Quantification of collagen organization in 3D engineered tissue
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Quantification of Collagen Orientation in
3D Engineered Tissue
Master’s thesis
by
Florie Daniels
June 2006
Student id: 491937
Commissioned by:
Prof. dr. ir. B.M. ter Haar Romeny (Dep. of Biomedical Engineering, TU/e)
Dr. ir. H.C. van Assen (Dep. of Biomedical Engineering, TU/e)
Ir. M.P. Rubbens (Dep. of Biomedical Engineering, TU/e)
Dr. ir. G.J. Strijkers (Dep. of Biomedical Engineering, TU/e)
Dr. W. Engels (Dep. of Biophysics, UM)
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Abstract
Tissue engineered heart valves are a promising alternative for current valve replacements.
However, the mechanical properties of these valves are insufficient for implantation at the aortic
position. A promising way of improving the mechanical properties of tissue engineered valves is
to optimize collagen remodeling (i.e. changes in fiber content, thickness, length, and orientation)
via mechanical straining. Tissue engineered constructs were made to investigate the influence of
strain on collagen orientation. To quantify the effect of strain on the orientation of collagen twophoton laser scanning microscopy is used. Two-photon laser scanning microscopy allows 3D
imaging of tissue engineered constructs. An algorithm was developed to extract fiber orientations
from the 3D images. A method based on the 2nd order derivative of a structure was used to
determine the general orientation of the collagen fibers. This second order derivative gives the
best result when it matches to the underlying structure. The optimal match is determined
automatically. Furthermore two methods that take into account the context of a structure are
presented. The algorithm is validated with artificial images that have a ground truth of
orientations. The results of validation point out that the algorithm finds no significant difference
in mean orientations between the ground truth and the orientations determined by the algorithm.
This work also presents a preliminary study of the effect of strain on collagen orientation. The
data is analyzed using the orientation analysis algorithm. In the first experiment alignment in the
direction of the applied strain is seen while in the second experiment alignment perpendicular to
the direction of the applied strain is seen. This is considered a result of the imaging location. The
results of the algorithm are shown as histograms. Visual inspection of these histograms shows
that the distribution of orientations becomes smaller for increased strain. This indicates that the
collagen fibers align more. Variance is used as a measure for alignment. The variance does not
indicate an increased alignment with applied strain. However, not enough data was available to
obtain statistical significance.
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Samenvatting
Getissue-engineerde hartkleppen zijn een veelbelovend alternatief voor de hartkleppen die
momenteel gebruikt worden om een zieke hartklep te vervangen. Echter, de mechanische
eigenschappen van deze kleppen zijn niet toereikend om deze te implanteren op de positie van de
aorta. Een veelbelovende manier on de mechanische eigenschappen van getissue-engineerde
hartkleppen te verbeteren is door het optimaliseren van collageen remodelering (oftwel
veranderingen in hoeveelheid, dikte, lengte en orientatie van collagen vezels) via mechanisch
rekken. Getissue-engineerde weefsels zijn gemaakt om de invloed van rek op collageen orientatie
te bepalen. Om het effect van rek op collagen orientatie te kwantificeren is gebruik gemaakt van
twee-foton microscopie. Met twee-foton microscopie is het mogelijk om collagen in 3D te
visualiseren.
Een algoritme is ontworpen om de orientatie van collageen vezels uit de 3D beelden te halen. Een
methode gebaseerd op de 2e orde afgeleide van een structuur is gebruikt om de algemene
orientatie van de collageen vezels te bepalen. Wanneer deze 2e orde afgeleide fit op de
onderliggende structuur wordt de orientatie het beste bepaald. Deze optimale fit wordt
automatisch bepaald in het algoritme. Verder worden twee methods gepresenteerd die de
omgeving van de structuur meenemen in hun analyse.
Het algoritme is gevalideerd met behulp van artificiële beelden van collageen vezels waarvan een
ground truth van orientaties bekend is. De resultaten van de validatie wijzen uit dat de
gemiddelde orientaties gevonden door het algorithme niet significant afwijken van de gemiddelde
orientaties in hun ground truth.
Dit werk presenteert ook een onderzoek naar het effect van rek op collageen orientatie. De data is
geanalyseerd met het algorithme. Het eerste experiment toont aan dat de collagen vezels in de
richting van de rek gaan liggen, terwijl de vezels in het tweede experiment loodrecht op de
richting van de rek gaan liggen. Dit verschil wordt veroorzaakt door de locatie waar het tweefoton beeld is gemaakt. De resultaten van het algorithme worden getoond als orientatie
histogrammen. Visuele inspectie van deze histgrammen suggereert dat de collageen vezels
minder random georienteerd zijn (meer alignment). De piek in het orientatie histogram wordt
namelijk smaller wanneer er meer rek opgelegd wordt. Variantie is gebruikt als maat voor
collageen alignment. De variantie suggereert geen toename in alignment met toenemende rek. Er
was echter niet genoeg data beschikbaar om statistieke significantie aan te tonen.
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Table of Contents
CHAPTER 1 INTRODUCTION ..................................................................................................................3
1.1 GENERAL INTRODUCTION AND AIM .......................................................................................................3
1.2 PREVIOUS RESEARCH .............................................................................................................................5
1.3 OVERVIEW OF CONTENTS .......................................................................................................................6
CHAPTER 2 PHYSIOLOGICAL BACKGROUND .................................................................................7
2.1 THE NATIVE AORTIC HEART VALVE........................................................................................................7
2.2 COLLAGEN .............................................................................................................................................8
2.3 VISUALIZATION OF COLLAGEN WITH TPLSM ......................................................................................10
CHAPTER 3 ALGORITHM FOR ORIENTATION ANALYSIS ..........................................................12
3.1 PRINCIPAL CURVATURE DIRECTIONS IN 3D ..........................................................................................13
3.2 SCALE SELECTION ...............................................................................................................................15
3.3 COHERENCE ENHANCING DIFFUSION ...................................................................................................22
3.4 TENSOR VOTING ..................................................................................................................................28
CHAPTER 4 VALIDATION ......................................................................................................................32
4.1 ARTIFICIAL IMAGE ...............................................................................................................................32
4.2 VALIDATION OF MINIMAL PRINCIPAL CURVATURE DIRECTIONS AND SCALE.........................................33
4.2 VALIDATION OF CED ..........................................................................................................................37
4.3 VALIDATION OF TENSOR VOTING ........................................................................................................38
CHAPTER 5 EXPERIMENTS ..................................................................................................................40
5.1 INTRODUCTION ....................................................................................................................................40
5.2 EXPERIMENTAL SETUP .........................................................................................................................40
CHAPTER 6 RESULTS .............................................................................................................................45
CHAPTER 7: GENERAL DISCUSSION AND CONCLUSIONS ..........................................................49
7.1 ALGORITHM PERFORMANCE ................................................................................................................49
7.2 EXPERIMENTAL RESULTS .....................................................................................................................50
7.3 CONCLUSIONS......................................................................................................................................51
7.4 RECOMMENDATIONS FOR FUTURE RESEARCH ......................................................................................51
APPENDIX I RESULTS PAIRED T-TEST FOR ARTIFICIAL IMAGES 1-8 ....................................53
APPENDIX II RESULTS PAIRED T-TEST FOR ARTIFICIAL IMAGES 9-14 ................................55
APPENDIX III 3D HISTOGRAMS OF THE ORIENTATION FOR EVERY SCALE.......................57
REFERENCES ............................................................................................................................................64
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Chapter 1 Introduction
1.1 General introduction and Aim
The heart consists of four chambers, two atria (upper chambers) and two ventricles (lower
chambers). When blood leaves from each chamber of the heart it has to pass a valve. The four
valves of the heart are the mitral, tricuspid, pulmonary and aortic valve (figure 1.1). These valves
prevent the backward flow of blood. Valvular heart valve disease is a significant cause of
morbidity and mortality world-wide [1]. There are two prominent types of valve disease. Valvular
stenosis occurs when a valve opening is smaller than normal due to stiff or fused leaflets. The
narrowed opening may make the heart work very hard to pump blood through it. The other type is
valvular insufficiency, also called regurgitation, incompetence or "leaky valve". This occurs when
a valve does not close tightly. If a valve does not seal, some blood will leak backwards across the
valve. As the leak becomes worse, the heart has to work harder to make up for the leaky valve,
and less blood may flow to the rest of the body.
Figure 1.1: Anatomic representation of the positions of the heart valves.
The most common treatment for valvular heart valve diseases is the replacement of the valve.
Classical heart valve substitutes are mechanical prosthetic valves with components manufactured
of nonbiologic material (e.g. polymer, metal, carbon) and biological valves (figure 1.2a, b) which
are constructed, at least in part, of either human (homografts) or animal tissue (xenografts). The
mayor drawback of the mechanical valves is that they contain foreign materials, which increases
the risk of infections and tromboembolic complications. To prevent thromboembolism, the patient
needs to take medication that prevents or slows down the clotting of blood (anticoagulation
medication) for the rest of its life. Biological prostheses do not require such medication, because
of higher immunological competence. However they show a limited durability (10-15 years)
because of structural dysfunction due to tissue deterioration [2]. To overcome these shortcomings
another type of heart valve replacement has to be developed and preferably one that has the
ability to grow, repair and remodel. One approach that has been used to make such a valve is
tissue engineering. Autologous cells are seeded on a pre-shaped biodegradable scaffold and
cultured in a bioreactor under conditions that mimic the physiological environment of the tissue.
To date, tissue engineered heart valve replacements (figure 1.2c) show promising results when
implanted at the pulmonary position in animal studies [3]. However these valves lack the
33
Chapter 1 Introduction
mechanical integrity to withstand the pressures at the aortic side. A tissue engineered aortic heart
valve with mechanical properties similar to the native aortic valve would be the ideal prostheses.
Figure 1.2: Examples of heart valve protheses. a) Mechanical heart valve (Medtronic). b) Biological heart
valve of non living fixed tissue (Hancock). c) Living tissue engineered tri-leaflet heart valve based on
human marrow stromal cells [3].
The protein primary responsible for the load-bearing properties of heart valves is collagen. The
strength collagen provides to these tissues depends on collagen fiber content, thickness, length,
and orientation. A promising way of improving the mechanical integrity of tissue engineered
valves is to optimize collagen remodeling (i.e. changes in fiber content, thickness, length, crosslinks and orientation) via mechanical conditioning strategies. Collagen remodeling is a balance
between collagen synthesis and degradation. The way remodeling takes place is strongly
influenced by mechanical straining. In a study by Mol, 2005 [4] the influence of different strain
levels on the strength of tissue engineered constructs is shown. The Youngs modulus increases
with applied strain (figure 1.3).
Figure 1.3: Load bearing properties of heart valve tissue cultured with increasing amount of cyclic strain
(mean ± standard deviation, unstrained tissue valves are set at 100%)[4].
Previous research by Billiar, 2000 [5] has shown that collagen orientation rather then collagen
content determines the mechanical properties in aortic valve cusps. Determining the relationship
between mechanical straining and collagen orientation may be helpful to improve the strength of
tissue engineered aortic heart valves. Therefore the focus of this master project will be on
developing an algorithm to investigate this relationship.
