1 Parameter selection
The stiff 0.5 mm diameter inclusion was used to investigate the effect of the
correlation window size and weighting parameters on the accuracy of the velocity and
strain estimates in the presence of image noise. Only the first frame pair was considered,
corresponding to a gross downward strain of -0.86 %.
95
Multiplicative noise was added to each image of the frame pair according to
G
where G,7 L
= G+n*G
(5.5)
is the noisy image, G is the original image, and n is a uniformly
distributed random variable with zero mean and variance o-. The original noise free
image and two noise levels, o- = 0.001 and o-
0.005, were used (Figure 5-7).
Figure 5-7: Simulated images of stiff inclusion with noise. The original simulated image with no noise
(A) has a 0.5 mm inclusion diameter and an inclusion to background stiffness ratio of 5.
Multiplicative Gaussian noise with variances of 0.001 (B) and 0.005 (C) were added to each image of
the frame pair.
Correlation window size
For each pair of images, correlation coefficients were calculated for each of four
correlation window sizes: 21 x 7, 41 x 7, 61 x 7, and 81 x 7 pixels (length x width). The
maximum correlation coefficient of each reference block is plotted in Figure 5-8. As
expected, correlation coefficients decrease with increasing image noise. As the window
size increases, maximum correlation coefficients decrease negligibly. As a measure of
image noise, the average maximum correlation coefficient is calculated for each frame
pair. In the noise free case, this average is p
cases, the averages are p
96
-
-
0.84 and p - 0.57.
0.97, and in the intermediate and noisiest
1
0.9
-ea
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 5-8: Maximum correlation coefficients for varying noise levels and correlation window sizes.
Correlation coefficients were calculated from images with no added noise in the first column, added
noise with variance 0.001 in the second column, and added noise with variance 0.005 in the third
column. Progressively larger correlation window sizes are used in each row (size in pixels: 21 x 7
(row 1), 41 x 7 (row 2), 61 x 7 (row 3) and 81 x 7 (row 4)).
Conventional elastography
Velocities were calculated using the conventional elastography method of
maximizing correlation only (Figure 5-9). In the noise free image pair, the velocity field
is qualitatively accurate for all window sizes, and quantitatively, the velocity RMS error
is less than 5% for all window sizes. Velocity estimates in the intermediate noise case
appear similar to those of the noise free case, but the velocity field is less smooth overall.
Using the smallest correlation window size, erroneous values appear in the velocity field,
97
but as window size becomes larger, the velocity field becomes smoother. The same trend
is more pronounced in the noisiest frame pair, where the largest correlation window
results in a smoother velocity field than the smallest window. This results in a velocity
field that is peppered with incorrect values. Noise in the image pair can create false peaks
in the correlation surface, leading to incorrect velocity estimates. The use of a larger
correlation window averages out the effect of noise, reducing the occurrence of false
peaks and leading to a smoother velocity field. However, as correlation window size
increases, effective velocity resolution decreases as the window becomes too large to
detect velocity gradients within the window. Thus, when choosing the optimal correlation
window size, there is a tradeoff between using a larger window for noise suppression and
a smaller window for high velocity resolution.
98
I
-_-26
I
28
24
22
20
18
16
14
12
10
Figure 5-9: Conventional algorithm velocity for varying noise levels and correlation window sizes.
Axial velocity fields were estimated using the conventional elastography algorithm for noise-free
images (first column) and images with noise variances of 0.001 (second column) and 0.005 (third
column). Correlation window size increases with each row (size in pixels: 21 x 7 (row 1), 41 x 7 (row
2), 61 x 7 (row 3), and 81 x 7 (row 4)).
While the velocity estimates are qualitatively accurate using the conventional
elastography algorithm, the axial strain estimates (Figure 5-10) are not. Because the axial
strain is the derivative of the velocity field, any noise present in the velocity image is
augmented in the strain image. Thus, by producing smoother velocity estimates, larger
correlation windows also produce smoother, more accurate axial strain estimates. In the
noise free image pair, the contrasting strain of the stiff inclusion is discernible from the
background strain when the smallest window is used, and the contrast increases as the
99
window size becomes larger. The intermediate noise frame pair follows the same trend,
though the strain contrast is lessened. Strain contrast in the noisiest image pair is never
sufficient to determine the presence and location of the stiff inclusion.
0
-0.2
-0.4
-0.6
- -- 0.8
-1
-1.2
-1.4
-1.6
-1.8
-
-
-
-2
Figure 5-10: Conventional algorithm strain for varying noise levels and correlation window sizes.
Axial strains were calculated as the derivative of axial velocity for noise-free images (first column)
and images with noise variances of 0.001 (second column) and 0.005 (third column). Correlation
window size increases with each row (size in pixels: 21 x 7 (row 1), 41 x 7 (row 2), 61 x 7 (row 3), and
81 x 7 (row 4)).
Variational algorithm
Using the same image pairs, the effect of correlation window size on the velocity
and strain accuracy of the variational algorithm was investigated. The weighting
parameters of equation (4.7) were arbitrarily chosen to be a = 1, b = 20, and c = 0.1.
100
The resulting velocity fields are qualitatively similar to the true velocity field (Figure
5-11). Increasing the correlation window size does not significantly affect the smoothness
of the velocity field, suggesting that the side constraints incorporated into the variational
energy function have more impact for noise suppression than the correlation window
size.
28
26
24
22
20
18
16
14
12
10
Figure 5-11: Variational algorithm velocity for varying noise levels and correlation window sizes.
Axial velocity estimates were calculated using the variational algorithm from noise-free images (first
column) and images with noise variances of 0.001 (second column) and 0.005 (third column).
Correlation window size increases with each row (size in pixels: 21 x 7 (row 1), 41 x 7 (row 2), 61 x 7
(row 3), and 81 x 7 (row 4)).
The smoother velocity fields estimated by the variational algorithm also lead to
smoother and more accurate axial strain estimates. Similar to the velocity estimates, the
101
strain fields are not sensitive to variations in correlation window size. In the noise-free
and intermediate noise level cases, the "X" pattern of low strain found in the true strain
field appears in the variational estimate. This characteristic strain pattern is absent in the
strain fields from the conventional motion tracking algorithm.
C
---
..-
0.2
-0.4
-0.6
-0.8
-1.
Figure 5-12: Variational algorithm strain for varying noise levels and correlation window sizes. Axial
strains from the variational algorithm were estimated for noise-free images (first column) and images
with noise variances of 0.001 (second column) and 0.005 (third column). Correlation window size
increases with each row (size in pixels: 21 x 7 (row 1), 41 x 7 (row 2), 61 x 7 (row 3), and 81 x 7 (row
4)).
Comparison of velocity RMS errors (Figure 5-13) confirms that for each noise
level, error in the conventional algorithm results decreases as the correlation window size
becomes larger, but results from the variational algorithm are independent of window
102
size. For all noise levels, velocity RMS errors are consistently smaller for the variational
algorithm than for the conventional algorithm.
45
-e- p - 0.97
-
......
-.-.-.
.-.-.-.-.-..-.-.--.-.-..-.-.-.--e
-p~0.84
40 --.
-e- p -
35
-
0)
30
-
p-
-
-.-.-.-.-.-.-.-.-.-.-.
-.
-.
--.
.
. . ... . . .
25
-.
20
--.
.
-.
-...-....
--.-.-.-.-.-.--.-.-.-
0
0
-
15
-~~~.
-.
.--... . . .
-.--.--.
..--.
..
-- p- 0.57
.
- -
-p
-$-
-0
0.57
0 97
.
0 84
.
10
5
U
20
30
40
50
60
70
Correlation window length [pixels]
80
90
Figure 5-13: Conventional and variational algorithm velocity RMS error for varying noise levels and
correlation window sizes. Increasing correlation window length decreases velocity RMS error for
velocities calculated using the conventional algorithm (denoted by circles), while the variational
algorithm's velocity RMS errors (denoted by diamonds) are independent of the correlation window
size.
The strain RMS error exhibits a similar trend (Figure 5-14). For the conventional
algorithm, the noisiest images result in the noisiest strain field, and increasing the
correlation window size decreases the strain RMS error. Strain errors from the variational
algorithm are consistently lower than errors from the conventional algorithm.
