High angular resolution diffusion imaging : processing & visualization

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High angular resolution diffusion imaging : processing &

visualization

Citation for published version (APA):

Prckovska, V. (2010). High angular resolution diffusion imaging : processing & visualization. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR690073

DOI:

10.6100/IR690073

Document status and date: Published: 01/01/2010 Document Version:

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High Angular Resolution Diffusion Imaging

Processing & Visualization

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The front of the cover represents the brain from the author and her thoughts. The rendering of the T1 brain data is done with the tool MRIcroGL and the HARDI and DTI glyphs with the DTITool that the author has been developing. The DW-MRI data for the glyph rendering is from the actual DW-MRI scan of the author’s brain. The cats in the brain (author’s pets Pixie and Kenzo) hold the metaphor of the thoughts that, obviously, frequently occupy the mind of the author.

On the back of the cover the most famous Serbian inventor, mechanical, and electrical engineer is shown, Nikola Tesla. The International System of Units unit measuring magnetic field the tesla, was named in his honor. It is the author’s favorite scientist.

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Colophon

This thesis was typeset by the author using LA_TEX2_ε_{. The main body of the text} was set using a 11-points Times Roman font.

Advanced School for Computing and Imaging

This work was carried out in the ASCI graduate school. ASCI dissertation series number 213.

This project was supported by the Netherlands Organization for Scientific Re-search (NWO) as a project number 643.100.503 “Multi-Field Medical Visualiza-tion”.

Financial support for the publication of this thesis was kindly provided by the Advanced School for Computing and Imaging (ASCI), and the Technische Uni-versiteit Eindhoven.

Printed by Kiro Dandaro, Bitola, Macedonia

A catalogue record is available from the Eindhoven University of Technology Li-brary

ISBN: 978-90-386-2347-4

© 2010 Vesna Prˇckovska, Eindhoven, The Netherlands, unless stated otherwise on chapter front pages, all rights are reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechani-cal, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the copyright owner.

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High Angular Resolution Diffusion Imaging

Processing & Visualization

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de

rector magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen op woensdag 20 oktober 2010 om 16.00 uur

door

Vesna Prˇckovska

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prof.dr.ir. B.M. ter Haar Romeny Copromotor:

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I dedicate this thesis to my family, who, I believe, is proud of it much more than I. For their endless support and love.

Ja posvetuvam ovaa teza na mojata familija. Na tato i mama, koi sekogax mi davale bezrezervna poddrxka, ǉubov i sigurnost, no

istovremeno strogost i disciplina.

Na baba Gina i dedo Mile, koi gi sakam kako svoi roditeli, i qija ǉubov i toplina mi davaat sila za ponatamoxni uspesi.

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List of abbreviations

3D 3 dimensional 5D 5 dimensional 6D 6 dimensional

ADC apparent diffusion coefficient CC corpus callosum

CHARMED composite and hindered restricted model of diffusion CP cortico-pontine

CR corona radiata

CSD constrained spherical deconvolution CST cortico-spinal tract

CT computerized tomography CTR cortico-thalamic radiations DBS deep brain stimulation DFT discrete Fourier transform DOT diffusion orientation transform

DSI diffusion spectrum imaging DT diffusion tensor

DTI diffusion tensor imaging DW diffusion weighted

DW-MRI diffusion weighted magnetic resonance imaging DWI diffusion weighted images

FA fractional anisotropy

fODF fiber orientation distribution function FRT Funk-Radon transform

GA generalized anisotropy

GFA generalized fractional anisotropy HOT high-order tensors

IC internal capsule

ILF inferior longitudinal fasciculus MR magnetic resonance

MRI magnetic resonance imaging NG number of gradients

NMR nuclear magnetic resonance ODF orientation distribution function PAS-MRI persistent angular structure MRI

PDF probability density function PGSE pulsed gradient spin echo-sequence

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q-ball numerical q-ball Qball analytical QBall

SAC solid angle consideration

SAC Qball Qball with solid angle consideration SD spherical deconvolution

SDF spherical diffusion function SPF spherical probability function

SH spherical harmonics

SLF superior longitudinal fasciculus SNR signal-to-noise ratio

STN subthalamic nucleus

List of symbols

τ diffusion time d distance vector

µ number of molecules that undergo a certain displacement r0 initial position of displacement

r final position of displacement D diffusion constant

q q-space position vector D diffusion tensor D diffusion coefficient

p diffusion propagator (diffusion PDF for fixed τ ) δ gradient pulse duration time

∆ diffusion time (the separation time between two gradient pulses) G Gaussian distribution

p a position

G(p) the surface of a glyph representing the data at position p R rotation matrix

V view ray

v eye position / start point of view ray w direction of view ray

m intersection point of view ray and glyph s a position on the view ray V

n normal vector

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SUMMARY 1

Summary

Diffusion tensor imaging (DTI) is a recent magnetic resonance imaging (MRI) technique that can map the orientation architecture of neural tissues in a com-pletely non-invasive way by measuring the directional specificity (anisotropy) of the local water diffusion. However, in areas of complex fiber architecture, e.g., crossing fibers, it fails to adequately depict the local diffusion process due to the crude assumption of the underlying diffusion process as Gaussian. To overcome these limitations of DTI, high angular resolution diffusion imaging (HARDI) was introduced enabling a more accurate description of the local diffusion process. However, HARDI has many disadvantages in the acquisition, processing and vi-sualization pipeline, that prevents it, for now, from consideration for clinical re-search practice and application. In particular, for a more accurate description of the underlying diffusion process, HARDI modeling techniques require acquisi-tions that make the total measurement time 4-5 times longer than for DTI. These acquisitions result in a large amount of data, that is difficult to process and vi-sualize as they require a great deal of computational resources. Additionally, the resultant output of these models is difficult to interpret for clinicians without technical pre-knowledge.

In this thesis some of the drawbacks are addressed and solutions provided: • To find the optimal acquisition schemes for a modern 3T scanner, we

per-form series of synthetic data experiments, as well as corresponding real data MRI scans. We evaluate the angular error of the simulated crossings in the reconstructed diffusion profiles by analytical decomposition tech-niques Qball [32] and the DOT [80]. We come to a conclusion that for a modern 3T MRI scanner given considerable time constraints important for clinical practice (10-20 minutes) a b-value of 2000s/mm2and a number of around 70 gradients are sufficient to accurately recover angles of crossings about 60◦.

