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Mathematical Modeling for Multi-Fiber Reconstruction from Diffusion-Weighted Magnetic Resonance Images

Permanent Link: http://ufdc.ufl.edu/UFE0021691/00001

Material Information

Title: Mathematical Modeling for Multi-Fiber Reconstruction from Diffusion-Weighted Magnetic Resonance Images
Physical Description: 1 online resource (118 p.)
Language: english
Creator: Jian, Bing
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: deconvolution, diffusion, dtmri, dwmri, hardi, mow, nnls
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Diffusion-weighted magnetic resonance imaging (DW-MRI) is a non-invasive imaging technique that allows neural tissue architecture to be probed at a microscopic scale in vivo. By measuring quantitative data sensitive to the water molecular diffusion, DW-MRI provides valuable information for neuronal connectivity inference and brain developmental studies. The broad aim of this dissertation is to develop mathematical models and computational tools for quantifying and extracting information contained in diffusion MR images. One of the fundamental problems in DW-MRI analysis is the mathematical modeling of the MR signal attenuation in a voxel in the presence of multiple fiber bundles. In this dissertation, we present a novel mathematical model and accompanying efficient algorithms for this problem. Our model uses a continuous probability distribution over the space of symmetric positive definite matrices and is general enough to model water molecular diffusion in a variety of situations involving complex tissue geometry including single and multiple fiber bundle occurrences. We show that the diffusion MR signals and the probability distributions for positive definite matrix-valued random variables are related by the Laplace transform defined on the space of symmetric positive definite (SPD) matrices. Another interesting observation is that when the mixing distribution is parameterized by Wishart distributions, the resulting close form of Laplace transform leads to a Rigaut-type fractal expression. This Rigaut-type function exhibits the expected asymptotic power-law behavior and has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until the development of the proposed model. Furthermore, both the traditional diffusion tensor model and the multi-tensor model can be interpreted as special cases of this continuous mixture of tensors model. In tackling the challenging problem of multi-fiber reconstruction from the diffusion MR images, we further develop the mixture of Wisharts (MOW) model, as a natural parametrization of the desired tensor distribution function, to describe complex tissue structure involving multiple fiber populations. The multi-fiber reconstruction using the proposed MOW model essentially leads to an inverse problem. Computational methods for solving this inverse problem are investigated under a unified deconvolution framework which also includes several existing model-based approaches. Finally, the theoretical framework we have developed for modeling and reconstruction of diffusion weighted MRI has been tested on simulated data and real rat brain data sets. The comparisons with several competing methods empirically suggest that the proposed model combined with a non-negative least squares deconvolution method yields efficient and accurate solution for the multi-fiber reconstruction problem in the presence of intra voxel orientational heterogeneity.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Bing Jian.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Vemuri, Baba C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0021691:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021691/00001

Material Information

Title: Mathematical Modeling for Multi-Fiber Reconstruction from Diffusion-Weighted Magnetic Resonance Images
Physical Description: 1 online resource (118 p.)
Language: english
Creator: Jian, Bing
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: deconvolution, diffusion, dtmri, dwmri, hardi, mow, nnls
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Diffusion-weighted magnetic resonance imaging (DW-MRI) is a non-invasive imaging technique that allows neural tissue architecture to be probed at a microscopic scale in vivo. By measuring quantitative data sensitive to the water molecular diffusion, DW-MRI provides valuable information for neuronal connectivity inference and brain developmental studies. The broad aim of this dissertation is to develop mathematical models and computational tools for quantifying and extracting information contained in diffusion MR images. One of the fundamental problems in DW-MRI analysis is the mathematical modeling of the MR signal attenuation in a voxel in the presence of multiple fiber bundles. In this dissertation, we present a novel mathematical model and accompanying efficient algorithms for this problem. Our model uses a continuous probability distribution over the space of symmetric positive definite matrices and is general enough to model water molecular diffusion in a variety of situations involving complex tissue geometry including single and multiple fiber bundle occurrences. We show that the diffusion MR signals and the probability distributions for positive definite matrix-valued random variables are related by the Laplace transform defined on the space of symmetric positive definite (SPD) matrices. Another interesting observation is that when the mixing distribution is parameterized by Wishart distributions, the resulting close form of Laplace transform leads to a Rigaut-type fractal expression. This Rigaut-type function exhibits the expected asymptotic power-law behavior and has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until the development of the proposed model. Furthermore, both the traditional diffusion tensor model and the multi-tensor model can be interpreted as special cases of this continuous mixture of tensors model. In tackling the challenging problem of multi-fiber reconstruction from the diffusion MR images, we further develop the mixture of Wisharts (MOW) model, as a natural parametrization of the desired tensor distribution function, to describe complex tissue structure involving multiple fiber populations. The multi-fiber reconstruction using the proposed MOW model essentially leads to an inverse problem. Computational methods for solving this inverse problem are investigated under a unified deconvolution framework which also includes several existing model-based approaches. Finally, the theoretical framework we have developed for modeling and reconstruction of diffusion weighted MRI has been tested on simulated data and real rat brain data sets. The comparisons with several competing methods empirically suggest that the proposed model combined with a non-negative least squares deconvolution method yields efficient and accurate solution for the multi-fiber reconstruction problem in the presence of intra voxel orientational heterogeneity.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Bing Jian.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Vemuri, Baba C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0021691:00001


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3bf2859fc09076f11138abf7d726f1c8
d4297af2954d1b0a8e8313c970a59b39a44ca085







MATHEMATICAL MODELING FOR MULTI-FIBER RECONSTRUCTION FROM
DIFFUSION-WEIGHTED MAGNETIC RESONANCE IMAGES


















By
BING JIAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2009

































S2009 Bing Jian



































To my parents and Geri










ACKNOWLEDGMENTS

This dissertation would not have been possible without the help and support front

many people. First and foremost, I would like to express my ininense gratitude to my

advisor, Dr. Baha Venturi, who introduced me to the field of medical image analysis

and assisted me in each step throughout my PhD study. The research described in this

dissertation was mostly horn front the lively meetings and intense discussions with Dr.

Venturi. Without his patient guidance and strongest support, I could never have reached

this important point in my life. I feel so fortunate to have him as a good friend and kind

mentor.

I would also like to thank my coninittee nienters, Dr. Sartaj Sahni, Dr. Anand

Rangaredi .Il Dr. Arunava Banerjee, and Dr. Siriphong L li.--111,.'.1-l I'!;I1.1 for being

very generous with their time and knowledge and ahr-l-:- heing supportive during the

completion of my PhD study. The five years I have spent in the University of Florida

is definitely one of the most nienorable experiences in my life. I am very grateful to all

the professors who have taught me many interesting and rewarding courses in computer

science, niathentatics and statistics. Especially, I would like to thank Dr. Baha Venturi,

Dr. Siriphong L li.--111,.'.1-l I'!;I1.1 Dr. .Jay Gopalakrishnan, Dr. Arunava Banerjee, Dr.

Yunnmei C'I. in~ Dr. .Jeffrey Ho, Dr. Tint Davis, Dr. Xianfeng Gu, Dr. David Groisser,

Dr. Sergei Shabanov, Dr. Brett Presnell, and Dr. Richard ?-. i.--us! Ia for all the time and

energy they have invested into my education and research. Nevertheless, any errors in this

dissertation are solely my own.

Most of my doctoral research activities were carried out in the Center for Vision,

Graphics and Medical Imaging (CVGMI) where I have learned a lot from Dr. Baha

Venturi, Dr. Anand Rangaredi .Il Dr. Arunava B lia. j. Dr. .Jeffrey Ho and my fellow

students during weekly seminars and daily interaction. I thank former and current

CVGMI lah nienters: Zhizhou Wang, Eric Spellnman, Timothy McGraw, Hongyu Guo, .Jie

Zhang, Fei Wang, Nicholas Lord, Santhosh K~odipaka, Adrian Peter, Angfelos Barnipoutis,










O'Neil Smith, Ajit Rajwade, Ozlent Subakan, Ritwik K~umar, Ting C'I. >.~ Guang C'I. 19_

Meizhu Liu, and Yuchen Xie, for making our CVGMI lah an intellectually thriving and

socially enjoi-l l,h environment. I consider it a big fortune of me to have the opportunity

to know these wonderful persons. I also want to take this opportunity to thank all my

friends in the CISE department, CVGMI SIAM Gators, ITSTC-ITF alumni, and many old

classmates.

The diffusion AIRI data used in this dissertation were provided by the McE~night

Brain Institute at the University of Florida. Thanks go to Dr. Thomas Mareci, Dr.

Paul Carney, and other nienters in their groups for the collaboration on this aspect. In

particular, I would like to thank Dr. Evren Ojzerslan for helping me better understand

the diffusion AIR imaging and interpret the key niathentatical model developed in this

dissertation. Evren also kindly provided me some simulated data used in the experiments

and the IDL visualization program to render the surfaces paranletrized by spherical

harmonics.

Across the campus of the University of Florida, there are many warnthearted staff

nienters whose excellent work and services make the study at ITF very pleasant. They

are the wonderful staff at the CISE department, the International Center, the libraries,

and many unsung heroes who deserve my deep thanks. Outside the ITF, I want to thank

Dr. Behzad Dariush at the Honda Research Institute ITSA for giving me the opportunity

to work with him as an intern in suniner 2006 and for his continued help. I would also

like to extend thanks to Dr. Arun K~rishnan, Dr. Sean Zhou, Dr. Gerardo Hermosillo, Dr.

Yoshihisa Shinagawa, Dr. Jiannming Liang, and many other colleagues, for their kind help

and support during my early career in the Computer Aided Diagnosis group at Siemens

Medical Solutions ITSA.

Finally, I would like to express my deepest gratitude to nly parents in Chan!~ I and my

girlfriend Geri, for their unwavering love, tremendous support, and firm belief in me. This

dissertation is dedicated to them.










The research for this dissertation was financially supported by NIH grants R01-

EB007082 and R01-NS42075 to Dr. Baha Vemuri. I also have received travel grants from

the CISE department, the Graduate Student Council at the University of Florida, and

the IEEE Signal Processing Society. I gratefully acknowledge the permission granted by

IEEE, Elsevier, and Springer for me to reuse materials from my prior publications in this

dissertation.










TABLE OF CONTENTS


page


ACK(NOWLED GMENTS


LIST OF TABLES.


LIST OF FIGURES

LIST OF ABBREVIATIONS

LIST OF SYMBOLS ....


ABSTRACT

CHAPTER


1 INTRODUCTION


Motivation.
Alain Contributions.
Outline .


2 DIFFUSION AIR FUNDAMENTALS BACKGROUND REVIEW

2.1 The Basics of Diffusion Physics
2.2 The Basics of Nuclear Magnetic Resonance ..........
2.2.1 Dynamics of Nuclear Spins.
2.2.2 Magnetic Resonance, Relaxations, and Bloch Equations .. ..
2.3 Measuring Diffusion using NMR .. ........ ..
2.3.1 Spin Echlo anld Diffusion Effects in NMRl. ........
2.3.2 The Fourier Transform Relationship .. ......

:3 DIFFUSION AIR MODELING AND DIFFUSION PROPAGATOR RECON-
STRITCTION -CLASSICS AND THE STATE OF THE ART ......

:3.1 From the Bloch-Torrey Equation to the S' i-1: I1-Tanner Equation
:3.2 Diffusion Tensor Imaging .
:3.2.1 Tensorial Steil1: I1-Tanner Equation
:3.2.2 Estimation of the Diffusion Tensor.
:3.2.3 Fiber Orientation and Anisotropy Measures Derived from the Diffu-


sion Tensor .
Problems of the Diffusion Tensor Imaging.
Angular Resolution Diffusion Imaging (HARDI).
Modeling Diffusivity Profiles.
:3.3.1.1 Spherical harmonics series.
:3.3.1.2 Generalized diffusion tensor imaging
:3.3.1.3 The limitation of ADC profile.


:3.2.4
:3.3 High
:3.3.1











:3.3.2 Multi-Conipartniental Models .. .. 45
:3.3.3 Deconvolution Approaches ..... .. . 46
:3.3.4 Model-independent Q-Space Imaging Approaches .. .. .. .. 46
:3.3.4.1 Diffusion spectrum imagfingf . . 47
:3.3.4.2 Q-hall imaging . ...... .. .. 48
:3.3.4.3 Diffusion orientation transform ... .. . .. 50
:3.4 Conclusion ......... .. .. 51

4 METHODS .... ._ ... .. 54

4.1 Some Mathematics on ~P, ......... ... .. 56
4.1.1 Measure and Integration on ~P, ..... .. . 56
4.1.2 The Laplace Transform on ~P, .... .. .. . 57
4.1.3 Wishart and Matrix-Variate Ganina Distributions .. .. .. .. 60
4.2 The Expected AIR Signal from Wishart Distributed Tensors .. .. .. 6:3
4.3 Methods for Multi-Fiber Reconstruction ... ... .. 68
4.3.1 The Mixture of Wisharts Model .... .. .. 68
4.3.2 A Unified Deconvolution Framework .... .. .. .. 70
4.4 Computational Issues ......... .... 74
4.4.1 Regularization and Stability . ..... .. 75
4.4.2 Nonnegativity and Sparsity Constraints .. .. .. .. 81
4.4.2.1 L1 nminintization methods .... .. .. 81
4.4.2.2 Non-negative least squares (NNLS) ... .. . .. 82

5 EXPERIMENTAL RESITLTS ......... ... .. 86

5.1 Simulations ......... .. .. 86
5.2 Real Data Experiments ......... .. .. 92

6 DISCUSSION AND CONCLUSIONS . ..... .. .. 101

6.1 Suninary ......... .. .. 101
6.2 Open Problems ......... . .. .. 102
6.2.1 Nonparanletric Inverse Laplace Transform ... .. .. 102
6.2.2 Adaptive Sparse Dictionary Learning .... .. .. 10:3
6.2.3 Sub-voxel Fiber Bundles Classification ... .. .. 104

REFERENCES .. .......... ........... 105

BIOGRAPHICAL SK(ETCH ..... ._. . .. 118










LIST OF TABLES


Table page

3-1 A list of diffusion anisotropic measures that can be derived from the eigfenvalues
of the diffusion tensor. (D) = trace(D)/3 = (At + X2 + 3) 3 is known as the
mean di~f: .It;, which indicates the average diffusivity over all directions. .. 41

4-1 A list of previously published fiber reconstruction methods expressed in the pro-
posed unified deconvolution framework. See text for meaning of symbols. Re-
produced from with permission. @[2007] IEEE. .. .. .. 72

5-1 Imaging parameters used for the optic chiasm dataset. ... .. .. 94

5-2 Imaging parameters used for the rat brain dataset. ... .. .. .. 98










LIST OF FIGURES


Figure page

2-1 The diffusion propagators for cases as the free diffusion are probability density
functions of Gaussian distributions. The diffusion coefficient and diffusion ten-
sor are closely related to the random displacement of particles. .. .. .. 25

3-1 An illustration of subvoxel fiber configurations arising from the intro-voxel ori-
entation heterogeneity (IVOH). ......... ... .. 42

3-2 From the diffusion data to orientation distribution function (ODF) via the spher-
ical random transform with q-ball imaging (QBI). ... ... .. 49

3-3 A schematic illustration of the diffusion orientation transform (DOT). .. .. 51

3-4 Various quantitative profiles derived from diffusion weighted signals simulated
from 1-fiber, 2-fiber, and 3-fiber geometries. ..... .. . 53

4-1 Plots of density functions of gamma distribution Y4,1 W.r.t the non-invariant
measure and scale-invariant measure. . ..... .. 63

4-2 The Wishart distributed tensors lead to a Rigaut-type signal decay. Reproduced
with permission from [75] @[2007] Elsevier. ..... .. . 65

4-3 Sphere tessellations using an icosahedron subdivision model with different itera-
tion numbers. ......... ... .. 69

4-4 The ill-conditioning problem in the spherical deconvolution approaches. Repro-
duced from [73] with permission. @2007 IEEE. .. .. 77

5-1 HARDI simulations of 1-, 2-, and 3-fibers (b = 1500s/mm2) VISualized in Q-ball
ODF surfaces. Reproduced with permission from [73] @2007 IEEE. .. .. .. 87

5-2 Results of w on 1-fiber HARDI simulation data using different deconvolution
methods. Reproduced from [73] with permission. @2007 IEEE. .. .. .. .. 89

5-3 Mean and standard deviation of (a) angular correlation coefficient and (b) er-
ror angles for the two-fiber simulation. Reproduced from [73] with permission.
@2007 IEEE. ........ . ... . 93

5-4 The statistics of angular correlation coefficients and error angles for the 2-fiber
simulation. Reproduced from [77] with permission. @2009 Springer. .. .. .. 93

5-5 Probability maps computed using (a) damped least squares with GCV; (b) Min-
Li with quadratic constraints (e = 1) initialized from (a); (c) damped least squares
with fixed regularization parameter (a~ = 0.6);( d) non-negative least squares
from a rat optic chiasm data set overlaid on axially oriented GA [114] maps.
Reproduced from [73] with permission. @2007 IEEE. .. .. .. 96