103
14000
-- ~~~
12000
10000
0.97
ep~
0.84 _
0.57
0.97
0.84
0.57
-
0
-
0
-P
8000
Cl)
6000
-
--
-
-
Cc
4000
2000
04
20
30
40
50
60
70
Correlation window length [pixels]
80
90
Figure 5-14: Conventional and variational algorithm strain RMS error for varying noise
levels and
correlation window sizes. Increasing the correlation window length decreases strain RMS
error for
the conventional algorithm (denoted by circles) but has no effect on strain RMS
error for the
variational algorithm (denoted by diamonds).
Weighting parameters
The three image pairs were also used to investigate the performance of the
variational algorithm with different weighting parameters, using a correlation window
size of 81 x 7 pixels. Five sets of weighting parameters were used, one in which all terms
were given equal weight, three in which a single term was emphasized, and one in which
the two side constraint terms were emphasized over the data fidelity term (Table 5-1).
Table 5-1: Weighting parameters used for the variational algorithm in the presence
of noise.
Data fidelity
a
1
5
1
1
1
104
Strain smoothness
b
1
1
5
1
5
Incompressibility
c
1
1
1
5
5
The smoothness of the resulting velocity fields is highly dependent on the ratio of
the weighting parameters (Figure 5-15). When the data fidelity weighting parameter, a,
is greater than the two side constraint parameters, the velocity field exhibits noise, due to
an increased dependence on correlation maxima. A high b value, corresponding to the
strain smoothness term, results in a smoother velocity
field, but when the
incompressibility term is emphasized (high c value), more dramatic velocity smoothing
occurs. The circular shape of the inclusion is seen in the velocity fields with low
incompressibility weighting, but it becomes more ellipsoid when c is increased.
105
I
28
26
24
22
20
18
16
14
12
10
Figure 5-15: Variational algorithm velocity for varying noise levels and weighting parameter values.
The variational algorithm was used to calculate velocity for noise-free images (first column) and
images with noise variances of 0.001 (second column) and 0.005 (third column). Each row features a
different ratio of weighting parameter values (a:b:c): 1:1:1 (row 1), 5:1:1 (row 2), 1:5:1 (row 3), 1:1:5
(row 4), and 1:5:5 (row5).
In the variational algorithm, velocity RMS error depends on both the weighting
parameter values and the degree of noise present (Figure 5-16). For the noisiest images, a
106
high weight on the incompressibility term decreases velocity error due to its dramatic
smoothing effect. However, in less noisy images, high incompressibility increases the
velocity error by smoothing over already correct velocity values. The strain smoothness
term, by providing only moderate smoothing of the velocity field, decreases error for
images of all noise levels.
2.8
--
--
.-.
- .--.
.- .2 .4 -..
-........
--..
.............
2.p
-.
--.
--.
p - 0.9 7
p~084
0
.5 7 -
-
2.6--
--
-0
C',
C.)
1 .2 --
1
1:1:1
---
- -
- - - -
- --
-5:1:1
-
-
-- ---.
. -- -- - --..
-- ----.
1:5:1
1:1:5
---.. ..---
1 .4
-
-
-..-.-.
.-.
.-.
.-.
..--.
.......-.
..-.
-0 1 .6 -.. . . . . . .--.
1:5:5
Parameter ratio [a:b:c]
Figure 5-16: Variational algorithm velocity RMS error for varying noise levels and weighting
parameter values. Velocity RMS error of the variational algorithm is dependent on the weighting
parameter values.
Similar trends are exhibited in the strain fields (Figure 5-17). When the data
fidelity term has the highest weight, the strain results are very noisy. Emphasis of the
strain smoothness term results in less noisy, more accurate strain fields, but emphasis of
the incompressibility term results in very smooth strain fields.
107
-0.2
r7
-
-
-0.4
-0.6
-0.8
-1.2
Figure 5-17: Variational algorithm strain for varying noise levels and weighting parameter values.
The variational algorithm was used to calculate strain for noise-free images (first column) and
images with noise variances of 0.001 (second column) and 0.005 (third column). Each row features a
different ratio of weighting parameter values (a:b:c): 1:1:1 (row 1), 5:1:1 (row 2), 1:5:1 (row 3), 1:1:5
(row 4), and 1:5:5 (row5).
The strain RMS error confirms these findings (Figure 5-18). In low noise images,
the moderate smoothing provided by a high b value results in the lowest strain error,
108
while noisy images benefit most from the increased smoothing provided by the
incompressibility term.
140
.............
120..............
120 - - ......
- -- ------------~0.57
--.--.---------. .-.----...
-
"'1 00 20
p - 0.97
p ~ 0.84
1::151:
:511:1:5
1:5:5
Parameter ratio [a:b:c]
Figure 5-18: Variational algorithm strain RMS error for varying noise levels and weighting
parameter values. Strain RMS error of the variational algorithm is dependent on the weighting
parameter values.
For the noisiest images, the incompressibility weighting parameter, with its strong
velocity and strain smoothing effect, has the greatest effect on velocity and strain
accuracy. In the absence of image noise, however, high incompressibility leads to
smooth, erroneous velocity and strain estimates. 'When little image noise is present,
emphasis on the strain smoothness term provides for the most accurate results.
In the remaining experiments of this chapter, noise-free images were used to
investigate strain resolution, modulus sensitivity, and strain sensitivity. The same
parameters of the previous section (a =1, b =20 , and c =0.1) were used. The high b
value was appropriate for the noise-free images, and the low c value prevented
smoothing over small gradients in the velocity and strain fields that might indicate the
presence of a small inclusion.
109
5.3.2 Strain resolution
One method of measuring strain resolution is to determine the smallest inclusion
size that can be detected in the strain image. Two series of simulated images were used,
one stiff and one compliant, each consisting of six images with a single inclusion, ranging
in diameter from 0.075 to 0.5 mm (Figure 5-19).
Figure 5-19: Simulated images of varying diameter inclusions. Stiff and compliant inclusions were
simulated with varying diameter inclusions: 0.075 mm (A), 0.1 mm (B), 0.125 mm (C), 0.15 mm (D),
0.25 mm (E), and 0.5 mm (F). Stiffness does not affect image appearance, thus only stiff simulations
are shown in the figure.
The presence of the stiff inclusion is difficult to determine by visual inspection of
the velocity fields alone (Figure 5-20). In fact, the inclusion is only distinguishable at the
largest diameter of 0.5 mm.
110
U
28
26
24
22
20
18
14
12
10
Figure 5-20: Velocity for stiff inclusions of varying diameter. Inclusion diameters are 0.075 mm (A),
0.1 mm (B), 0.125 mm (C), 0.15 mm (D), 0.25 mm (E), and 0.5 mm (F). Only the largest diameter
inclusion is detectable by inspection the velocity field alone.
The presence of the stiff inclusion is more easily detected by visually inspecting
the axial strain estimates (Figure 5-21). High strain at the inclusion location and the
characteristic "X" pattern in the background strain are both evident at inclusion diameters
of 0.15 mm and larger.
-0.2
-0.4
-0.6
-0.8
-1.2
Figure 5-21: Axial strain for stiff inclusions of varying diameter. Inclusion diameters are 0.075 mm
(A), 0.1 mm (B), 0.125 mm (C), 0.15 mm (D), 0.25 mm (E), and 0.5 mm (F). The inclusion can be
distinguished at diameters of 0.15 mm and larger.
111
A plot of the strain CNR vs. inclusion diameter (Figure 5-22) shows the same
trend. For a strain CNR of less than 0.5, the inclusion strain is not visually distinguishable
from the surrounding strain, but for a CNR greater than 1, the inclusion strain is clearly
distinguishable.
3
- -~~.
.. .-.
.-.
-----
2.5P--
-.
2
..-...
zc
1.5
. . . . .-.
. .-.
.....
.
-.
1
0.5 F
0
0
0.1
0.2
0.3
0.4
0.5
Inclusion diameter [mm]
0.6
0.7
0.8
Figure 5-22: Strain CNR for stiff inclusions of varying diameter. For a strain CNR greater than 1,
the inclusion is detectable in the strain image.
In the compliant inclusion velocity fields, the inclusion is distinguishable from the
background at a minimum diameter of 0.25 mm (Figure 5-23).