We will investigate the orientation of collagen fibers by using tissue engineered constructs and
visualizing collagen with a two-photon laser-scanning microscope (TPLSM) (figure 1.4). A new
fluorescent probe, CNA35 [6], will be used to specifically mark collagen. This probe has a high
affinity for collagen type-I. TPLSM produces large three-dimensional datasets and the complexity
4
Chapter 1 Introduction
of the three-dimensional fibrous networks makes observation and extraction of quantative
information from these datasets very difficult. An automatic method for extraction of orientation
information will provide a faster, more objective and more accurate way to analyze the data
compared to analysis by hand. TPLSM shows collagen fibers that are curved and vary in
thickness. This indicates that the notion of scale is an important parameter for our analysis.
Figure 1.4: Example of a TPLSM image of collagen in a porcine heart valve stained with CNA35 and
TPLSM.
The aim of this master project consists of two parts:
1)
To design an image analysis tool for automatic 3D orientation estimation of collagen
fibers in TPLSM images.
2)
To quantify collagen orientation in 3D unstrained, statically and dynamically strained
heart valve tissue engineered equivalents using TPLSM.
1.2 Previous research
Previous research takes either collagen remodeling into account or the mechanical properties of
collagen. However these two are highly dependent; collagen remodeling will result in a change in
mechanical properties, and changes in mechanical properties will lead to different stress-strain
behavior of the tissue and hence different cues for tissue remodeling. In order to obtain
information about the structural arrangement of collagen fibers previous studies make use of
small angle light scattering (SALS) [7] and polarized light microscopy (PLM) [8,9]. These
methods use the birefringent optical properties of collagen fibers, which results in images that
contain coded orientation information.
More recently, laser scanning microscopy has experienced an increasing interest for the analysis
of collagen fibers. Confocal laser scanning microscopy (CLSM) is routinely used in 3D tissue
studies [10,11,12]. In contrast to the above-mentioned techniques (SALS, PLM), CLSM is nondestructive and produces volume data directly. It should be noted however that CLSM suffers
from a limited penetration depth. Axer et al., 2001 [13] perform a coarse analysis by hand of the
3D architecture of the collagen fibers in linea alba and rectus sheets.
These studies make use of manual analysis and do not use automatic algorithms for analyzing the
structure in question. There are some studies that do make use of automatic analysis software. Wu
et al., 2003 [14,15] demonstrate an algorithm designed to extract quantitative structural
information about individual collagen fibers (orientation, length and diameter). CLSM was used
to visualize collagen fibers in gels of bovine collagen type I. In contrast to the image data used by
5
Chapter 1 Introduction
Wu et al. our data contain collagen networks that are very complex, which makes it almost
impossible to extract and analyze single fibers. Elbischger et al., 2004 [16] developed automatic
analysis software to analyze collagen fibers based on transmitted light microscopy of thin
histological tissue samples. Eigenanalysis of the Hessian matrix was used to determine fiber
orientation.
Research concerning the orientation of other fibrous or line-like structures may also be useful to
design an appropriate automatic method for orientation analysis. One way to analyze the
orientation of structures is by Fourier analysis [17, 18].The magnitude plot of the Fourier
Transform gives an indication of the general orientation and the variance of this orientation. This
is a very global analysis and no information about the location of a certain orientation is available.
Freeman and Adelson, 1991 [19] proposed a method for making steerable filters to allow
evaluation of the filter response over an infinite number of angles with a limited number of
convolutions. The response of the filter with a certain orientation is highest when the filter
matches the structure. In 3D this method may be very computationally expensive. A local
structure descriptor seems a better approach. In Niessen et al., 1997 [20] trabecular bone
orientation in 2D and 3D CT and MR data is defined in a robust, quantative and automatic
manner. The orientation of the trabeculae is determined by the Gaussian structure tensor. By
eigenanalysis of the structure tensor the orientation of the structure is determined. They also
define a confidence measure, which is a measure of anisotropy of the structure. When the
confidence measure is below a certain threshold, the orientations that are found are not
determined accurately and excluded from the final result. Furthermore many studies concerning
extraction of line-like (tubular) structures like blood vessels can be found in literature [21, 22,
23]. These studies use the Hessian matrix to describe the structure in the images.
Driessen et al., 2003 [24] presented computational models to study mechanically induced
collagen remodeling and resulting mechanical properties in the aortic heart valve. They assumed
that collagen architecture aligns with the strain field within the tissue and collagen content
increases with fiber stretch. Predicted fiber architectures resembled that of native tissue. Even
better results were obtained when collagen fibers were assumed to align with preferred directions,
situated in between the principal strain directions. The orientation of these preferred directions
depend on the magnitude of the principal strains. This results in the hypothesis that when collagen
fibers are strained, reorientation in the direction of the strain or re-synthesis occurs. Information
about the orientation of collagen will help these models to better predict whether tissue
engineered valves are strong enough for implantation.
1.3 Overview of contents
This report will start in chapter 2 with some background information to provide insight into the
physiology of the native aortic heart valve and collagen. Also the basics of TPLSM are explained
in short. In chapter 3 the theory that is used to design the algorithm for automatic 3D orientation
analysis is explained. In paragraph 3.1 the way orientations are determined by using principal
curvature directions is presented. Then in paragraph 3.2 it is explained what is meant with scale
and the way this was implemented in the algorithm. In the last two paragraphs two different
methods that improve the results of the orientations found by the principal curvature directions by
taking into account the context are presented. Chapter 4 gives the results of the validation of the
algorithm. The experiments done to study the organization of collagen fibers are presented in
chapter 5. In chapter 6 the results of the analysis by the algorithm of collagen orientation in 3D
engineered tissue are shown. Finally a general discussion with conclusions and recommendations
for future research is presented in Chapter 7.
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Chapter 2 Physiological background
Knowledge of the organization and load bearing properties of collagen in the native aortic heart
valve and knowledge about the structure, synthesis, degradation of collagen will help understand
the requirements the tissue engineered aortic heart valves have to fulfill. Therefore this will be
explained in short in the first two paragraphs. In paragraph 2.3 the basics of two-photon laserscanning microscopy will be explained.
2.1 The native aortic heart valve
The aortic heart valve is the valve situated at the outlet of the left ventricle. It consists of three
leaflets, three sinuses and the aortic ring (figure 2.1). The load-bearing part of the leaflets shows a
layered architecture consisting of the fibrosa, spongiosa and ventricularis (figure 2.2 a).
(a)
(b)
(c)
Figure 2.1: Schematic representation of the aortic valve: a) side view of the complete valve, b) after
removal of one leaflet and the corresponding sinus, and c) view from aortic side.
The fibrosa is considered the mean load-bearing layer. This layer is predominantly composed of
circumferentially (from commisure to commisure) aligned densely packed collagen fibers (figure
2.2 b). The centrally located spongiosa consists of loosely arranged collagen. The few collagen
fibers in this layer are oriented radially. The ventricularis, containing elastin, provides the tensile
recoil necessary to retain the folded shape of the fibrosa. The valve cusps contain about 50%
collagen and 13% elastin by dry weight.
Figure 2.2 Left: Histological cross section of porcine valve cusp showing the three mayor layers. From top
to bottom are fibrosa, spongiosa and ventricularis. Right: Typical collagen fiber structure of natural aortic
valve leaflet. The commissures are denoted by ‘c’.
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Chapter 2 Physiological Background
Collagen type I is most abundant (74%, thick bundles), furthermore type III (24%, finer bundles)
and type V can be identified.
The layers are very mobile and can easily compress and shear during opening and closing of the
valve. The cusps must withstand large cyclic deformations with changes as high as 50% as well
as the pressure differences arising during the cardiac cycle [2]. This difference is at its maximum
at the beginning of diastole and decreases almost linear. In figure 2.3 the structural and
biomechanical features of the aortic heart valve during systole and diastole and the stress-strain
relationship of elastin and collagen are shown. During opening of the valve, elastin extends at
minimal load during extension of collagen crimp and corrugations. When the valve is nearly
closed and the collagen is fully unfolded, the load-bearing element shifts from elastin to collagen
and stress rises while coaptation (i.e. contract of adjacent cusps) is maintained to prevent prolaps.
During systole, elastin restores the contracted configuration of the cusps.
Figure 2.3: Structural and biomechanical features of the aortic heart valve. Top: Schematic representation
of cuspal configuration and architecture of collagen and elastin during systole and diastole. Bottom:
Schematic representation of biomechanical cooperativity between elastin and collagen during valve
motion [2].
2.2 Collagen
Collagen is distinct from other proteins in that the molecule comprises three polypeptide chains
(α-chains), which form a unique triple-helical structure. For the three chains to correctly wind into
a triple helix they must have the smallest amino acid, glycine, at every third residue along each
chain. The chains contain Gly-X-Y repeats in which X and Y can be any amino acid, but are
frequently proline or hydroproline. At this time, 25 distinct collagen types have been identified.
Collagen type I is the most abundant of these. Collagen types I, II, III, V and XI are fibril forming
collagens. The major collagen fibrils are mixtures of several collagen types.
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Chapter 2 Physiological Background
Fibrillogenesis of collagen is defined as deposition of soluble collagen molecules (procollagen)
and the subsequent organization into structural fibers [25]. The endoplasmatic reticulum of the
cell is the place where collagen formation starts. A fundamental feature of fibril-forming
collagens is that they are synthesized as soluble procollagens (figure 2.4).
Figure 2.4: Extracellular events leading to collagen formation. Procollagen consists of a 300 nm long
triple helical domain flanked by a trimeric globular C-peptide domain and a trimeric N-propeptide domain.
Procollagen is secreted from cells and is converted into collagen by removal of the N- and C-propeptides
by procollagen N-proteinase and procollagen C-proteinase respectively. The collagen generated in the
reaction spontaneously self assembles into cross-striated fibrils. The fibrils are stabilized by covalent
cross-linking [26].
These procollagens contain telopeptides, which are important in collagen assembly and the
formation of crosslinks. The combination of three pro-α-chains forms procollagen triple-helices.
The procollagen triple helices are then transported to the cellular membrane by secretory vesicles
and secreted into the extracellular matrix. These triple helices are soluble in the extracellular
matrix. Procollagens are then converted into collagens by specific enzymatic cleavage of terminal
propeptides by the procollagen C-proteinase and/or procollagen N-proteinase. After cleavage the
solubility drops and polymerization of collagen fibrils is initiated. The collagen molecules
polymerize longitudinally and aggregate laterally. The collagen fibrils are stabilized by covalent
cross-links initiated by the enzyme lysyl oxidase.
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Chapter 2 Physiological Background
The procollagen triple-helices are 1.5 nm in diameter and 300 nm in length [26]. An assembly of
five triple helices, called ‘five stranded Smith microfibril’, is a filamentous structure with a 4 nm
diameter. By staggered aggregation of the triple helices in microfibrils, D-periodic cross-striated
fibrils are assembled with D = 67 nm, the characteristic axial periodicity of collagen. When
hundreds of fibrils aggregate they form a fibril, which has a diameter ranging from 10 to 500 nm
and a length of approximately 10 to 30 μm. Subsequently these fibrils aggregate into collagen
fibers (figure 2.5). Fiber bundles have been reported as large as several hundred micrometers in
length.
Figure 2.5: Hierarchy of collagen: α-chains, triple helices, fibrils and fibers.
Degradation of collagen is performed by degrading enzymes. Matrix metalloproteinases (MMP)
form the mayor class of collagen degrading enzymes. The MMPs can be devided into four groups
based on their specificity and structure. The first group, the collagenases, can cleave fibrillar
collagens (e.g. collagen type I, II and III). Mammalian collagenases cleave collagen after the Gly
residue of the partial sequence Gly-[Ile or Leu]-[Ala or Leu] in the triple helix at ¾ from the
terminal end. This result in ¼ and ¾ collagen fragements which unfold their triple helix and fall
apart into fragmented single α-chains, called gelatins. The gelatins are degraded by the second
group of MMP, the gelatinases. Group three are the stromelysins. They are active against some
types of collagen and a broad spectrum of other extra-cellular matrix (ECM) components. Group
four MMPs degrade several ECM components and are able to activate other MMPs.