5-6 Probability surfaces computed from a rat optic chiasm image using (a) QBI-
ODF, (b) DOT, (c) MOW+Tikhonov regularization, and (d) MOW+NNLS.
Reproduced from [77] with permission. @2009 Springer. .. .. .. 97

5-7 Probability maps of coronally oriented GA images of a control and an epileptic
hippocampus. Reproduced with permission from [75] @2007 Elsevier. .. .. 100









LIST OF ABBREVIATIONS


ADC

CNS

DLS

DOT

DSI

DTI (DT-MRI)

DWI (DW-MRI)
EAP

FA

FOD

FT (FFT)
GA

GCV

GDTI

HARDI

IVOH

MLE

MRI

MOG

MOW

NNLS

NMR

ODF


apparent diffusion coefficient, or simply diffusivity, in units of m2 -1

central nervous system

damped least squares
diffusion orientation transform

diffusion spectrum imaging

diffusion tensor (magnetic resonance) imaging

diffusion weighted (magnetic resonance) imaging

ensemble average propagator

fractional anisotropy

fiber orientation distribution

(fast) Fourier transform

generalized anisotropy

generalized cross validation

generalized diffusion tensor imaging

high angular resolution diffusion imaging

intravoxel orientational heterogeneity

maximum likelihood estimate

magnetic resonance imaging
mixture of Gaussian distributions

mixture of Wishart distributions

non-negative least squares

nuclear magnetic resonance

orientation distribution function










PAS

PDF

PGSE

RBF

RF

SH (SHS, SHT)
SNR

SPD

SVD

TE

TR

QBI

QSI


persistent angular structure

spin displacement probability density function, probability profile

pulsed gradient spin echo
radial basis functions

radio frequency

spherical harmonics (series, transform)

signal-to-noise ratio

symmetric positive definite

singular value decomposition
echo time

repetition time

q-ball imaging

q-space imaging









LIST OF SYMBOLS


b-value, b = 22G( /) taeB

B-matrix, B m bggT

duration of diffusion gradient

Dirac delta function

time between diffusion gradients

(apparent) diffusion coefficient, or diffusivity

diffusivity profile as a function of directions g

(apparent) diffusion tensor

a prolate tensor with eigfenvalues At > X2 = 3 and dominant eigfenvec-
tor v

Fourier transform

inverse Fourier transform

gyromagnetic ratio

a Wishart (matrix-variate Gamma) distribution over ?P, with shape

parameter p > 0 and scale parameter C E 7P,

diffusion gradient

magnitude of diffusion gradient, G = |G|

direction of diffusion gradient, g = G/|G|

initial spin density

diffusion propagator giving the probability of a particle traveling from

position z to zt in the diffusion time t, cPDF

ensemble average propagator, PDF

probability profile, P(r) := P(r, t = -r)

the manifold of 3 x3 symmetric positive definite matrices


b

B



6(-)


D

D(g)
D

Dy












G

G





P(z/|x, t)



P(r, t)

P(r)










q displacement reciprocal vector, or wave vector, q = 7~6G



{qi}=1 a diffusion imaging scheme with NV different wave vectors sampled in q-space

r relative displacement

R3" 3-dimensional Euclidean space

S2 2-dimensional unit sphere in R3"

S(q) spin echo signal, diffusion MR signal

So spin echo signal in the absence of an applied diffusion gradient

S(q)/So diffusion MR signal attenuation

-r effective diffusion time -r = a 6/3

v a unit vector.

{v~K~ k Sphere tessellation containing K sample directions.

YEm spherical harmonics of degree I and order m

w w = {wkKIS a vector, of nonnegtive weights. i,.e., > 0.

(-) ensemble average









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MATHEMATICAL MODELING FOR MULTI-FIBER RECONSTRUCTION FROM
DIFFUSION-WEIGHTED MAGNETIC RESONANCE IMAGES

By

Bing Jian

August 2009

C'I ny~: Baba C. Vemuri
Major: Computer Engineering

Diffusion-weighted magnetic resonance imaging (DW-MRI) is a non-invasive imaging

technique that allows neural tissue architecture to be probed at a microscopic scale in

vivo. By measuring quantitative data sensitive to the water molecular diffusion, DW-MRI

provides valuable information for neuronal connectivity inference and brain developmental

studies. The broad aim of this dissertation is to develop mathematical models and

computational tools for quantifying and extracting information contained in diffusion MR

images.

One of the fundamental problems in DW-MRI analysis is the mathematical modeling

of the MR signal attenuation in a voxel in the presence of multiple fiber bundles. In

this dissertation, we present a novel mathematical model and ....c.. I .nvII~ing efficient

algorithms for this problem. Our model uses a continuous probability distribution over

the space of symmetric positive definite matrices and is general enough to model water

molecular diffusion in a variety of situations involving complex tissue geometry including

single and multiple fiber bundle occurrences. We show that the diffusion MR signals

and the probability distributions for positive definite matrix-valued random variables

are related by the Laplace transform defined on the space of symmetric positive definite

(SPD) matrices. Another interesting observation is that when the mixing distribution

is parameterized by Wishart distributions, the resulting close form of Laplace transform

leads to a Rigaut-type fractal expression. This Rigaut-type function exhibits the expected










.I-i-up!lletic power-law behavior and has been phenomenologically used in the past to

explain the AIR signal decay but never with a rigorous niathentatical justification until the

development of the proposed model. Furthermore, both the traditional diffusion tensor

model and the niulti-tensor model can he interpreted as special cases of this continuous

mixture of tensors model.

In tackling the challenging problem of niulti-fiber reconstruction front the diffusion

AIR images, we further develop the mixture of Wisharts (\lOW) model, as a natural

paranietrization of the desired tensor distribution function, to describe complex tissue

structure involving multiple fiber populations. The niulti-fiber reconstruction using the

proposed MOW model essentially leads to an inverse problem. Computational methods

for solving this inverse problem are investigated under a unified deconvolution framework

which also includes several existing model-based approaches. Finally, the theoretical

framework we have developed for modeling and reconstruction of diffusion weighted AIRI

has been tested on simulated data and real rat brain data sets. The comparisons with

several competing methods enipirically -II---- -1 that the proposed model combined with

a non-negative least squares deconvolution method yields efficient and accurate solution

for the niulti-fiber reconstruction problem in the presence of intra voxel orientational

heterogeneity.









CHAPTER 1
INTRODUCTION

1.1 Motivation

The desire to discover the .Inr I~linin and functionality of the human body, especially

the brain structure and central nervous system (CNS), has been one of the driving forces

behind efforts to develop sophisticated medical imaging technologies. In the mid 1940s, a

breakthrough achievement was made involving the discovery of nuclear magnetic resonance

(NMR). This eventually led to the invention of magnetic resonance imaging (jl RI) whose

imaging capability due to the spatial localization of NMR signal was first demonstrated

and implemented in the 1970s. Since then, MR imaging has advanced tremendously

and become an indispensable diagnostic tools in modern medicine that is used everyday

for clinical and research applications. Built on technologies enabling the fast acquisition

time and high image quality, MR continues to pIIli .a important role in the diagnosis and

treatment of a number of diseases associated with the brain, heart, liver, and other organs

in the human body. Furthermore, "the ability to use MR imaging to noninvasively probe

the individual regions of the brain that control vision, sensation, motor function, memory,

language, and other processes has made this an extremely valuable modality to those

engaged in virtually any kind of brain-related research.[128]"

Among the many types of MRI modalities, diffusion Weighted MRI (Diffusion MRI,

DW-MRI or DWI) is a unique MRI technique that permits in-vivo measurement of the

diffusion of water molecules within tissue samples being imaged [88]. By exploiting the

sensitivity of the MR signal to the random motion of water molecules, diffusion MRI is

able to quantify different water diffusion characteristics in tissue samples locally. Because

these diffusion characteristics may be substantially altered by diseases, neurologic disor-

ders, and during neurodevelopment and aging, diffusion MRI is now recognized as a very



1 See [40] for a beautifully written history of NMR and MRI.










important clinical tool for brain-related diseases. For example, it has been successfully

applied to the evaluation of early ischemic stages of the brain [108]. Furthermore, the

directional dependence of water diffusion in fibrous tissues, like muscle [39] and white-

matter in the brain [107], provides an indirect but powerful means to probe the anisotropic

microstructure of these tissues. Like the idea of reconstruction the map of higfh--li in

a geographical region from the direction and frequency information of vehicle traffics in

that area, the motion of water molecules in neuro tissues can he potentially used to draw

inference about neuronal connections between different regions of the central nervous

system. Because of this powerful potential, diffusion AIRI has a distinct position in the

field of neuroscience research.

The broad aim of this dissertation is to develop mathematical models and computa-

tional tools for quantifying and extracting information contained in diffusion AIR images.

As opposed to a straightforward qualitative study, the process of producing accurate

quantitative results necessarily involves substantially more time and effort. However,

"the benefits of quantification are that fundamental research into biological changes in

diseases, and their response to potential treatments, can proceed in a more satisfactory

way. Problems of hias, reproducibility and interpretation are substantially reduced. [141]"

1.2 Main Contributions

The most significant original contributions of this dissertation are summarized

below. First, we present a novel mathematical model for diffusion-attenuated AIR signal

which involves a continuous probability distribution over the space of symmetric positive

definite matrices. This model is general enough to model water molecular diffusion in

a variety of situations involving complex tissue geometry including single and multiple

fiber bundle occurrences. We make the interesting observation that the diffusion AIR

signals and probability distributions for positive definite matrix-valued random variables

are related by Laplace transform defined on the space of symmetric positive definite

(SPD) matrices. We further show that in the case of Wishart distributions or mixtures










of Wishart distributions, a closed form expression for the Laplace transform exists and

can he used to derive a Rigaut-type .I-i-mptotic fractal law for the AIR signal decay

behavior which has been observed in the past [84] hut never with a rigorous mathematical

justification until now. This theoretical result depicts surprising consistency with the

experimental observations. Moreover, Diffusion Tensor Imaging (DTI), currently the most

widely used technique, can he viewed as a limiting case of the proposed model.

The measurements from diffusion AIRI provide unique clues for extracting orientation

information of brain white matter fibers and can he potentially used to infer the brain

connectivity in vivo using tractography techniques. Nowced .va- the widely used DTI tech-

nique fails to extract multiple fiber orientations in regions with complex microstructure.

In order to overcome this limitation of DTI, a vali'. iv of reconstruction algorithms have

been introduced in the recent past. One of the key ingredients in several model-based

approaches is deconvolution operation which is presented in a unified deconvolution

framework in this work. Additionally, some important computational issues in solving

the deconvolution problem that are not addressed adequately in previous studies are

described in detail here. Further, we investigate several deconvolution schemes towards

achieving stable, sparse, and accurate solution. Experimental results on both simulated

and real data are presented. The empirical comparisons -II---- -1 that non-negative least

squares method is the technique of choice for the multi-fiber reconstruction problem in the

presence of intra voxel orientational heterogeneity.

1.3 Outline

This dissertation is organized as follows:

ChI Ilpter 2 provides the background knowledge for understanding models and methods

used in diffusion magnetic resonance imaging. The first two sections of this chapter

briefly review hasic concepts in diffusion physics and nuclear magnetic resonance (NMR),

respectively. Then the key principles for measuring the diffusion phenomena using NAIR,










together with the fundamental relationship between the measured NMR signal and the

statistical properties of molecular diffusion, are presented.

('!s Ilter 3 reviews existing models and methods used in diffusion magnetic resonance

imaging. The first section in this chapter presents the classical Bloch-Torrey equation

and the Stei-l: I1-Tanner equation which are the foundations of the diffusion AIR imaging

modeling. The second section in this chapter introduces the now widely used diffusion

tensor imaging (DTI) method. The third section in this chapter is dedicated to the more

recent high angular resolution diffusion imaging (HARDI) methods with a focus on the

multi-fiber reconstruction techniques.

The first two sections in C'! Ilpter 4 present the technical details of the proposed

continuous tensor distribution model with a brief introduction to Laplace transforms

and Wishart distributions on the manifold of symmetric positive definite matrices. This

part builds the mathematical foundation of the proposed research. The last two sections

in C'!s Ilter 4 focus on the multiple fiber reconstruction from diffusion AIRI using the

proposed mixture of Wisharts model. In this chapter the proposed reconstruction method

and several existing approaches in literature put into a unified deconvolution framework.

Additionally, some important computational issues in solving the deconvolution problem

that are not addressed adequately in previous studies are described in detail here. Further,

we investigate several deconvolution schemes towards achieving stable, sparse, and

accurate solutions.

Experimental results on both simulations and real data are presented in ('! .pter

5. The comparisons with other approaches demonstrate the merits of the proposed

continuous tensor model together with the deconvolution reconstruction framework solved

using non-negative least squares method for the multi-fiber reconstruction problem in the

presence of intra voxel orientational heterogeneity. Finally in ('! .pter 6 we summarize

the main contributions of this dissertation and discuss a few open problems for further

research.










CHAPTER 2
DIFFUSION AIR FUNDAMENTALS BACKGROUND REVIEW

This chapter provides the background knowledge for understanding models and

methods used in diffusion magnetic resonance imaging. The first two sections of this

chapter briefly review hasic concepts in diffusion physics and nuclear magnetic resonance

(NMR), respectively. Then the key principles for measuring the diffusion phenomena using

NAIR, together with the fundamental relationship between the measured NMR signal and

the statistical properties of molecular diffusion, are presented.

2.1 The Basics of Diffusion Physics

On a macroscopic level, diffusion results from the microscopic random thermal

agitation of the particles in a medium. This phenomenon of random thermal agitation is

ubiquitous and known as "Brownian motion", named after the botanist Robert Brown

who first discovered and described it in 1827. In the context of this study, diffusion

refers specifically to the perpetual random translational motion of water molecules in any

part of a human or animal .Inr Irh ow.J Diffusion of water molecules in biological tissues,

observed as a macroscopic manifestation of Brownian motion, highly depends on many

factors including restrictions due to cell membranes, and a variety of microstructure

properties. The ability to measure the diffusion process of water molecules in tissues and

the understanding of how it is affected by these factors are extremely useful for studying

the biological microstructure.

It is natural to model the random motion of particles using a probabilistic framework.

A starting point of studying diffusion processes is the so-called self-diffusion propagator.

Generally, the propagator P,(z/|x, t) denotes the conditional probability density that a

particle initially located at position coordinate z moves to a position in a volume element

dz at zl after a time interval t.









The classical treatment of the diffusion propagator is to describe it as the Green's

function for the diffusion equation via the Fick's law


vls~= V D(VP,(2/|x, t)) (2-1)


subject to the initial condition


P,(z/|x, 0)) = 6(xt 2) (2-2)


and proper boundary conditions. In (2-1), D is the molecular self-diffusion tensor, a

physical quantity describing the diffusion processes in a more compact form.

In the case of unrestricted self-diffusion, also known as free diffusion, the boundary

condition of (2-1) is simply P,(z/|x, t) 0 as z/ c o which yields

1 (2t 2)TD-1(2t 2)
Psx/xt)=exp(- )(2-3)
zdet(D)(4xt) 4t

combined with the initial condition (2-2).

An interesting observation is that Ps in (2-3) depends only on the displacement r

but not the initial position 2, which reflects the Markov property of Brownian motion

statistics. Another important observation is that the self-diffusion tensor D is related to

the time dependence of covariance matrix (rrT) by


D = (rr T). (2-4)

On the basis of the central limit theorem, the above results may be derived either from a

formal Markov's method as in [33] or from an elementary random walk model as presented

in [55, pp. 325].
In an isotropic medium, the diffusion tensor D is proportional to an identity matrix.

Accordingly, the diffusion propagator (2-3) simplifies to

1 |xt |21
P(z/|x, t) = exp(- )(2-5)
(4xDL) 4Dt









where D is the self-diffusion coefficient given by the diagonal of the diffusion tensor D. In

this case, the corresponding diffusion equation via Fick's laws is


=DV2ps(z1/|x t) (2-6)

and we obtain

D = trace(D)/3 = ~(rTr) (2-7)

which agrees with the well known Einstein relation [51].

Because it is unrealistic to measure the random motion of a single particle, we shall

take an in,~ in!!. --ave I, ii view to depict the macroscopic behavior of a large number of

molecules on a statistical basis. Starting from the self-diffusion propagator, one can define

the total probability, W(zt, t) of finding a particle at position z/ at time t as


W~m/ t) W~, 0)s~x|x, )dz(2-8)


where W(2,0O) is just the initial particle density p(z). Using the notation of the displace-

ment vector, we can modify (2-8) and define the so-called ensemble average l".-y..;yl.>r

(EAP), also known as the displacement ly .-?..:?-9i;, distribution function (PDF), by inte-

grating the diffusion propagator over all initial positions [29, 79]


P~ ) P~x+rlz, t)p(z)dz (2-9)


where p(z) is the initial particle density. As we shall see later, this ensemble averaged

propagator which cumulates all the microscopic contributions distributed in a voxel at the

macroscopic scale may be measured directly by NMR. This is the fundamental principle

exploited in the diffusion MR and it will be the topic investigated later in this chapter.