112
28
26
I
24
22
20
18
16
14
12
10
Figure 5-23: Velocity for compliant inclusions of varying diameter. Inclusion diameters are 0.075 mm
(A), 0.1 mm (B), 0.125 mm (C), 0.15 mm (D), 0.25 mm (E), and 0.5 mm (F). The inclusion can be
detected at diameters of 0.25 mm and larger.
In the strain field, however, the compliant inclusion appears as a high strain
region for all diameters (Figure 5-24).
-0.2
-0.4
-0.8
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
Figure 5-24: Axial strain for compliant inclusions of varying diameter. Inclusion diameters are 0.075
mm (A), 0.1 mm (B), 0.125 mm (C), 0.15 mm (D), 0.25 mm (E), and 0.5 mm (F). High strain in the
inclusion is detectable at all diameters.
113
Strain CNR for the compliant inclusion is always greater than 1, confirming the
visually observed result that the inclusion strain is distinguishable from the background
for all inclusion diameters.
5
4
cc0
................
.
..
..
...
..
...
...... ....... ....... ....... .......
................
3.5
3
.. . .. ... .. .... ...
.
....... ...
4.5 .........
............ ... .......
....... .......
..........
.......
2.5E
2
................ ....... ........
1.51F .......
1
10
....... ....... ............... .......
0.1
0.2
0.3
0.4
0.5
Inclusion diameter [mm]
0.6
0.7
0.8
Figure 5-25: Strain CNR for compliant inclusions of varying diameter. For a strain CNR greater
than 1, the inclusion is detectable in the strain image.
Strain resolution depends on both the size and stiffness of the inclusion. In
simulations of a stiff inclusion, the inclusion can only be distinguished from the
background material in the strain field at a diameter of 0.25 mm. However, the compliant
inclusion can be distinguished at a diameter as small as 0.075 mm. This seeming
discrepancy is due to the stiffness ratio of the stiff inclusion being close to unity
compared to that of the compliant inclusion, which results in a small strain difference
between the stiff inclusion and its background compared to the difference between the
compliant inclusion and its background.
5.3.3 Modulus sensitivity
The sensitivity of the variational algorithm to stiffness ratio was probed using
simulated images featuring inclusions of varying modulus value. Four simulations were
114
performed, each with a 0.5 mm diameter inclusion of modulus value 0.5, 0.75, 1.25, or
1.5.
Except for the most compliant inclusion, the velocity fields for each simulation do
not suggest the presence of an embedded inclusion (Figure 5-26). The most compliant
inclusion manifests as a region of elevated downward velocity.
28
26
24
22
20
18
16
14
12
10
Figure 5-26: Velocity for 0.5 mm diameter inclusions of varying stiffness. Velocity fields were
calculated for 0.5 mm diameter inclusions of varying stiffness ratios: 0.5 (A), 0.75 (B), 1.25 (C), and
1.5 (D).
In the axial strain field (Figure 5-27), the compliant inclusion is discernible for a
stiffness ratio of 0.5. The stiff inclusion of modulus value 1.5 exhibits low strain that is
marginally distinguishable from the background strain, though the characteristic "X"
pattern is absent.
115
-0.2
-0.4
-0.6
-0.8
-1.2
Figure 5-27: Axial strain for 0.5 mm diameter inclusions of varying stiffness. Strain fields were
calculated for 0.5 mm diameter inclusions of varying stiffness ratios: 0.5 (A), 0.75 (B), 1.25 (C), and
1.5 (D).
Examination of the strain CNR corroborates this finding (Figure 5-28). For the
most compliant case, in which the inclusion is identifiable, the CNR is greater than 1.2.
However, for the stiffest inclusion, the CNR is approximately 0.9, and the inclusion strain
does not significantly stand out from the background.
116
1.3
.
1 .2 ... . . . . . . . . . . . . .
.
1 .1 .. .. . . . . . . . . .
0 0 .9
z
-- - -.-.----.-.-.---
-.--.-.-
CO0.7
0.5
-
- - - --.-.- -.-- --.-
- - - - --.- -.-.-.-.-
- -
-
- - -- --
-
- - -- -
0.6 -- -
0.4
0.3
0.5
1
Stiffness ratio (inclusion/background)
1.5
Figure 5-28: Strain CNR for 0.5 mm diameter inclusions of varying stiffness. For a strain CNR
greater than 1, the inclusion is detectable in the strain image. The stiffest inclusion, with a strain
CNR of 0.9, is only marginally distinguishable from the surrounding material.
5.3.4 Strain sensitivity
The ability to distinguish inclusion strain from background strain depends not
only on the inclusion's size and stiffness, but also on the magnitude of the applied strain.
This strain sensitivity was investigated using simulated images of the complaint 0.15 mm
diameter inclusion at applied downward gross strains of 0.86%, 1.72%, 2.58%, 3.44%,
and 4.3%.
As applied strain increases, the inclusion becomes more easily identifiable in the
velocity and strain fields (Figure 5-29).
117
24
-0.5
22
20
4L
18
16
-1.5
50
-1
45
-1.5
40
-2
35
-2.5
-3
70
-2
60
-3
50
-4
100
90
-3
80
-4
70
-5
120
-3
110
-4
100
-5
90
-6
80
Figure 5-29: Velocity and axial strain for a compliant inclusion with varying applied strain
magnitude. Velocities (first column) and axial strains (second column) are estimated for a 0.15 mm
compliant inclusion subjected to varying applied strain magnitudes: 0.86% (row 1), 1.72% (row 2),
2.58% (row 3), 3.44% (row 4), and 4.3% (row 5).
118
However, the strain CNR (Figure 5-30) does not exhibit a similar increase with
increasing applied strain magnitude, due to the fact that different applied strains are being
compared. Recall from equation (5.4) that CNR is calculated as the difference in mean
strains of the inclusion and background divided by the standard deviation of the
background strain. As the applied strain increases, the variance of the background strain
also increases as a matter of course. The difference in mean strains increases as well, but
not as rapidly as the background strain variance. Additionally, the denominator of the
CNR is meant to be a measure of "noise", but in this case, the background strain is not a
normally distributed noise process, nor is it independently distributed. In fact, this
"noise" is actually skewed away from the mean value of the inclusion strain, making the
inclusion even more distinguishable. For these reasons, while the CNR is useful for
measuring strain contrast for varying inclusion size and modulus value at a given strain
level, it is not an ideal measure for comparing strain contrast between different strain
levels.
1.5
1
1.5
2
2.5
3
35
Gross applied strain [%])
4
45
5
Figure 5-30: Strain CNR for a compliant inclusion with varying applied strain magnitude. Strain
CNR for an inclusion diameter of 0.15 mm decreases with increasing applied strain, despite the fact
that visual strain contrast actually increases.
119
5.4 Summary
Analysis of simulated OCT images shows that the variational motion tracking
algorithm offers significant improvements in velocity and strain accuracy over the
conventional elastography algorithm, particularly when image noise is present.
Correlation window size is a common parameter to both algorithms; in conventional
elastography increasing the window size decreases velocity and strain errors, but in the
variational algorithm, window size has minimal effect on accuracy. The weighting
parameters of the variational energy function wield much greater influence on velocity
and strain accuracy. Placing a heavy weight on the data fidelity term is most similar to
using the conventional algorithm and results in the most inaccurate velocity and strain
fields. Increasing the weighting on the strain smoothness term has a subtle smoothing
effect on both velocities and strains, resulting in the most accurate estimates in images
with low noise. When the incompressibility term is emphasized, more dramatic
smoothing occurs in the velocity and strain estimates, resulting in improved accuracy for
the noisiest images. The variational algorithm builds directly upon the conventional
algorithm by incorporating assumptions about tissue motion, which are necessary for
OCT elastography. Using these side constraints, the variational algorithm is capable of
estimating velocity in cases where the conventional approach would surely fail. However,
the disadvantage of the variational algorithm is that optimization of three additional
parameters is required, a task made more complex by the optimal parameters'
dependence on image quality.
Simulated OCT images also provide a framework for probing strain resolution,
modulus sensitivity, and strain sensitivity. In the variational algorithm, strain resolution
depends on the size of the feature being tracked as well as the stiffness ratio of the feature
to the surrounding material. For simulated stiff inclusions, with an inclusion to
background stiffness ratio of five, the smallest detectable inclusion is 0.25 mm, but
compliant inclusions, with stiffness ratio of 0.0001, are detectable as small as 0.075 mm.