2.3 Visualization of collagen with TPLSM
To visualize collagen architecture in 3D use can be made of two-photon laser-scanning
microscopy (TPLSM). Van Zandvoort et al., 2004 [27] demonstrated the use of two-photon laser
scanning microscopy, for ex vivo experiments to visualize cellular structural details for the
carotid artery. In TPLSM two photons are used to excite a molecule. Each of these photons has
approximately half the energy needed to excite a molecule. Therefore only two photons that are
absorbed by the molecule simultaneous can result in excitation. When two photons with sufficient
combined energy are absorbed by a molecule, the molecule is excited from the ground state to a
high vibrational level in the excited state (figure 2.6). Within the excited energy level the
10
Chapter 2 Physiological Background
molecule can relax to the lowest vibrational level in the excited state by internal conversion
(energy loss). After relaxation, the molecule can return to the ground state by radiationless
internal conversion or by emitting a photon with specific energy. Compared to the excited
wavelength the emitted photon will have a lower energy and the fluorescent light will thus have a
longer wavelength.
Figure 2.6: Jablonski diagram. An electron is excited from S 0, the ground state, to one of the vibrational
levels of S1, the excited state (blue lines). For S1 the electron returns to S0 by emitting a photon (green
lines) or radiationless internal conversion (purple lines).
In two photon microscopy, optical sectioning results from the fact that the probability of a two
photon event occurring (i.e. excitation), happens only at the focal plane where there is an
extremely high photon density. As a result, in two-photon imaging excitation occurs only at the
plane of focus. The advantages of TPLSM are that, because there is only excitation in the plane of
focus, no pinhole is needed, less scattering occurs in the surrounding tissue, also because of the
larger wavelength that can be used, the phototoxity and bleaching is reduced and it is possible to
image deeper into thick tissue then possible with for example CLSM. A disadvantage is that
TPLSM has a lightly lower resolution compared to CLSM.
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Chapter 3 Algorithm for Orientation Analysis
In this chapter an algorithm for automatic 3D orientation analysis of collagen is presented. After
considering the many possible approaches found in previous studies concerning automatic
orientation analysis, we propose to estimate the local orientation in the images by determining
principal curvature directions from the Hessian matrix. This will be explained in the first
paragraph. If we look through a small window as a local feature detector does, structures are
hardly visible. If we see a larger part of an image, suddenly the structure of interest emerges.
Apparently, for our visual system the spatial context is an important clue for object detection. To
improve the orientations extracted from the TPLSM data some methods taking into account the
context of the feature will be explained. The aim of context enhancement is to improve local
feature data using knowledge of the spatial neighborhood. In paragraph 3.2 a method which
determines the scale of the 2nd order Gaussian derivatives is presented. A preprocessing step to
enhance the collagen fibers, called coherence enhancing diffusion, is explained in paragraph 3.3.
In paragraph 3.4 tensor voting will be explained. Tensor voting can be used to obtain more robust
orientations by taking into account the orientations in the neighborhood. A schematic overview of
the algorithm is given in figure 3.1.
Figure 3.1: Schematic overview of the algorithm for robust orientation analysis.
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Chapter 3 Algorithm for Orientation Analysis
3.1 Principal curvature directions in 3D
One way to determine the orientation of a point in an image is by examining the local
neighborhood of that point. Curvature is a local feature, which can be determined at any point on
a curve. The general definition for curvature in mathematics is the rate of change of the angle (at
a point) between a curve and a tangent to the curve [28]. In the case of a line in 2D there is only
one direction along the curve for which curvature is defined. Curvature in any point on that line is
defined as the inverse of the radius of a circle fitted to the line at that point (figure 3.2a).
At any point in a 2D or 3D image one can step into an infinite number of directions away from
that point, and in each direction a curvature is defined. This results in an infinite number of
curvatures for each point. It turns out however, that curvatures in opposite directions are always
the same. At a point on a surface (2D image) the maximum and minimum curvature are called the
principal curvatures and the directions in which these occur are called the principal directions
(figure 3.2b). At a point on an object (3D image) however three principal curvatures and principal
directions can be defined. These principal directions are the directions in which the local structure
of the image can be decomposed (figure 3.2c). The minimal curvature direction is oriented along
the general orientation of the object. This method therefore may be used for determining the local
orientation of a tubular structure.
λ1
λ2
(a)
(b)
(c)
Figure 3.2: Curvature presented for a 1-dimensional, 2-dimensional and 3-dimensional function. a: curve
in 2D which shows the definition of curvature as the inverse radius of a circle touching point P. b: In 3D
the curve is a surface and the principal directions are perpendicular to each other. c: In a 3D image the
curve can also be an object and three principal directions, that are perpendicular to each other, are
present.
A common approach to describe the local behavior of an image, L, is to consider its Taylor
expansions in the neighborhood of a point x0,
1
L(x0 x0 , ) L(x0 , ) x0T L(x0 , ) x0T 2 L(x0 , ) x0 (3.1)
2
where 2 denotes the vector product of the Nabla operator with itself given by
x
2 T
y
x
z
y
2
2
x
2
z yx
2
zx
13
2
xy
2
y 2
2
zy
2
xz
2
yz
2
z 2
(3.2)
Chapter 3 Algorithm for Orientation Analysis
This expansion approximates the structure of the image up to second order. L ( x 0 , ) and
2 L(x0 , ) are the gradient vector and the Hessian matrix, respectively, of the image computed in
x0 at scale σ. The components of the Hessian matrix consist of second order derivatives and
describe the curvature of the image L.
Lxx
L(x, ) Lyx
Lzx
Lxz
Lyz
Lzz
Lxy
Lyy
Lzy
2
(3.3)
Here differentiation is defined as a convolution with Gaussian derivatives
L(x, ) L(x) G(x, )
x
x
(3.4)
Where the m-dimensional Gaussian is defined as
G (x, )
1
2
2
m
e
x2
2 2
(3.5)
Figure 3.3.Left: The second order derivative of a Gaussian function. Right: The second order Gaussian
kernel at scale σ =10.
This method slightly blurs the image with a low pass filter so that it becomes robust to noise, even
for higher order derivatives. The second order derivative of a Gaussian kernel at scale σ generates
a kernel (figure 3.3) that measures contrast between the regions inside and the regions outside the
range (-σ, σ) in the direction of the derivative. The eigenvalues and the eigenvectors of the
Hessian matrix correspond to the principal curvatures and principal directions of the image [29].
The principal directions are in 3D commonly represented by an ellipsoid which is scaled by the
eigenvalues (figure 3.4).
Figure 3.4: Graphical representation of principal direction by a 3-dimensional ellipsoid with its
eigensystem.
14
Chapter 3 Algorithm for Orientation Analysis
The orientation of each principal direction is given by two angles, θ and φ (equation 3.7). Every
vector, v={x , y , z}, in space can be described by θ and φ (figure 3.5). Here θ is defined to be the
angle in the xy-plane from the x-axis with 0 < θ < 2π, φ to be the polar angle from the z-axis with
0 < φ < π, and r to be the length of the vector. Since we use normalized vectors for our analysis r
is always equal to 1. The Cartesian coordinates can be described by
x = cos sin
y = sin sin
z = cos
(3.6)
When rewriting these equations it follows that
cos 1 ( z )
y
(3.7)
tan 1
x
Because in our analysis we do not want to distinguish between vectors pointing in the opposite
direction θ can also be defined between 0 and
Figure 3.5.A representation of the angles of a vector in 3D [30].
3.2 Scale Selection
In the real world all objects are only meaningful at a certain scale. For example, when considering
a tree, it is obvious that the leaves, branches and the stem cannot be described by the same scale.
Our visual system is an expert at looking at the right scale to detect objects and structures of
interest. This should also be the case in image analysis. It is possible to detect structures of
various sizes according to the scale at which they give a maximal response. Also the Hessian
matrix ‘looks’ at a certain scale because it is made up of second order Gaussian derivatives. In
figure 1.4 it can be seen that the collagen fibers are curved and vary in thickness. Also several
fibers in the image seem to have merged. When the scale of the Gaussian derivatives matches the
scale of the collagen fiber in the image the Gaussian will give an optimal response and this will
result in more accurate parameters derived from the Hessian, e.g. orientations. In figure 3.6 the
influence of scale on the fiber orientations is shown. When a small scale is used to calculate the
principal directions the directions that are found for the small fibers are in line with the fiber
15
Chapter 3 Algorithm for Orientation Analysis
orientation, while at larger fibers the directions vary quite a lot. When larger scales are used the
directions found for the large fibers are more in line with the fiber orientation.
Figure 3.6: The principal direction corresponding to the minimal principal curvature projected in red onto
a slice of TPLSM data of scales 2, 4 and 8.
A stack of images taken at a range of scales is called a scale space. Figure 3.7 gives an example
of scale space analysis of a two-dimensional image. At fine scales mainly noise and small
textures are detected, while at a larger scale the main objects in the image are detected and appear
as blobs until at even coarser scales these blobs merge to one object.
Figure 3.7: Different levels in scale space of a two-dimensional image at scale levels ½σ2= 0, 2, 8, 32, 128
and 512 together with grey-level blobs indicating local minima at each scale [31].
A property of scale space representation is that the amplitude of the spatial derivatives decreases
with scale, i.e. the response given by a signal after smoothing with the spatial derivatives gives
lower numerical values for larger scales. Lindeberg, 1998 [31] showed the necessity of
normalized derivatives of the image. Therefore we use normalized Gaussian derivatives (equation
3.8) to compute the Hessian at different scales.
G(x, )normalized 2G(x, )
(3.8)
We assume that every voxel has a preferred orientation. To determine how good this preferred
orientation is found a measure has to be defined that becomes maximal for fiber-like structures.
The eigenvalues of the Hessian are an indication of the type of structures that are present in the
image. A voxel belonging to a fiber region will ideally have its smallest eigenvalue close to zero
and the other two of large magnitude and almost equal. Whereas when the eigenvalues are
16
Chapter 3 Algorithm for Orientation Analysis
similar, this indicates that there is an isotropic structure present. In table 3.1 different situations of
eigenvalues are summarized. The eigenvalues are ordered from small to large 1 2 3 .
The collagen fibers appear as bright tubular structures in a dark environment. This prior
knowledge related to the imaging modality can be used as a consistency check to discard
structures present in the data with a polarity different then the one sought. So the conditions for
an ideal bright tubular structure in the 3D TPLSM-images are
1 0
1 2
(3.9)
2 3
and the signs of 2 and 3 should be negative.
Two measures, that give a maximum response when the conditions of equation 3.9 are met, are
explained in the following paragraphs.
Table 3.1: Possible structure types depending on the eigenvalues.