In cases such as the free diffusion where P,(z + rlz, t) is independent of the initial

position 2, the ensemble average propagator is also a Gaussian

1 rTD-lr
P(r, t) =exp(- ). (2-10)
d et(D 4x) 41










The diffusion propagators for the free diffusion as well as the corresponding diffusion

coefficient and the diffusion tensor are summarized in Fig. 2-1.

Diffusion tensor =-DP)1rD^ r
SP(r, t) = exp(- )
D = (rr T dtD(4) 4t

isotropic isotropic

Diffusion coefficient =D~V2P 1 |7| T
,, P(r, t) = exp-
D = (r Tr) 4Dtt

Figure 2-1. The diffusion propagators for cases as the free diffusion are probability density
functions of Gaussian distributions. The diffusion coefficient and diffusion
tensor are closely related to the random displacement of particles.


It is worth noting that the diffusion tensor framework and the Gaussian propagators

described above are derived from the free diffusion which only represents a very limited

class of diffusion phenomena. Commonly observed are a vast range of phenomena, such

as restriction, heterogeneity, anomalous diffusion, finite boundary permeability. Generally,

the diffusion equations are imposed with nontrivial boundary conditions depending on

the nature of confining geometries and other physical properties. Solutions to diffusion

equations in a nr1I~1ii of simple geometries are available in [44]. It has been shown that

if the diffusion boundaries are closed, unrelaxing, and completely impermeable, then in

the long run (long time of diffusion), the propagator will assume the shape of the space

accessible to particles [29, 150]. Even though the limiting conditions are rarely satisfied in

biological applications, the diffusion-structure relations nevertheless provide useful clues

for probing microstructures, which has fostered a vast promising research field.

2.2 The Basics of Nuclear Magnetic Resonance

In this section, we present the basic physical principles of nuclear magnetic resonance

(NMR). The main references we consult are [29, 92, 94, 133].










2.2.1 Dynamics of Nuclear Spins

The fundamental concept of NMR is the spin aq,:(;,lar momentum, an intrinsic prop-

erty possessed by certain atomic nuclei. In atomic physics, the spin angular momentum p

of an atomic nucleus is quantized by the nuclear spin quantum number I according to the

formula

2x1p| ( (2-11)

where & is the Planck's constant. The spins possessed by an atomic nucleus are combined

by the spins of the protons and the neutrons inside the nucleus. Since the spins of protons

and neutrons may interact in various configurations, e.g., parallel or anti-parallel to each

other, the value of nucleus spin quantum number I is chosen as the one in the lowest

energy nuclear state, also called the iI,. ;,,../ state nuclear spin. Depending on the nuclear

structure, the number I can only be an integer, half-integer, or zero with the following

rules[92]: (1) if the numbers of protons and neutrons are both even, the ground state

nuclear spin I is zero; (2) if the numbers of protons and neutrons are both odd, the

ground state nuclear spin I is a positive integer; (3) if the total number of protons and

neutrons is odd, then I is given by a half-integer. Note that the nucleus of 1H, which is

most abundant in nature and human body, contains a single proton, hence its nucleus

quantum spin number I is 1/2; while the nuclei 12C 160, and 56Fe all have a ground state

spin I = 0. A complete listing of nuclear isotopes and their NMR properties may be found

at http: //www web elements com.

Another intrinsic property possessed by fundamental particles is the magnetic

moment. The spin angular momentum p and the magnetic moment p are proportional to

each other as given by

I- = 7p (2-12)

where y is called the ~i;, tion 7,: It.: H: ratio, a characteristic value of a particular atomic

nucleus. According to the classical mechanical model, when a nucleus with spin angular

momentum is subject to an external magnetic field, the interaction between the magnetic









moment p and the field Bo will generate a torque force L trying to align the two:


L = dp/dt = p x Bo.


The outcome of this torque force is the precessional motion of the nucleus, in which the

magnetic moment vector p rotates around the direction of the external magnetic field

Bo in addition to the spinning motion of the nucleus around its own axis. The equation

describing the precession is


dp/ldt = ydp/dt = yL = wox p- (2-13)


where wo = -7Bo is the so-called Larmor /, ~i;, ,: ;t of precession. Note that the minus

sign in (2-13) implies that the rotation obeys the left-hand rule.

According to the quantum mechanical model, the nuclear magnetic moment p can

only have 2I + 1 orientations in a magnetic field Bo, corresponding to 2I + 1 energy levels:


E = -p Bo. (2-14)


Here the minus sign in (2-14) indicates that the magnetic energy is lowest when p is

parallel to the Bo field while the magnetic energy is highest when p is anti-parallel

(spin-down) to the external field Bo. The difference between the two energy levels is

proportional to the strength of the applied field Bo:


AE = yBo. (2-15)


With I = 1/2, the 1H nucleus can either align with (spin-up) or against (spin-down) the

applied field, corresponding to low and high energy states respectively. When an ensemble

of nuclei are immersed in an external magnetic field, the distribution of the allowed

orientations is described by Boltzmann statistics. The resulting macroscopic magnetization









vector M~ can be defined as the vector sum of all the microscopic magnetic moments


M = pi (2-16)
i= 1

where ps is the individual magnetic moment of the i-th nuclear spin, and NV is the total

number of spins. At equilibrium, two macroscopic effects can be observed: (1) the overall

transverse component of M~, i.e., the component perpendicular to the applied magnetic

field direction, is zero because of the random phases introduced by the processing magnetic

moments, (2) more nuclei relax into the parallel orientation (lower energy state) than the

antiparallel orientation (high energy state), resulting in a bulk magnetization vector M~.

The direction of this bulk magnetization vector M~ is aligned with the applied magnetic

field direction, and its strength depends on the proton 1 1

2.2.2 Magnetic Resonance, Relaxations, and Bloch Equations

The principle of NMR is that the alignment of nuclear spins can be perturbed by

applying a circularly polarized magnetic field B1, called a radio fr .;, e.'.;i excitation

pulse, which is rotating about Bo at the Larmour frequency wo. A typical B1 field is

perpendicular to Bo and its generation takes the following form:


Bl(t) = Bl(t)e-i("Ot+4) (2-17)


where 4 is the initial phase angle and can be assumed to be zero as it has no significant

effect on the excitation result. The excitation property of an RF pulse is specified by the

shape and duration of the envelop function Bl(t).Two most widely used RF pulses are the

r Litty(;J.;<, pulse and the since pulse.




1 The term RF pulse is so called because wo/2xr is normally in the frequency range
of radio waves. The field usually lasts for a few microseconds or milliseconds. Also, the
strength of the field B1 is much weaker than the static magnetic field Bo.









On the microscopic level, the energy absorbed by nuclei during the RF excitation

makes some nuclear spins jump from the lower energy state to the higher energy state and

hence changes the overall magnetization vector M~. Although the microscopic behaviors

of individual spin magnetic moments are quantized, the observed bulk magnetization as a

macroscopic manifestation may be treated classically. The excitation induced by B1 tips

the bulk magnetization vector M~ away from the direction of Bo by a spiral precession

around Bo described by the following equation


dM/ldt = wox M~ = ylM x Bo. (2-18)


Note that the processing magnetization creates a periodically changing magnetic field in

the transverse plane. This effect may be detected and measured by a closely placed wire

coil according to Faradovl~i's law of induction. The resulting signal then can be exploited in

NMR spectroscopy and magnetic resonance imaging.

At the time when the RF pulse is turned off, M~ makes an angle a, called flip nl

with the static field Bo. In general, the flip angle a~ depends on the strength By (t) and the

duration -r of the applied RF:



~o (2-19)
= B1-r (for a constant B1 amplitude).

Because the amplitude of the alternating voltage induced in the receiver coil is propor-

tional to the transverse magnetization component, a 900 pulse is commonly used as an

excitation pulse as it produces a maximum transverse magfnetization component equal to

the equilibrium magfnetization.

Once the RF pulse is removed, the nuclear spins will tend to restore the equilibrium

energy levels distribution by releasing excess energy into the surroundings. This process is

called relaxation and has two-fold effects: (1) the transverse magnetization component My,

decays to zero exponentially, which is referred as spin-spin relaxation, (2) the longitudinal









magnetization component 14 returns to the equilibrium value ii1, gradually, which is

referred as spin-lattice relaxation.

These two types of relaxations are phenomenologically described by the following

equations


dt T~
1 ~(2-20)

dt T2
with solutions


(2-21)
It) = 1Av(0) exp(-t/T2)

where the parameters Ti and T2 are known as lory~ll.:;J~.:..al relaxation time and the trans-

verse relaxation time, respectively. Note that both Ti and T2 depend on the molecular

environment and thus can be used to characterize different samples.

In the rotating reference frame where the x-y axis keeps rotating with the precession

frequency wo, the phenomenological descriptions of spin-lattice and spin-spin relaxation

can be combined together into a system known as the Bloch equations [27]

dM~(t)
= TM(t) x B(t) R(M~(t) MI,) (2-22)

where B(t) includes both the static field and the RF pulse, i.e., B(t) = Bo + Bl(t), and R

is the relaxation matrix:


R = 0 /T2 0(2-23)



and the vector M.i = (0, 0, il,)T. The Bloch equations provide a valuable reference

in describing macroscopic phenomena in NMR imaging. Note that if there were no

relaxation, i.e., both Ti and T2 were approaching infinity, the Bloch equations simplify

to Eq. (2-18), as expected. With the relaxation effect after the single RF pulse B1,









the subsequent NMR signal induced by the (. 1;11 Ir;1). decaying magnetization in free

precession is then detected in the time domain. It is therefore known as the free induction

I/ .,ru (FID). The measured signals are conveniently represented by complex numbers

where the real part and the imaginary part correspond to the x-direction and the y-

direction in the rotating frame respectively. By Fourier transformation the signals may be

represented in the frequency domain. For a detailed discussion, the reader is referred to

[29].

It is important to understand that the Bloch equations (2-22) only describe the

dynamics of the magnetization under the effects of spin-lattice and spin-spin relaxation in

an NMR experiment. By introducing a diffusion term, Torrey modified the original Bloch

equations to reflect the effects of molecular diffusion. The modified equations are known

as the Bloch-Torrey equations [142] and plI li- a fundamental role in modeling the diffusion

imaging data that we will discuss later.

2.3 Measuring Diffusion using NMR

2.3.1 Spin Echo and Diffusion Effects in NMR

One assumption of the Bloch equations is the homogeneity of the magnetic field Bo

in the absence of B1, which is never true in practice. The variations in the local magnetic

field contribute to the gradual reduction of the transverse magfnetization in addition to

the spin-spin relaxation. Consequently, May decays more rapidly than the exponential

decay May = il.*:-tl/ predicted by the Bloch equations. However, if a 1800 pulse is

applied at time t following the initial 900 pulse, then the additional decay due to the field

inhomogeneity can be reversed at time t after the application of that 1800 pulse. This

phenomenon of the signal restoration is called spin echo. The time between the initial 900

pulse and the echo formation is called TE (echo time) which is twice the time interval

between the two RF pulses.

Another factor affecting the signal reduction is the random thermal motion of the

spins, or the molecular diffusion at the macroscopic level. Simply speaking, the molecular










diffusion plus spatial variation of the magnetic field will induce a phase distribution in the

transverse plane and hence result in signal attenuation. This signal attenuation is closely

related to the amplitude of the spin displacements caused by the diffusion. Intuitively,

the faster the diffusion, the more random the phases distributed, and a larger signal

attenuation will be observed. This is the exact idea used in the classical S' i-. H:.-Tanner

pulse-gradient spin-echo (PGSE) experiment [138], the first NMR experiment specifically

designed for quantitatively measuring diffusion in a sample.

In the standard PGSE pulse sequence for diffusion in.! I_;oy a pair of two identical

field gradients are placed before and after the 180" pulse, to perform the "diffusion

encodingt In the remainder of this dissertation, these two gradients are denoted by G

with strength G = |G| and direction g = G/G. Each gradient pulse will last a time 6, and

the pair is separated by a time a between the leading edges of the two gradient pulses.

Because that the 180" pulse only cancels the phase shift of spins induced hv the first field

gradient, the final detected signal will take into account the phase shift of spins induced

by the molecular diffusion during the time period a which separates the two gradient

pulses. Note that for a complete AIR image formation the applied diffusion gradients must

he combined with a sequence of .slice .selection, S,. it;,. m e; encoding, and phase encoding for

spatial localization of AIR signals. However, the details of these so-called k-space sampling

techniques are beyond the scope of this dissertation and we refer the reader to [94].

2.3.2 The Fourier Transform Relationship

Let 1 be the time when the first gradient is applied, the phase shift #1 of the spin

transverse magnetization induced by this gradient pulse with duration 6 is given by


I1 = ]* GTzdt = q*6GTz (2-24)


where ]* is the gyro-magnetic ratio and z is the spin position supposed to be constant

under the narrow pulse assumption (6
gradient pulse is applied, suppose the spin moves to a different position a + r, then the net









phase-shift induced by this pair of gradients will be:


S= y6GTr. (2-25)

Note that if spins were stationary, i.e. r = 0, a perfectly refocused echo will occur.

The NMR signal measurement is proportional to the total transverse magfnetization

My, from a very large number of spins. Consider the millimetric resolution provided by
NMR devices and the micro-metric scale of the molecular diffusion, it is reasonable to

neglect the inter-voxel diffusion effects on a voxel scale. Let E(G, a) denote the amplitude
of the "echo signal", then it can be expressed as the ensemble average


E(G, a) =(exp(i )) = (exp(iyGGTr)). (2-26)

It is important to note that here E(G, a) is 1...)~ II!. .1. ii so that E(0, a) = 1, in other

words, E(G, a) should be considered as the signal decay induced only by the molecular

diffusion but excluding the attenuation of the echo due to the T2 T68XatiiOn.

The net phase distribution that weights the ensemble average of individual phase term

exp(iyGGTT) iS the probability for a spin to travel from z to z + r during time a. Note
that this probability is precisely p(z)P,(z + r)|x, a) where p(z) is the spin density and
Ps (x + | t) is the, diffusion, propagator westudied~ in the,,,;, prviussetio.Inetngti

probability into (2-26) and using the substitution zt = z + r, we obtain

E(G, A) = psP~ ~, x~gG~t-x)md

= ( p(ms: D~~)i)Psl(/7 + ~r~x~sc) epi)Trt-x) (2-27)

= P(r, A\) exp)(6igbGl))dr

where P(r, t) is exactly the ensemble average propagator (EAP) defined in Eq. (2-9).

By introducing a displacement reciprocal vector, q, defined as


q = 6G, (2-28)
2xr









we can now rewrite (2-27) in a more so-----~ -lHi.- form


E(q, a) = ~ ,A)epi7rT~ -1[P(r, A). (2-29)


As clearly expressed in (2-29), there is a simple Fourier reciprocal relationship between the

spin echo signal E(q, a) and the EAP P(r, a). This Fourier relationship is fundamental

in the direct reconstruction of the EAP P(r, a). The space of all possible q vectors is

called q-space. By varying either of diffusion gradient G or the gradient duration 6, the

signal can he measured at many sampled points in q-space and the EAP P(r, a) can he

obtained by taking an inverse Fourier transform of (2-29)


P(r, a) = FT[E(q, a)] = F[S(q)/So]. (2-30)


Employing Eq. (2-30) to calculate the EAP is called "q--p. .. analysis[29, 42]. Since

in practice, the diffusion time a is treated as an experimental constant, the notation

P(r) = P(r, a), as the displacement probability distribution function (PDF) defined on

RW", is the central mathematical quantity being studied in this dissertation.









CHAPTER 3
DIFFUSION MR MODELING AND DIFFUSION PROPAGATOR RECONSTRUCTION
-CLASSICS AND THE STATE OF THE ART

In this chapter, existing models and methods used in diffusion magnetic resonance

imaging are reviewed. The first section presents the classical Bloch-Torrey equation and

the Stei-l: I1-Tanner equation which are the foundations of the diffusion MR imaging

modeling. The second section introduces the now widely used diffusion tensor imaging

(DTI) method. The third section is dedicated to the more recent high angular resolution

diffusion imaging (HARDI) methods with a focus on the multi-fiber reconstruction

techniques. For published review articles on these topics, we refer the reader to [2, 89,

104].

3.1 From the Bloch-Torrey Equation to the Stejskal-Tanner Equation

For free diffusion in an isotropic medium, combining Eqs. (2-5) and (2-27) yields


S(q)/So = exp (-(y6G)2DA). (3-1)

There are two problems associated with this simple model. First, in commercial MRI

units, long gradient durations 6 are required to produce observable dephasing effects,

hence diffusion occurring during the application of the gradient pulses may not be ignored.

Second, it does not take into account the effects induced by other additional gradient

pulses, including imaging pulses and background residual gradients [24]. To address these

issues, we resort to the following Bloch-Torrey equation




which adds a free diffusion term to the original Bloch equations (2-22). Detailed math-

ematical derivations [24, 113, 150] show that in infinite and homogeneous media the

solution to (3-2) with a spin-echo sequence is given by


dL,(t) = !1,(0) exp(- k(u) TDk(u)dzL) (3-3)









where

k(t) = 7 G(u)du. (3-4)

Note that the sign of G in Eq. 3-4 has to be inverted for all gradient pulses following the

refocusing 1800 RF pulse.