• To simplify the complex HARDI output and reduce the costly post-processing and visualization, we propose a classification scheme based on DTI and HARDI scalar measures. We evaluate the classification power of different measures using a statistical test of receiver operation characteristic (ROC) curves applied to an ex-vivo ground truth crossing phantom. The evaluation shows that this classification scheme allows the use of the simple diffusion tensor model where it is justified and uses HARDI only when is necessary. • We use enhancing processing techniques for DTI data that extrapolate cros-sing information of the neighbouring linear profiles. We perform experi-ments demonstrating that the quality of the extrapolated crossings is similar

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to the regularized Qball.

• We present new interactive visualization tools for HARDI data, where we fuse DTI and HARDI profiles. This visualization reduces the complexity and increases the interactivity with the HARDI data.

• At the end, an application of HARDI in neurosurgical research, we present a study of the reconstruction profiles in the area around the subthalamic nucleusthat shows the additional value that HARDI acquisitions and mo-deling can give for deep brain stimulation (DBS).

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1

Introduction

”Dimidium facti qui coepit habet.” - Horace

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Motivation

For over centuries the human brain remained one of the greatest mysteries of the human body. Luckily, the new technical developments in the late 20th century have provided the humanity with equipment and knowledge of several crucial concepts of physics that opened new opportunities for discoveries of the brain structure, connectivity and functioning. The invasive techniques for studying the brain white matter such as dissections from the 19th century [52, 26], antero-grade autoradiographic techniques [24] from the 1980s, chemical markers [98] in the late 1990s and others have now been replaced for many applications by a completely non-invasive imaging technique known as diffusion weighted ma-gnetic resonance imaging (DW-MRI) that gives a unique view to the brain white matter. This technique probes one very fascinating process that happens in all biological tissues, the process of the water diffusion. In fibrous tissues, the dif-fusion process is hindered by the boundaries of the fibers forcing the majority of water molecules to diffuse along these fibers. However, this process happens on a very small scale, few orders of magnitude smaller than the imaging scale. Therefore, DW-MRI captures the average diffusion of water molecules and gives important information about the underlying fibrous structure, i.e., the neuronal architecture. There has been numerous advents in the acquisitions and the mo-deling of the DW-MRI data. Among the most relevant for this thesis are the dif-fusion tensor imaging (DTI), established in the 1990s by Peter Basser et al. [10] and its successor high angular resolution diffusion imaging (HARDI) established a decade later by David Tuch [112]. DTI models the underlying measured dif-fusion process with a 3D Gaussian probability function. Typically a symmetric and positive definite diffusion tensor (DT) is calculated and attributed per ima-ging voxel. With eigenanalysis of the DT the principal direction of the diffusion can be extracted in order to reconstruct the underlying fiber tracts. With DTI and fiber tractography, the understanding of the several neurological and psychiatric disorders, such as schizophrenia, traumas, stroke and edemas has been increased and they have also been applied clinically to aid the presurgical planning be-fore intracranial mass resections [43, 73, 77]. However, one major limitation of DTI has been widely acknowledged, the failure to accurately model the diffusion process in areas of complex fiber heterogeneity (often referred as crossing fiber areas). Therefore, HARDI techniques pioneered by Tuch [112] were introduced, that model the complex diffusion process more accurately at a cost of more de-manding acquisitions with respect to the acquisition time and gradient strengths of the MRI scanners. An emerging research in the modeling of HARDI data followed [6, 44, 79, 108, 113], leaving open the practical questions about the re-quirements for acquisitions and post-processing of the data desirable in a clinical setting. This thesis builds upon these questions. Now that HARDI has proved

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5

its benefits over the DTI, the question whether it can be successfully applied in a clinical setting still remains open. This thesis attempts to answer exactly this question in several steps of the data-transformation pipeline: from acquisition to visualization.

Individual details of the content of each chapter and its contribution follow.

Organization of the thesis

The general pipeline of diffusion imaging, from the acquisition part, to the visua-lization and application, is roughly illustrated in the diagram of figure 1.1. The rounded rectangles on the left describe the different processes of the pipeline, and the rectangles on the right some important parameters related to them and partially addressed and answered in this thesis. Let us describe them in details here.

In chapter 2 (figure 1.1 - purple) we describe the brain white matter, from the main building elements, to the major fiber bundles connecting different parts of the gray matter. Understanding the underlying anatomy is important when applying different modeling, post-processing and visualization techniques. It is important that the recovered output matches the underlying anatomy, and there-fore in this chapter we briefly cover the basis of the white matter anatomy and connectivity.

Chapter 3 (figure 1.1 - red) covers the main principles of diffusion weighted MRI. We start by describing the free diffusion process of water molecules (in-tuitively and mathematically) and how this process is affected in fibrous tissues. Furthermore, we describe the main principles of pulse gradient spin echo (PSGE) sequence and the relation of the measured signal and the probability density func-tion of water molecules displacement we aim to reconstruct. Different modeling techniques of the DW-MRI data follow, classified by their acquisition require-ments, with special accent on the HARDI modeling techniques mostly used in this thesis: the analytical Qball [32] and the diffusion orientation transform [80]. We additionally explain the different outputs from the described modeling tech-niques. This chapter is the basis for understanding the studies and techniques developed in this thesis.

Since we do not know the “ground truth” in the human brain, one of the most difficult, yet extremely important aspects of DW-MRI is the validation. In or-der to validate the accuracy of the modeling, post-processing and visualization techniques, software and hardware phantoms are often created. These models are an over-simplified version of the fiber configurations that appear in the brain white matter. Different models are used in literature without justification which one is more appropriate than other, nor how the properties of the derived

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mo-Validation CH 5 CH 7 CH 8 CH 9 CH 6 CH 2 CH 3 CH 4 Application - b-value - # gradient - spatial resolution - acquisition time

- white matter connect. - STN

- stroke

- multiple-sclerosis - tensor glyphs (CPU, GPU) - HARDI glyphs (CPU, GPU)

- DTI - DSI

- Qball, DOT, SAC Qball - SD - - regularization - sharpening Measured tissue DW-MRI acquisitions Modeling Techniques Post-processing techniques Visualization

Figure 1.1: Overview of the main pipeline in diffusion imaging (rounded rectangles), from acquisition to visualization and application, with the main research problems (rec-tangles) marked with the thesis chapters that tackle the problems with dashed rounded rectangles in different colors per chapter.

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7

dels/features change using one model or another. In chapter 4 we address exactly these issues using the most common synthetic data models the multi-tensor mo-del [6] and S¨oderman’s [101] momo-del. Moreover, we add to this analysis hardware phantom and in-vivo data. This study is aimed to help scientists choosing the most appropriate synthetic data model when conducting DTI and HARDI expe-riments.