Structure type
blob
blob
tubular
tubular
plane
plane
Polarity
bright
dark
bright
dark
bright
dark
Eigenvalues
λ1<<0, λ2<<0, λ3<<0
λ1>>0, λ2>>0, λ3>>0
λ1≈0, λ2<<0, λ3<< 0
λ1≈0, λ2>>0, λ3>>0
λ1≈0, λ2 ≈ 0, λ3<<0
λ1≈0, λ2 ≈ 0, λ3>> 0
3.2.1. Confidence measure
We assumed that every point has a preferred orientation. To check whether this is the case a
confidence measure can be associated to a certain orientation. This measure was introduced by
Niessen et al., 1997 [20] to measure whether the orientation of trabecular bone had a high enough
confidence to be included in the final result. The confidence measure is defined as
if λ2>0 or λ3>0,
0
C ( , )
( )2
1 e 2 c2
with 2 1 2
2
2
(3.10)
otherwise,
3
2
3
1
2
, 1 2 3 and c a predefined threshold.
This predefined threshold ensures that only a response is given when is large enough to exceed
this threshold. The value of c is chosen to be Max(im)/4, because the intensities in the image
range from 0 to 1 and we assume that voxels with a values larger than 0.25 belong to a fiber. The
measure becomes 0 for regions with no preferred orientation and has a maximum of 1 for regions
with a high preference for one orientation.
17
Chapter 3 Algorithm for Orientation Analysis
3.2.2. Vesselness measure
The vesselness measure defined by Frangi et al., 1998 [21] is mainly used to enhance blood
vessels (figure 3.8). The measure consists of three components. The first component is a ratio that
Figure 3.8: Illustration of vesselness measure. Left: Original MIP of a MRA image. Middle: MIP of vessel
enhanced image. Right: Closest vessel projection.[Frangi et al].
expresses how much a structure deviates from a blob but cannot distinguish between a line- and a
plate-like structure
RB
Volume /(4 / 3))
( L arg est Cross Section Area / )3/ 2
1
2 3
(3.11)
This ratio has a maximum for blob-like structures and is zero when 1 0 or 2 and 3 tend to
zero.
The second component refers to the largest area cross section of the ellipsoid. It distinguishes
between plate-like and line-like structures because only in the latter case it will be zero.
RA
( L arg est Cross Section Area) / 2
( L arg est Axis Semi length)2
3
(3.12)
These two ratios are invariant under intensity scaling. Therefore only geometric information is
taken from the image
A way to distinguish between background pixels and fiber pixels is by looking at the triplets of
eigenvalues. When all have a small magnitude the pixel belongs to the background. This is
possible because the magnitude of the second order derivatives (and thus the eigenvalues) is small
for the background pixels. To quantify this, the Frobenius matrix norm is used. This gives the
third component of “second order structureness”
S H
F
j m
2
j
(3.13)
where m is the dimension of the image.
This measure will be low in background where no structure is present and the eigenvalues are
small. In regions with high contrast and thus at least one large eigenvalues the norm will become
larger. This results in the final vesselness measure
18
Chapter 3 Algorithm for Orientation Analysis
0
if λ2>0 or λ3>0,
2
2
2
V ( , )
(3.14)
RA
RB
S
1 exp 2 2 exp 2 2 1 exp 2c 2 otherwise,
where α, β and c are thresholds which control the sensitivity of the line filter to the three
components RA, RB and S. Here for α and β a default value of 0.5 is chosen corresponding to the
default values in Frangi et al., 1998, because no criteria could be found on how to determine these
parameters. For c a value of Max(im)/4 is chosen for the same reasons as the c in the confidence
measure.
3.2.3. Implementation and analysis of scale selection method using a test-image
For simplicity scale space analysis was implemented and tested using an artificial image of size
40x100x40 (figure 3.9). Artificial fibers of varying thickness were created drawing lines with one
pixel in diameter and then blurring each line with a Gaussian function at a certain scale.
Figure 3.9: Left: Plot of the twentieth slice of test-image with selected points in different colors. Right:
Cross-section displaying the different thicknesses of the fibers in the test-image.
The Hessian of this image was analyzed for scales ranging from 0.5 to 8 pixels with steps of 0.5.
In the image several points were selected and each given a different color (figure 3.9). This was
done to verify if indeed a maximum over scale occurred. That this was indeed the case for both
measures can be seen in figure 3.10.
Figure 3.10: The confidence measure (left) and the vesselness measure (right) plotted over scale for the
points in figure 2.7 in the corresponding color.
A plot of the scale index as a function of location in the image (figure 3.11) indicates what
happens. At locations were there are fibers a scale is found corresponding quite well with the
fiber thickness. At locations where the conditions for an ideal tubular structure are not fulfilled
there is no structure of interest and no scale is selected.
19
Chapter 3 Algorithm for Orientation Analysis
Figure 3.11: Plots of the scale index in color for every slice of the image containing fiber information. Top:
Legend for the scale from σ = 0 to σ = 6.Left: For the maximum response of the confidence measure. Right:
For the maximum found of the vesselness measure.
20
Chapter 3 Algorithm for Orientation Analysis
The vectors corresponding to the optimal scale are plotted in 3D over the test-image in a color
corresponding to their scale in figure 3.12.
Figure 3.12: For every second pixel of the twentieth slice of the test-image, the vectors are plotted in the
color corresponding to their scale over the original image. Top: After scale selection with the confidence
measure. Bottom: After scale selection with the vesselness measure. Right: Legend for the scale from σ = 0
to σ = 6.
The response of the confidence measure is larger then the response of the vesselness measure
(figure 3.10). It can be seen in figures 3.11 and 3.12 that the vesselness measure finds small scales
at the edges of the fibers in the artificial image. The directions, corresponding to these small
scales, are pointing in a direction, which is not in line with the fiber direction. The confidence
measure also finds orientations at the edges of the fibers that mainly point in false orientations. In
Chapter 4 we will evaluate the result obtained with the two measures and select the one that will
be used in the final algorithm.
21
Chapter 3 Algorithm for Orientation Analysis
3.3 Coherence Enhancing Diffusion
The quality of the collagen fiber images from the TPLSM is poor. Because the Hessian is a local
feature detector the poor quality will result in false orientation estimations. Therefore it would be
desirable to have a tool, which improves the quality of structures in the image without destroying,
for instance, the boundaries between the fibers. Coherence-enhancing diffusion filtering
(abbreviated as “CED”) is a method, which enhances the structure of an image according to its
surroundings. When applying coherence-enhancing diffusion, smoothing occurs along the
preferred orientation of the structures but not perpendicular to the structures in the image.
Weickert, 1998 [32] uses the structure tensor to describe the orientation of the structures. The
structure tensor will be discussed in paragraph 3.3.1. However, also other methods for
determining the orientation can be used, for example the Hessian matrix as shown in the last
paragraph of this chapter. In contrast to most nonlinear diffusion filters, coherence-enhancing
diffusion uses an approach where the process of filtering is steered by a diffusion tensor instead
of a scalar-valued diffusivity. This enables direction-dependent diffusion and not only adapts
diffusion to the location. The principle of CED will be explained in paragraph 3.3.2. In paragraph
3.3.3 the way CED is implemented using an additive operator splitting (AOS) stabilized scheme
will be explained and in the last paragraph some examples of CED will be shown.
3.3.1 The structure tensor
Given an m-dimensional image L, the structure tensor is based on the gradient L of the image,
which is usually calculated by means of Gaussian derivative filters (equation 3.4) with standard
deviation σ as the noise scale. Edges smaller then this noise scale will be ignored. From
differential geometry it is well known, that the gradient always points into the direction normal to
the steepest edges. The gradient is not able to detect parallel structures, because gradient
smoothing averages directions instead of orientations. In order to get an appropriate
representation of local orientation that is invariant under rotations by 180° the gradient is replaced
by its tensor product
J 0 (L ) L L L LT
(3.15)
This tensor is symmetric and positive semidefinite. The eigenvectors which can be calculated
from this tensor matrix are parallel and orthogonal to L . The orientations can be averaged by
applying a componentwise convolution with a Gaussian G :
J (L ) G (L L )
(3.16)
This matrix is called the structure tensor and is useful for many different applications. The
standard deviation ρ denotes the size of the texture and is usually large compared to σ. The
eigenvector corresponding to the smallest eigenvalue is the orientation with the lowest
fluctuations. The orientation is called the coherence orientation.
The eigenvalues, as mentioned before in paragraph 3.1, determine the amount of anisotropy of the
structure. A measure of coherence can therefore be defined in 3D as
k (1 2 )2 (2 3 )2 (3 1 )2
(3.17)
with 1 2 .... m .
22
Chapter 3 Algorithm for Orientation Analysis
For very anisotropic structures, i.e. different eigenvalues, it becomes large, while for isotropic
structures it tends to zero.
3.3.2. Coherence-enhancing diffusion filter
In the previous paragraph we determined the orientation of the coherent structures with the
structure tensor. By embedding the structure tensor (or the Hessian matrix) into a nonlinear
diffusion filter the structures can be enhanced. The basic equation which governs nonlinear
diffusion filtering is
t u div( D u )
on
(0, ).
(3.18)
where u(x,t) is a filtered version of the original image with scale parameter t ≥ 0. The original
image is given as the initial condition
u( x,0) L( x)
on
,
(3.19)
(0, ).
(3.20)
And the boundary conditions are
nu 0
on
where n denotes the outer normal to the image boundary ∂Ω. The diffusion process has to be
adapted to the image itself. This can be achieved by choosing the symmetric positive definite
diffusion tensor D (dij ) R mm as a function of the local image structure.
A smoothing process, which mainly acts along the flow direction, is needed for enhancing
coherence in images with flow-like structures. The smoothing should also increase with the
strength of its orientation given by the coherence k. This can be achieved by constructing the
diffusion tensor such that it has the same eigenvectors as the structure tensor and its eigenvalues
are given by
i :
(3.21)
for i = 1,…,m-1, and by
m :
C
(1 )exp k
if k = 0,
else.
(3.22)
C > 0 serves as a threshold parameter: For k >>C λm ≈ 1, and for k<<C λm ≈ α. The exponential
function and α were introduced by Weickert for two theoretical reasons: First, the smoothness of
the structure tensor is guaranteed to carry over to the diffusion tensor. Second, the process never
stops. This means that when the structure becomes isotropic there remains some small linear
diffusion with diffusivity α > 0, thus α keeps the diffusion tensor uniformly positive definite.
More information on why these are useful requirements and theoretical properties of CED can be
found in the paper of Weickert, 1998 [32].
3.3.3 Implementation using the additive operator splitting (AOS) stabilized scheme
Coherence enhancing diffusion filtering is a continuous process. Computer processing can only
handle discrete processes. Therefore CED has to be implemented as a numerical approximation.
23
Chapter 3 Algorithm for Orientation Analysis
To accomplish this, the derivatives are replaced by finite differences. The continuous CED is
given by
m
t u xi (dij xi u )
(3.23)
i , j 1
To explain the AOS stabilized scheme the 2-dimensional diffusion process is considered. The 2dimensional diffusion process is described by
d xx ( x ) d xy ( x ) x u ( x )
t u ( x ) ( x y )
d ( x ) d ( x ) y u ( x )
yy
xy
(3.24)
This equation can be written in the full form as
d xx x u d xy y u
( x y )
d u d u
yy y
xy x
x (d xx x u d xy y u ) y (d xy x u d yy y u )
x (d xx x u ) y (d yy y u ) x (d xy y u ) y (d xy x u )
x d xx x u d xx xx u y d yy y u d yy yy u x (d xy y u ) y (d xy xu )
Lxx
Lyy
(3.25)
Lxy
This leads to three elements; One element which contains only x components, one element that
contains only y components and one element which contains a combination of the two.