Furthermore, for an isotropic medium, the signal decay at echo time TE in a spin-

echo experiment is


,(E)= ,() xp-D k(t)Tk(t~dt). (3-5)
12,ti.) IIY(I)C*0/TET

By substituting the so-called ,9 II! 1.0~ factor", b-value, defined as in [88]

0/TE


into (3-3), we obtain a simpler expression for the signal attenuation


14,(TE) = 14,(0) exp(-bD) (3-7)

which is strictly valid for diffusion in unrestricted, homogenous, and isotropic media. Note

that the b-value is a useful quantity that characterizes the sensitivity of NMR sequences to

diffusion [24].
For the most widely used PGSE sequence [138], an exact solution to Eq. (3-5) exists

and the corresponding signal attenuation is given by

S(q) = So exp(-72 2G2(A 6/3)D) = So exp(-bD) (3-8)

where q = 276G and

b=4xr2 q2( 2/) y2G2 _9).

The time constant (A 6/3) in Eq. (3-8) is therefore known as the effective diffusion time

where the 6/3 correction accounts for the diffusion that occurs during the application of

gradient pulses.









It is important to note that the S' i- 1-1 .-Tanner equation (:38) is derived from

the Bloch-Torrey equation (:32) which is a microscopic, free-diffusion physical model;

while the observed diffusion AIR signal measured at a millimetric voxel is an ensembled

contribution from all the microscopic displacement distribution of the water molecules

in this voxel, as shown in (2-27). To reflect and partially bridge the gap between these

two scales, one can consider replacing the physical diffusion coefficient, D, with its global,

statistical counterpart, Do>,, termed the apparent diffusion coefficient (ADC) [88], in the

Steil1: I1-Tanner equation, which now reads


S(q) = So exp(-bDo;;). (:310)


Although the concept of ADC requires some intra-voxel homogeneity assumption, ADC

has been largely used in the literature since it was introduced. [87]

The Steil1: I1-Tanner equation (:38) also assumes free diffusion as no boundary

conditions were imposed. However, the water molecular diffusion in tissues is obviously

both hindered and restricted. This fact inevitably leads to a deviation of the observed

signal from the monoexponential behavior implied by Eq. (:310).

More sophisticated models, such as the hiexponential model[:38, 111], the Laplace

transform model [164], the Rigaut-type .I-i-mptotic fractal expression [129, 1:30], and the

stretched exponential model [22], have been investigated in literature to depict the non-

exponential signal decay. A brief discussion on these models with comments can he found

in [11:3, Ch.2].

3.2 Diffusion Tensor Imaging

The scalar Steil1: I1-Tanner equation (:38) depicts the dependence of the diffusion AIR

signal on the gradient strength, however it can not describe the diffusion anisotropy which

is observed in fibrous biological tissues, for example, in muscles [:39], the spinal cord [108],

and the brain white matter [:37]. Diffusion tensor AIRI (DT-AIRI or DTI), introduced by

Basser et al. [17, 18], provides a relatively simple way of quantifying diffusion anisotropy









as well as extracting fiber directions locally from multidirectional diffusion MRI data.

The present section provides a brief description of DTI, more complete treatments can be
found in recent review articles on DTI, for example, [25, 83, 120].

3.2.1 Tensorial Stejskal-Tanner Equation

In Section 3.1, the solution to the Bloch-Torrey equation with an anisotropic diffusion

tensor term (3-2) for free diffusion in homogeneous media measured by a spin-echo

sequence is given by


i,(t) = !I,(0) exp(- k(u)TDk(u)dzL) (3-11)

where

k~t) 7 G~~da.(3-12)

Introducing the matrix

B =k(t~k(t)Tdt (3-13)
0/~TE
which is referred to as the B-matrix [18], we obtain the tensorial Stei-l: I1-Tanner relation:


S(q) = So exp (-trace(BD)) (3-14)

which is the foundation of the diffusion tensor imaging (DTI). Note that to be meaningful

at the voxel scale, the D in (3-14) should be considered as an apparent diffusion tensor

Day, different from the original microscopical diffusion tensor in (3-2) based on the same
argument provided for apparent diffusion coefficient (ADC) in Section 3.1.

Let g be the unit vector representing the direction of the diffusion gradient, i.e.,
G = Gg, the B-matrix can be quite accurately approximated by a pure outer product of

the gradient direction scaled by the b-value, i.e.,

B mbggT (315)

where b = y2 2G2( 6 3) aS defined in Eq. (3-9).









Defining the apparent diffusion coefficient D(g) along a certain direction g


D(g) = gTDg (3-16)

reveals the connection between the two versions of Steil1: I1-Tanner equation as follows

S(G) = S(Gg) = So exp(-bD(g)) = So exp(-bgTDg) or
(3-17)
S(q) = So exp(-74xr2 TDq)

where -r = a 6/3 is the effective diffusion time and q = 276G is the displacement

reciprocal vector.

Substituting S(q)/So from (3-17) into the Fourier relationship (2-29) leads to the

Gaussian propagator
1 -rTD-lr
P(r, a) =exp[ ]. (3-18)
|/D|(4xA)3) "
The comparison of (3-18) and the previous result in (2-3) demonstrates the consistency of

the Steil1: I1-Tanner equation and the diffusion propagator formalism [14].

3.2.2 Estimation of the Diffusion Tensor

Taking the natural logarithm at both sides of Eq. (3-14) yields the simple relation


trace(BD) log(So) = log(S(q)). (3-19)


The left-hand side of Eq. (3-19) is linear with respect to D and log(So), therefore, the

tensor components are obtained by solving a linear system of equations formed by stacking

(3-19) with a set of measurements S(q) and corresponding B-matrices.

It is worth noting that the linear system approach is equivalent to maximum likeli-

hood estimate (j!1.11) when assuming the Gaussian noise-model on the log-transformed

signals. However, in practice, the more realistic noise model should be Rayleigh dis-

tributed rather than Gaussian distributed. Consequently, the Gaussian MLE will be biased

if high q-values are used since Gaussians are no longer good approximation to Rayleigh










distributions at the tails. For more detailed discussion on modeling noise in diffusion 1\RI

data, see [1:36].

Since there are six independent elements in diffusion tensor, image acquisitions

from at least six linearly independent directions, together with a reference image So, are

required. Usually, a more reliable measurement of the diffusion tensor requires more than

six directions. Even though a diffusion tensor should be positive definite theoretically, the

positivity of the estimated diffusion tensor may be destroyed by the signal noise, therefore,

one might consider methods that can preserve the positivity of the estimated tensors, for

example, [158].

3.2.3 Fiber Orientation and Anisotropy Measures Derived from the Diffusion
Tensor

Represented by a :3 x :3 symmetric matrix, the diffusion tensor can he decomposed

into:

D = &QAQT (:320)

where Q = (ele283) is an orthogonal matrix of eigenvectors and A = diag(Ay, X~, 3S) is a

diagonal matrix of real eigenvalues ordered by At > Xa2 X3-

Intuitively, the fiber orientation can he estimated by taking the directions along the

peaks of the probability profile given in Eq. (:318). Finding the peaks of the probability

profile in Eq. (:318) is equivalent to finding the minima of the quadratic form: rTD-lr

for r E S2. It iS easy to show that TTD-lr has a uique minimum when r takes the

direction of the dominant eigfenvector, el, provided that X1 > X2 3.XS In this case the

ellipsoids generated from the isosurfaces:


rTD-lr = C


also align to this dominant direction, hence they are widely used to visualize the estimated

diffusion tensors.









Table 3-1. A list of diffusion anisotropic measures that can be derived from the eigfenvalues
of the diffusion tensor. (D) = trace(D)/3 = (At + X2 + 3) 3 is known as the
mean di~ffr.: .It;, which indicates the average diffusivity over all directions.
Name Definition

Relative Anisotropy (RA) C ~(As (D))

Frac~tional Anisotfropy~~ (F) 2 1(s- D)
Volume Ratio X1 2 3
(D) 3
Prolateness Metric X1-X
3 (D)
Oblateness Metric 2(Ag-X3)
3 (D)
Sphereness Metric


While the largest eigenvalue and its corresponding eigenvector of the diffusion tensor

describe the quantity and direction of the principal diffusion, its eigenvalues are also

exploited to derive some scalar measures that are meaningful for studying the nature of

diffusion anisotropy exhibited in the tissue of interest. Table 3-1 lists several anisotropy

metrics that have been studied in the literature [15, 162].

Note that all metrics defined in Table 3-1 are dimensionless and rotation-invariant.

Among these metrics, the fractional anisotropy (FA) has been most widely used because

it is relatively insensitive to noise [15] and does not require the sorting of the eigenvalues.

Additionally, FA is automatically normalized to the unit interval, FA takes the value of 0

when the diffusion tensor is totally isotropic, i.e., At = X2 3 X, Whereas it takes the value

of 1 when the diffusion tensor is extremely anisotropic, i.e., X2 = 3 = 0.

Anisotropy metrics are quite useful in quantitatively assessing the orientational

coherence of the diffusion compartments within a voxel [120]. For example, the high FA

values are typically associated with strongly aligned fibers such as axons in white matter,

while the FA values are expected to be relatively low in regions of fiber intersections or in

dense tissues where diffusion is restricted equally in all directions.

3.2.4 Problems of the Diffusion Tensor Imaging

As previously mentioned, the diffusion tensor framework can only describe a very

limited class of diffusion phenomena. Although promising results have been achieved using









DTI to study regions of the brain and spinal cord with significant white-matter coherence

and to map anatomical connections in the central nervous system [19, 41, 105, 106, 156]

Sthe major drawback of diffusion tensor MRI is that it can only reveal a single fiber

orientation per voxel and fails in regions where fiber populations cross, kiss, splay, branch

or twist as shown in illustrated in Fig. 3-1. In a recent study [21], it was estimated that

this intra-voxel orientational heterogeneity (IVOH) problem affects one third of white

matter voxels. In those regions, tractography applications based on the diffusion tensor

model may result in artefactual reconstructions of pathi- .va~ [19]. In recent years, high

angular resolution diffusion imaging (HARDI) techniques have been able to address the

challenges inherent in diffusion tensor imaging and will be the topic of the next sections.













(a) kissing fibers (b) crossing fibers (c) splaying fibers

Figure 3-1. An illustration of subvoxel fiber configurations arising from the intro-voxel
orientation heterogeneity (IVOH).


3.3 High Angular Resolution Diffusion Imaging (HARDI)

Despite the promising results achieved by the diffusion tensor it! s. ./). the diffusion

tensor model is known to be inadequate for resolving complex neural architectures,

particularly in regions with complicated intra-voxel fiber patterns [4, 56, 152, 154].

This limitation of the diffusion tensor model has stimulated exploration of both more

demanding image acquisition strategies and more sophisticated reconstruction methods.

Tuch et al. [151] developed a clinically feasible approach in which apparent diffusion

coefficients are measured along many directions distributed almost isotropically on a










spherical shell in the diffusion wave vector space. Since then, many methods that augment

the angular resolution of the diffusion model have been used in the literature and are

commonly referred to as high r,:ll;larr resolution diffusion :l,,nryl.e..y (HARDI).

3.3.1 Modeling Diffusivity Profiles

In the original HARDI method [151], with diffusion gradients applied along many

directions, the diffusivity profile is calculated by using the scalar version of the Steil1: I1-

Tanner equation (:38) along each direction but does not assume any particular model. In

diffusion tensor in.! I_;by the diffusivity profile is assumed to take the form of Eq. (:316)

which is intrinsically a quadratic model. Since it has been shown that the diffusivity

function exhibits complex local geometry in voxels with orientational heterogeneity

(IVOH) [152, 154] and the diffusion tensor model is inadequate in such situations, various

higher order models have been proposed to approximate the underlying diffusivity profile.

3.3.1.1 Spherical harmonics series

Fr-ank [56] introduced the use of spherical harmonics (SH) series [60] to model the

diffusivity profile
00 1

l=0 z= -1

In (:321) the coefficients airn can he calculated using the spherical harmonics transform

(SHT)

at,>= Dg)*>gd. (:322)

The spherical harmonics series (SHS) in (:321) is truncated so that only the most sig-

nificant terms are included in the expansion. Furthermore this SHS should only include

even-order spherical harmonics due to the positivity (D(g) > 0) and antipodal symmetry

(D(g) = D(-g)) of diffusivity profile. Instead of directly applying SHT, Alexander et al.

[4] used linear regression to estimate air, and also -II_0-r-- -1. a hypothesis testing method

to determine up to which order the SHS should be truncated. If the highest order of SHS

is L, then the number of aine for all even I from 0 to L is (L + 1)(L + 2)/2, which reduces










to 6 when L = 2. Descoteaux et al. [47] emploi- a the real-valued spherical harmonics basis

and proposed to regularize spherical harmonics coefficients using the Laplace-Beltrami

operator.

3.3.1.2 Generalized diffusion tensor imaging

Starting with an extension to the Bloch-Torrey equation, Ojzarslan and Mareci [114]

proposed to use Cartesian tensors of rank higher than 2 to model the measured diffusion

coefficients
33 3
D)(g) = o?- D9ixi..izixi 9 (3-23)
ii=0 iz=0 i3=0
where Dixi,...i, are the components of the Cartesian, rank-1 tensor. To ensure the positivity

and antipodal symmetry of D(g), I must be an even number and Diiz...il have to be

realized as a totally symmetric tensor which contains (l+ 1)(l+ 2)/2 independent elements.

In their work [114], the correspondence between the coefficients in the SHS and Cartesian

tensors of higher ranks was derived as well, which reveals the equivalence between these

two approaches. To generalize the anisotropy measures for DTI, Ojzarslan et al. [117]

also proposed scalar measures in terms of variance or entropy derived from the higher

order tensor coefficients. Recently, Barmpoutis et al. [10] represent a 4th-order tensor as

a homogeneous polynomial of degree 4 in 3 variables, the so-called ternary quartic, and

then impose the positivity of the 4th-order tensors in the estimation from diffusion MRI

data based on the Hilbert's theorem which states that any non-negative ternary quartic

can be expressed as a sum of squares of three quadratic forms. Barmpoutis et al. [12]

further introduced a novel parametrization of the 4th-order tensors for the simultaneous

estimation and regularization of the 4th-order field while preserving the positivity of the

estimated tensors.

It is worth noting that another formulation of generalized diffusion tensor imaging

was independently proposed by [96, 97] where the diffusion process is quantitatively

characterized by a series of diffusion tensors with increasing orders. Similarly, diffusion









kurtosis imaging (DK(I) has also been proposed to characterize the non-Gaussian property

of water diffusion by a so-called diffusion kurtosis '~~-~ is- [71, 86, 98].

3.3.1.3 The limitation of ADC profile

Although the measured diffusivity file can be used to indicate the complexity of

the fiber structure within the voxel, it is important to point out that the maxima of

the diffusivity profile may not correspond to the underlying distinct fiber orientations.

von dem Hagen and Henkelman [154] observed that the peak of the ADC profile measured

from a voxel containing two perpendicular fibers occurs at an angle in the middle of the

two fibers but not in the direction of either. Similar observation was also reported by Tuch

et al. [152], Zhan and Yang [165] in vivo. Due to this fact, the diffusivity profile can not

be used directly for extracting fiber orientations, and one might still need to investigate

the average diffusion propagator by taking the Fourier transform of the signal attenuation

implied by the diffusivity profile as done in [115, 119].

3.3.2 M ult i- C ompart me ntal M models

A direct extension of the DTI model proposed by Tuch et al. [152] assumes that the

diffusion propagator takes a form of Gaussian mixture densities. Under this assumption,

the signal can be modeled as a finite mixture of Gaussians as well:


Scq) = So my~ exp(-bgTDyg) (3-24)


where my is the apparent volume fraction of the compartment with diffusion tensor Dj.

Behrens et al. [20] introduced a simple partial volume model where the diffusion sig-

nal is expressed as the combination of an infinitely anisotropic component and an isotropic

component. A B li-. -1 Ia inference is then used to estimate the model parameters. This

partial volume model was further extended in [21, 67] to allow the estimation of multiple

fiber orientations. However, both extensions require complicated solution techniques to

address the model selection problem properly, for example, the Markov C'I I!1, Monte Carlo










(\!C\!lC) eI~! 1i--; used in [67] and the automatic relevance determination (ARD) used in

[21].

A slightly more complex multi-compartment model, called composite and hin-

dered restricted model of diffusion (CHAR MED) was described in [7, 8]. Similar to the

multi-Gaussian model, CHARMED also interprets the signal as a weighted sum of the

contributions from a highly restricted compartment and a hindered compartment. Note

that the diffusion in the hindered compartment is approximated by a Gaussian while the

diffusion in the restricted compartment is described by a Neuman's model for restricted

diffusion in a cylinder 110 .

3.3.3 Deconvolution Approaches

1 To avoid determining the number of components in the modeling stage and possible

instabilities associated with the fitting of these models, Tournier et al. [14:3] emploi-, I1 the

spherical deconvolution method, assuming a distribution, rather than a discrete number,

of fiber orientations. Under this assumption, the diffusion AIR signal is the convolution

of a fiber orientation distribution (FOD), which is a real-valued function on the unit

sphere, with some kernel function representing the response derived from a single fiber. A

number of spherical deconvolution based approaches have followed [:3, 5, 145] with different

choices of FOD parameterizations, deconvolution kernels and regularization schemes.