Tuch [112] proposed spherical sampling to decrease the high diffusion spatial resolution requirements for the dense cartesian sampling in diffusion spectrum imaging (DSI). This spherical sampling is now known as HARDI. However, se-veral question still remained unsolved. How dense should the sampling be? How high should the b-values be to have good trade-off between angular resolution of the reconstructed profiles and a good signal-to-noise ration (SNR)? How should the spatial resolution be? And, one of the most important questions, how will all these parameters affect the acquisition time which is precious in a clinical setting? These questions are addressed in chapter 5.

After answering the questions about the possibilities of application of HARDI protocol in a clinical setting, in chapter 5, several other problems that make HARDI unattractive from the post-processing side are addressed. Normally ap-plying complex HARDI models results in a complex output, that has demanding memory and processing speed requirements. However, large part of the white matter is composed of single fiber bundles [6, 13, 112] where the DTI model is justified. Given that this model has modest processing requirements, and visuali-zation that is more intuitive to interpret compared to the complex HARDI output, using the DTI model where it is justified is beneficial. Therefore, in chapter 6, we make a study of the classification power of DTI and HARDI anisotropy mea-sures, applied as a classification criteria for detecting non-Gaussian profiles in the white matter. This classification is fast and easy to apply. We validate our real data results and we come to an interesting conclusion for clinicians, where datasets acquired with lower acquisition parameter combination provide better basis for our classification compared to the ones with higher acquisition parame-ter combination.

An interesting question is addressed and answered in chapter 7. Can we over-come the limitation of the DTI model in the areas of composite fiber structure by looking at the neighborhood information? In other words, can the crossings be extrapolated from data modeled by the DT model and acquired with typical cli-nical acquisitions? In chapter 7 we give these answers and perform a comparison with the crossings recovered from data modeled with HARDI techniques.

The following chapter 8 is closely depended on the findings from chapter 6. Since simple and fast classification applied on the DTI or HARDI data is proven to be possible, merging the outputs of these techniques together is brought into practice in chapter 8. Here, we exploit the capabilities of modern GPUs to

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addi-tionally accelerate the visualization performance of the glyphs. Furthermore, by using simple diffusion ellipsoids or even fibers in the areas classified as single fi-ber, we reduce the overwhelming information that the complexity of the HARDI output introduces. We propose several color-coding schemes and enhancements on the output of the HARDI techniques, to furthermore simplify the visualiza-tion complexity of the rendered scene. Finally, we present a user evaluavisualiza-tion for several different HARDI research applications, that gives a first indication of the potentials of the proposed visualization tools.

At the end, in chapter 9, we study the reconstruction profiles of DTI and HARDI in the area around the subthalamic nucleus (STN) and show that HARDI gives an additional value for this application, thus stressing the need for further research in this direction.

Software contribution

Most of the algorithms used or presented in this thesis have been integrated in the software tool for processing and visualization of DTI and HARDI data: the DTITool. This tool has been developed in collaboration with Tim Peeters (www.timpeeters.com), Paulo Rodrigues (www.paulorodrigues.org) and other mem-bers of the BMIA group (www.bmia.bmt.tue.nl). The tool and the implemented algorithms are available on request.

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2

Human brain white matter

”If the human brain were so simple that we could understand it, we would be so simple that we couldn’t.”

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Contents

2.1 Overview . . . 10 2.2 Short history of brain connectivity research . . . 10 2.3 White matter microstructure . . . 11 2.4 Brain organization . . . 13 2.5 Conclusion . . . 18

2.1 Overview

Diffusion-weighted magnetic resonance imaging (DW-MRI) is a clinical medi-cal imaging technique that provides an unique non-invasive view on the struc-ture of brain white matter in-vivo. DW-MRI, right from its early stages, in the early 1990s, perceived immediate value for the evaluation of neuropathologies such as acute ischemic stroke. Since then, numerous advents in diffusion ima-ging technology have greatly augmented image quality unraveling new clinical applications. Moreover, the debut of diffusion tensor imaging (DTI) and fiber tractography, enabled completely new noninvasive view on the brain, and its in-formation pathways, i.e., the white matter. With DTI and fiber tractography, the understanding of several neurological and psychiatric disorders, such as schizo-phrenia, traumas, stroke and edemas, has been increased and they have also been applied clinically to aid the presurgical planning before intracranial mass resec-tions [43, 73, 77]. The main application of DW-MRI is the study of the brain white matter connectivity. Therefore, in this chapter, we will briefly summarize the main brain anatomical structures and its organization.

2.2 Short history of brain connectivity research

The nervous system with the brain as its headquarters, is the most complex and mysterious system in the human body. Even though for over a century the brain anatomy has been intensively studied, the secrets of the functioning of the brain still remain unsolved and are subject to profuse research. The study of the brain anatomy, however, is still not finished and much is to be discovered. For over hundreds of years, this process of trials and errors has been emerging and refi-ning. Finally, the current state of knowledge is an integration of a century of discoveries that often contradict each other. Notwithstanding, we are still far away from discovering the link between the underlying anatomy and functioning of the human brain.

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2.3. WHITE MATTER MICROSTRUCTURE 11

Figure 2.1: Photograph of Dejerine.

Let us take a short tour of the history of the brain anatomy research. In the early 19th century, the initial crude ideas about the organization of the white matter were formed by rough dissection [52]. In the late 19thcentury Dejerine (figure 2.1) pioneered myelin stains [26]. From simply observing these myelin stains, Dejerine managed to draw very precise images of the major fiber bundles of the human brain. Later on, in the 20thcentury, tracing techniques were propo-sed like the silver and nauta stains, that allow tracing in excipropo-sed brains. And in the 1980’s the invasive anterograde autoradiographic techniques [24] for tracing in the animal’s brains appeared. Nowadays, we can non-invasively reconstruct the wiring of the white matter of the brain by the novel imaging technique DW-MRI. However, validation techniques are crucial to determine the accuracy of the reconstructed fiber tracts.

2.3 White matter microstructure

The main building elements of the brain are the neurons (see figure 2.2). They are built out of a cell body, dendrites, an axon and terminal buttons. The cell body of the neuron contains mostly cytoplasm and a nucleus. From the body of the neu-ron the dendrites are branching out enabling contacts with dozens or thousands of neighbors. At the end of the axon, the terminal buttons form synapses with other neurons or muscle cells. The main function of the neurons is to carry electrical signals or impulses through their output fibers (axons), that can be long up to a meter. The dendrites receive signals from the previous nerve cell, and the axons pass it to the following cell. However, bare axons do not carry impulses on their

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Figure 2.2: Left: Illustration of a nerve packed with neurons. Right: Zoom in of a single neuron with myelin stain in the upper right corner. The neuron consists of cell body with nucleus, dendrites, axon and terminal buttons that connect with other neurons. Image adapted from http://www.adam.com/ and Allyn and Bacon, 2001.

own efficiently due to the leakage of their electrical charge. To avoid the leakage, the neurons are insulated in a fatty substance produced by the Schwann cells, cal-led myelin. The Schwann cells wrap themselves around the axons, forming layers like in a jelly roll. They are separated by small gaps called nodes of Ranvier, and nerve impulses move by jumping from one node to the next (figure 2.2).