Central difference approximation can be used to approximate the diffusion equation. The central
difference for the first order derivative of a function f with grid size h is given by
x f
f ( x h) f ( x h)
2h
(3.26)
For the second order derivative this is
xx f
f ( x h) f ( x h ) 2 f ( x )
h2
(3.27)
When substituting the partial derivatives in equation 3.25 by these central differences we obtain
the following approximations
24
Chapter 3 Algorithm for Orientation Analysis
d xxk ( x 1, y ) d xxk ( x 1, y ) u k 1 ( x 1, y ) u k 1 ( x 1, y )
u k 1 ( x 1, y ) u k 1 ( x 1, y ) 2u k 1 ( x, y )
d xxk ( x, y )
2h
2h
h2
d yk ( x, y 1) d yyk ( x, y 1) u k 1 ( x, y 1) u k 1 ( x, y 1)
u k 1 ( x, y 1) u k 1 ( x, y 1) 2u k 1 ( x, y)
Lyy
d yyk ( x, y)
2h
2h
h2
u k ( x, y 1) u k ( x, y 1)
u k ( x 1, y) u k ( x 1, y)
Lxy Dx (d xyk ( x, y )
) D y (d xyk ( x, y )
)
2h
2h
Lxx
The simplest discrete approximation of the diffusion process is given by the finite difference
scheme
u k 1 u k
m
L u
i , j 1
k
ij
k
(3.28)
where u is a vector which contains the grey values at each pixel, the index k denotes the time
level, discrete times tk : k , with a time step size of τ, are considered and Lkij is a central
difference approximation to the operator xi (dij xi ) . Equation 3.28 can be rewritten such that we
observe that u k 1 can be calculated directly from u k without solving a system of equations
m
u k 1 ( I Lkij )u k
(3.29)
i , j 1
where I Rm is the unit matrix. This scheme is called an explicit scheme. Nonlinear diffusion
filtering is commonly performed with explicit schemes. However they are only stable for very
small time steps. This leads to time-consuming filtering and therefore limits their practical use.
That is why a semi-implicit scheme is considered. The slightly more complicated semi-implicit
discretization of the diffusion is given by
u k 1 u k
m
L u
i , j 1
k
ij
k 1
(3.30)
This scheme does not give the solution u k 1 directly, but solving a linear system first is required.
For this reason it is called linear- implicit or semi-implicit scheme. Lkij can be written as
Lkij u k 1 Lkxx u k 1 Lkyy u k 1 Lkxy u k
(3.31)
Substituting Lkij in equation 3.30 leads to
u k 1 u k ( Lkxx u k 1 Lkyy u k 1 Lkxy u k )
(1 ( Lkxx Lkyy ))u k 1 (1 Lkxy )u k
u k 1 (1 ( Lkxx Lkyy )) 1 (1 Lkxy )u k
semi implicit
(3.32)
explicit
When replacing the explicit part by V k this results in:
25
Chapter 3 Algorithm for Orientation Analysis
(1 ( Lkxx Lkyy ))u k 1 V k
(3.33)
The x and y parts are split:
1
((1 2 Lkxx ) (1 2 Lkyy ))u k 1 V k
2
(3.34)
This results in the 2-dimensional AOS scheme:
u k 1
1
( I 2 Lkll )1 V k
2 l x , y
(3.35)
When approximation an m-dimensional function, the 2 is replaced by m leading to the mdimensional AOS scheme:
u k 1
1 m
( I m Lkll )1 V k
m l 1
(3.36)
Stabilization is achieved by the non-negative matrices ( I m Lkll )1 , which describe a semiimplicit discretization of the diffusion caused by the l-th diagonal entry of the diffusion tensor.
For the approximation of central derivatives, the matrix inversions come down to solving
diagonally dominant tri-diagonal systems of linear equations. The Thomas algorithm can
accomplish this. All coordinate axes are treated in the same manner by additive splitting. AOS
schemes have been introduced in Weickert et al, 1998[33]. They show that AOS schemes are an
efficient and reliable method for nonlinear diffusion filtering.
3.3.4 Examples
The most common used example for showing the effect of CED is a 2D image of a fingerprint. It
is observed from figure 3.13 that diffusion along the coherence orientation takes place and
interrupted lines are closed. The influence of the integration scale parameter, ρ, is also
demonstrated in figure 3.13. It can be seen that a value for ρ, which doesn’t match with the
structure present in the fingerprint image, does not lead to dominant coherent orientation. The
image is improved significantly when increasing the value for ρ. However when a value for ρ is
increased even more, not so much improvement can be seen.
Figure 3.13 Influence of the integration scale on coherence-enhancing diffusion. Top Left: original of
fingerprint image. Top Right: ρ = 1. Bottom Left: ρ = 4. Bottom Right: ρ = 8. Constant parameter settings;
α = 0.001, C=1, σ = 0.5 and t = 16.
26
Chapter 3 Algorithm for Orientation Analysis
Also the influence of the time t was investigated. A sketch of a woman with curly hair, as shown
in figure 3.14, was used. Increasing t shows that the coherence of structures becomes coarser and
small structures disappear.
Figure 3.14: Influence of time, t, on coherence-enhancing diffusion. From left to right: Original image,
coherence enhancing diffusion after t=4, 8, 16. Constant parameter settings; α = 0.001, C=1, σ = 0.5 and
ρ =2.
For all these examples the orientation of the coherent structures was determined using the
structure tensor.
The first results when applying CED in 3D to the TPLSM collagen images seem promising
(figure 3.15 middle). The images appear less noisy and the fibers are enhanced.
Figure 3.15: Left: Original slice of a TPLSM image of the native heart valve. Middle: Image after CED
with α = 0.001, C=1, σ = 2, ρ = 8 and t = 4. Right: Image after CED with Hessian as structure descriptor
and with α = 0.01, C=1, σ = 2, ρ = 8 and t = 4.
Replacing the structure tensor with the Hessian gives worse results (Right figure 3.15). An
explanation for this may be that the structure tensor is more capable of finding edges and the
Hessian is more capable of finding ridges. This will result in less diffusion across edges when the
structure tensor is used.
27
Chapter 3 Algorithm for Orientation Analysis
3.4 Tensor Voting
Tensor voting (TV) is a technique for robust extraction of lines and curves from images. In noisy
images, local feature measurements, i.e. measurements of local edges or ridges, are often
unreliable. TV aims at making these local feature measurements more robust by taking into
account the measurements in the neighborhood. The name “tensor voting” comes from the fact
that information is encoded in tensors and these tensors communicate by means of a voting
process. This means that neighbors are telling each other what their observation is. For example,
when all the pixels belonging to an oriented structure have the same direction except for one
pixel, this one pixel receives orientation information from its neighbors as votes and it will adjust
itself accordingly. The more tensors are likely to belong to a structure that is of importance in the
image the more votes they receive.
Tensor voting was first introduced by Medioni et al., 2000 [34]. Previous work, dependent on the
type of input, includes extraction of object shapes out of noisy 3D data, shape from stereo and
motion estimation. In the coming paragraphs the different elements of the method are explained
together with how it is applied to enhance orientations.
3.4.1 Tensor representation and decomposition
In tensor voting, the input data is encoded into a field of second order symmetric and real-valued
tensors. In 2D this tensor can be visualized as an ellipse and in 3D as an ellipsoid. The shape of
the tensor defines the type of information it contains. Each tensor has the following form:
t11
T t12
t
13
t12
t22
t23
t13
t23 (e1 e2
t33
1 0
e3 ) 0 3
0 0
0 e1T
0 e2T
3 e3T
(3.37)
where 1 , 2 and 3 are nonnegative eigenvalues ( 1 2 3 ), and e1 , e2 and e3 are the
orthonormal eigenvectors ( e1 e2 e3 , e1 e2 e3 1 ). The eigenvectors correspond to the
principal directions and the eigenvalues encode the size and shape of the ellipsoid (figure 3.4).
Thus, the tensor contains both the orientation information and its confidence, or saliency.
In 3-D, a point on a smooth surface is represented by a tensor in the shape of an elongated
ellipsoid (stick tensor) with its major axis along the surface normal. A junction of two surfaces (a
curve) is represented by a tensor in the shape of a circular disk (plate tensor) which is
perpendicular to the curve's tangent. Alternatively, the plate can be thought of as spanning the 2dimensions subspace defined by the two surface normals, in the case of the surface junction.
Finally, an isolated point or a junction of curves has no orientation preference and is represented
by a tensor in the shape of a sphere (ball tensor). Any second order symmetric tensor, therefore,
can be expressed as a linear combination of three cases (figure 3.16); stickness, plateness and
ballness, i.e.
T (1 2 )e1e1T (2 3 )(e1e1T e2e2T ) 3 (e1e1T e2e2T e3e3T ) .
28
(3.38)
Chapter 3 Algorithm for Orientation Analysis
Figure 3.16: Tensor decomposition into stick, plate and ball components
3.4.2 Vote Analysis
After the voting procedure, the eigensystem of the resulting tensor which encapsulates all the
information propagated to the location can be computed. The information that is contained in a
tensor is interpreted as follows [34]:
Stickness: orientation e1, saliency is λ1-λ2
Plateness: orientation is e3, saliency is λ2-λ3
Ballness: no orientation, saliency is λ3
The orientation of the feature contained in the tensor is describes by
e1 y
e1 x
tan 1
(3.39)
cos (e1 z )
1
with γ and β between 0 and π. Each tensor is uniquely determined by these properties. The stick
component, which is parallel to the eigenvector corresponding to the largest eigenvalue and its
saliency is equal to the difference of the largest and the second largest eigenvalue, encodes the
likelihood of the point belonging to a smooth surface i.e. a measure of anisotropy of the ellipse.
The plate component spanned by the eigenvectors corresponding to the two largest eigenvalues
and whose saliency is the difference of the second and third eigenvalue encodes the likelihood
that the point belongs to a smooth curve or a surface junction. Finally, the ball component that has
no preference of orientation (isotropic) and its saliency is equal to the smallest eigenvalue
encodes the likelihood of the point being a junction. Assuming that noisy points are not organized
in salient perceptual structures they can easily be identified by their low saliency.
The TV technique takes a tensor field (T) as input, and generates a similar tensor field (U) as
output. The way the input tensor field should be encoded depends on the application. There are
many ways to generate an input tensor field from an input image. For the application of this
project the response from one of the anisotropy measure can be used as the saliency and the
angles corresponding to the principal direction with the largest eigenvalue give the orientation
information. So the output field is a context enhanced version of the input field, achieved by
communication between spatially neighboring tensors, as will be described in the next section.
29
Chapter 3 Algorithm for Orientation Analysis
3.4.3 Tensor voting process
The communication between tensors (tokens) is performed through a voting process, where each
tensor token casts a vote at each site in its neighborhood. The size and shape of this
neighborhood, and the vote strength and orientation are encoded in predefined voting fields
(kernels), one for each feature type: the ball, stick and plate voting fields. Those voting fields
have the same representation as the input data; they are tensor fields as well.
Figure 3.17: Shape of a stochastic stick voting field in 3D
For the continuation of oriented structures stick tensor voting fields are used. Because this is the
case for our application only sticks tensor voting will be explained. The stick tensor voting field
can be depicted as a tensor field template. It consists of stick tensors whose stickness describes
the likelihood that a feature at position x belongs to the same orientated structure as the feature
positioned in the centre of the voting field. The orientation of the tensor at x describes the most
probable orientation of a feature in that point. Medioni et al. [34] state that there is only one free
parameter: scale. The scale is the parameter that essentially controls the size of the voting
neighborhood and the strength of the votes. Their stick voting field is a model based on
assumptions about curves in images. The magnitude of the vote decays with distance and
curvature. The voting field can also be considered as the model used in tensor voting [35]. The
voting field can therefore be different from the one defined by Medioni et al. In our algorithm a
voting field based on stochastic completion fields (figure 3.17), introduced by Williams and
Jacobs, 1997 is used [36, 37]. The theory of Williams and Jacobs is probabilistic and models the
contour as the motion of a particle performing a random walk (figure 3.18). Particles decay after
every step, thus minimizing the likelihood of completions that are not supported by the data or
between distant points.