For a relatively large inclusion diameter of 0.5 mm, the algorithm can detect differences
in modulus values as small as 50% in either direction (either stiffer or more compliant).
Thus there is a tradeoff between the smallest detectable size of a feature and its stiffness
120
ratio. For intravascular elastography, this tradeoff may not be a serious limitation. While
it may not be possible to detect small developing lesions, vulnerable lesions consisting of
large, very compliant lipid pools should still be distinguishable. Thus, as a diagnostic
tool, intravascular elastography may prove viable.
121
122
Chapter 6: Elastography Validation in
Phantoms
6.1 Introduction
In the previous chapter, simulated OCT images were used to evaluate the
performance of the variational motion tracking algorithm. However, simulated images
utilize a simplified OCT model and provide for ideal imaging conditions, neglecting
effects such as imaging noise and motion artifacts. Real-world validation of the
elastography algorithm is still necessary and is best performed using phantoms of known
material properties for which velocity and strain can be predicted.
This chapter describes the use of polyvinyl alcohol (PVA) as a phantom material
appropriate for OCT elastography. Phantom construction and experimentation is
described. FEA is used to predict phantom velocity and strain, and elastography and FEA
results are compared. Finally, limitations regarding accuracy are discussed and possible
algorithm improvements are suggested.
6.2 Polyvinyl Alcohol (PVA)
An ideal phantom material for OCT elastography must have appropriate optical
scattering properties for OCT imaging and sufficient mechanical strength to sustain
applied loads. Commonly used phantom materials include agar and gelatin, ideal for their
easily adjustable imaging properties and their 3D deformation patterns, which mimic the
incompressible behavior of biological soft tissues [79]. However, these materials exhibit
low tensile strength, making them unsuitable for phantoms representing vascular tissue,
which are subjected to tensile loads. As an alternative, this chapter explores the use of a
polyvinyl alcohol (PVA) cryogel, which has suitable mechanical strength and
controllable optical scattering properties.
PVA has many desirable characteristics for pharmaceutical and biomedical
applications. It becomes a physically crosslinked hydrogel when prepared through
123
repeated cycles of freezing and thawing, and gels crosslinked in this manner exhibit a
high degree of swelling in water, a rubbery and elastic nature, and high mechanical
strength. These properties are controllable through the concentration of the aqueous
solution, temperature and time of freezing and thawing, and the number of freeze-thaw
cycles [80].
The gel structure of PVA consists of three distinct phases: a water phase with low
PVA concentration, an amorphous phase, and a crystalline phase. When placed in the
freezer, the water in the PVA solution freezes, with an accompanying large volume
expansion. This leads to the formation of large pores in the cross linked hyrdogel. Optical
scattering of the hydrogel results from fluctuations in refractive index due to these pores
[81].
6.3 Methods
Planar and cylindrical phantoms were constructed. Planar phantoms were used to
investigate the effect of varying phantom parameters and for elastography of phantoms
subjected to stretching. Cylindrical phantoms were inflated for elastography experiments.
6.3.1 Phantom construction
Phantoms were constructed using LentiKat@Liquid (geniaLab@, Germany), a
commercially available preparation of PVA. LentiKat@Liquid was diluted in water and
boiled at 100'C for at least 20 minutes, while being stirred periodically. The hot solution
was poured into either a planar or cylindrical mold, sealed with wax paper to prevent
evaporation, and left to sit at room temperature for several hours to allow any air bubbles
in the solution to rise. Each phantom was subjected to at least two freeze/thaw cycles,
where each cycle consisted of 12 hours in a -20'C freezer followed by 12 hours in a 4'C
refrigerator.
PVA phantom parameters
LentiKat@Liquid is an aqueous solution of 10% PVA and various proprietary
additives. Preliminary experiments were conducted to determine the effect of solution
concentration and number of freeze-thaw cycles on the resulting elastic modulus and
124
optical properties of the hydrogel. Elastic modulus was determined via a tension test
performed on the Pyris Diamond Thermomechanical Analyzer (TMA) (PerkinElmer,
Norwalk, CT). Optical scattering was qualitatively assessed by visual inspection of OCT
images. Phantoms were constructed in a planar geometry to facilitate imaging and
modulus measurements.
First a series of homogeneous phantoms was created with a range of solution
concentrations to determine the effect on stiffness and optical scattering. A suitable
concentration was chosen and another series of phantoms was created with a varying
number of freeze-thaw cycles. These results were used to determine an acceptable
combination of solution concentration and freeze-thaw cycles for the creation of
phantoms of suitable mechanical integrity and optical properties.
Planar geometry
Planar phantoms were easily constructed using a planar phantom mold created by
gluing glass slides onto sheet metal to create wide, shallow channels. Prepared PVA
solution was poured into the channels, and the mold was covered with another piece of
sheet metal and sealed with wax tape to prevent evaporation. After completion of the
freeze-thaw cycles, the resulting phantoms were rectangular sheets 1 mm high, 1 cm
wide, and 8 cm long. Planar phantoms were used to test the phantom construction
parameters and for stretching experiments. While only homogeneous phantoms were
used for these experiments, heterogeneous phantoms can easily be created by embedding
different materials in the PVA solution prior to the start of freezing.
Cylindrical geometry
Cylindrical phantoms were used to simulate vascular geometry and loading
conditions. Cylindrical phantom molds, consisting of two concentric stainless steel tubes
held in place by acrylic end caps, were used to create each phantom. The inner and outer
diameters of the resulting phantom were determined by the diameter of the stainless steel
tubing. The inner diameter ranged from 0.127 to 0.165 cm, and the outer diameter ranged
from 0.269 to 0.381 cm, yielding a wall thickness range of 0.104 to 0.254 cm.
Heterogeneous phantoms can be easily constructed by inserting materials of
different elastic modulus into the phantom wall prior to the first freeze/thaw cycle. In
125
these experiments, heterogeneous phantoms were created by inserting an optical fiber
into the wall. After the freeze/thaw cycles were completed, the optical fiber was removed,
leaving a hollow cylinder in the phantom wall. To model a compliant inclusion, either
water or a UV curing adhesive was injected into the cylinder prior to the inflation
experiment (Figure 6-1). Subsequent exposure to UV light cured the adhesive,
transforming the compliant inclusion phantom into one with a stiff inclusion.
Figure 6-1: Photograph of heterogeneous cylindrical phantom. Heterogeneous phantoms were made
by inserting an optical fiber into the phantom wall before the first freeze/thaw cycle. After the PVA
was fully cured, the fiber was pulled out of the wail and the remaining gap was filled with water or
UV curing adhesive to create an inclusion of differing material properties. In the figure, blue water is
injected into the phantom, highlighting the shape and location of the inclusion.
6.3.2 Loading experiments
Planar phantoms were stretched at a predefined displacement rate, while
cylindrical phantoms were inflated with an intraluminal pressure load.
Planar phantom stretching
Two computer controlled stages were used to stretch the planar phantoms. Each
end of the phantom was secured onto a stage, and the stages were simultaneously moved
in opposite directions at identical speeds. The XY OCT scanner was used to acquire
images of the deformation. This is an ideal setup for elastography validation for two
reasons: 1) the strain rate can be directly and precisely controlled by the motorized stage
speed, and 2) both the XY scanner and the phantom can be securely fastened to a stable
surface, reducing the possibility of motion artifacts. A static phantom image sequence
(with no applied strain) was used to assess the possible effect of Brownian motion on the
validity of the frozen speckle model. Different stretching rates were used to assess
limitations in strain rate with respect to imaging speed.
126
Phantom
Figure 6-2: Schematic of planar phantom stretching apparatus. Each end of the planar sample is
secured to a motorized stage, and strain is applied by moving the stages in opposite directions at
equal velocity. The XY OCT scanner acquires images of the deformation.
Cylindrical phantom inflation
Each cylindrical phantom was imaged while undergoing inflation (Figure 6-3),
mimicking the deformation pattern of a coronary artery. Each end of the phantom was
stretched onto a barbed tubing connector. One connector was attached to a water
reservoir that was manually raised and lowered to control the phantom's intraluminal
pressure. The OCT catheter was inserted into the phantom through the other connector.