More detailed discussion on the deconvolution approach is presented in OsI Ilpter 4 of this

dissertation.

3.3.4 Model-independent Q-Space Imaging Approaches

In contrast to previous approaches which assume diffusion tensor model or multi-

compartmental models, the so-called q-space :lr.:,:ny: (QSI) technique directly employs

the Fourier relation between the diffusion measurements in q-space and the probability

profile in displacement space without invoking any assumption on the underlying diffusion



1 This subsection is reprinted with permission from [7:3]









process. Originally, the q-space imaging principle was used to vield the one-dimensional

density autocorrelation function in real space from time-scale echo attenuation data in

structural imaging of inanimate mate ~ 1 .h [.$, 42]. After the the emerging of the high

angular resolution diffusion imaging (HARDI) [151], several in vivo q-space imaging

techniques have been recently proposed to approximate the ensemble averaged diffusion

propagator in :3D displacement space by performing the full :3D Fourier transform on

q-space measurements using different sampling schemes.

3.3.4.1 Diffusion spectrum imaging

In diffusion spectrum imaging (DSI) [159, 160], the diffusion signal is obtained by

sampling a dense three-dimensional Cartesian lattice in q-space with a very high number

of gradient directions and different h-values. Then the full displacement probability

density function (PDF) of the diffusion, termed diffusion .spectrum in [159, 160] for each

voxel is reconstructed directly based on the Fourier transform relation (2-29)

S(q)/So = P(r)ol iexp'i2rq~ i-
(:325)
P(r) = .F[|S(q)|/So].

It has been shown in [150, 160] that in an isolated system with time invariant (i.e. homo-

geneous) diffusion properties the Fourier transform of the EAP in the stationary state is

real and positive. Thus the signal magnitude information is sufficient to reconstruct the

EAP due to a homogeneous diffusion process. For the purpose of determining the fiber ori-

entations and better visualization, the resulting PDF is usually reduced to an orientation

density function (ODF) by a weighted radial projection:


ODF(u) = (u~T3_26)

with |u| = 1, ru = r, and Z is a normalization constant[160].

Note that in order to make q-space imaging feasible in vivo, DSI adapts the twice-

refocused balanced echo (TRBE) sequence to reduce the eddy- current-induced artifacts









created hv the 180" refusing pulses of the PGSE[127]. In addition, effectively constant

gradients are emploi- II in the TR BE experiment [150]. This modification violates the

narrow pulse condition assumed in the conventional PGSE experiment, and actually

measures the probability of a spin departing from its mean position over time 0 to TE/2

to its mean position over time TE/2 to TE. According to Tuch [150], this so-called

apparent center-of-mass (CO1\) propagator still preserves the orientational structure

of the originally desired diffusion propagator. Recently, diffusion spectrum images have

been successfully acquired on both human and small animal subjects [62, 95, 150]. These

experiments have demonstrated the ability of DSI to resolve complex tissue microstructure

such as intravoxel fiber crossing and divergence; However, the heavy sampling burden of

DSI still makes the acquisition time-intensive and limits the wide spread application of

DSI.

3.3.4.2 Q-ball imaging

The ODF reconstruction by using radial projection in DSI captures the salient

angular contrast of the diffusion function but discards all of the radial information

contained in the diffusion function. Inspired by this observation, Tuch [149, 150, 153]

proposed a model-independent sampling and reconstruction scheme termed q-hall imaging

(QBI). QBI samples the diffusion signal S(q) only on a single sphere and then directly

reconstructs a function defined on the unit sphere by taking the spherical Radon transform

of the diffusion signal 2

The spherical Radon transform, also known as Funk-Radon transform, sends a

spherical function to another spherical function hv the following integration


(R [f]) (u)= f (x)6(uTx)dx (3-27)




2 QBI can also be extended to multiple shell HARDI data. K~hachaturian et al. [82]
describes a two-shell acquisition scheme to improve the sampling efficiency and signal-to-
noise ratio of QBI.









where |u| = 1. Tuch [150] showed that the spherical Radon transform of the diffusion data

sampled on the sphere, (R [S])(u), closely resembles the ODF(u) obtained by the radial

projection of the PDF

ODF(u) = ~ u .(3-28)

Recent studies have expressed QBI's Funk-Radon transform in a spherical harmonic

basis [5, 48, 66]


ODF,(0, ~)= (-)/ ~(,)(3-29)
even I m=-l
where

sim = S(, V)IMm(0,1 4)in Od~idc (3-30)

is calculated from the spherical harmonics transform of the diffusion signal. In addition


difusin sgnl S0, -spherical radon transform -qbl D

difsimn = Jn S(04) (, 4)- sinl Odd >(1)



Figure 3-2. From the diffusion data to orientation distribution function (ODF) via the
spherical random transform with q-ball imaging (QBI).


to the spherical harmonics, other mathematical models for representing spherical functions

or distributions have also been used to express and compute the q-ball ODF, including the

mixture of von-M~ises[102], the mixture of Watson densities[126], spherical wavelets[112],

and spherical ridgelets[103].

The QBI reconstruction has advantages of being efficient and model-independent,

which made it a popular high angular resolution reconstruction scheme in recent works

[30, 122, 123, 148], however, the end result obtained by the radial projection of the three-

dimensional displacement PDF via a line integral is a convolution of the real probability

values with a 0-order Bessel function [149] but not the probability values themselves. This









convolution gives rise to an undesirable "contamination" of the probability along one

direction with probabilities from other directions and induces spectral broadening of the

diffusion peaks. To address this problem, recently alternative approaches attempting to

yield a more accurate approximation of the ODF using the spherical radon transform have

been investigated[1, 147].

3.3.4.3 Diffusion orientation transform

The diffusion orientation transform (DOT), introduced in [116, 119], is a robust and

fast model-independent method. The key mathematical tool used in DOT is the Rayleigh

expansion (plane-wave expansion) which expands a plane wave in terms of the product of

derivatives of spherical Bessel functions and spherical harmonics:


exp(iigr) = xid~rM(/8)E~/r)(331)
1,m

where ji(-) is the 1-th order spherical Bessel function and YEm is the spherical harmonic

function. Inserting the Rayleigh expansion in the Fourier transform relation (2-29), we

obtain

J m~*4~d S"3

where

Iz~~p) 4x q2 (2qr| ) exp(- 4xr22tD(p)) (3-33)

and D(p) is the diffusivity profile (angular distribution of apparent diffusivities) that can

be obtained from the Stei-l: I1-Tanner expression. Note that the integral in Eq. (3-33)

can be evaluated analytically by using the formulas derived in Ojzarslan et al. [119]. With

these powerful tools, DOT is able to transform the diffusivity profiles into probability

profiles either directly or parametrically in terms of a spherical harmonic series (Figure

3-3). The estimated probability function from DOT is also "impure" in the sense that

the end result is the true probability values convolved with a function which cannot be

specified analytically. It is worth noting that much of the blurring in the DOT is due to















Mono- or Multi-exponential
-I I-I .I-Tanner Equation


Transform


Figure 3-3. A schematic illustration of the diffusion orientation transform (DOT).


the monoexponential decay assumption of the MR signal, hence using the extension of the

transform to multiexponential attenuation as described in [119] can alleviate the blurring,

however, it in turn would necessitate collecting data on several spherical shells in the

q-space.

3.4 Conclusion

To conclude this chapter, we simulate diffusion weighted MR signals using a restricted

diffusion model [137] and compute different quantitative profiles described above. The

signal simulation is based on the exact form of the MR signal attenuation derived from the

diffusion propagator for particles diffusing inside cylindrical boundaries [137] which can be

considered as a simplified model for diffusion inside real neural tissues.

For diffusion within a cylinder of length I and radius r, the signal attenuation with

diffusion coefficient D obtained by Soderman and Jansson [137] is as follows:


E(q, 8, a) = Kmr Sn22)
n=0 k=1 m=0 [(12r/T1) (2;7qlr cos 8)2]

x[1 (-1)" cos(2;ql cos 8)] [J (2;qr sin 8)]2
12 [am -, (2xrqpsinO)2] 2~(~, m 9
x~~~k ex -2 7 D










In Eq.(:3-34), 8 is the angle between the direction of the magnetic field gradient and

the symmetry axis of the cylinder, a is the time separation, Jr~ is the mth order Bessel

function, Gkm, is the Akuz solution of the equation J,,(ca) = 0 with the convention colo = 0,

and K,an's are constants defined by K,an = 21,,>.21-,,> where 1A is the indicator function

on a set A.

The gradient directions were chosen to point toward 81 vertices sampled on a unit

hemisphere from the second-order icosahedral tessellation. The orientations in our

1-, 2- and :$-fiber configurations are specified by the azimuthal angles of $1 = :300,

2a = {200, 100"} and #:3 = {200, 750, 1350} respectively. Polar angles for all fibers were

taken to be 8 = 900, so that a view from the x axis will clearly depict the individual fiber

orientations. as illustrated in Fig. 3-4.

Three .straightforward observations can be maade from this Agure: (1) the di~fu~son

tensor model is not able to cheeracterize the IV/OH; (2) the di~f;,. .it;, 14.e..01. does not ;;.:. I1

the correct Aber orientations; (3) the diffusion y-e~..;yl.r'l~l derived profile~s are calpuble of

resolving comp~lex: tissue microstructure~s .such
the profiles presented in the right three columns correspond to the targeted results of three

model-independent techniques: DOT, QBI, and DSI, respectively



















SThe actual results of these three profiles were computed from the method proposed in
this dissertation. See details in later chapters.





Figure 3-4. Various quantitative profiles derived front diffusion weighted signals simulated
from 1-fiber, 2-fiber, and 3-fiber geometries.


Simulated
Fiblers


Traditional
DTI


Diffusivity
Profile


Plr = ro)


SP(r)dr


SP(r)r dr


)tC









CHAPTER 4
METHODS

The fundamental formula in the diffusion tensor imaging (DTI) is the tensorial

Stei-l: I1-Tanner relation [18]:


S(q) = So exp (-trace(BD)) (4-1)

It is derived as the solution to the Bloch-Torrey equation with an anisotropic diffusion

tensor term (3-2) for free diffusion in homogeneous media


i,(t) = !I,(0) exp(- k(u) TDk(u)dzL) (4-2)

where

k(t) = 7 G(u)dia (4-3)

and

B =k(t~k(t)Tdt. (4-4)

As discussed before, the Bloch-Torrey equation only describes the microscopic

phenomena and the measured diffusion MR signal at the voxel level is an macroscopic

quantity. Hence Eq. (4-1) is only valid by assuming the homogeneity of diffusion tensors

in a voxel. And the diffusion tensor estimated from the Eq. (4-1) is called the apparent

diffusion tensor.

In this work, we model the measured diffusion MR signal in a voxel using the ensem-

ble sum of the individual transverse magnetization vectors:




where each individual 14, is modeled by Eq. (4-2) with its own diffusion tensor. Suppose

the diffusion tensors in a voxel are distributed according to a density function f(D), then

combining (4-1) and (4-5), we propose the following tensor distribution model for the









diffusion weighted MR signal:


S(q) =So ex~'.p (-trace(BD)) f (D)dD (4-6)


where ?P, denotes the manifold of n x n symmetric positive-definite matrices, and

by default, refers to the the manifold of 3 x 3 symmetric positive-definite matrices

throughout this work. The key postulation in the proposed model (4-6) is that each voxel

is associated with an underlying probability distribution defined on the space of diffusion

tensors. Clearly, Eq. (4-6) is a more general form of multi-compartmental models and

simplifies to the diffusion tensor model when the underlying probability measure is the

Dirac measure.

In this chapter, we will first review the necessary mathematics for studying the

integration on ~P,. Interestingly, Eq. (4-6) is exactly the Laplace transform of a prob-

ability distribution on ?P, whose formal definition is given later in this chapter. Since

diffusion tensors are used to depict the time dependence of the covariance matrices of

random molecular displacements, it is natural to choose this distribution as the Wishart

distribution [163] on which we also present a short discussion in this chapter. The signal

decay associated with a Wishart-distributed random tensors is no longer a Gaussian, but

a Rigaut-type .I-i-~!lp)tic fractal expression given by the closed form Laplace transform

for Wishart distributions. The technical details of this closed form Laplace transform will

be derived later in this chapter. This Rigaut-type .I-i-mptotic fractal expression has been

used in previously published literature [84] to explain the MR signal decay phenomenolog-

ically. To the best of our knowledge, it is our statistical model that first gives a rigorous

mathematical justification of this Rigaut-type .I-i-mptotic fractal expression. Furthermore,

our formulation is readily extended to a mixture of Wishart distributions to tackle the

multi-fiber reconstruction problem. In fact, DTI and the multi-compartmental models

are limiting cases of our method when the tensor distribution is chosen to be a Dirac

distribution or a mixture of Dirac distributions. In the last section of this chapter, out









continuous tensor distribution model is then put into a unified deconvolution framework

[73], and several deconvolution schemes are further designed and investigated to achieve

stable, sparse and accurate solutions.

4.1 Some Mathematics on ?P,

4.1.1 Measure and Integration on ~P,

To define integration on ~P,, which is not a vector space, we need to introduce some

fundamental facts about the geometry and the measure on ~P,. Consider the general linear

group G = GL, of nonl-sing~ular Ix n rLeal mlatric~es andu define lthe auction of geG c on

Ye ?, by

Y Y [g] = gTYg. (4-7)

It is easy to show that G acts transitively on ?P, according to the action [g] defined in

(4-7), whichl imlies that~ ?, is a homogUenou~s spacet of ~lthe general Ilinear group GL,. (e

[131, 139] for the definition and meaning of homogenous spaces.)

In the following, we will also see that ?P, has a GL,-invariant measure (volume

element). By GL,-invariant measure (volume element), we basically means that if dp, is

a G~L,-invariant volume element on ~P,, then dp,(Y) = dp,(Y [g]) should hold for any

g E GL,. Since for any Y, WE ~P,, there is ag E GL, such that W = Y [g], then we

should have dp,(Y) = dp,(W) for any Y, We p, which is a quite useful property for

doing integration on ~P,.

As a convex cone embedded in the space of symmetric matrices, ?P, has a natural

induced Lebesgue measure defined as the direct product of the Lebesgue measures over the

independent elements of the matrix variable, that is, for Y = (yij) E pn,


dY =dy (4-8)


where dayi is the Lebesgue measure on RW. Though dY is commonly used for integrals

involving functions of matrix argument, it is not a GL,-invariant volume element on ?P,










which can be shown by the fact that


Y = aX 4 dY = an(n+1)/2dX (4-9)


where a is a scalar quantity.

To find the relation between a GL,-invariant measure and the dY, we first consider

the Jacobian of the mapping induced by the group action with respect to dY.

Theorem 1. [101, p.82],{189, p.19] Let J(g) be the Jacobian of the I'trry.::ll


Y H Y[g] = gT~g


with respect to dY for g e GL,, i.e. J(g) = |dW/dY | for Y e Sym, and W = Y[ g].

Then

J(g) = | det(g)|i" (4-10)



Theorem 2. [189, p.18] Let dp, be the measure on ?, I.lb:

n-E1
dpl = dpl(Y) = (det Y)- z dY. (4-11)


The&IUn dp, is th G~L,-invariant volume element on ~P,.

Note that a special case of this measure when n = 1 is just the scale-invariant

dp-(x) = (1/x)dx = d log x for x e (0, 00).

4.1.2 The Laplace Transform on ?P,

In this section we first give the definition of the Laplace transform on pn which pIIl we

a central role in our model, and then we briefly discuss two kinds of matrix argument

special functions that are used later for describing our model.

For definition of Laplace transforms on ~P,, we follow the notations in [139, p.41].









Definition 1. [189, p.41] The Lap~lace I,,r,,,4,rm of f : ~P, C,@ denoted by 2f, at the

/ iiiili. matrix: Z E Cox" is 1, Ji, .1 by:


ff (Z) f i(Y) ex -rnaceYZ)]dY (4-12)

where dY = dayi 1 < i < j < n.

An equivalent definition of the Laplace transform for matrix-variate functions is also

given in [101, p.255].

The Laplace transform converges in the right half plane, Re(Z) > Xo, for a sufficiently

nice function f, where Re(Z) denotes the real part of Z and A > B means A Be ~P,.

Theorem 3. [65, pp.479-480]. The inversion formula for this Lap~lace I,,r,:,,,,rm is:


(2i-~+)2ff(Z) exp [trace(YZ)] dZ f (YfrY r (4-13)
ne=xo 0, otherwise.

Here dZ = n dzzy and the :,.J Iyeal is over ;; ./ i~ii. matrices Z with ix~ed real part.

If f is the density function of some probability measure FT on ?P, with respect to the

dominating measure dY, i.e. dPT(Y) = f(Y)dY, then Eq. (4-12) also defines the Laplace

transform of the probability measure FT on ?P, which is denoted by MFT. Note that the

L/aplace tr t-fr, tsI, can also be 7. fr...l ~ by replacing the dY with the invariant volume ele-

ment dp,(Y) discussed in the preceding section and I,;.;;.9. y1 f (Y) to f (Y) (det Y) il" )

accordingly. In the rest of this work, the Lap~lace i,r,,,, 4,>rm & of a matrix-variate function

1~ :;,.. on ~P, will be interpreted as integration with respect to the invariant measure dp,

unless il.. 11/l~ stated.