Figure 3.18: Left: An example of a random walk. Middle: 1000 random walks.
For every (nonzero) tensor in the tensor field, the stick field is centered at the position of the
tensor and rotated to align with the first eigenvector e1 of the tensor. All tensors in a certain
neighborhood (determined by the context scale) of this tensor receive a weighted contribution,
called a vote, by addition. In other words, the information contained by the tensor is broadcasted
30
Chapter 3 Algorithm for Orientation Analysis
to the neighborhood. The way the voting field broadcasts its votes is illustrated in figure 3.19 for
a sparse number of tensor tokens.
Figure 3.19: The stick voting communication within TV [38].
31
Chapter 4 Validation
In the previous chapters the methods that were evaluated to design the algorithm for orientation
analysis are explained. In figure 3.1 the algorithm was presented schematically. To evaluate
whether the proposed algorithm is an appropriate and accurate way to determine collagen
orientation in TPLSM images, the algorithm is validated. No existing validation method for these
types of algorithms and applications has been found in literature. Therefore a validation method
had to be designed. Artificial images were created. The way these artificial images were made is
explained in paragraph 4.1. The basis of the algorithm is the determination of the minimal
principal curvature directions at the correct scale. Therefore this step was analyzed first. In the
paragraph 4.3 the signal-to-noise ratio (SNR) is determined before and after coherence enhancing
diffusion. And in the last paragraph tensor voting was validated using the input of the scale
selection step and generating new output.
4.1 Artificial image
To be able to validate the method developed for orientation analysis with scale selection an
artificial 3D image was created in Mathematica [39]. A starting position is chosen for the fiber.
This position can be chosen freely. From this position on, a neighboring voxel is selected by
stepping into a predefined direction. This direction can be chosen by defining the two angles,
and . Repeating this step will result in a fiber that is one voxel in width and has a certain
orientation. For every step the orientation in which the fiber is elongated is saved as the ground
truth. A range of orientations can also be defined. This will result in a fiber that is curved,
however to ensure that the fiber follows its general direction the resulting orientation is calculated
by weighting it with the two previous orientations.
The fiber is then blurred with a Gaussian filter. This gives fibers with a diameter resulting from
the standard deviation of the Gaussian filter. A threshold is set that includes values above ¼ of the
maximum intensity of the image. To obtain images that take into account the partial volume
effect, images of 256 x 256 x 100 are created and down-sampled with a factor of 4 (figure 4.1).
This was done by calculating the mean of every 4 x 4 pixels and placing the resulting value at the
corresponding pixel in the new image.
Figure 4.1 Left: A slice of an artificial image (256x256x100) with two fibers of 1 voxel in diameter. Middle
left: Same image after Gaussian blurring. Middle right: The image after setting a threshold (max[image]/4)
for the Gaussian fibers. Right: The final artificial image (64x64x25) after sub-sampling.
The partial volume effect is the effect that occurs at the boundaries of contours. This can be
solved by filling the voxel equivalent to the physical integration of the intensity over the area of
the detector (figure 4.2). Down-sampling is an estimation to solve the partial volume effect.
323
Chapter 4 Validation
Figure 4.2: Image without and with taking into account the partial volume effect at the boundaries of the
contours.
The ground truth has to undergo the same manipulations. Blurring the orientation angles with a
Gaussian is possible by converting the orientation angles to vectors and blurring each component
of this vector. Then only the vectors on the pixels that were included after the threshold are
selected and down-sampled.
Now it is possible to validate the algorithm with images of fibers at different orientations and of
different diameter and to compare the results of the algorithm with a ground truth.
4.2 Validation of minimal principal curvature directions and scale.
In total 14 different artificial images with two fibers, were created with the parameter settings
listed in table 4.1.
Table 4.1: Overview of artificial images.
Image
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Orientation in radians
θ
φ
1/2 π
1/2 π
0
0
0
1/2 π
1/6 π
1/4 π
1/4 π
1/3 π
1/3 π
2/3 π
5/6 π
2/5 π
3/4 π
3/8 π
5/8 π
3/4 π
1/7 π
2/3 π
3/5 π
1/2 π
1/2 π
1/8 π
2/7 π
4/10 π
2/3 π
π
Gaussian scale
σ1
σ2
4
4
8
8
12
12
6
6
10
10
2
2
4
4
10
6
8
4
4
12
2
3
14
14
5
10
8
8
Intensity of the fiber
I1
I2
1
1
1
3
1
4
1
5
1
2
1
1
1
1
4
1
1
1
1
1
1
1
1
1
1
1
1
1
To determine whether there is a significant difference between the ground truth and the
orientations found by the algorithm, a paired t-test is used. In this test every angle found by the
algorithm is compared to its ground truth. SPSS 14.0 [40] was used to compute the result. First
images 1 to 8 were analyzed. For many images a significant difference in orientation angles was
found (see Appendix I). For image 5 a very large t-value for φ was found. When plotting the
vectors in the orientation found by the algorithm it can be seen that most vectors are in line with
the fiber orientation (figure 4.3). The ground truth for this image was, however, not correct.
33
Chapter 4 Validation
Figure 4.3: Plots of the 3-dimensional vectors found by the algorithm using the confidence measure. The
vectors are plotted on the pixels belonging to the fiber in the artificial images 1-8. To obtain a clear
representations vectors were plotted only every 2 or 3 pixels.
34
Chapter 4 Validation
Therefore image 5 is not taken into account in the total evaluation. The results for the total
evaluation can be seen here (figure 4.4).
Figure 4.4: Statistical result from SPSS of artificial images 1-8, with the p-value in the last column (Sig.).
No significant difference was found for θ while for φ there was a significant difference according
to the statistical analysis. One reason for this could be the vectors found at the end of the fibers
(figure 4.3). Therefore 6 new artificial images (table 4.1 images 9-14) were made with fibers
running from one end of the image to the other (figure 4.5). The orientations found within 5
pixels from the boundaries of the 3-dimensional image were excluded from the analysis
(Appendix II).
Figure 4.5: Statistical result from SPSS of artificial images 9-14, with the p-value in the last column (Sig.).
This analysis shows that there is a significant difference in all four angles (p<0.05).
Because it is most important for the algorithm to find an appropriate mean orientation, we
calculated the mean orientation of images 1-14 (image 5 excluded) separately and compared this
with the mean of their ground truth (figure 4.6). The mean is defined as the angle with the
maximum in the histogram. The maximum was used because orientations are continues, so 0 is
the same as π and this continuity provides a problem for determining the mean.
Figure 4.6: Statistical result from SPSS of the mean orientation of artificial images 1-14 (not image 5).
35
Chapter 4 Validation
No significant difference was found for both measures and angles (p>0.05). The confidence
measure has p-values of 0.301 and 0.408 for the mean θ and mean φ respectively and the
vesselness measure has p-values of 0.081 and 0.083 for the mean θ and mean φ respectively,
which indicates that the confidence measure is statistically better at finding the mean orientation
of the fibers.
To validate scale selection, fibers with their principal axis in the z-direction, created the same way
as described in paragraph 4.1, are analyzed. The diameter of the fiber is measured by hand,
ranging from 4 to 16 pixels. These fiber diameters are compared to the diameters found by the
two measures (figure 4.7). The confidence measure finds scales that are consistent with the
change in manual determined diameter. The vesselness measure gives worse results.
Figure 4.7: Plots of the manual determined diameter versus the scale found by the confidence measure
(left) and vesselness measure (right).
The scales were plotted in color over the fiber diameter (figure 4.8).
Figure 4.8: Plots of the scale in color over the middle slice of a 3-dimensional artificial image with a fiber
in the z-direction.
36
Chapter 4 Validation
The validations results show that the confidence measure finds a better scale and finds
orientations that are statistically better then the vesselness measure and will therefore be used for
analyzing the TPLSM data.
4.2 Validation of CED
Noise in a fluorescence microscope has different causes. The quantum nature of light results in
Poisson noise in the TPLSM images. Light can be considered as a series of particles called
photons. Photon production by any light source is a statistical process governed by the laws of
quantum physics. The source emits photons at random time intervals. The number of photons in a
fixed observation interval will result in a number that obeys Poisson statistics. The uncertainty in
photon counting yields a Poisson distributed signal. This photon detection induced Poisson noise
is sometimes referred to as intrinsic noise [41], and is unavoidable when acquiring an image.
However, light detectors and sensors contribute extrinsic noise to the detected signal. This
extrinsic noise can be neglected if the detector is photon-limited. A light detector is said to be
photon-limited if the extrinsic noise is negligibly small compared to the amount of intrinsic noise
induced by the detection of photons. Scientific CCD cameras and photomultiplier tubes (PMT),
which were used in our case, can be regarded as photon-limited [42] and therefore the extrinsic
noise is neglected. The third image of table 4.1, with Poisson noise added, was used to evaluate
CED. The SNR is used to determine whether the fibers become more apparent compared to the
background after CED. The SNR [29] is given by
SNR
m2 m1
var1 var 2
(4.1)
with m1 the mean of the background, m2 the mean of the signal from the fiber, var1 the variance
of the background and var2 the variance of the signal from the fiber. The signal is defined as the
difference of the means, the noise as the sum of the variances of the intensity values. In figure
4.9, the regions that were selected to determine the means and variances are indicated by the red
squares. In reality these squared are 3D cubes, one placed in the background and one on the fiber.
The fiber profiles before and after CED (figure 4.9 bottom) indicate that the noise is substantially
reduced and the edges in the orientation of the fiber are preserved.
Figure 4.9: Top left: input image. Top right: input image after coherence enhancing diffusion with
α=0.001, C=1, σ = 2, ρ = 6 and t = 25. Bottom images: intensity profile of middle row of pixels in middle
slice of 3D images stack for both images.
37
Chapter 4 Validation
With α = 0.001 the SNR increases substantially during the iterations (figure 4.10). When a value
for α of 0.01 is chosen, so when more diffusion is allowed in isotropic regions, a maximum can
be found for the SNR.
Figure 4.10 Left: The signal-to-noise ratio (SNR) increases with evolution time (α = 0.001). Right: A
maximum is found for CED with increasing evolution time (α = 0.01).
4.3 Validation of Tensor Voting
The reason to use tensor voting is to improve the orientations found by minimal principal
curvature directions with scale selection analysis. The results of artificial images 1-4 were used as
input. To validate tensor voting the vector fields before and after tensor voting are compared
(figure 4.11).
Figure 4.11: Plots of the 3-dimensional vectors found after tensor voting. The vectors are plotted on the
pixels belonging to the fiber in the artificial images 1-4. To obtain a clear representations vectors were
plotted only every 2 or 3 pixels.
38
Chapter 4 Validation
These plots show no improvement in the orientation of the vectors.
The influence of the saliency (or confidence of the vector) is tested using an artificial vectorfield
(figure 4.12). One vector has an orientation that deviates from the rest. This vector was assigned a
saliency value of 2, 4, 6 and 8 respectively while the other vectors had a saliency of 1. Figure 4.12
shows that a vector with salience of 8 does not align with the other vectors.