For simplicity, intraluminal pressure was applied linearly, rather than using a
physiological time-varying pressure function. Intraluminal pressure was simultaneously
measured during imaging using a manometer and images and pressure data were recorded
by the OCT system.
127
Water
reservoir
Digital Pressure
Meter
(Change height to
change pressure)
OCT system
(Acquires images and
pressure readings)
Silicon
tubing
44
Valve
Barbed
connector
Phantom
OCT
catheter
Figure 6-3: Schematic of cylindrical phantom inflation apparatus. The cylindrical phantom is
stretched onto barbed connectors, forming a watertight seal. The water reservoir is raised to increase
the intraluminal pressure and inflate the phantom. The OCT system acquires OCT images and
digital pressure readings simultaneously.
6.3.3 Numerical modeling
FEA was used to find the "true" velocity and strain for the cylindrical phantom,
similar to the methodology presented in Chapter 3. First, the phantom geometry was
manually extracted from the initial image of the inflation sequence to create the finite
element mesh. The PVA was modeled as a linear elastic material, and each mesh element
was assigned a single, experimentally measured Young's modulus and a Poisson's ratio
of 0.495. The boundary conditions were a pair of nodes at 3 and 9 o'clock constrained to
horizontal motion only as well as a pair of nodes at 12 and 6 o'clock constrained to
vertical motion. These boundary conditions allowed the mesh to expand outward from
the center of the lumen. The finite element displacements were output to a text file.
Velocity estimates from elastography describe motion relative to the catheter
position, which does not always coincide with the stationary reference point in the FEA
model. Thus, to facilitate comparisons between the FEA-predicted and elastography128
A
estimated velocities, the FEA velocities were transformed to describe motion relative to
the catheter position.
6.3.4 Elastography
Elastography was performed on each image sequence using the variational
algorithm. The correlation window for the planar stretching experiments was 81 pixels
long by 7 pixels wide, while the correlation window for cylindrical inflation experiments
was 61 pixels long and 61 pixels wide. The weighting parameters for all experiments
were a
-
1, b = 30, and c = 20. High values for b and c were chosen because of the
presence of image noise. The Green-Lagrange strain tensor was calculated for each image
sequence, and its eigenvalues were calculated, yielding principal strains.
6.4 Results and Discussion
6.4.1 PVA phantom parameters
To investigate the relationship between phantom stiffness and PVA concentration,
a series of planar phantoms was created with PVA concentrations of 20%, 40%, 60%,
and 80%. Each phantom was subjected to two freeze-thaw cycles, and the elastic
modulus of each was measured. As PVA concentration increases, the stiffness of the
resulting hydrogel also increases (Figure 6-4).
129
300
250
-
co
C,
200
E
150
- . . . . ... . .
..--..-.-.-.
.
.--..
-
0
0> 100
0-
50
n
0
20
40
60
PVA concentration [%]
80
100
Figure 6-4: Dependence of PVA stiffness on solution concentration. The elastic modulus of the PVA
hydrogel increases with increasing PVA concentration.
Each planar phantom was then imaged with OCT (Figure 6-5) to observe the
dependence of the optical scattering properties on concentration. A denser speckle pattern
is seen with the 40% PVA solution than with the 80% solution. Light attenuation also
increases as concentration increases.
Figure 6-5: OCT images of PVA phantoms compared to human aorta. In OCT images, 40% PVA (A)
has a denser speckle structure than 80% PVA (B). The speckle pattern of PVA is similar to that of
human aorta (C).
High PVA concentration produces high mechanical stiffness but results in low
OCT signal penetration depth. Conversely, low PVA concentration results in very long
penetration depths, but mechanical stiffness is low. Therefore, as a compromise, 60%
PVA concentration was used in the remaining experiments.
130
Using a 60% concentration solution, the number of freeze-thaw cycles in phantom
preparation was varied to determine the effect on the elastic modulus (Figure 6-6). With
an increasing number of freeze-thaw cycles, the phantom becomes stiffer, but the effect is
gradual. Since each cycle requires 24 hours and the number of cycles does not
significantly affect stiffness, the remaining experiments were conducted with the
minimum two freeze-thaw cycles.
150
CZ
o 100 --..............
..................
E
0
50'
0
2
6
4
Freeze thaw cycles [#]
8
10
Figure 6-6: Dependence of PVA stiffness on number of freeze-thaw cycles. As the number of freezethaw cycles increases, PVA stiffness gradually increases.
While PVA phantom construction seems to be a simple process, it is not easily
reproducible experimentally. For a single batch of LentiKat@Liquid solution, several
phantom molds were filled and subjected to identical freezing and thawing conditions.
However, within each batch, some phantoms failed to form a hydrogel, while others
became too stiff or exhibited too much light attenuation to be useful for OCT
elastography. The results presented in this section provide a general guideline for
phantom construction parameters, but the process has low yield overall, requiring
significant trial and error to construct suitable phantoms.
6.4.2 Planar phantom stretching
A planar homogeneous phantom (Figure 6-7) was used to investigate the effect of
Brownian motion in the sample and the relationship between strain rate and imaging
speed. The phantom was imaged under static conditions and at various stretching rates.
131
Figure 6-7: Planar homogeneous phantom. The planar homogeneous phantom was imaged at no load
and while being stretched equally from both the left and right ends.
To investigate the effect of Brownian motion, the phantom was imaged under
static conditions. Correlation coefficients were then calculated between the first frame
and each subsequent frame to determine the rate of image decorrelation due to both
intrinsic motion of the phantom material and imaging noise. Three regions in the initial
image were defined, and the mean maximum correlation coefficient was calculated for
each region (Figure 6-8). The blue region maintains a high correlation coefficient over
the entire 40 frame image sequence (10 seconds) because of its location near the top of
the phantom. Correlation windows in this region include the strongly reflecting
air/phantom interface and thus remain well correlated through the entire sequence. The
green and red regions display faster decorrelation rates because their correlation windows
do not contain any large, sharply defined features. Correlation values in the red region are
lower than those in the green region, due to its lower depth and higher signal attenuation.
Imaging noise competes with the low signal magnitude in the red region to rapidly
decorrelate the images, but the effect is less severe in the green region. In the first 2
seconds of imaging, all correlation coefficients remain above 0.9, suggesting that the
frozen speckle model is not violated by Brownian motion at an imaging speed of 4 fps.
132
1
B
S095
0
0
0.85
0
2
6
4
8
10
Time [s]
Figure 6-8: Correlation coefficients for static homogeneous planar phantom. The homogeneous
planar phantom was imaged as it was held static. Maximum correlation coefficients were calculated
and averaged for the blue, red, and green regions (A). The average maximum correlation coefficient
for each region is plotted in (B).
Applying strain to the phantom may increase the magnitude of Brownian motion,
so the same analysis was performed while the phantom was being stretched at various
rates. The phantom was stretched at stage speeds of 100 and 200 tm/sec, corresponding
to 1 and 2 pixels per frame, per stage, respectively. Note that the separation distance
between the two stages is larger than the imaging site, so the maximum velocity within
the OCT images is less than 1 pixel per frame. These speeds were chosen because slower
speeds correspond to very small interframe velocities, which are difficult to track, and
higher speeds are likely to suffer from rapid decorrelation.
Correlation coefficients were calculated between the first frame and each
subsequent frame for a total of 8 frames (Figure 6-9). In both cases, the blue region has
the highest correlation coefficients, followed by the green and red regions. Additionally,
133
correlation coefficients decay more rapidly when the phantom is subjected to the faster
stretch rate. Coefficients remain high for the first several frames, suggesting that
deformation rates of 1 to 2 pixels per frame are acceptable for elastography.
1
B
0.9
100 gm/s
200 gm/s
--
...........................
0.8
0.
.
. . ..... . . . . . . .
0.51
0
0.5
1
1.5
2
Time [s]
Figure 6-9: Correlation coefficients for the stretched homogeneous planar phantom. The
homogenous phantom was imaged as it was being stretched from both ends at 100 and 200 pm/s.
Maximum correlation coefficients were calculated and averaged for the blue, red, and green regions
(A). The average maximum correlation coefficient for each region is plotted in (B).