If we look at a coronal slice of a brain (see figure 2.3(a)), or a T1-weighted magnetic resonance image (MRI) (see figure 2.3(b) ) we can see the differences between the white and gray matter. In terms of neurons, the nucleus and the dendrites always lie in the gray matter, whereas the white matter is composed only of the axons. It is important to stress that the axons do not compose the white matter chaotically, but on the contrary, they are very well organized in brain nerves or nerves if they leave the brain. In a nerve (see figure 2.2 left), groups of axons called fascicles are wrapped in tough but elastic tissue called perineurium. An outer layer, epineurium, encloses several fascicles together with blood vessels and fat-containing cells. In most (brain) nerves, the cell bodies of the neurons are kept separately in swellings called nuclei (if situated near the brain) or ganglia (if situated near the spinal cord) and compose the gray matter. The brain itself contains more than 100 billion neurons that sort and sift incoming information and control the body. This thesis mainly tackles questions on white matter connectivity of the human brain. Therefore, we will proceed to explain the human brain structure and the organization of the white matter within.

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2.4. BRAIN ORGANIZATION 13

Figure 2.3: Coronal slices of the brain (a) Taken from human specimen from the brain museum (www.brainmuseum.org) (b) T1 image of a coronal slice generated by EduVIS tool developed by the BMIA group (www.bmia.bmt.tue.nl).

2.4 Brain organization

If we look under the microscope at the white matter from a slice of the brain stai-ned by Nissl (see figure 2.4), we are struck by the fact that the white matter is not homogenous. Compartments with different cellular distribution that corresponds to different fiber bundles in the brain white matter can be hardly distinguished from each other (figure 2.4).

The white matter connects different gray matter structures and transmits infor-mation between them in a form of nerve impulses. There are many important gray matter structures. One of the most important structure is the cerebral cortex, which is in charge for the cognitive functions. Other important structures are the striatumwhich is a subcortical part of the forebrain. It contains structures like the putamen, nucleus caudatus, accumbens and others.

The striatum which is a subcortical part of the forebrain is best known for its role in the planning and modulation of movement pathways but is also invol-ved in a variety of other cognitive processes involving executive functions. The thalamusis another important gray matter structure that takes charge of relaying sensation, special sense and motor signals to the cerebral cortex, along with the regulation of consciousness, sleep and alertness. The subthalamic nucleus (STN) (which is a deep brain structure) is a small lens-shaped nucleus in the brain. It is a part of the basal ganglia system involved in three circuits with different functions, namely motor, cognitive, and emotional (see figure 2.5).

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ori-Figure 2.4: Nissl stain of the white matter slice immediately under the cortex. We can spot that there is no visible distinction between the U-fibers and the inferior longitudinal fasciculus(ILF). Image adapted from Schmahmann and Pandya [95].

Figure 2.5: Coronal view of a brain slice with markings of several gray matter struc-tures. Image provided by Ulrike von Rango, Department of Anatomy and Embryology, Maastricht University.

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2.4. BRAIN ORGANIZATION 15

Figure 2.6: Association fibers. 1. Short arcuate bundles (U-fibres) 2. Superior lon-gitudinal fasciculus(SLF) 3. External capsule 4. Inferior occipitofrontal fasciculus 5. Uncinate fasciculus6. Sagittal stratum 7. Inferior longitudinal fasciculus. Images adap-ted from Williams et al. [123].

ginating from it. The fibers in the white matter can be divided in five different categories:

1. Association fibers 2. Commissural fibers

3. Projection fibers to thalamus 4. Projection fibers to brainstem 5. Striatal fibers

The first two types of fibers are bidirectional, whereas the projection and stria-tal fibers are unidirectional.

Association fibers connect different cortical areas within a hemisphere of the brain. They can be short or U-fibers that connect the cortical region to the same or adjacent gyrus (which is a ridge on the cerebral cortex). There are also neigh-borhood association fibers that are slightly longer and connect cortical regions to a nearby gyrus inside the same lobe. Finally, there are long association fibers that connect distant cortical regions in a different lobe. Among the most important association fibers are the cingulum, the superior longitudinal fasciculus (SLF), inferior longitudinal fasciculus(ILF) and others (see figure 2.6).

Commissural fibers connect regions of different hemispheres in the brain. The main commissural fiber bundle is the one of the corpus callosum (CC). But also very important are the anterior commissural and the hippocampal commissural fiber bundles (see figure 2.7).

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Figure 2.7: Commissural fibers. 1. Frontal forceps 2. Corpus callosum 3. Short arcuate fibres4. Occipital forceps 5. Indusium griseum 6. Medial longitudinal stria 7. Lateral longitudinal stria. Images adapted from Williams et al. [123].

Projection fibers connect the cortex with the spinal cord, the cerebellum and subcortical structures like the thalamus and the basal ganglia. They start as a large fiber bundle at the pyramidal tract from the cortex and go through the corona radiata(CR), the internal capsule (IC), the cerebral peduncule, and reach the cerebellum. Major bundles composing the projection fibers are: the cortico-thalamic radiations (CTR), connecting the thalamus with the cortex and the cortico-pontine fibres (CP), connecting the pons with the cortex including the cortico-spinal tract(CST) (see figure 2.8).

Striatal fibers are subdivided in two bundles. The Muratoff bundle that connects the cortex to the caudate, a tail-shaped basal ganglion located in a lateral ven-tricle of the brain. The external capsule is the other bundle that connects to the putamenand the accumbens.

If we look at an individual gyrus (see figure 2.9), we notice that in the center the nerve fibers (i.e., cord fibers) are the most densely packed. These fibers then spread into commissural fibers, thalamic fibers and pontine fibers. Below the cortical ribbon there are short U-fibers, and in between them and the cord fibers lie the neighborhood association fibers and striatal fibers. The long association fibers on the other hand, are more diffusively distributed inside the gyrus and travel perpendicular to the cord fibers.