Figure 4.12: Influence of the saliency on tensor voting. The open pieces are a result of opposite directions.
39
Chapter 5 Experiments
5.1 Introduction
This chapter presents the experiments that were done to study the influence of strain on the
collagen orientation. The structural properties of collagen in 3D for unattached, attached and
strained heart valve tissue engineered equivalents were examined and analyzed. Because
conventional microscopic techniques are limited with respect to studying structural changes in
collagen organization in tissue engineered constructs, collagen was visualized using TPLSM.
5.2 Experimental setup
5.2.1 Tissue engineering protocol
The model system consists of a flexercell FX-4000T straining system, on which four 6 well plates
with flexible membrane bottoms are mounted (Figure 5.1). By applying a vacuum these
membranes are sucked inwards and stretched over a loading post. The scaffolds, which serve as a
cell carrier, were made by cutting polyglycolic acid (PGA) strips in a rectangular shape of 25 x 4
x 1 mm3. These samples were coated with a solution of 1% of Poly-4-hydroxybutyrate (P4HB)
and attached to the flexible membrane of the flexercell FX-4000T straining system. Two different
protocols were used to attach the samples. In the first experiment the samples were attached with
velcro. Many samples detached when velcro was used. Therefore in the second experiment the
samples were gently pressed into silicone gel. Then the scaffolds were seeded with human
Venous Saphena Cells with fibrin as cell carrier. The samples were cultured for one week in a
static condition at 37°C.
Figure 5.1 Overview of flexercell FX-4000T straining system with the 6 wells plate with flexible membrane
and scaffold.
In the first experiment five sets containing each 6 tissue samples were developed. Four of which
were strained uniaxially at straining levels of 0% (A1) (static strain), 4% (B1), 8% (C1) and 12%
(D1) and one was unstrained (E1). The samples were strained for three weeks at 37°C and then
sacrificed. The samples that were strained with 8% and 12% detached from the flexible
membrane and were therefore not analyzed. In the second experiment five sets containing each 8
403
Chapter 5 Experiments
tissue samples were developed. Four sets were strained uniaxially at straining levels of 0% (A2)
(static strain), 4% (B2), 8% (C2) and 12% (D2) and one was unstrained (E2). The sample with
12% strain detached from the flexible membrane and was not analyzed. The samples were
strained for three weeks and then sacrificed. The sacrificed tissue samples were imaged the same
day and kept on ice to slow down degradation.
5.2.2 Image acquisition
The microscope setup consisted of a BioRad Radiance 2100MP in combination with a Spectra
Physics Tsunami Ti: Sapphire laser and a Nikon E600FN microscope (Figure 5.2). As a standard
an excitation wavelength of 800 nm and a 60x 1.0 NA water-dipping objective with 1.2 x optical
zoom was used. Three photomultipliers were used to detect the fluorescence in the specimen.
Figure 5.2 Image of the microscope setup at the department of Biophysics in Maastricht.
The collagen in the constructs was stained with the fluorescent probe CNA35 (-OG488)
Krahn[6]. CNA35 has a high affinity for collagen type-I. The probe consists of a collagen binding
domain (bacterial collagen receptor) and a fluorescent dye, which is conjugated to the binding
domain. In general the procedure for staining was as follows: the constructs were incubated at
37 °C overnight using 0.1µM CNA35.The emission spectra of CNA35 contain a peak at 515 nm.
An emission bandpass filter of 500 to 530 nm was applied so that CNA35 was present in the
green channel.
5.2.3 Image analysis
Automatic image analysis was performed on the CNA35 image data to determine the scale and
collagen orientation. The methods described in chapter 3 were implemented in Mathematica [39].
Table 5.1 provides the set of parameters that was used for all the datasets. Several preprocessing
steps were performed before the TPLSM- data were analyzed.
41
Chapter 5 Experiments
Table 5.1: Overview of parameters.
Method
CED
Scale selection
Intensity threshold
Parameter
σ
ρ
α
Value
0.5 μm
1.5 μm
0.001
C
h
τ
endt
σ-start
σ-eind
σ-step
c (confidence measure)
α, β (vesselness measure)
c (vesselness measure)
Threshold
1
1
0.25
6
0.3 μm
2 μm
0.05μm
0.25 x Max(im)
0.5
0.25 x Max(im)
0.25 x Max(im)
Definition
Noise scale
Structure scale
Isotropic diffusion
parameter
Threshold parameter
Grid size
Time step
Time
Minimal scale
Maximum scale
Scale step
Threshold parameter
Threshold parameter
Threshold parameter
Threshold parameter
Memmory issues
Because the amount of memory required to process the data was too large the images were first
down-sampled by a factor 4 in the x- and y-dimension. This was done by calculating the mean of
every 4 x 4 pixels and placing the resulting value at the corresponding pixel in the new image.
Figure 5.3: Slice (206 x 206 μm) of TPLSM-data (left) and image after down-sampling (right).
The voxels in the 3D stack are of equal size in the xy-plane but differ in size in the z-direction.
This means for example that an ellipse in the image is in reality a circle. Furthermore, the size of
the voxels in the z-direction differs between datasets, depending on the distance between the
slices that was chosen during imaging. Therefore, by interpolating in the z-direction, this
proportion is taken into account to obtain isotropic voxels.
Intensity correction
In some stacks it was clear that the slices made deeper in the tissue are relatively darker (figure
5.4). This can for example be caused by absorption, scattering of excitation and fluorescent light
and photobleaching.
42
Chapter 5 Experiments
Figure 5.4: Part of a slice in the x, z-direction of a porcine heart valve.
An exponential decay of the intensity (I) with depth (d) is expected [44, 45] and therefore an
exponential function (equation 5.1) was fitted to the intensity decay curve. The intensity can
decrease because of a decrease in brightness and because of a decrease in contrast.
I=I0e d
(5.1)
with μ the decay coefficient and I0 the initial intensity. The decay in brightness and contrast can
be corrected for by multiplying the intensity of each pixel in a slice with a correction factor. This
factor can be determined by calculating the decay of the mean intensity as a measure for
brightness and the variance of the intensity as a measure of contrast for every slice and then
fitting an exponential function to their decay (figure 5.5).
Figure 5.5: Left: the mean intensity versus depth in black and the fitted exponential function in red. Right:
the variance of the intensity versus depth in black, the fitted exponential function in red
After the intensity correction it shows that the means are now on one line and that the difference
in variance has decreased. The maximum value however increases (figure 5.6).
Figure 5.6: Plots of the maximum and minimum in red, the mean in green and the variance in blue for
every slice. Left: before correction. Right: after correction.
The difference between before and after the correction can be seen in figure 5.7.
43
Chapter 5 Experiments
Figure 5.7: Last slice of TPLSM z-stack (206 x 206 x 74 μm) before and after depth correction.
This step is not included automatically in the algorithm, because the more recent microscope has
a built-in correction for this. Thus, the user can choose whether this is necessary or not.
44
Chapter 6 Results
Microscopy images
Selected image slices from the TPLSM data are shown for two of the constructs of the first
experiment (figure 6.1), one attached construct of 0% strain and one with 4% strain. In both
experiments the constructs were strained horizontally. Note how the collagen fibers appear more
aligned in the direction of the strain for the 4% strained construct then in the attached construct.
A1
B1
Figure 6.1 Selected image slices (206 x 206 μm) from the TPLSM data. Two constructs of the first
experiment one attached construct of 0% strain (A1) and one of 4% strain (B1).
Two of the constructs of the second experiment (figure 6.2), one attached construct of 0% strain
and one of 8% strain are shown in figure 6.2. The orientation of the collagen fibers after strain is
perpendicular to the direction of the strain and they appear more aligned.
A2
C2
Figure 6.2 Selected image slices (172 x 172 μm) from the TPLSM data. Two constructs of the second
experiment, one attached construct of 0% strain (A2) and one of 8% strain (C2).
453
Chapter 6 Results
Orientation analysis
The orientation analysis algorithm as described in chapter 3 was used to analyze the TPLSM data.
In figure 6.3 the histograms of both angles are given for all the samples of experiment 1. The
distribution of orientations becomes smaller when the constructs are strained compared to the
unattached construct. An even smaller peak in φ can be seen when the strain is increased.
Figure 6.3: Orientation analysis results of attached construct A1, B1 and E1. Histograms of the orientation
are given for every angle with both angles divided into 50 bins each bin representing 0.02 π. N is the
number of counts. The histograms are shifted so that the maximum response is centered.
In figure 6.4 the histograms of both angles are given for all the samples of experiment 2. It can be
seen that the distribution peak of θ becomes smaller when the constructs are strained compared to
the unattached constructs. Sample E2 φ has a small peak and the width of the peak increases
when strain is applied. Comparing the distributions of φ for samples A2, B2 and C2 the peaks of
the distributions become smaller with increased strain.
46
Chapter 6 Results
Figure 6.4: Orientation analysis results of attached construct A2, B2, C2 and E2. Histograms of the
orientation are given for every angle with both angles divided into 50 bins each bin representing 0.02 π. N
is the number of counts. The histograms are shifted so that the maximum response is centered.
The mean orientation was calculated to determine the general orientation of the collagen fibers.
The mean orientation is the orientation with the maximum response in the histogram. The
variance in orientation from this mean is used as a measure for alignment.
47
Chapter 6 Results
In table 6.1 the mean orientations that were determined for both angles and their variance are
shown. Many small scales were selected. This can be seen in Appendix III where the results are
given as 3D histograms of the orientation for every scale. The small scales are mostly found at
the boundary between tissue and background and give many false orientations. Therefore the
orientations found at the smallest scale are not included in the calculation of the mean and
variance.
Table 6.1: The mean and variance of collagen orientation determined from the TPLSM data
TPLSM-data
Mean
orientation
of θ
(in degrees)
Mean
orientation
of φ
(in degrees)
Variance in Variance in
θ
φ
(in degrees2) (in degrees2)
Experiment 1
E1 (unattached)
A1 (0% strain)
B1 (4% strain)
46,8
90,0
90,0
90,0
90,0
90,1
31,9
22,8
30,4
5,4
4,3
11,7
Experiment 2
E2 (unattached)
A2 (0%strain)
B2 (4%strain)
C2 (8% strain)
21,6
90,1
176,5
169,2
90,0
93,6
90,0
90,2
34,7
34,3
23,6
22,6
4,7
7,0
13,3
7,8
The variances in θ indicate more alignment with applied strain when E1 is compared to A1,
however, B1 show only a little decrease in variance compared to E1 and a higher variance
compared to A1. In the second experiment the variance in θ decreases when 4% and 8% strain is
applied compared to the unattached sample and 0% strain sample. The variance in φ increases
when strain is applied in both experiments. The variances in table 6.1 do not always correspond
with the histograms in figures 6.3 and 6.4.
48
Chapter 7: General Discussion and Conclusions
In this chapter the results will be analyzed and discussed. Firstly, the performance of the
algorithm for orientation analysis will be analyzed and possible improvements will be presented.
Secondly, changes in collagen orientation as a result of strain will be discussed based on the
experimental results. Finally, the general conclusions which may be drawn from this study will be
presented and recommendations for future research are given.
7.1 Algorithm performance
An algorithm was designed to analyze the orientation of collagen fibers in 3D TPLSM images.
The methods that form this algorithm are discussed separately here.
The minimal principal curvature directions and scale selection.