In order to understand the property of the model being discuss later, we digress for

moment to study two important matrix argument special functions on ~P,, namely, the

power function and the gamma function.

The most basic special function on ?P, is a generalization of the complex power

function y*~, y7~E P1 = RW+, s EC and is defined as follows:









Definition 2. [189, p.89] The power function p,(Y) for Ye EP,, and a = (sl, .. s) E



ps(Y = (et Y)" ,(4-14)
j=1
where Yi is the jth leading principal minor of Y.

Proposition 1. [189, p.89] If Ye E and U is an upper t,..attyul;,lrr matrix;, then

ps (Y[U]) = p,(Y)p,(I[U]).

The proof for this proposition is given in [139] where other interesting properties of

power functions are also discussed, for example, power functions are eigfenfunctions of

invariant differential operators on ~P,.

The ordinary gamma function of a complex number z with positive real part is

defined by:

F(z) = tz- etdt= to (-u)dlog(t) = 2(tz)(1) (4-15)

which can be viewed as the Laplace transform of a power function evaluated at the

identity. Similarly, the multivariate Il.::::::;.; function for ~P,, denoted by F,(s), is defined

by:

Definition 3. [189]


En~) P(Y)ex(-race(Y))dyl(Y), (4-16)




Re( si) > (j )2 ,.,n.


In fact, the multivariate gamma function En(s) can be expressed as a product of

ordinary gamma functions 01 = 0 as given in the following theorem:

Theorem 4. [61, pp.19/, [189, pp.41]



j= 1 i= j











Let 2(p,)(Z) be the Laplace transform of the power function p,(Y), note that

the multivariate gamma function F,(s) is defined as (p,) (I). Herz [65] proved that

M(p,)(Z) is absolutely convergent for any complex symmetric matrix Z with Re(Z) > 0
and leads to the following quite useful identity:

Theorem 5. [61, pp.19/, [189, pp.41], [101, pp.254]


2 (ps)(Z) =~ p. Y) e~xp(-tmrrac(YZ7))d,() (Z )u() (4-18)

for Re(Z) > 0 and Re(C" s;) > (j 1)/2, j=1,.. n.
4.1.3 Wishart and Matrix-Variate Gamma Distributions

In Theorem 5, let n = 1, a = k > 0, Z = 0 > 0, the fumetion


f (Y)= Y"e~le (Y > 0) (4-19)
Okf (k)

integrates to 1 with respect to the invariant measure Y- dY and actually gives the density
ftmetion of a gamma distribution of a shape parameter k and scale parameter 8. For

general n, let s = (0,. ., p) and Z = E-l > 0, then we have p,(Y) = p,,, !,) (Y) = (det Y)"
and

ii, (etC)P,()I (det Y)Y exp(-trace (YE- ))dp(Y) = 1 (4-20)

where



Hence, the ordinary gamma distribution can be naturally generalized to the matrix variate

case as follows [91]:

Definition 4. {91]l For EE E ailnd for p i~n AZ = { 1, "-1 } U (" ,0 ther W~ishart

distribution y,,c with scale parameter E and shape parameter p is [n 1s


dy,,r (Y) = F,(p)-l (det Y)p-(n+1)/2 (det E) exp(-trace(E- Y)) dY. (4-22)









It is easy to see from the above definition that the matrix variate gamma distribution

is closely related to the Wishart distribution, a probably more well known name, in

multivariate statistics [6, 61, 109].

Definition 5. [61] A random matrix: Ye E is said to have the (central) Wishart

distribution W,(p, E) with scale matrix: E and p degrees of freedom, a < p, if the joint

distribution of the entries of Y has the following 1 ,.'-.11/ function with respect to the

L/ebesgue measure dY.


f (Y) = c (det Y)(p-a-1)/2 (det E)-p/2 exp -taeE ),(-3


with E E ?P, and c = 2-up'/20a(p/2)-]

Note that the correspondence between the two notations (4-22) and (4-23) is simply

given by yp/2,2E = W,(p, E). In the rest of this dissertation, we will keep using the

notation y,,c but will refer to Wishart distribution and matrix-variate gamma distribution

interchangeably.

The Wishart distribution is one of the most important probability distribution

families for nonnegative-definite matrix-valued random variables. It has been typically

used for describing the covariance matrix of multivariate normal samples in multivariate

statistics [109]. Its importance in the study of multivariate statistics can be seen from the

following property:

Theorem~ ~ ~~I 6. {61, p.8,19 p.88 L i be a random variable in R", for i = 1, 2, .. ., n,

where n < p. And suppose that X1,...,X, are I,,;,linall;i independent and distributed

according to NV(0, ); that is, normal with mean 0 and covariance matrix: EE ?P, Let

X = (X1,...X,) E Rnxp and Y = XXT. Then with ] <..7.11,*/,*///l 086, Y iS in ~Pn Gnd iS

distributed I. I;,.../ by (428).

As a natural generalization of the gamma distribution, the Wishart distribution

preserves the following two important properties of the gamma distribution [61, 91, 101]:









Theorem 7. [61] The Lap~lace I,,r, J~rm of the (generalized) Il.::::::;.; distribution y,, is


exp(-trac( 8Y)) dyl:(Y) (det(I, + OE))-" (4-24)

where (8 + E-l) E ~P,.

Theorem 8. [61] Let Y be a random variable (matrixc) with a (matrix;-variate) Il.::::::;.;

distribution y,,, then its expected value Eax C(Y) is pE.

However, it should be pointed out that the expected value pE does not yield the max-

imum value of the density function in (4-22) or (4-23). For instance, the mode of gamma

density function expressed in (4-22) occurs at (p 1)E. Interestingly, if the dominating

measure is chosen to be the GL,-invariant measure, dp(Y) = (det Y)-(n+1)/2dY, then the

corresponding density function, which is


dy,rc(Y) = F,(p)-l (det Y)" (det E)-" exp(-trace(E- Y)) dp (4-25)

does reach its maximum at the expected point pE.

Proof. Let X = YE-1, then

dy,rc d(det X)" exp(-trace(X))
dX dX

=p(dlet X)"X- exp(-tIrace(X)) (dlet X)"exp(-trac~e(X))I (426i)

=(det X)" exp(- trace(X) )(pX I)

When X = YE-1 = pl, the above derivative becomes zero which implies that the density

function reaches maximum when Y = pE. O

The difference between the density functions of gamma/Wishart distribution with

respect to the two different carrier measures is illustrated in Figure 4-1.









density functions of y4


density functions of Y1


Figure 4-1. Plots of density functions of gamma distribution Y4,1 W.r.t the non-invariant
measure and scale-invariant measure. The x-axis in the right figure is on a log
scale. Clearly, the expected value 4 corresponds to the peak of the density
function w.r.t. invariant measure but not for the case of the non-invariant
measure .


4.2 The Expected MR Signal from Wishart Distributed Tensors

Substituting the general probability distribution in Eq. (4-6) by the Wishart (matrix-

variate gamma) distribution dy,,s, we obtain


S(q)/So = |1, + BE|--


(4-27)


by applying the closed-form Laplace transform of Wishart distribution stated in Theorem
7. If the B-matrix is approximated by B = by g T, then we further have


S(q)/So = |1, +BE|-"= (1 + (bgT S)-p


(4-28)


1 + (bgTCg), we first prove the following


To establish the second equality, |1, + BE|


useful results on the Schur

Theorem 9. [124] Let P

Then


complement in linear algebra and theory of matrices.

AB" ~ ~;----r


(1) if |A| / 0 then |P| = |A| |D CA-1IB.

(2) if |D| / 0 then |P| = |A BD-1C| |D|.









Note that in Theorem 9, the matrix D CA-1B (or A BD-1C) is called the Schur

complement of A (or D) in P [166].

Let PI = Then 1 + by TCg is the Schur complement of I in P, and we

have


det P= det Idet(1 + bgTCS STC


In an analogous manner, I + b(Eg)gT is the Schur complement of 1 in P which yields


det P = det(I + b(Eg)gT)det@1).


Combining the above two equations and noting that B = by gT, we have


|1, + BE|= |1, + EB|= |1, + bCggT|= 1 +(bgTC)

Note that the well known identity |1,2 + BE| = |1,, + EB| can also be derived by applying

Theorem (9) on the Schur complements of -e

Consider the family of Wishart distributions 7;>z and let the expected value he

denoted by D = pE. In this case, Eq. (4-28) takes the form:


S(q) So (1 +(bg TDglp)-"


(4-29)


This is a familiar Rigaut-type2 .Iinji**lle'~ fractal expression [129]. The important point

is that this expression implies a signal decay characterized by a power-law in the large-|q|,

hence largfe-b region exhibiting .I- inia.,l'tic behavior. This is the expected .I-i-in!llani s

behavior for the 1\R signal attenuation in porous media [135]. Note that, although this



2 The phrase "Rigaut-type" is used to distinguish this function from Rigaut's own for-
mula [129] function. Although slightly different, the Rigaut-type function shares many of
the desirable properties of Rigaut's own function such as concavity and the .I-i-inia.,'tic
linearity in the log-log plots.









form of a signal attenuation curve had been phenomenologically fitted to the diffusion-

weighted MR data before [84], to the best of our knowledge, the proposed Wishart
distribution model is the first rigorous derivation of the Rigaut-type expression that was

used to explain the MR signal behavior as a function of b-value. Therefore, this derivation

could be useful in understanding the apparent fractal-like behavior of the neural tissue in

diffusion-weighted MR experiments [84, 118].

Here a question naturally arises. How to choose the parameter p? In our model,

we choose p to be 2 based on the following theoretical consideration. The .-i-mptotic

form of the signal attenuation due to diffusion/scattering from inhomogeneous materials

obeys the well known Debye-Porod law [46] which states that is a power-law. And in
three-dimensional porous media the signal decays as E(q) ~ q-4. Since the p-value is

defined to be the exponent of the b-value in Eq. (4-29) and b is proportional to q2, the

most meaningful choice should be p = 2.


Rigaut-type
asymptotic fractal_
Gaussian








expected sigrmnal;
intermediate surns
all 10000 tensors


OO


Figure 4-2. The Wishart distributed tensors lead to a Rigaut-type signal decay.
Reproduced with permission from [75] @[2007] Elsevier.


6i5


0 .1 00 0


0.0100o



0. 0010



0. 0001

0.0


0. 10

Y r(in-1')









The relation (4-29) can be empirically validated by the following simulation. From a

Wishart distribution with p = 2, we first draw a random sample that contains a random

rank-2 tensors {D1,...,D,} and then simulate the corresponding multi-exponential signal

decay using a discrete mixture of tensors:


E.(q) =S(q)/So = exp (-bgTDig). (4-30)
i= 1

For each sample size n, we plot the signal decay curve by fixing the direction of diffusion

gradient q and increasing the strength q = |q|. The relation between signal decay behavior

and the sample size is illustrated in Figure 4-2. The left extreme dotted curve depicts

the signal decay from a mono-exponential model, where the diffusion tensor is taken to

be the expected value of the Wishart distribution. The right extreme solid curve is the

Rigaut-type decay derived from (4-29). Note that the tail of the solid curve is linear

indicating the power-law behavior. The dotted curves between these two extremes exhibit

the decay for random samples of increasing size but smaller than 10,000. The dashed curve

uses a random sample of size 10,000 and is almost identical to the expected Rigaut-type

function. As shown in Figure 4-2, a single tensor gives a Gaussian decay, and the sum

of a few Gaussians also produces a curve whose tail is Gaussian-like, but as the number

of tensors increases, the attenuation curve converges to a Rigaut-type .-i-mptotic fractal

curve with desired linear tail and the expected slope in the double logarithmic plot.

If p is an integer, then a well known result is that the gamma distribution y,,

also describes the distribution of the sum of p independent exponentially distributed

random variables with parameter a. It follows from the central limit theorem that if p

(not necessarily an integer) is large, the gamma distribution y,~, can be approximated

by the normal distribution with mean pa and variance pa2. More precisely, the gamma

distribution converges to a normal distribution when p goes to infinity. A similar behavior

is exhibited by the Wishart distribution. Note that when p tends to infinity, we have


S(q) iSo exp(-bgTg ( 1









which implies that the mono-exponential model can be viewed as a limiting case (p

00) of our model. Furthermore, the Taylor expansion of (4-29) in the long wavelength or

low-q regime leads to a quadratic decay:


S(q)/So = 1 4xr2 TDq(A 6/3) + O(q4) _432)


which can also be derived from the diffusion tensor imaging model [14]. Therefore, Eq.

(4-29) can be seen as a generalization of Eq. (3-17). By the linearity of the Laplace

transform, the bi-exponential and multi-exponential models can be derived from the

Laplace transform of the discrete mixture of Wishart distributions as well.

When a Wishart distribution used as the mixing distribution in Eq. (4-6) with a

pre-specified p parameter, Eq. (4-28) can be rewritten as


S(q)So)- ,,, 43


Let K be the number of diffusion measurements acquired at each voxel, the above

equation can be further expressed in a matrix form:

1 \

(S2)- B, -- 2Bz C,, 1
(4-34)


(SK)- Bz Ex -- 2Bzz / Ezz 1

where Bij and Egy are the six components of the matrices B and E, respectively. The lin-

ear system formulated in (4-34) clearly -II_0-1-;- -a linear regression method for estimating

diffusion tensors from diffusion-weighted images, which is very similar to the traditional

diffusion tensor estimation methods [16]. Note that in this model the final estimation

of diffusion tensor D is given the expected value of the Wishart distribution y,,4, i.e.

D = pE.

It is worth noting that ]rn Ilw alternative methods which involve nonlinear optimiza-

tion and enforce the positivity constraint on the diffusion tensor, as in [34, 158], can be










applied to the direct nonlinear problem:


mmi (S(q) So(1 + trace(BE))")2 (4-35)

formulated from the Wishart model proposed here. Similarly, the resulting diffusion

tensor field can also be analyzed by numerous existing diffusion tensor image analysis

methods [161].

4.3 Methods for Multi-Fiber Reconstruction

4.3.1 The Mixture of Wisharts Model

The density of a simple Wishart distribution as a function of diffusion tensors reaches

one single diffusion maximum at its expected value, therefore, a single Wishart model

can not resolve the intra-voxel orientational heterogeneity. Clearly, the Laplace transform

relation between the MR signal and the probability distributions on ?P, naturally leads

to an inverse problem: to recover a distribution on ?P, that best explains the observed

diffusion signal. The difficulties entailed in inverting the Laplace transform are well known

(see [54] for "cogent reasons for the general sense of dread most mathematicians feel about

inverting the Laplace it~ lI!-1its~! ), especially for noisy data in high dimensional space,

which is the case in our application domain. In order to make the problem tractable, the

following simplifying assumptions are made:

First, just as a discrete mixture of tensors model can be adapted, so can a discrete

mixture of Wisharts model where the mixing distribution in Eq. (4-6) is given as


f (D) =~ 7, (D) (4-36)
i= 1

where r; .: (D) is the density function of the matrix-variate gamma distribution

dr; Note that in this model the set of (pi, Es) is treated as a discretization of the 7-

dimensional parametric family of Wishart distributions. Hence, the number of components

in the mixture, NV, only reflects the resolution of this discretization and should not be

interpreted as the expected number of fiber bundles.










Secondly, it is further assumed that all the pi take the same value, pi = p = 2, which

is a reasonable assumption based on the analogy between Eq. (4-29) and the Debye-

Porod law of diffraction [135] in 3D space. Since the fibers have an approximate axial

symmetry, it also makes sense to assume that the two smaller eigenvalues of diffusion

tensors are equal. In practice, the eigfenvalues of Di = p~i are fixed to specified values

(Ai, A2, 3S) = (1.5, 0.4, 0.4)p2m MSCOnSIStent With the values commonly observed in white-

matter tracts [152]. Due to this rotational symmetry, the discretization of ?P, forming the

mixture of Wisharts is reduced to a spherical tessellation (Figure 4-3). Accordingly, the

prominent eigenvector of each Es can be taken from the unit vectors uniformly distributed

on the unit sphere. Because of the antipodal symmetry, the sampling is actually performed

on the projective plane, i.e. Only half of a normal spherical tessellation is used.













(a) 92 vertices (b) 162 vertices (c) 252 vertices

Figure 4-3. Sphere tessellations using an icosahedron subdivision model with different
iteration numbers.


Note that all the above assumptions leave us with the weights w = (I,) as the

unknowns to be estimated. Given K measurements with qj, the signal model equation:


Scq) = SoC it -(1 + trace(BEi))-P. (4-37)
i= 1

leads to a linear system Aw = s, where s = (S(q)/So) contains the normalized measure-

ments, A is the matrix with Aji = (1 + trace(Bj~i))- and w, = (I, ) is the weight vector









to be estimated. The details of solving Eq. 4-37 will be the central topic of the upcoming
sections .