The variational elastography algorithm was tested on a single frame pair (frames 4
and 5) of the 100 gm/sec sequence. The OCT image spans the central 1 cm of the entire 5
cm length of the phantom (measured between the two stages). Thus the horizontal
velocity at the edges of the image is expected to be 0.2 pixels in magnitude, and a
velocity of zero is expected at the center. As the phantom was stretched, tension caused
the phantom to vertically compress and rise slightly from its initial position, with the
bottom edge of the phantom undergoing higher magnitude vertical motion than the top
edge. The elastography-estimated velocity fields (Figure 6-10) are a poor match for the
134
predicted velocities. Horizontal velocities are generally in the correct direction, but the
magnitudes are lower than expected. The vertical velocity correctly indicates that the
phantom moves upward. However, in both cases, the velocity fields vary nonlinearly and
indicate nonphysical motion for a homogeneous phantom. For example, the right side of
the horizontal velocity field features two regions of high magnitude motion towards the
right separated by a region of lower magnitude motion to the right. This would indicate
that a portion of the phantom folded upon itself, which is not possible given the loading
condition.
0.04
0.02
0
0.0 2
0.0 4
0.1
0.1 5
0.2
0.2 5
0.3
0.3 5
Figure 6-10: Stretched planar homogeneous phantom velocity from elastography. Horizontal (A) and
vertical (B) velocity fields are shown in pixel units.
The principal strain fields from elastography are also inaccurate (Figure 6-11).
The first principal strain field corresponds to horizontal extensional strain, which is
predicted to be uniform in magnitude. Instead, the estimated strain field is nearly uniform
overall in magnitude and sign, with pockets of high strain throughout. This strain field
also indicates nonphysical motion, where high strain areas are stretched more than the
135
adjacent material. Similar nonphysical strains are seen in the second principal strain field,
which corresponds to vertical compressive strain.
1.5
1
0.5
0
0
0.5
1.5
2
2.5
3
Figure 6-11: Stretched planar homogeneous phantom strain from elastography. The first principal
strain (A) is extensional strain in the horizontal direction, while the second principal strain (B) is
compressive strain in the vertical direction. The strain fields are shown as percentages.
The failure of the variational elastography algorithm may be due to the
homogeneity of the PVA phantom's speckle structure. The lack of well defined features
in the speckle pattern makes motion tracking particularly difficult. Additionally, phantom
motion is large compared to the OCT optical wavelength, so the speckle pattern may not
translate linearly with the scatterers in the tissue.
6.4.3 Cylindrical phantom inflation
Elastography performed on cylindrical phantoms may yield better results than the
planar phantom because the phantom geometry is small enough and the correlation
136
window is large enough that the strongly reflecting air/phantom and water/phantom
boundaries boost correlation coefficients at all points in the image.
Homogeneous phantom
A homogeneous cylindrical phantom was inflated from 10.5 to 23.3 mmHg while
being simultaneously imaged for eleven frames (Figure 6-12). The first and fourth frames
of the image sequence were chosen for velocity estimation, corresponding to a pressure
change of 3.5 mmHg.
137
25
D
10
0
2
4
6
Frame
8
.. . .
.
2 0 -.. . . . . . . . . . ...
10
12
Figure 6-12: OCT images of homogenous phantom inflation. The phantom was inflated from 10.5
mmHg (A) to 23.3 mmHg (C). Elastography was performed using the first and fourth frames (B),
corresponding to a pressure change of 3.5 mmHg. Intraluminal pressure rises nearly linearly
throughout the inflation sequence (D).
Numericalvelocity and strain
The phantom geometry from the initial image was used to create a 2D finite
element mesh (Figure 6-13). Plane strain elements were used with mesh density defined
by an edge length of 0.05 mm. The boundary conditions were chosen to allow radial
expansion of the mesh from the lumen centroid. The FEA model was inflated to an
internal pressure load of 3.5 mmHg.
138
z
_Y
U2 U 3
Figure 6-13: Undeformed FEA mesh of homogenous cylindrical phantom. The mesh boundary
conditions ensure that the phantom expands radially outward from the lumen centroid when the
model is subjected to an internal pressure load.
To verify the model geometry, the pre- and post-inflation boundaries of the FEA model
were overlaid onto the corresponding OCT images (Figure 6-14). The inflated FEA
geometry shows close agreement with the OCT image of the inflated phantom.
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I
Figure 6-14: Phantom FEA model boundaries overlaid onto corresponding OCT images. The initial
FEA model boundary (shown in blue) was derived from the initial phantom OCT image (A). The
inflated FEA model boundary (shown in red) matches the OCT image of the inflated phantom (B).
Velocity results from FEA show that the phantom motion is predominantly
radially outward (Figure 6-15).
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I
6
I
4
3
2
0
1
2
Figure 6-15: Inflated homogeneous phantom predicted velocity from FEA. Horizontal (A) and
vertical (B) velocities predicted by FEA show that the phantom expands radially outward (C).
The FEA-predicted principal strain fields were calculated (Figure 6-16). The first
principal strain is tensile hoop strain and decreases with increasing radius. The second
principal strain is compressive radial strain, which also decreases in magnitude with
increasing radius.
141
I
I
40
5
35
1 0
1
30
2350
10
5
25
Figure 6-16: Inflated homogeneous phantom predicted strain from FEA. The first principal strain is
tensile hoop strain (A), and the second principal strain is compressive radial strain (B). Strains are
shown as percentages.
Elastography
The variational elastography algorithm was used to estimate velocities in the
phantom (Figure 6-17). Velocities are qualitatively accurate and point in the correct
direction, but the velocity magnitudes are underestimated. The root-mean-square error,
calculated for the velocity magnitude, is 37%.
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I
3
I
3
2
2
0
1
0
1
2
Figure 6-17: Inflated homogeneous phantom velocity from elastography. Horizontal (A) and vertical
(B) velocities indicate that the phantom expands radially outward (C).
The strain fields are also inaccurate (Figure 6-18). The concentric ring pattern
seen in the FEA predicted strains is not present in the estimated strain fields.
Furthermore, both strain fields are positive, erroneously indicating that the phantom is
increasing in volume.
143
60
10
50
5
40
0
30
5
20
10
10
Figure 6-18: Inflated homogeneous phantom strain from elastography. The first (A) and second (B)
principal strains are both positive, indicating that the phantom increases in volume as it inflates.
Heterogeneous phantom
A heterogeneous phantom with a water-filled inclusion was also imaged under
intraluminal pressure for a total pressure change of 1.5 mmHg (Figure 6-19). At another
cross-sectional location of the phantom, the wall between the inclusion and the lumen had
broken, so water was able to freely flow between the two cavities.
-Nod
Figure 6-19: Inflated heterogeneous phantom images. The first frame was imaged at 1.2 mmHg (A),
while the second frame was imaged at 2.7 mmHg.
144
Numerical prediction
The phantom was modeled using the geometry of the initial OCT frame. The
inclusion was treated as a second lumen and assigned a pressure load equal to that of the
true lumen. The mesh density and boundary conditions were similar to the example in the
previous section.
FEA predicted velocities indicate that the phantom moves radially outward
(Figure 6-20). Additionally, vertical velocity magnitudes immediately surrounding the
inclusion are higher than the corresponding velocities on the opposite wall of the lumen
because the inclusion's compliance does not withstand the intraluminal pressure load
well.
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4.
3
4
31
2
1
11
2
10
0
2
2
3
4
4
3
Figure 6-20: Inflated heterogeneous phantom velocity from FEA. The horizontal (A) and vertical (B)
velocity fields reflect the phantom's radially outward motion (C).
Strains predicted by FEA (Figure 6-21) show the effect of the inclusion on the
phantom's deformation. The tensile hoop strain surrounding the inclusion is elevated
because there are only two thin strips of phantom material available to maintain the
structural integrity of the phantom in the region. Similarly, the compressive radial strain
is high near the inclusion because there is less material to absorb the intraluminal
pressure load.
146
A70
5
60
1 0
50
1 5
40
20
30
25
20
30
10
35
Figure 6-21: Inflated heterogeneous phantom strain from FEA. Tensile hoop stress (A) and
compressive radial stress (B) are elevated in the region surrounding the inclusion.