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inter-2.4. BRAIN ORGANIZATION 17

Figure 2.8: Projection fibres (mid-saggital view). 1. Corona radiata 2. Anterior thalamic radiation3. Internal capsule 4. Anterior commissure 5. Optic tract 6. Cerebral peduncle 7. Longitudinal pontine fibres (cortico-spinal and cortico-nuclear tracts) 8. Pyramidal tract of medulla oblongata9. Hilus of olivary nucleus 10. Olivary nucleus. Images adapted from Williams et al. [123].

Figure 2.9: Scheme of individual gyrus. Image adapted from Schmahmann and Pan-dya [95].

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Figure 2.10: Coronal slice of the cerebral hemispheres. The red rectangle marks the centrum semiovalewhere fibers of the corpus callosum, corona radiata, and superior longitudinal fasciculusform a three-fold crossing. Image provided by Ulrike von Rango, Department of Anatomy and Embryology, Maastricht University.

mixed with each other. Therefore on a certain scale different fiber configurations are to be expected. Sometimes some fiber bundles diverge, like in the case of CST diverging through CR to the cortex. Or cross, like the CC and CR in the centrum semiovale, and depending on the region or partial volume effects sometimes fi-bers appear to be kissing. All these are areas with composite fiber heterogeneity and knowing the underlying anatomy is very important for the algorithms and visualizations developed in this thesis.

Therefore, an important region for illustrating the benefits of the processing and visualization techniques developed in this thesis is the centrum semiovale (see figure 2.10). There, fibers of the corpus callosum, corona radiata, and su-perior longitudinal fasciculusform a three-fold crossing.

2.5 Conclusion

In this chapter, we made a short overview of the white matter architecture from microscopic scale (scale of axons) until macroscopic scale of different brain structures and connectivity. Knowledge of the structure and the properties of the measured underlying (neural) tissue is important when applying algorithms for local reconstruction or techniques for recovering the global connectivity of the human brain.

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3

Principles of diffusion weighted magnetic

resonance imaging. From measurements

to fibers.

”If there is magic on this planet, it is contained in water.” - Loren Eiseley

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Contents

3.1 Overview . . . 20 3.2 Short history of magnetic resonance imaging (MRI) . . . 20 3.3 Diffusion: the physics and representation . . . 21 3.3.1 Physics of water diffusion . . . 22 3.4 Probing the water diffusion via MRI . . . 26 3.5 Understanding the derived structural information in relation

to the measured tissue . . . 32 3.6 Different diffusion modeling techniques . . . 36 3.6.1 Techniques that require low q-space sampling . . . 37 3.6.2 Techniques that require dense q-space sampling -

diffu-sion spectrum imaging (DSI) . . . 41 3.6.3 Techniques that require modest q-space sampling - high

angular resolution diffusion imaging (HARDI) . . . 42 3.7 Conclusion . . . 50

3.1 Overview

In the previous chapter, we gave a short introduction of the anatomy of the brain white matter. In this chapter, we will describe the basic principles of imaging, as well as the techniques that transform the measured DW-MRI signal into a mea-ningful functions that describe the local diffusion pattern. These functions can have different representations and different relations to the properties of the mea-sured tissue. Moreover, most of them correspond to the fiber directions that we aim to reconstruct. Finally, we will explain different modeling and reconstruction techniques for DW-MRI data.

3.2 Short history of magnetic resonance imaging (MRI)

In the 1973 the first MRI image was published. This novel technique, develo-ped by the nobel prize winners Mansfield and Lauterbur, became widely used for neurological imaging due to the ability to provide detailed images, with grea-ter contrast between the different soft tissue parts of the body than compugrea-terized tomography (CT), and non invasive due to the lack of radiation exposure. The main principle of MRI is to probe the diffusion of water molecules in a comple-tely non-invasive way, thus enabling detailed visualization of internal structures of the body.

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3.3. DIFFUSION:THE PHYSICS AND REPRESENTATION 21

In the past decades many improvements and applications of the magnetic reso-nance imaging appeared. One of them is the DW-MRI, where the acquisition is made sensitive to the microscopic movement of the water molecules (diffusion) restricted by the local structure. With some of the modeling techniques for the DW-MRI data, evaluation of the structural integrity of fibrous tissues, such as the white matter in the brain is made possible. One of these techniques, the diffusion tensor imaging (DTI), was established as a viable imaging method in the 1990s by Peter Basser et al. [10]. However, the DTI model was found to have limitations in recovery of complex fibrous structures. To overcome some of the limitations of this rather simple model, a decade later, more demanding acquisitions coupled with more complex models known as high angular resolution diffusion imaging were introduced by David Tuch [112].

In this short paragraph we introduced several concepts crucial for the unders-tanding of this thesis. Therefore, we will now proceed with an in depth expla-nation of the main principles of DW-MRI imaging, the resulting complex output and its interpretations, as well as the relations to the underlying tissue that is measured.

3.3 Diffusion: the physics and representation

One very fascinating fact about the human body is that it is vastly composed of water. About 60% of the body weight of an average human belongs to water1. Water molecules agitated by the thermal energy diffuse inside the body hindered by the boundaries of the surrounding tissues and different structures. If somehow, this movement can be probed, one would be able to reconstruct the boundaries that hinder this motion. Hence, we intuitively deduct that for the purpose of the reconstruction of these boundaries (that can be fiber bundles in the brain, muscle fibers in the heart and other) we need to understand three key concepts:

1. Physics of the water diffusion; 2. Probing the water diffusion via MRI;

3. Understanding the derived structural information in relation to the measu-red tissue.

Let us now describe, in details, all these concepts.

1

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Figure 3.1: Illustration of the diffusion of water molecules inside a glass of water. a) free b) restricted diffusion.

3.3.1 Physics of water diffusion

Molecular diffusion, or diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of the particles. Let us first start explaining the concept of free diffusion and its mathematical modeling. To illustrate this concept, let us for a moment think of a glass full of water. In this scenario (see figure 3.1(a)) the molecules will be randomly displaced within the boundaries of the glass. Given a short time interval τ , the probability that the molecules undergo short distances is higher than the one for longer distances. If we want to mathematically describe this process, we can use the concept of a probability displacement distribution function that describes the proportion of molecules that undergo a displacement in a certain direction for a certain distance.

For instance, if all the molecules (M ) start the displacement at time t = 0, after a given time t = τ (which would be the diffusion time in our case) they would randomly be displaced and the number of molecules per displacement can be measured.