After obtaining the vector w, the diffusion weighted MR signal attenuation model

E(q) = S(q)/So can be analytically expressed using the general form as in Eq. (4-43).
As a result, the displacement probability P(r) can then be computed by the Fourier

transform P(r) = f(S(q)/So) exp(-iq r) dq where r is the displacement vector. The

P(r) function defined above describes the probability for water molecules to move a fixed
distance and has been emploi- II in the DOT method [119].

Note that when the mixture of Wisharts model is used, the resulting P(r) can be

approximated as a mixture of oriented Gaussians

P(r, -) =~ exprl(-qTUqr) dF(D) exp(-iq r) dq

exr"p(-qT) qr)ep(-iq r)dq dF(D)
1 (4-38)

eC xp(Ta-r Dr/47) d(D
i' -1~l~


where Di = p~i are the expected values of r;

In this case, many of the quantities produced by other methods including the radial

integral of P(r) in QBI [149] and the integral of P(r)|r|2 in DSI [160] are analytically
computable. This fact provides us the opportunity to understand these quantities and
evaluate their performances in resolving complicated local fiber geometries.
4.3.2 A Unified Deconvolution Framework

Interestingly but not surprisingly, the proposed mixture of Wisharts model (Eq. 4-37)
can be cast into a general deconvolution framework

S~q)/o = ~ q, ) f x~dx(4-39)

that unifies several model-dependent HARDI reconstruction approaches in literature.









The signal in Eq. (4-39) is expressed as the convolution of a probability density

function f(x) and a kernel function R(q, x). The integration is over a manifold MZ/, each

point, x, of which contains information indicating fiber orientation and anisotropy. The

convolution kernel, R(q, x) : RW3 X iZ/1 HWmodels how the signal receives response from

a single fiber. In order to handle the intra-voxel orientational heterogeneity, the volume

fractions represented by a continuous function f(x) : MRR I models the distribution of

fiber bundles. Hence, the deconvolution problem is to estimate f(x), given the specified

R(q, x) and measurements S(q)/So.

A common approach in practice is to represent f(x) as a linear combination of NV

basis functions: f(x) = E wyfj(x). The choice of convolution kernels and basis functions

often depend on the underlying manifold MZ/. A simple example is to let MZ/ be the unit

sphere (or more precisely, the projective plane due to the antipodal symmetry), which

leads to the spherical deconvolution problem [3, 143]. Several other approaches start from

the manifold of diffusion tensors, but again reduce to the spherical deconvolution problem

since only rotationally symmetric tensors are considered. [5, 125, 143].

Following [5, 125, 143], we choose the standard diffusion tensor kernel in our model.

However, it is the Wishart basis function that distinguishes our method from these related

methods. It is worth noting that the Wishart basis reduces to the Dirac function on ?P,

when p = 00 and thus leads to the tensor basis function method [125] as well as to the

FORECAST method [5], which both estimate fiber orientations using the continuous

axially symmetric tensors and hence resemble our method very closely.

Table 4-1 summarizes some of the existing multi-fiber reconstruction methods in the

above described unified deconvolution framework. The first approach listed in Table 4-1

corresponds to the well known diffusion tensor imaging method Basser et al. [18] where the

mixing distribution in the convolution integral is taken to be a single Dirac distribution


S(q)/BSo~ = ep(-qTDq)6(DD l^)dD exp(-qT ^) (4 40)









Table 4-1. A list of previously published fiber reconstruction methods expressed in the
proposed unified deconvolution framework. See text for meaning of symbols.
Reproduced from with permission. @[2007] IEEE.

Method MZ/ x f (x) R(q, x) unknown
Basser et al. [18] ?,n D 6(D; D) exp(-qTDq) D
Tuch et al. [152] ~P, D I,, JC(D; Dk) exp(-qTDq) { Tournier et al. [143, 146] S2 U Clm~m exp(-(U q)2) (Clm
Anderson [5] S2 Uv Clm~m exp(-qTDyq) {czm)
Alexander [3] S2 U, ,i, e? ex(( ) t
Jian et al. [75] pn D Ramirez-Manzanares et al. [125] S2 U t23 k exp(-qTDyq) w={<*}


where ?P, is the manifold of 3 x3 positive definite matrices, 6(x; xo) denotes the Dirac delta

function centered at point xo and the problem is to estimate the unknown D that best fits

the diffusion tensor model.

Instead of the single Dirac delta function used in the diffusion tensor model, all other

methods listed in Table 4-1 express the fiber orientation distribution f(x) as a linear

comblination of K basis functions f(x) C~= I E to fk(x). For. example, the second approach

listed in Table 4-1 corresponds to the multi-tensor model [152]
K K
S(q)/So = exp(-qTDq) C (D; Dk)dD expl(-qTDkf) 1

In the original multi-tensor model proposed by [152] the objective is to find a set of

Kt tensors Dk and corresponding fractions no by performing conventional nonlinear

optimization algorithms. Even though the eigfenvalues of Dk are Set tO Specified values

to favor physiological solutions, it is usually desirable to fix the number K to be a small

number, 2 or 3, to ensure the stability of the nonlinear optimization, which in turn

introduces a model selection problem.

Due to its close relation to the diffusion physics, the diffusion tensor response kernel


R(q, D) = exp(-qTDq) (4-42)









is still widely used in a number of recently proposed reconstruction methods. However,

unlike the traditional diffusion tensor model and multi-tensor model where K is set to a

small number (1,2,or 3), these extended diffusion tensor based models tend to let K be a

relatively large number in order to gain a quite good coverage of the underlying manifold

MZ/. In this sense, the number K only depends on the resolution of the discretization

on the manifold MZ/ and should not be interpreted as the number of expected fiber

populations.

In order to avoid the difficulty of directly working on the manifold of diffusion tensors

~P,, many models take the similar assumptions used by our model. Some models [75, 125],

only consider a set of K rotationally symmetric tensors, DI, with specified eigenvalues ,

A = { At > a2 =3 }, but varying dominant eigenvectors, {vl, .. ., vK } obtained from a

sphere tessellation. For this reason, the underlying domain of deconvolution is reduced

to the sphere from the manifold of diffusion tensor, ~P,. It is also worth noting that the

response kernel exp(-(v q)2) used in [3, 143] is just the special case of the exp(-qTDyq)

when X2 = 3 = 0. However, it is the Wishart basis function that distinguishes our method

from these related methods since Yp,Dy aS a COntinuOUS distribution on ?P, implicitly takes

contributions from the entire ?P, even there are only K discrete Dy. It is also interesting

to note that the Wishart basiS p,~Dyx reduces to the Dirac function on 7P, when p 00o and

thus the models in [5, 125] can be treated as special cases of our model.

Finally, except for the single diffusion tensor model and the multi-tensor model

[152] where both the volume fraction I, and the parameter of basis functions (Dk) are

unknown, all the other methods lead to a problem that can be expressed in the general

form of

Aw = s + rl, (4-43)

where, on the righthand side, s is a column vector containing NV multi-directional mea-

surements, {si} = {S(qi)/So}, and rl represents the noise, while on the lefthand side,










A = {aik) is an 1VX K matrix given by aik, = MI RGi, x) k(x)dx and w is the un-

known weight vector. Note that the integral to compute the entries of A may have an

analytical solution as in [75] or needs to be numerically approximated as in [3], depending

on the choices of convolution kernels and basis functions. But once the response kernel

R(q, x) and the basis function are specified, the matrix A can be fully computed (or

approximated) and only w, a column vector containing K unknown coefficients, remains

to be estimated. It is worth noting that a further spherical harmonics (SH) expansion on

w in [5, 143, 146] leads to a parametric deconvolution of w in terms of SH coefficients.

However, it is straightforward to change this parametric deconvolution back to the direct

nonparametric deconvolution of w. In our opinion, it is more natural and easier to inter-

pret and handle the weight vector than the SH coefficients vector. For this reason, we will

only discuss the nonparametric deconvolution of w, in this work.

4.4 Computational Issues

In the terminology of signal representation theory, the NVx K matrix A in Eq. (4-43)

as a collection of vectors ai E R"N, i = 1... K is usually referred as a "J..i.o ,:.re;, and the

column vectors (ai) are called "rlia;,, ". The deconvolution problem is then equivalent to

representing the signal s = Ci f as a linear combination of atoms in the dictionary A.

If we require all the weights to be non-negative, then the linear combination is restricted

to a conic combination. In addition, a parsimonious representation for signal is preferred if

only a few atoms are able to produce the signal s.

In the normal clinical setting, the number of diffusion MR image acquisitions, K, is

rarely greater than 100. On the other hand, a high resolution tessellation with NV > 100 is

usually preferred for an accurate reconstruction. This situation yields a II rectangular

system matrix A in (4-43) and in this case the dictionary A is called overcomp~lete. An

under-determined system of equations may have infinitely many solutions in the least

squares sense. To make things even worse, most deconvolution models used in literature

result in extremely ill-conditioned linear systems whose standard least squares solutions










may be -r I---- l in-Oy unstable. This disturbing consequence of the numerical stability issue

can he a real concern as was shown in [5].

Since the weights w = {m, } K1 correspond to volume fractions, they are expected to

be non-negative. Negative weights are not physically meaningful and should be penalized

by adding a regularization term or excluded by imposing an explicit non-negativity

constraint. In addition, it is reasonable to assume that most white matter voxels only

contain contributions from relatively few fiber bundles. Therefore, apart from a few

significant peaks, we expect w has a sparse support, i.e. most entries of w are expected

to be zero (or very small). It is important to note that the sparsity property also has

advantages in the optimization process required to find the maxima of the water molecule

displacement probability function, which represent the fiber orientations in the multi-fiber

reconstruction problem.

4.4.1 Regularization and Stability

The first question one needs to consider before solving the deconvolution problem

is how to measure the goodness of a solution, in our case, the discrepancy between Aw

and s. Commonly used measures include the L2 norm, the L1 norm, or more generally,

an L, norm for 1 < p < 00. If all the elements of A,w,s are assumed to be nonnegative,

then, some information divergence e.g. K~ullback-Leibler (K(L) divergence can also be used

to measure the distance between Aw and s after appropriate normalization. Under the

assumption that the measurement errors are independent and follow an identical normal

distribution, the maximum likelihood estimate of w naturally leads to the L2 Ilorm as a

goodness measure and is equivalent to the corresponding least squares (LSQ) problem [26]

that minimizes the residual sum of squares


(PI) min ||Aw s||2 4 )









The solution of (P1) in the least squares sense is given by w, = A+s where A+ is the

pseudo-inverse of the A given by


A+ = (ATA)-1AT ifA has full colunin rank (-5

SAT(AAT -1 ifA has full row rank. 45

Direct methods are adequate here since in our application the size of the linear system in

Eq. (4-44) is not that large to require iterative methods. 1\oreover, since the matrix A is

independent of the spatial location, the pseudoinverse is only computed once, and hence

the computational burden is light. Despite its simplicity and efficiency, a direct solution

to the linear system is highly susceptibility to noise, especially when A is ill-conditioned,

which is usually the case in our application.

To illustrate this ill-conditioning phenomena, we study two different convolution

models under the unified framework of Eq. (4-39). The first model involves the use of

radial basis functions and a Gaussian kernel with a degenerated rank-1 covariance matrix,

as in [:3]. The second model uses a standard diffusion tensor kernel weighted by a mixture

of Wishart distributions as discussed in [75]. The reason for choosing these two models

for the purpose of comparison is due to the fact that the delta function basis as used in

[5, 125, 14:3] is actually a special case of the Wishart basis when p c o. The further use

of spherical harmonics expansion on w in [5] is equivalent to a parametric deconvolution of

w which is different front the nonparanletric deconvolution of w as discussed in this work.

Consider an example situation wherein we are given 81 gradient directions chosen to

point toward the vertices sampled on a unit hemisphere front a second-order icosahedron

tessellation of the sphere. To fully specify the matrix A, we still need to specify its column

space which is dependent on the number of discretization points on the sphere, i.e., the

discretization resolution on the sphere. We will illustrate the ill-conditioning with two

values of resolution (column size) namely, 81 and :321. Thus, the matrix A will be of size

(81, 81) in one case and (81, 321) in another. As seen from Figure 4-4, both the models









x17
2x 0


12x 0

10~ P



6t

--size(A): 81 x 821


E




o


-a-size(A): 81 x 81
|+size(A): 81 x 321


20 40 60 80 100 ,0 100 200 300
a in the radial basis functions p in the Wishart distributions

The linear system constructed in (4-43) is often extremely ill-conditioned with
very high condition number. The plot on the left shows the profile of condition
numbers when the A matrices are constructed from the radial basis function
and the tensor kernel model as in [3]. The plot on the right shows the case
with a standard diffusion tensor kernel weighted by a mixture of Wisharts. In
both scenarios there are K = 81 diffusion gradient directions. Two tessellation
schemes of different resolution levels (NV = 81 and NV = 321) are considered for
each model. Reproduced from [73] with permission. @[2007] IEEE.


Figure 4-4.


result in extreme ill-conditioning of matrix A. The plot on the left shows the case where

the A matrices are constructed from the radial basis function model. Note the condition

numbers are of the order of le + 06 and steadily increase with increasing o-, the value

of radial basis function parameter. The plot on the right shows the case where the A

matrices are constructed from the mixture of Wisharts model [75]. The condition numbers

in this case are an order of magnitude less than those for the radial basis function model

and quickly converge to an upper limit. Note that when the parameter p in Wishart model

goes to infinity, the resulting system converges to the one obtained in [5, 125, 143].

II I.!y methods aiming at reliable multi-fiber reconstruction in the presence of noise

have been emploi-v I including low-pass filtering [143], minimum entropy [144], and the

maximum entropy spherical deconvolution [3]. Recently, Tournier et al. [145] proposed

to combine the spherical deconvolution with Tikhonov ,ell;,lar,.:~r..rHn, a very popular

technique for solving ill-conditioned problems [140]. In the general framework of Tikhonov









regularization, the original problem in Eq. (4-43) is reformulated as finding the estimate
w that minimizes

(P2) min ||Aw s||2 + I~l2 _46)

where T is a suitably chosen regularization operator that imposes a constrain function

||Tw||2 On W), and a~ is a non-negative scaling factor that controls the balance between the

residual term and the penalization term. In the statistical literature, the method using

Tikhonove regularization is also known as ridge regression.

Typically the matrix operator T is chosen as a discrete approximation to some

derivative operator if smoothness is the desired property of the underlying solution. In

our case, we take T to be the identity operator, I, which leads to a zeroth order Tikhonov

regularization, giving preference to solutions with smaller manitudes. K~awata et al. [81]

described an application of the zeroth order Tikhonov regularized least-squares method to

3-D reconstruction of optical microscope tomography data. They also pointed out that the

Wiener-Helstrom filtering [64] or the minimum mean-square error (1111510;) deconvolution

[80] can also be derived from the zeroth order Tikhonov regularization by letting the

signal-to-noise ratio be a~ and assuming that the signal and the noise are uncorrelated

stochastic processes. Minimization of the objective function in Eq. (4-46) with T = I

yields the following normal equation:


ATs = (ATA + a~TTT~ 47


with an explicit solution given by


w = (ATA + a~TT -1ATs 48


The so-called damped least squares (DLS) method[157], which is equivalent to the

zeroth order Tikhonov regularization technique, is emploi- II in [75] to reconstruct the

mixture of Wisharts representing a continuous distribution over diffusion tensors, where









the solution is given by


w = (ATA + alI)- ATS
(4-49)
=AT(AAT + I-18

In equation (4-49), (ATA + alI)-1AT (or AT(AAT + ~)-1) iS called the damped least

squares inverse of A and a~ is called the damping factor. The damped least sqaures inverse

can be expressed in terms of the truncated Singular Value Decomposition (SVD) [58] as:


(ATA + l)- ATi a cGi UiiT _50)

where r is the rank of A, ai's are singular values of A, and vi(ui)'s are the associated

left-(right-) singular vectors of A.

The non-negative damping factor a~ controls a trade-off between accuracy and

stability. The determination of a suitable a~ to prevent the solution from the under-

regularization or the over-regularization is often am rl i difficulty in the use of Tikhonov

regularization. A similar damping factor appears in the well known and widely used

Levenberg-Marquardt algorithm for solving nonlinear least squares problems and can be

adjusted at each iteration.

Recently, a Damped Least Squares method was used to regularize the fiber orientation

distribution [132], where the optimal damping factor a~ is determined by minimizing the

Generalized Cross Validation (GCV) criterion [155]. The underlying statistical model

used in the GCV method is that the components of s are subject to random errors of zero

mean and covariance matrix a2I, Where a may or may not be known. Let A,+ denote the

damped least squares inverse of A with damping factor a~. The predicted values of s from

the solution in (4-49) is then


li = Aw = AA,+s = Pos (4-51)


where the symmetric matrix P, = A(ATA + alI)-1AT iS called the f'illa. 0..: matrix;.









When a2 is known, it was -II_t-r-- -1.. in Craven and Wahba [45] that a~ should be

chosen as the minimizer of an unbiased estimate of the expected true mean square error

l1 81 2 a2 (4-52)


where ni is the number of rows in A. If a2 is unknown, then the so-called generalized

cross-validation (GCV) function given by


m||s s|| /[trace(I P,)]2 45:3)


may be used to estimate the proper value of c0. The main reason of using this GCV

function is in that the minimizer of (4-53) is .I-i-ingdo' tically the same as the minimizer of

(4-52) when nz is large [59].