Elastography
The variational elastography algorithm was applied to the image pair to estimate
velocities (Figure 6-22). The velocity field appears qualitatively accurate, but it is not
quantitatively accurate, with a velocity magnitude error of 39%.
147
22
1.5
1.5
11
0.5
0.5
0
0
0.5
0.5
1.5
1 .5
2
Figure 6-22: Inflated heterogeneous phantom velocity from elastography. Horizontal (A) and vertical
(B) velocities indicate radial outward motion (C).
The estimated strain field (Figure 6-23) is inaccurate. High magnitude tensile
hoop strain, which is expected near the inclusion, occurs in the opposite wall of the
lumen instead. The compressive radial strain field does show high magnitude strain near
the inclusion, but it also shows tensile radial strain in other parts of the phantom wall.
Tensile hoop and radial strains occur simultaneously at many locations in the figure,
suggesting that the phantom is increasing in volume.
148
I
I
50
40
10
5
400
30
20
5
1 0
15
Figure 6-23: Inflated heterogeneous phantom strain from elastography. The first (A) and second (B)
principal strains are both positive at many locations, a nonphysical result indicating that the
phantom volume is increasing.
6.5 Discussion
The examples shown in this chapter illustrate the challenges to applying OCT
elastography to real imaging data. The planar phantom results are inaccurate, possibly
because the PVA speckle structure is too homogeneous for accurate motion tracking.
Results from the cylindrical phantom are more accurate because the presence of the
air/phantom and water/phantom boundaries provides distinct structures that facilitate
motion tracking. However, in all cases, the velocity and strain fields are often
nonphysical and violate geometric compatibility constraints. Improvements to both the
algorithm and the imaging hardware may be necessary before OCT elastography can be
performed in real imaging applications.
6.5.1 Decorrelation
Requirements for the size scale of the underlying tissue structure are interrelated
with the magnitude of applied strain relative to the imaging speed and Brownian motion.
The applied strain needs to be large enough to track, but not so large that the speckle
pattern becomes completely decorrelated. Unfortunately, the small size of the OCT PSF
relative to the underlying motion makes speckle tracking difficult as speckle patterns may
change dramatically even for a small motion. The presence of larger scattering structures
149
may improve elastography estimates by creating a more stable speckle pattern.
Fortunately, diseased arteries generally have less homogeneous structure than the PVA
phantoms.
Despite decorrelation effects, there are a few steps that may improve velocity
estimation. Strongly reflecting image boundaries are easy to track, and incorporating
velocity estimates at boundaries as a constraint in the variational energy function may
improve the accuracy of the algorithm. Additionally, frame averaging and image blurring
may stabilize rapidly decorrelating speckle patterns by emphasizing large structures in
the image and features that persist for several frames.
6.5.2 Motion artifacts
Catheterized imaging may prove to be problematic for elastography because of
the presence of motion artifacts. Typically, the catheter is not constrained within the
lumen of the sample, and the rotational motion of the core inside the flexible catheter
sheath may cause the entire sheath and core assembly to wobble within the lumen crosssection. This non-stationary frame of reference manifests in the image sequence as
shifted A-lines. For example, imaging a perfectly cylindrical sample with the catheter in
the center of the lumen would result in a perfect circle in the image. However, a sudden
movement of the catheter for a portion of the acquisition time would result in an image
featuring a shifted sector in an otherwise perfect circle.
Similarly, the free space between the outer diameter of the imaging core and the
inner diameter of the sheath may cause the same type of motion artifact. Artifacts caused
by core movement within the sheath are distinguishable from sheath movement in the
sample by the visual presence of the A-line shift in the sheath portion of the image. That
is, a sudden movement of the core inside the sheath can cause the entire A-line to
suddenly "jump" closer to or farther from the center of the image. Because the sheath is
visible in the image, it may be possible to correct for this artifact by shifting A-lines so
that the sheath image matches some predetermined sheath geometry.
Another, more damaging artifact arises from the non-uniform rotation of the
catheter. Under the ideal conditions of uniform rotation and properly calibrated rotation
speed, each A-line depicts the same location from frame to frame. If the rotational speed
150
I"
is uniform but not correctly calibrated, the image will appear to precess. At small
precession rates, this artifact does not significantly affect motion tracking accuracy.
However, the effect of non-uniform rotation rate is far more dramatic. For example, if the
rotation rate were too slow for a portion of one frame, five consecutive A-lines may
depict the same region that three A-lines depicted in the previous frame, and vice versa
for an increase in rotation rate. Over an image sequence, this manifests as false
circumferential motion, where sectors of the image appear to move back and forth
circumferentially in a non-physical manner.
6.5.3 Energy function
Further improvements can be made to the variational algorithm by adding and
modifying terms in the energy function. For example, the strain smoothness term is
calculated at all points in the image, encouraging a smooth strain field throughout the
image. Real tissues,
however,
have piecewise
continuous
strain fields, with
discontinuities occurring at the locations where the underlying tissue stiffness changes.
Imposing the strain smoothness term over the entire image makes it difficult to detect
regions of inhomogeneous stiffness. A more sophisticated refinement to the energy
function would be to incorporate a strain "edge" field to specify boundary locations over
which the strain smoothness term should not be calculated. This edge field would become
an argument of the energy functional, so minimization would occur over both the velocity
field and the edge field.
The variational algorithm can also be made more robust by adding more prior
information to the energy function. Boundary movement is relatively easy to estimate and
can be incorporated into the energy function. Often the general direction and approximate
magnitude of the loading conditions are known, and this information can be incorporated
into the energy functional as a second data fidelity term by penalizing deviations from the
expected principal strain directions.
Geometry compatibility is a very strong prior that can be incorporated into the
variational energy function to improve velocity estimation accuracy. This could be
accomplished by introducing a deformable mesh to ensure geometry compatibility at
every point, thus requiring a one-to-one mapping between material points at time t to
151
material points at time t + At. Instead of estimating velocities at each image pixel, the
image would be subdivided into a mesh, with each term of the energy function redefined
in terms of the mesh node coordinates. In this scheme, the strain smoothness term would
be automatically enforced by compatibility of the mesh geometry, and the modified data
fidelity term would be based on correlation between the image data in each original mesh
element and the image data in the corresponding deformed mesh element.
6.6 Summary
PVA can be used to create phantoms with suitable optical scattering and
mechanical properties for validation of OCT elastography in both planar and cylindrical
geometries. A high concentration of PVA results in phantoms with high stiffness and
strong light attenuation. Conversely, low PVA concentration phantoms have low
mechanical stability and are highly optically scattering throughout their cross-section. A
60% concentration of LentiKat@Liquid, a readily available commercial preparation of
PVA, subjected to two freeze-thaw cycles of 24 hours each provides an ideal compromise
between mechanical stiffness and light attenuation.
Phantoms were constructed in planar and cylindrical geometries for validation of
the variational elastography algorithm. Analysis of a stretched planar phantom showed
that despite noise and Brownian motion, high correlation coefficients can be obtained
when low strain rates are applied. Velocity estimates for the stretched planar phantom,
however, were inaccurate and nonphysical, possibly due to the lack of well-defined
features in the speckle structure. A homogeneous and a heterogeneous cylindrical
phantom were inflated with intraluminal pressure. FEA models were created for the
cylindrical phantoms to predict velocity and strain. Velocity and strain estimates from
elastography appeared qualitatively accurate, but they still described a nonphysical result.
There are several possible ways of improving the velocity and strain estimates
from elastography. Speckle decorrelation may be reduced by imaging samples and
biological tissues with large, distinct features and by using frame averaging or image
blurring to emphasize large features in the speckle pattern. Several modifications can be
made to the variational energy function to improve estimation. An edge field can be
152
added so that strain smoothness is enforced over distinct regions instead of over the
whole image, reducing the possibility of smoothing over regions of inhomogeneous
stiffness. Velocity at boundaries, which is easy to estimate, can be incorporated into the
energy function. In cases where loading conditions are known, approximate strain
directions can be predicted and used to create a second data fidelity term. Finally, a
deformable mesh framework can be used to enforce geometry compatibility, which
would prevent estimation of nonphysical velocity and strain estimates.