We can now introduce the displacement vector d, and count the number of molecules µ that randomly moved for this distance = d = |d| (see figure 3.1(a)). Then, we can describe the displacement distribution as a histogram of the relative

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3.3. DIFFUSION:THE PHYSICS AND REPRESENTATION 23

Figure 3.2: Histogram of a probability displacement distribution of molecules due to anisotropic diffusion in a one-dimensional model.

number of molecules (_Mµ) versus the displacement distance d (see figure 3.2). p(d, τ ) = p(r|r0, τ ) = µ M 3.1 where d = r − r0and r0and r are the starting and ending positions respectively. In the case of a free diffusion of water molecules inside a glass, the resultant histogram would be a Gaussian function (as in figure 3.2) according to the central limit theorem for independent and identically distributed variables. However, the histogram as in figure 3.2 is adequate for display of one-dimensional data but not well suited for representation of a three-dimensional displacement of the water molecules. Color coding the probability in 3D would be a better approach. Hence, if we assign a color that represents the values of the probability of a water molecule displacement at each position defined by the vector d, we get a better representation of the underlying diffusion process. In figure 3.3(a) this type of representation for the case of free diffusion is illustrated, where blue color stands for low probability values and red for high. As we mentioned previously, close to the origin the probability is higher, and most of the molecules will finish their displacement after the diffusion time τ in that area (in figure 3.3(a) the area defined with red color). We also notice that the underlying probability for a free diffusion of the water molecules does not present a preferred direction. In other words, the underlying probability function is an isotropic Gaussian.

The diffusion proces can be described in mathematical terms, and that is what Einstein did in 1956 [39]. He related the diffusion coefficient D that characterizes the viscosity of the medium (i.e., the mobility of the molecules), to the mean square displacement:

< dTd >= 6 · D · τ 3.2 where <> denotes an ensemble average. The diffusion coefficient for water at temperature of 37◦is approximately D ≈ 3 · 10−9m2/sec.

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Figure 3.3: Color coded probability of a 3D diffusion displacement of water molecules. a) Isotropic case where the diffusion does not have a preferred direction b) Anisotropic case where the diffusion has one preferred direction c) More complex diffusion pattern with two dominant diffusion directions. The red color states high probability values whereas the blue low. Adapted from Hangmann et al. [51].

Figure 3.4: Electron micrographs of a) axons perpendicular to fiber tracts b) axons along fiber tracts. Images adapted from Beaulieu et al. [12].

Now let us describe how this diffusion process relates to the restricted diffusion of water molecules within the brain white matter. In a typical MR acquisition, with diffusion time of about 40ms, the displacement that we are measuring is of the order of d ≈ 10µm. On the same scale, 1 − 10µm, are the diameters of individual axons and therefore the diffusion allows us to probe the underlying microstructure of the measured tissue (see figure 3.4). However, we still need to make assumptions about the sources of diffusion anisotropy in the brain white matter.

The simplest model that can be proposed is that the axons are densely packed, coherently oriented, impermeable, infinitely long cylinders as in figure 3.5(a). Here, the diffusion is obviously no longer free, but restricted by the boundaries of the cylinders. Therefore, the displacement is restricted by the barriers. In other words, this scenario is very similar to the previous example of the glass, but now with straws in it that restrict the diffusion of the water molecules (see figure 3.1(b)). If the diffusion time τ is long enough such that the molecules start

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3.3. DIFFUSION:THE PHYSICS AND REPRESENTATION 25 D_along D_normal Dintra fintra Dextra fextra Dinter finter a) b)

Figure 3.5: a) Illustration of approximation of the axons within a nerve as cylinders. b) Zoomed in view of different diffusion in different compartments including exchange diffusion between compartments.

colliding with the surrounding barriers, then the diffusion process is no longer free and the diffusion coefficient is a function of the time (see equation 3.3). This diffusion coefficient D(t), is known as apparent diffusion coefficient (ADC) (equation 3.3) and we will come back to it later.

< dTd >= 6 · D(t) · t 3.3 From chapter 2, we know that the axons are not simple cylinders, but on contrary they have a lot of structure like axonal membrane and myelin sheath. They also have structure inside, such as neurofilaments, microtubules and others. The diffusion can take place inside and outside the axons, in multiple compart-ments, each with potentially different diffusion coefficient and different volume fraction f [12] (illustrated in figure 3.5(b)). The scenario can be even more com-plex if we include exchange between the intra and extra compartments. This exchange can be assumed to be slow, where the molecules remain in the com-partments where they originally started to diffuse. The other possibility is fast exchange, where the molecules can pass through the barrier between compart-ments multiple times within the measured diffusion time. In this case the the resulting diffusion is not simply addition of both intra and extra diffusion from both compartments.

To summarize, the diffusion that is happening in the brain white matter is in-fluenced by several tissue parameters among which are: the axonal diameter and its packing density (the difference in volume fractions fintraversus fextra). Fur-thermore it is influenced by the presence and thickness of myelin sheaths as well as the permeability of the membrane, the temperature, presence of other cell types (e.g.,the glial cells) and many others. Nevertheless, the primary source of ani-sotropy comes from the restricted diffusion due to axonal membranes and can

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Figure 3.6: Illustration of areas of interference between fiber bundles.

be modulated by the presence or absence of myelin for about 20% [12]. If the axons are coherently oriented in one direction, then the probability distribution of this restricted diffusion of water molecules would be similar to the image on figure 3.3(b) which means it is not isotropic anymore and without preferred di-rection as in the unrestricted diffusion case (figure 3.3(a)), but rather cigar shaped with one prominent diffusion direction.

However, strictly coherently organized fibers in a single direction do not ap-pear everywhere in the brain. Areas where fiber bundles interfere with each other (figure 3.6) are also very frequent (1/3-2/3 of the white matter [6, 112, 13]) as we mentioned in chapter 2. In cases like this, the resultant anisotropy comes from different contributions of anisotropy originating from the different fiber bundles. Hence, the distribution of water molecules displacement can get even more com-plex (e.g.,figure 3.3(c)).

Now that we understand the concept of diffusion in the brain white matter, let us move to the main principles of diffusion magnetic resonance imaging.

3.4 Probing the water diffusion via MRI

In 1950, Hahn [53] discovered that there is a signal decay in the presence of a heterogenous magnetic field due to motion of spins. Even though he sugges-ted that there should be a way to measure the diffusion coefficient of a solution containing spin-labeled molecules, he did not propose a direct method for doing so. It was later in 1956, that Torrey [105] proposed the actual equations that describe the magnetization of spins in an MR spectroscopy experiment. Howe-ver, the most significant event in nuclear magnetic resonance (NMR) happened in 1965, when Stejskal and Tanner [102] established the first pulsed gradient spin echo-sequence (PGSE) with two short duration gradient pulses (see figure 3.7(a)). The main principle behind this sequence is to influence the phase of the spins by adequately applying magnetic fields with varying strengths.