Several assumptions were made which resulted in the choice of the Hessian matrix for
determining orientation. The first assumption is that the collagen fibers in the TPLSM data are
tubular structures. When inspecting the TPLSM image the collagen fibers appear as bundles of
wool sticking together. The second assumption is that the collagen fibers have a Gaussian profile.
The fibers have a blurred appearance due to the properties of the imaging system, but it could not
be verified what profile they have. The Hessian matches second order Gaussian derivatives to the
structures in the images. A match between the Gaussian second order derivatives from the
Hessian and the fibers may not be optimal as a result of these assumptions and will sometimes
result in false orientation estimations. The orientation analysis with minimal principal curvature
directions was validated using a set of artificial 3D images. A ground truth of orientations was
created for these images. In figure 4.4 the results of the paired t-test are presented. Significant
differences in orientation for the first 8 images are found when all the orientations of the voxels
belonging to a fiber are compared to the ground truth. To investigate whether this is caused by
false detections at the ends of the fibers 6 new images were developed and analyzed. That
boundary effects are not the reason for a significant difference in angles can be seen in figure 4.5.
The difference in angles can also result from the fact that the partial volume effect is estimated by
down-sampling, which causes false measurement along the entire fiber contours. The first three
artificial images do not suffer from this because they are in the x-, y- and z- direction. From the
plots of the principal directions found by the algorithm (figure 4.3) it becomes clear that in the
centre of the fiber correct orientations are found. Thin fibers have relatively more false detections
then large fibers because the edges have a larger influence. The significant difference in angles
can also be caused by the way the paired t-test evaluates the angles. Because many measurements
were available the standard mean error becomes very small and results in very narrow confidence
interval. The mean differences in figure 4.4 and figure 4.5 are in the order of hundreds of radians,
which indicate that only small differences appear between the ground truth and the measured
orientations. Because many measurements were available, the standard error of the mean
becomes very small and results in a very narrow confidence interval. We can conclude that the
differences may be significant but they are not relevant for our application. In figure 4.6 the
results are presented of the statistical analysis of the mean orientations of the fibers. There is no
significant difference between the means for both the confidence and the vesselness measure.
The choice for the confidence measure and the vesselness measure was also based on the
assumption that fibers are tubular. These measures are used to find the scale where the Hessian is
the most anisotropic. The confidence measure can not differentiate between a tubular structure
and a plate-like structure as well as the vesselness measure. When for example eigenvalues of
(0, -1, -1) (bright tube), are entered this gives the same response as (0, 0, -1) (bright plate).
493
Chapter 7 Discussion and Conclusions
Another factor that influences the results is the range of scales that was chosen to compute the
Hessian. These scales were based on the size of the fibers present in our data. When the scales are
chosen too large however, it could be that a maximum response is found for fibers sticking
together. Scale selection is validated by making test-images containing fibers which are oriented
in the z-direction. From figure 4.7 it is clear that the confidence measure is better at finding the
appropriate scale. Figure 4.8 shows that most mistakes in scale are made at the edges of the fiber.
Especially the vesselness measure seems to have difficulties with this.
Coherence enhancing diffusion
Coherence-enhancing diffusion is applied to enhance the fibers in the TPLSM data. The structure
tensor is used to describe the orientation of the structures, because the Hessian gives worse
results. One reason for this may be that the structure tensor is better at finding edges while the
Hessian is better at finding ridges. The parameters that are chosen for CED are based on visual
inspection of the results of the tissue engineered constructs after down-sampling. Fine-tuning
these parameters is expected to improve the results of CED. Whether coherence-enhancing
diffusion is appropriate for enhancing tubular structures is tested in paragraph 4.2. The SNR ratio
increases substantially with the number of iterations. Because not much information is present in
the artificial image, the parameter that determines the diffusion in isotropic areas, α, is increased
to 0.01. This leads to a maximum of the SNR with the number of iterations. The optimal choice of
t is the time where this maximum is found.
Tensor voting
Tensor voting does not improve the orientations found by the Hessian. The reason no
improvement can be seen may be that the response of the confidence measure is not capable of
separating the correct and false orientations or that too many deviating orientation are found. The
influence of the saliency (the strength of the vectors) is tested using a vectorfield with one vector
pointing in a false direction (figure 4.12). This vector has a saliency value of 2, 4, 6 and 8
respectively while the other vectors have a saliency of 1. After tensor voting the vector is rotated
to align with the rest of the vectorfield until the saliency becomes 8 times as large.
Memory issues
Another issue that reduces the performance is memory requirements. The algorithm was
implemented in Mathematica which does not always handle its memory efficiently. As a result
the algorithm could not run on the original datasets. Down-sampling the images by a factor of
approximately 4 (the voxels were also made isotropic so that the factor is not the same in each
direction) resolves the issue, but also decreases the resolution. This in effect increases errors due
to fibers that are close to one another. CED was also implemented in Mathematica and is a very
time-consuming preprocessing step. Implementing CED in C++ will speed up this process
enormous. Tensor voting was implemented in C++ and is very fast. It was not tested enough to
apply it on the TPLSM data.
7.2 Experimental results
Changes in collagen orientation can be observed in the TPLSM images shown in figures 6.1 and
6.2. The fibers of the constructs that were only attached but not strained show a more random
collagen orientation then the constructs that were strained. This can also be seen in the orientation
histograms (figure 6.3 and 6.4) that were computed by our algorithm for orientation analysis. The
collagen fibers in the first experiment reorient in the direction of the applied strain (mean
orientation in table 6.1). In the second experiment the collagen fibers are oriented perpendicular
to the direction of the applied strain. This is expected to be a result of the imaging location. A
recent experiment which has not been included in this work has shown that the orientation of
50
Chapter 7 Discussion and Conclusions
collagen at the surface of the samples is indeed perpendicular to the direction of the strain while
deeper into the tissue the fibers are oriented in the direction of the strain. Thus, the results depend
on the imaging location and imaging deeper into the tissue may result in a better representation of
the collagen fibers with TPLSM.
The smallest scale and the largest scale have the largest amount of counts in the histograms of
figure A3-1 to A3-7. Small scales were mostly found at locations where the tissue ends, while
large scales are found at locations were the fibers are very dense. These small scales were not
included in the analysis, because they correspond to mostly false orientations.
Figure 6.3 shows that the peak in the histogram is smaller for the tissue engineered constructs that
are strained compared to the unattached constructs. This indicates that strain results in more
aligned collagen fibers. Figure 6.4 also shows a smaller distribution in θ, however, this can not be
seen in φ. When the strain is increased the peak in the histogram for φ becomes even smaller for
A1 compared to B1 and A2 compared to B2 and C2. This indicates that applying more strain
results in more alignment. In the variance table 6.1 however does not always support the
observations made from the histograms. Only a very small amount of data was available therefore
no significant changes in collagen orientation could be found.
7.3 Conclusions
This work contains a method which can be used to obtain robust orientation information. The
method determines the minimal principal curvature directions from the Hessian matrix. Three
possible methods were described to improve the orientations found by the principal curvature
analysis by taking into account the context of the structures in the image. An algorithm was
developed based on these methods and validated. The algorithm was used to study collagen
orientation in 3D engineered tissue. The conclusions are summarized below.
3D principal curvature directions are an effective way to determine local orientation of
tubular structures.
CED can be used to enhance collagen fibers in TPLSM images.
Tensor Voting can in theory be used to improve the local orientation estimations; in
practice however this still has to be investigated.
TPLSM makes it possible to study collagen orientation in 3D tissue engineered
constructs.
This study indicates that there is an increase in collagen alignment with increased strain
magnitude based on the orientation histograms.
The variance in orientation does not support the observations made from the orientation
histograms.
7.4 Recommendations for future research
Algorithm for 3D orientation analysis
The algorithm is very computationally expensive and memory requirements are a major problem.
Implementation in C++ would speed up the process significantly and handle the memory more
efficiently. To analyze the alignment of collagen fibers in TPLSM images faster and in a more
global way also Fourier analysis could be considered. The interpretation in 3D however is
somewhat difficult. In figure 7.1 a 2-dimensional collagen image and its magnitude plot are
shown. It can be seen that the general orientation of the fibers is perpendicular to the largest axis
of the ellipse in the magnitude plot.
51
Chapter 7 Discussion and Conclusions
Figure 7.1: Slices of TPLSM-image of collagen and their Fourier magnitude plot.
Tensor voting to improve 3D orientation estimations has been introduced. This method however
should be investigated more before it can be applied.
Influence of strain of collagen organization
The results of this study cannot be considered conclusive and a logical step is therefore to
increase the number of experiments.
When the tissue engineered constructs are imaged with TPLSM only a small part of the construct
is visualized. Imaging deeper into the tissue is recommended, because at the surface collagen
orients in a direction perpendicular to the direction of the applied strain. Imaging with less
magnification (60x was used here) may result in a better representation of the entire construct.
Note however that this will lead to images of less resolution.
Future research should be done to investigate the relation between mechanical conditioning and
collagen remodeling. In this work collagen orientation was investigated. However, also for
example collagen fiber diameter is an important tissue property that has influence on the strength
of tissue engineered heart valve tissue. Another option is to quantify from day 1 of tissue
engineering how the collagen organization changes in time. This means imaging of the same
construct in time is necessary. Investigation of different straining methods to obtain the best
structural properties is also worth investigating. In the experiment presented in this work only
uniaxial strain was applied. For example the influence of changing the strain direction is also an
interesting subject.
52
Appendix I Results paired t-test for artificial images 1-8
533
Appendix I Results paired t-test for artificial images 1-8
543
Appendix II Results paired t-test for artificial images 9-14
553
Appendix II Results paired t-test for artificial image 9-15
56
Appendix III 3D histograms of the orientation for every
scale
Figure A3-1: Orientation analysis results of attached construct A1 (0% strain). Histograms of the
orientation are given for every scale with both angles running from 0 to π divided into 20 bins each bin
representing 0.05 π. N is the number of counts.
573
Appendix III 3D histograms of the orientation for every scale
Figure A3-2: Orientation analysis results of construct B1 (4% strain). Histograms of the orientation are
given for every scale with both angles running from 0 to π divided into 20 bins each bin representing 0.05
π. N is the number of counts.
583
Appendix III 3D histograms of the orientation for every scale
Figure A3-3: Orientation analysis results of unattached construct E1. Histograms of the orientation are
given for every scale with both angles running from 0 to π divided into 20 bins each bin representing 0.05
π. N is the number of counts.
593
Appendix III 3D histograms of the orientation for every scale
Figure A3-4: Orientation analysis results of attached construct A2 (0% strain). Histograms of the
orientation are given for every scale with both angles running from 0 to π divided into 20 bins each bin
representing 0.05 π. N is the number of counts.
603
Appendix III 3D histograms of the orientation for every scale
Figure A3-5: Orientation analysis results of construct B2 (4% strain). Histograms of the orientation are
given for every scale with both angles running from 0 to π divided into 20 bins each bin representing 0.05
π. N is the number of counts.
613
Appendix III 3D histograms of the orientation for every scale
Figure A3-6: Orientation analysis results of construct C2 (8% strain). Histograms of the orientation are
given for every scale with both angles running from 0 to π divided into 20 bins each bin representing 0.05
π. N is the number of counts.
623
Appendix III 3D histograms of the orientation for every scale
Figure A3-7: Orientation analysis results of unattached construct E2. Histograms of the orientation are
given for every scale with both angles running from 0 to π divided into 20 bins each bin representing 0.05
π. N is the number of counts.
633
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