153
154
Chapter 7: Summary and Future Work
Cardiovascular disease is the leading cause of death in industrialized nations, with
half of those deaths attributed to coronary heart disease. Atherosclerosis, an inflammatory
disease of the arteries, can cause heart attack and death through the rupture of vulnerable
plaques, which are characterized by the presence of a large lipid pool covered by a thin
fibrous cap. Rupture of the fibrous cap releases thrombogenic materials from the lesion's
core into the lumen, where they come into contact with coagulation factors in the blood,
forming a thrombus. Finite element analysis of vulnerable plaques have shown that the
thin fibrous cap is subjected to high stresses and strains, making it likely to rupture.
Diagnosis of vulnerable plaque may be enhanced by utilizing information about the
biomechanical behavior of an individual plaque. Elastography, a method of strain
imaging, provides a way to probe the biomechanics of a plaque.
OCT imaging is a recently developed imaging modality that is the optical analog
of ultrasound imaging, but utilizes light instead of sound. The use of optical frequencies
results in higher resolution images than ultrasound, at the expense of decreased depth
penetration. Ex vivo studies have shown that OCT is capable of characterizing the
structure and composition of normal and diseased arteries, and intravascular OCT
catheters have been developed and applied for imaging coronary arteries in vivo.
In Chapter 3, OCT imaging was used as a basis for finite element analysis of
realistic plaque geometries. Two plaques, one lipid-rich and one calcific, were first
imaged with OCT and then processed with histology. Each OCT and histology image was
segmented into distinct regions classified as fibrous plaque, lipid, calcium, or arterial
wall, and finite element meshes were created from the segmentation boundaries.
Additionally, since OCT's limited depth penetration makes segmentation of the outer
regions of the OCT images difficult, multiple OCT readers were used to segment the
images, and multiple FEA models were created from each image. Material properties
were assigned using values taken from previously published literature, and a pressure
load was applied to the lumen of each mesh, ramping from 0 to 120 mmHg. Stress and
155
strain results from OCT-based models were compared to "true" stress and strain from the
histology-derived models. OCT- and histology-based FEA result in similar stress and
strain fields, even when different OCT segmentation outer boundaries were used,
suggesting that the most biomechanically relevant geometry is near the lumen and thus
OCT's limited depth penetration is not a significantly limiting factor. Additionally, the
histology-based models occasionally suffered from artificially high stress concentrations
caused by sharp corners in the histology image due to folding of the histology sample
during processing. OCT is an ideal basis for FEA modeling because it does not introduce
artifacts similar to those found in histology and because it can be used repeatedly in vivo,
allowing for studies of disease progression and treatment efficacy from a biomechanics
standpoint.
The focus of Chapter 4 was the use OCT imaging as a basis for elastography.
Elastography entails imaging a sample undergoing deformation, tracking pixel motion
through each frame of the image sequence, and finally calculating strain. Conventional
motion tracking relies on maximizing correlation coefficients between sub-regions of
consecutive image frames, which assumes the validity of a frozen speckle model. In the
frozen speckle model, the displacement between sub-blocks of the two image frames is
small enough that the two speckle patterns retain the same distribution and are related by
a simple translation. This assumption is frequently violated in OCT because displacement
magnitudes are frequently on the same order or greater than the OCT PSF. Additionally,
imaging noise and Brownian motion can cause strain-independent image decorrelation. A
novel variational approach was described for OCT elastography in which the velocity
fields are estimated by minimizing a variational energy function. The energy function
features three terms: 1) a data fidelity term for maximizing correlation, 2) an
incompressibility term to ensure that the sample deforms incompressibly, and 3) a strain
smoothness term to ensure the strain field varies smoothly. Once velocities are estimated,
strains are calculated using either the linear small strain definition or the rotation
invariant Green-Lagrange strain tensor for large deformations.
Validation of the variational motion tracking algorithm was performed using
simulated images in Chapter 5 and phantoms in Chapter 6. Simulated images were used
for initial validation because they provide a means for testing the algorithm under ideal
156
conditions, imaging noise, motion artifacts, and Brownian motion are all neglected in the
simulation. Simulations also allow precise control and easy variation of geometry,
boundary, and loading conditions, while conducting the same experiments in phantoms
would be difficult and time consuming. The simulation model featured a block with an
inclusion of varying diameter and stiffness. First, a single pair of images was used with
varying levels of added noise to compare the performance of both algorithms under
different noise conditions and with different parameter values. The conventional
elastography algorithm has poor performance for images with any added noise, while the
variational algorithm is able to estimate velocities and strains for even the noisiest image
pair. Results from variation of the weighting parameters show that emphasizing the
incompressibility and strain smoothness terms over the data fidelity term is very useful
for noisy images, but performs too much smoothing for noise-free images.
Simulated images were also used to probe strain resolution, modulus resolution,
and strain sensitivity of the elastography algorithm under ideal conditions. There exists a
tradeoff between the smallest detectable inclusion size and its stiffness ratio. For a
relatively large inclusion of 500 pm diameter, modulus sensitivity is quite high, with
modulus values of 50% and 150% of the background modulus being detectable. As the
inclusion size becomes smaller, the stiffness ratio must be further away from 1 before the
inclusion is detectable. As a diagnostic tool for atherosclerosis, an elastography algorithm
with these limitations should be sufficient to detect vulnerable plaques that are the most
clinically relevant, i.e. when they contain a large, compliant lipid pool. However, OCT
elastography may not be sensitive enough for ongoing studies of gradual disease
progression.
Finally, elastography validation was performed using phantoms. PVA was used to
construct phantoms of known mechanical properties in planar and cylindrical geometries.
Correlation coefficients for a planar phantom both at rest and being stretched were
calculated to determine the effects on image decorrelation of image noise, Brownian
motion, and applied strain. For the static phantom, initial-to-final frame correlation
coefficients remain high over several seconds, suggesting that the levels of image noise
and Brownian motion present are acceptable. When strain is applied, correlation
coefficients drop within the first second, with a greater decrease in the faster stretch rate,
157
indicating that relatively slow strain rates are necessary for elastography. Velocity and
strain fields estimated from the stretched planar phantom qualitatively show the correct
trends overall, but indicate nonphysical motion.
Two cylindrical phantoms, one homogeneous and one heterogeneous, were
inflated with intraluminal pressure. FEA was used to predict velocity and strain for
comparison with the elastography results. In both cases, the velocity estimates are
qualitatively correct, benefiting from the presence of the strongly reflecting air/phantom
boundary in the image. However, estimated strain fields do not match the predicted strain
fields and in fact, the strain fields indicate that the phantom is increasing in volume, a
physically impossible result.
While the variational elastography algorithm offers significant improvement over
the conventional algorithm in velocity and strain accuracy, there are many possible
measures to further improve the algorithm. Speckle decorrelation, which affects the data
fidelity term of the variational energy function, can be improved by applying
elastography to samples with larger, more distinct speckle features, or by utilizing frame
averaging or image blurring to emphasize large or persistent features in the existing
speckle pattern. The side constraint terms in the variational energy function can also be
modified to improve the algorithm. The addition of an edge field over which strain
smoothness is not enforced would increase the probability of detecting smaller regions of
differing strain, thus increasing modulus sensitivity. The incorporation into the energy
function of velocity estimates at boundaries, which are easy to obtain, may also improve
velocity accuracy. Additionally, in many cases, loading conditions are known, so
approximate strain directions can be predicted and added to the energy function to
improve estimation. Finally, geometry compatibility may be enforced by using a
deformable mesh framework, preventing the estimation of nonphysical velocity and
strain fields.
This thesis has demonstrated that OCT elastography has potential as a viable
method of probing cardiovascular tissue mechanics. However, many obstacles need to be
overcome first. For more relevant assessment of algorithm performance, future
experiments should be performed on either diseased arteries or phantoms with scattering
properties more similar to diseased arteries, which have larger macrostructures than PVA
158
phantoms. Additionally, two important improvements could be made to the variational
energy function: the addition of the edge term for strain smoothing and the enforcement
of geometry compatibility. The presence of the edge term is crucial for detecting the
plaque components of a diseased artery, such as lipid pools and calcium nodules.
Geometry compatibility would improve estimation accuracy by preventing the estimation
of smooth but nonphysical velocity fields. With these improvements, OCT elastography
may become a useful tool for the study of atherosclerosis.
159
160
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