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3.4. PROBING THE WATER DIFFUSION VIAMRI 27

pulse as illustrated in figure 3.7(a). The gradient pulses should be short enough allowing a clear distinction between the encoding time (the gradient pulse du-ration time δ) and the diffusion time (the sepadu-ration time between two pulses ∆). However, in practice the gradient pulse duration time δ is never negligible compared to the separation time between two pulses ∆. The direction of applied gradients governs the direction of measured diffusion. If the diffusion gradient is applied along the x-axis, it will cause different changes in the phase of the spins depending on their location on the x-axis. The following 180◦ pulse inverts the phase of the spins. Therefore the last gradient pulse gives the opposite effect on the phase, compared to the first gradient pulse (figure 3.7(c)). The key principle follows: in the absence of motion or diffusion, the dephasing before the 180◦ pulse is exactly reversed after the second pulse because the spins experience the same magnetic field throughout. If instead, the spins have experienced a drift (fi-gure 3.7(b)), such as diffusive motion between the two diffusion gradients, there will only be a partial refocusing of the spins (figure 3.7(d)), and therefore the signal amplitude will be decreased. The amount of diffusion attenuation of the signal depends on several factors: the diffusion properties of the sample deter-mining the motion of the spins, the pulse sequence parameters deterdeter-mining the duration and amplitude of diffusion gradients and the time during which the dif-fusive motion takes place. If the displacement distance is longer, the phase shift is higher and therefore the signal is more decayed. This addresses one very im-portant issue when correlating the measured signal to the diffusion proces. If the diffusion is higher in the direction of the applied gradient, the measured signal would be low.

At this point we can introduce the normalized measured MR signal as E(q), where q is a 3D vector q = γδG/2π, whose orientation is in the direction of the diffusion we want to measure and length proportional to the gradient strength G and γ is the gyromagnetic ratio for water protons. Stejskal and Tanner [102] proved that if the duration of the applied gradient pulses is short enough (i.e., narrow pulse approximation), such that the diffusion of the water molecules is negligible during that time, the signal attenuation is related to the probability density function of water molecules displacement (i.e., the averaged diffusion propagator) p by the Fourier integral as in equation 3.4:

E(q, τ ) = S(q, τ ) S0 = Z R3 p(r|r0, τ ) exp(−2πiqTd)dr = F[p(r|r0, τ )]. 3.4

In equation 3.4, S(q, τ ) is the signal in the presence of the gradient field and S0 is the baseline or un-weighted signal, acquired in absence of diffusion gradients, p(r|r0, τ ) is the probability of finding a particle, initially at position r0, at r after

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time τ and d = r − r0is the displacement vector. Sometimes, instead of q, a unit vector g = q/|q| is proposed, and it is common to introduce another scanner parameter b = τ |q|2 = (∆ − δ/3)|q|2. Throughout this thesis, the terms of a probability density function (PDF) of water molecules displacement and the diffusion propagator (commonly used term in DW-MRI community [112, 34]) will be used interchangeably with the same meaning.

By applying gradients in different directions and with different amplitudes at specific moments, the measured signal is frequency and phase encoded. These signal values are stored in so called “k-space” (see figure 3.8). To reconstruct a position-encoded visual image a 2D inverse Fourier transform should be applied. In diffusion weighted imaging for each applied gradient a new position-encoded visual image DWIiis created. That means if we apply n gradients we get n DWI each of them (depending on the spatial resolution) with m small cubic volumes called voxels. Now for each voxel we have a new 3D space defined by the q vec-tor called “q-space”. In other words, for every voxel position yj(see figure 3.8), we get n signal values that fill in the q-space. This q-space data is subjected to a 3D Fourier transform in each brain position in order to reconstruct the diffusion propagator (equation 3.4). By applying more diffusion gradients, we can sample the q-space at a finer scale and in a wider region resulting with a more accurate reconstruction of the diffusion propagator.

We can move in q-space with the q vector that is proportional to the diffusion time ∆ and the gradient strength G. However, we normally fix the diffusion time (typically 40ms) and move through q-space only with the diffusion gradients, since we want to measure certain PDF for that specific diffusion time. This can be confusing when one thinks in terms of b-values. The main question here is whether the diffusion process is stationary or not. When the diffusion process is stationary, the PDF is diffusion-time independent; if the diffusion process is non-stationary, the PDF is diffusion-time dependent. By changing the diffusion time, you are able to determine the time dependency of the diffusion process. By changing the strength of the q-vector, either by increasing the diffusion time or increasing the gradient amplitude, you are improving the angular resolution of the reconstructed PDF. In other words, we can achieve the same b-value by either increasing the ∆ in one case, or increasing the G in other, but the reconstructed PDFs will be for different diffusion times. The longer the diffusion time, the lar-ger the displacement that the molecules undergo, thus the sharper the diffusion profiles. However, longer measurement times in general produce signals with low signal-to-noise ratio (SNR) (see figure 3.9(a) and 3.9(b)). Stronger gradients or more densely filled q-space will not only produce signals with high SNR but bet-ter capture more complicated diffusion patbet-terns and crossings under small angles (figure 3.9(c) and 3.9(d)).

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3.4. PROBING THE WATER DIFFUSION VIAMRI 29 G 90º ₁₈₀º δ δ Δ [t] B x B x y’ x’ Φ₁ y’ x’ Φ₂

+

y’ x’ Φ

=

a)

b)

c)

d)

Figure 3.7: a) PGSE sequence with two gradient pulses with short duration; b) Drift of the molecules during the time between the two gradient pulses; c) Opposite effect on the phase of the spin due to the change of the magnetic field; c) Illustration of the resulting phase shift in case of molecular displacement within the diffusion time.

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Figure 3.8: Illustration of the whole transformation process from the measurements that fill in the k-space, via inverse Fourier transform reconstruct the diffusion weighted images (DWI) and finally through 3D Fourier transform recovers the diffusion propaga-tor within each imaging voxel. n is the number of diffusion weighted volumes (which depends on the number of diffusion weighted gradients) and m the number of voxels per volume. Therefore there will be m q-spaces per location in the brain, filled with n points. yjis the jthvoxel. Central part of the image borrowed from Hangman et al [51].

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3.4. PROBING THE WATER DIFFUSION VIAMRI 31

Δ = short

G = const Δ = longG = const

a) b) Δ = const G = weak c) Δ = const G = strong d)

Figure 3.9: The effect of changing the diffusion time. In a) and b) we keep the gradient strength constant and change the b-value by changing the diffusion time. In c) and d) we keep the diffusion time constant and change the b-value by changing the gradient strength.

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a) b c) d)

Figure 3.10: Different fiber configurations. a) single fiber b) crossing fibers c) kissing fibers and d) splaying fibers.

3.5 Understanding the derived structural information in

relation to the measured tissue

The fundamental aim of DW-MRI is to recover the fiber orientations of the mea-sured tissue (e.g., the fiber bundles in the brain white matter). We learned so far, that via MRI measurement the water diffusion proces is probed. In terms of brain white matter, the diffusion process happens locally, on a very small scale of the axon diameters. However, the measurement scale is one order of magnitude lar-ger, and we are dealing with measuring averaged diffusion. On this scale several scenarios can happen, with different types of interference between the densely packed axons (as we discussed previously). Fibers might be oriented in a single directions, but they might as well cross, kiss or splay (see figure 3.10). These scenarios can be expected in different parts of the brain white matter, given that the brain is a heterogenous medium with highly compartmentalized cytoarchitec-ture in different locations. That gives the intuition that the brain area must be discretized in small cubic volumes i.e., voxels, and the diffusion must be mea-sured locally within these volumes. Since the diffusion can be represented by a probability density function as in figure 3.3, we can attach a function like that to each voxel location. The resulting image is a complex image that in every 3D location has a 2D function that can be a function on a sphere ψ(θ, φ) illustrated in figure 3.11 or a 3D volume of a probability density function. The 2D function is normally derived from the 3D PDF by integration in each direction and we will explain this in details later. In other words, we are dealing with a 5D or 6D image that characterizes the diffusion in the brain white matter. We could look at this representation as a 3D image that in each location has additional 3D (or 2D) image that encodes the molecular displacement. This is the basic concept of data representation in DW-MRI.

The visual information contained in this 5D or 6D image is overwhelming and confusing. Also the memory requirements for managing and displaying this data

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3.5. UNDERSTANDING THE DERIVED STRUCTURAL INFORMATION IN RELATION TO THE

MEASURED TISSUE 33

Figure 3.11: Illustration of the resultant 5D image in DW-MRI. Every 3D position in the brain yj (i.e., imaging voxel) is associated with a 2D orientation distribution function (ODF) that represents the diffusion pattern in the measured voxel, giving a complex 5D image. Color-coding for the ODF is done by mapping the [x,y,z] coordinates of each diffusion direction by the RGB coloring scheme.

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Figure 3.12: a) 2D illustration of the diffusion PDF as a continuous function b) examples of different extracted isosurface from the PDF.

are far from modest. One, hence wonders, whether a detailed diffusion profile adds an extra value when the goal is, in essence, determining the most prominent diffusion orientations. Therefore in literature, more simplified representations than the full PDF are common. An isosurface of the PDF is one option for sim-plifying the data (see the 2D correspondence of an isocontour in figure 3.12). However, this would require calculating the full PDF which has drawbacks on its own and will be addressed later. A very popular approach, and less sensitive to noise is computing the orientation distribution function (ODF). The ODF is initially a sphere, deformed in each direction proportionally to the integration of all the values of the PDF in that direction. If we come back to figure 3.11, the concept of 5D image gets more clear i.e., for each 3D position we attach a 2D function on a sphere.

To make it more clear, lets illustrate this on a 2D example. Let us consider the diffusion propagator of water molecules to be hindered by two crossing fibers in 2D, as presented in figure 3.13(a). Here a 2D PDF is illustrated sampled at different radii ri. The color coding stands for the probability values on a specific position on the PDF p(ri|0, τ ). For instance, all red points on the shells of the PDF represent very high probability, that a particle initially at the origin, ends up at that position. On figure 3.13(b) we illustrate this information in 3D where the heights of the PDF positions correspond with the PDF values. Again, we observe that the red points on the PDF shells have the highest probability. However these two representations in 2D (note that this gets more complex in 3D) are not very intuitive when one needs to visually grasp the most probable directions of water displacement, i.e., the underlying fiber directions. Additionally one often wants to determine and show the maxima of the PDF and this kind of representation is not very suitable for that neither.

To be able to make the orientation clearly visible without the need of an extra dimension a shell at radii ri of the PDF is taken and deformed according to the

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3.5. UNDERSTANDING THE DERIVED STRUCTURAL INFORMATION IN RELATION TO THE MEASURED TISSUE 35 a) c) d) b) d d’ min max

One shell of the PDF

Figure 3.13: a) 2D illustration of the diffusion PDF. Grey dashed lines depict underlying fiber bundle directions b) the PDF where the z-axis is used to illustrate the probability values c) the common way of visualizing the diffusion PDF d) and the ODF derived from the PDF.

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reconstruction time (min) output

DWI 1-3 gray-scale image ADC, Trace 2-4 gray-scale image

ADC modeling q-ball, Qball

DOT SD PAS-MRI

DSI 15-60 3D PDF

diffusion tensor

10-30 2D functions on a sphere 3D PDF

Modest q-space sampling

b=1000-3000s/mm2,

NG =60-120

DTI 3-6

Low q-space sampling

b=700 - 1000s/mm2,

NG =1-20

Dense q-space sampling b>8000s/mm2, NG >200

Table 3.1: Taxonomy of different DWI modeling techniques. NG stands for number of gradients. The averaged measured time is estimated assuming that 30 axial sections are acquired, each with a thickness of 3 mm.

probability values in that point. Now the colors in figure 3.13(a) or the heights in figure 3.13(b) would represent the same information as the deformations in each point of this new reconstructed PDF illustrated in figure 3.13(c). This is a common way of visualizing the local diffusion characteristics in every imaging voxel. Even though the full diffusion PDF provides important information about the structure of the material at microscopic scale, for applications like fiber tra-cking only the orientation is important. Therefore, an ODF is commonly derived. In figure 3.13(d) we show the ODF derived from the PDF from figure 3.13(a) and deformed in the same way as we explained for figure 3.13(c).

A common standard for coloring is mapping the [x,y,z] coordinates of each diffusion direction by the RGB coloring scheme.

3.6 Different diffusion modeling techniques

Over the past decades there has been growing research in the area of modeling the diffusion of water molecules by transforming the measured signal in order to deduct important structural information about the underlying measured tissue. Many of the modeling techniques have a set of specific acquisitions and therefore we can make an initial taxonomy by their requirements for q-space sampling. On table 3.1 several common modeling techniques are listed, classified by the q-space requirements, and followed by a brief description of their average mea-surement time and their main type of output.