The GCV criterion is a one-dimensional function of the regfularization parameter

(damping factor c0) and can he computed from the SVD of A and Pa~. 1\ore efficient

numerical methods based on the bidiagonalization of 4 [5:3, 6:3, 68] have also been devel-

oped; see [26] for more discussions on the minimization of GCV function. However, GCV

does not guarantee the real optimal solution due to the theoretical limits of the GCV

method, as pointed out in [155], "the theory jll-r;fi-;in-4 the use of GCV is an .I-i-ini!!lle'~

one. Good results cannot he expected for very small sample sizes when there is not enough

information in the data to separate signal from in~ ss--

Additionally, it is important to note that the regularization scheme discussed here is

performed voxel by voxel, which is quite different from some other regularization schemes

in the context of smoothing diffusion tensor field [4:3, 121] where a spatial coherence

term is usually added in the minimization problem. The presented unified deconvolution

framework could be combined with such spatial regularization schemes as well (e.g, [125]).

However, the discussion of spatial regularization schemes is not the main focus of this

work.










4.4.2 Nonnegativity and Sparsity Constraints

4.4.2.1 L1 minimization methods

The so-called Lo norm is the most straightforward measure of sparsity by counting

the number of nonzero entries. However, the naive exhaustive search for the sparsest

solution with smallest Lo norm is known to be NP-hard. A recent strand of research has

established a number of interesting results, both theoretical and experimental, on stable

and efficient recovery of sparse overcomplete representations in the presence of noise (see

[32, 49] and references therein). One such important result is that, under appropriate

conditions on A and s, minimizing L1 norm of the solution often recovers the sparsest

solution. This phenomenon is referred to as LI/Lo equivalence. It is shown in [50], that

when the dictionary A has a property of mutual incoherence (defined in [50]) and that

when the ideal noiseless signal s has a sufficiently sparse representation with respect to

A, minimizing the L1 norm of the solution, (|| w ||1 = Ci i' I ) often recovers the sparsest

solution and is locally stable, i.e., under the addition of small amounts of noise, the result

has an error which is at worst proportional to the input noise level. Unfortunately, it

turns out that our systems consistently have very high mutual coherence values, in the

range of 0.95 to 1, which makes these nice theoretical hounds inapplicable to our problem.

To investigate the performance of L1 minimization based methods in the context of our

problem, we experimentally tested the L1-MAGIC package a collection of MATLAB

routines for solving the convex optimization problems central to sparse signal recovery

[31]. Among the several programs implemented in L1-MAGIC package, we are particularly

interested in the following two programs:

*Min-L1 with equality constraints. The program can he mathematically stated as


(P3) min ||w||I1 subject to Aw = s. (4-54)




1 http://www.11l-magic.org










This program is also known as the ba~si~s pursuit [:36] and the solution is the vector

with smallest L1 norm (||w||I1 = Ci 'II |) that explains the observation s. Candes

et al. [:31] have shown that if a sufficiently sparse wi exists such that Awi = .s, then

this technique will find it and when s, A, w have real-valued entries. Note (IS) can

he recast as an linear-programming (LP) problem.

*Min-L1 with quadratic constraints. In this formulation, one finds the vector

with minimum L1 norm that comes close to explaining the measurements s, i.e.,


(P4) min I~11 subject to ||Aw s||2 < e (4-55)


where e is a user specified parameter. Candes et al. [:31] have shown that if a

sufficiently sparse wi exists such that s = Aw + e, for some small error term ||e||2

e, then a solution w, to (P4) Will be clOSe to wi in the sense that || w wil || < C ,
where, TC is asmll ,,cnsant It has1, also~,,, been shown, that,,, (P,,, eTCRtR

second order cone programming (SOCP) problem.

The experimental results for these two programs are reported in Chapter 5. Other

programs implemented in L1-MAGIC but not investigated here include minimum L1 error

approximation, minimum total variation (TV) with equality constraints and minimum TV

with quadratic constraints, etc.

4.4.2.2 Non-negative least squares (NNLS)

A direct solver may produce a solution with many negative-valued components, which

is not physically meaningful. For example, the zeroth order Tikhonov regularization is

able to suppress the large spurious negative spikes in the weight vector w as expected,

however, there are still many negative entries in the resulting w, vector. Note that the

above L1 minimization methods (P3) and (P4) do not explicitly enforce the nonnegativity

constraints either. It has been shown in image reconstruction literature [80, 81] that

constraining the solution to the nonnegative space could drastically reduce the ambiguity

of the solution and hence improve the final reconstruction results. It has been recently









reported in [146] that a spherical deconvolution technique imposed with a non-negativity

constrain on the estimated FOD does not require a low-pass filtering which was previously

----I _-r. .1 in [143].

The least squares problem subject to a nonnegativity constraint is formulated as


(Ps) min ||Aw s||2 Subject to w > 0. (4-56)

The non-negative least squares (NNLS) problem is an important special case of least

squares problems with linear inequality constraints (LSI) [85, Ch. 23]:


min ||Ax b||2 Subject to CTx > d. (4-57)

The LSI problem is essentially a quadratic programming problem that minimizes a

concave quadratic function in a linearly bounded convex feasible hyperspace. Let ce denote

the i-th column vector of the matrix C. The separating hyperplane defined by the i-th

constraint is {x : cTx = di}. Note that the vector ce is orthogonal to this separating

hyperplane and points to the feasible halfspace {x : cTx > di}. Given a feasible point x, it

may exactly sit on some separating hyperplanes, the indices of these hyperplanes form the

set 8 (equality) and the complementary set of 8 is called S (slack). The optimal solution

Ai to the LSI problem should meet the following K~uhn-Tucker conditions [99]:

jri > 0 if ie E i.e. cN X= di
(4-58)
jri = 0 if i E S, i.e. TcN > di

where jr defined by Cjr = AT(AAi b) is the dual vector for this problem.

Proof. [85, Ch. 23] Let g = AT(AAi b) denote the gradient vector of the quadratic cost

function ~(x) = ||IAx b||2 at ~i. The K~uhn-Tucker conditions in (4-58) lead to


fii (4 59)
iLES









which implies that the negative gradient -g at the solution is expressed as a nonnegative

(ysi > 0) linear combination of the outward pointing normals -ci to the constraint

hyperplanes on which ~i lies (ie E ).

Consider an arbitrary perturbation n of 1i, which yields a new value of the cost

function: #(Ai + u) = #(~i) + UTg + ||AU||2/2. Note that for &i + n to remain feasible,

U~cs > 0 for i e must hold, hence a~g = Ci, eg ye c > 0 and it follows that

~(A + u) > ~(i) for any feasible perturbation n of ~i. O

Based on the K~uhn-Tucker conditions in (4-58), Lawson and Hanson [85, Ch. 23]

developed an active set strategy [57] to find the optimal solution to the NNLS problem.

In this classic algorithm, a series of least squares problems without constraints are solved

sequentially according to the tentative status of an active set p, which contains only

positive components and is initialized to an empty set. A positive component is added

to P in the main loop, followed by a possible deletion of some components from P in the

inner loop. On termination, x will be the solution vector and y will the dual vector, both

satisfying the K~uhn-Tucker conditions in (4-58) with C = I and d = 0.

An interesting observation is that the active set strategy emploi- II in this algorithm

tends to find a sparse solution if there exists one, even though the sparsity constraint

is not explicitly imposed. This behavior was also reported in many other applications

[28, 69, 78, 93]. Furthermore, even though the algorithm finds the solution to (P5) it-

eratively, it has been proved in [85] that the iteration alr-ws- converges and terminates

in a finite number of steps. Hence, this algorithm does not require the tuning of cutoff

parameters and the output is not sensitive to the initial guess. The number of iterations

to reach the full convergence, as expected, depends on the amount of noise in the measure-

ments. However, a fairly satisfactory solution can often be achieved well before the full

convergence, since the solution gets improved smoothly with iteration.

Since the number of the insertion/deletion operations is proportional to the size of

the solution vector, an active set algorithm can be slow for large scale problems. In the










Algorithm 1: The active set algorithm for NNLS [85, Ch. 23]
Input : AE RmX", bE Rm
Output: Ai > 0 such that ^< = arg min||IAx b||2
1 begin
2 x := 0; y := AT(b Ax)
3 Z:= {1,2,..., n}; p:= 0
4 while Z / 0 and maxiez yi > 0 do
.5 3 = arg maxiez yi
B Move index j from Z to P
7 Solve the least squares || Cj,? zyAj b|| where Aj is the column j of A
s zj =0Ofor j EZ
9 while min(zy) < 0 do
to a = miniap Xi/(Xi ze)
11 x = x + a~(z x)
12 Find index j in P such that xj = 0
13 MOVe index j from P to Z
14 Solve the least squares || C ,~ zy~ A b||
1s zj = 0 for j EZ

17 y = AT(b Ax)
Is end


context of our reconstruction problem, the size of the matrix is relatively small, at most

hundreds by hundreds, which makes the active set method still a reasonable choice over

some other algorithms for large scale problems [35]. However, in real reconstruction of

volume data, we do have to solve the NNLS problem voxelwise. Unlike the unconstrained

least squares and regularized least squares where the pseudoinverse or the damped inverse

can be computed only once and reused for multiple right-hand sides, the active set method

usually has to solve different sequences of subproblems in its inner loop. To alleviate this

problem, a recent variant of NNLS algorithm [23] is able to avoid the many unnecessary

recomputations by rearranging the calculations in the standard active set method on the

basis of combinatorial reasoning. This so-called fast combinatorial NNLS (FC-NNLS) has

been tested in our experiments and proved much faster than the standard NNLS algorithm

in the real MR volume data.









CHAPTER 5
EXPERIMENTAL RESULTS

In this chapter, we present experimental results on both simulations and real data to

validate the proposed model and methods we have investigated in Chapter 4. Portions in

this chapter are reprinted, with permission, from [73, 75, 77].

5.1 Simulations

In order to compare the performance of the deconvolution methods described in

the previous chapter, a series of experiments were first performed on simulated data.

Throughout this chapter, the signals were simulated by using the exact form of the MR

signal attenuation derived from the diffusion propagator for particles diffusing inside

cylindrical boundaries [137] which can be considered as a simplified model for diffusion
inside real neural tissues.

For diffusion within a cylinder of length I and radius r, the signal attenuation with

diffusion coefficient D obtained by Soderman and Jansson [137] is as follows:

OC 0 002Kamr2(2?iqr)4Sin2(20)a m
E(q, 8, A)=22
n=0 k=1 m=0 [(nr/1)a a(2r cos 8)"2
x[1 (-1)" cos(2;ql cos 8)] [J (2;qr sin 8)]2(51
12 [am, 2xqpsin)!2] 2a: m 2



In Eq.(5-1), 8 is the angle between the direction of the magnetic field gradient and

the symmetry axis of the cylinder, a is the time separation, Jm is the mth order Bessel

function, akm, is the kth solution of the equation J (a) = 0 with the convention a~lo = 0,

and Kam's are constants defined by Kam = 21">o21m>o where 1A iS the indicator function

on a set A.

As in [119, 154], in our simulations we used the following parameters: 1 = 5 mm,

r = 5 p-m, D = 2.02 x 10-3 mm2 S, a = 20.8 ms, 6 = 2.4 ms, b = 1500 s/mm2 and

the infinite series were truncated at n = 1000 and k, m = 10. To simulate 2- and 3-fiber






















Figure 5-1. HARDI simulations of 1-, 2-, and :$-fibers (b = 1500s/nnn2) ViSualized in
Q-hall ODF surfaces using [5, (21)] with spherical harmonics expansion
terminated up to I = 8. The orientation configurations used in the simulation:
azimuthal angles 01 = :300 2 = { 200, 1000 }, 3 = { 200, 750, 1:350 }; polar angles
are all 900. Reproduced with permission from [7:3] @[2007] IEEE.


geometries, we simply used an additive model based on the Eq.(5-1) by mixing signals

simulated from two or three cylinders with known orientations.

The gradient directions were chosen to point toward 81 vertices sampled on a unit

hemisphere from the second-order icosahedral tessellation. The orientations in our

1-, 2- and :$-fiber configurations are specified by the azimuthal angles of $1 = :300,

2a = {200, 100"} and #:3 = {200, 750, 1350} respectively. Polar angles for all fibers were

taken to be 8 = 900, so that a view from the x axis will clearly depict the individual fiber

orientations. Figure 5-1 shows the corresponding orientation distribution function profiles

computed from a model-independent q-hall ODF formula derived in [5, Eq. (21)] with

spherical harmonics expansion up to I = 8.

For the experiments using simulation data, the deconvolution methods described in

the previous chapter were all implemented using MATLAB as follows:


The pseudo-inverse method w = pinv(A)s as in (P1) where piny is MATLAB's
built-in function for computing pseudo inverse of a matrix;
The damped least squares method as a solver of the Tikhonov regularization
problem (P2) aS Well aS the GCV criterion were implemented using the MATLAB
regularization toolbox developed by Hansen [6:3];
The two different L1 minimization methods (?3) ndil (P4) were imp~lll~lemete using
the L1-MAGIC MATLAB package;










*The nonnegative least squares method was implemented by the the 1\ATLAB
optimization toolbox function, Isquonneg.

The first experiment was done on the 1-fiber HARDI simulation data shown in Figfure

5-1 without noise. The deconvolution problem to be solved was formulated with the

matrix A being constructed by using the Wishart model with p = oc and the tessellation

containing :321 unit vectors sampled from a hemisphere. Since the given signal, on the

righthand side, is 81 dimensional, the matrix A is of size 81 x :321 and the unknown of this

under-determined system is a :321 dimensional weight vector.

Figure 5-2 plots the results of w obtained from these methods. The initial guess

for w was set to the zero vector for all the iterative methods (P3), (P4) and (P5). A

qualitative impression of these methods is clearly indicated from Figure 5-2. The least

squares solution to (P1) contains a large number of negative weights and has a relatively

large magnitude. A zeroth-order Tikhonov regfularization is able to reduce the magnitude

significantly but does not help achieve the sparsity and nonnegativity. By minimizing the

L1 norm with equality or quadratic constraints, (P3) yields a relative sparse solution, but

the magnitude and the negative values are not well controlled. The result obtained by (P4)

has better sparsity and nonnegativity. Evidently, the best result is produced by solving

(P5) using the nonnegative linear least squares. Among the :321 components, only two are

significant spikes, both of which lie in the neighborhood of true fiber orientation (:$Oo 90o).

It is important to note that the true fiber orientations do not necessarily occur at the

maxima of the discrete w vector. Although all of these different results of w actually lead

to very similar displacement probability functions P(r) after taking the Fourier transform,

a sparse positive representation of w is much more desirable as it provides very good

initial guess for estimating the extrema of P(r) which correspond to the fiber orientations.

A more realistic comparison on noisy 2-fiber simulation profiles is presented in Figure

5-1. The noisy profiles were created by adding Rician-distributed noise with increasing

noise levels (o- = .01,.02,...,.10). And for each noise level, we generated 100 random











(P1) Pseudoinverse solution 0.2 p3) Min-L1
(Pg) Tikhonov regularization
0.15 with equality constraints

001



-1 -0.05 -4
0 100 200 300 0 100 200 300 0 100 200 300

1.5 0.15
--(31.7, 90.0)
(P4) with t = 0.1 (P4) with t = 0.5 0.8
1~ 0.1
0.61 (P5)
0.51 0.05~11 Non-negative least squares
0.4
0 =- -- 1- C / II p 'I 0.2
(23.8, 90.0)
-0.5 -0.05 0
0 100 200 300 0 100 200 300 0 100 200 300

Figure 5-2. Results of w on 1-fiber HARDI simulation data using different deconvolution
methods. The x-axis shows the indices of w while the y-axis presents the
corresponding numerical values of w. The matrix A is of size 81 x 321 and is
built from the tensor kernel model and Wishart basis with
p c o, A = {0.0015, 0.0004, 0.0004}. Reproduced from [73] with permission.
@ [2007] IEEE.


samples. Then, we formulate the problem as in Eq. (4-43) using the 321 tessellation

vectors and the Wishart model with p= 2. In this experiment, the following numerical

methods were tested to recover the weight vector w:

(a) the pseudo-inverse method as in (P1);

(b) the damped least squares as in (P2) (the damping factor is empirically chosen to be

So- which gives quite satisfactory results);

(c) the damped least squares method with the damping factor determined using the

GCV criterion;

(d) the Min-L1 norm with equality constraints as in (P3);

(e) the Min-L1 norm with quadratic constraints as in (P4) (e = 1.0) ;

(f) the nonnegative least squares as in (Ps)


2









In all these methods, the deconvolution formulation is derived from the mixture of

Wisharts model [75]. The GCV solution was used as the initial guess in both the methods

(d) and (e). For the nonnegative least squares (NNLS) approach, the initial guess for w
was ahr-l- .- set to the zero vector.

As discussed in OsI Ilpter 4, once the deconvolution problem formulated from the

mixture of Wisharts model is solved, quantities including the probability displacement

function restricted on a sphere, P(|r| = r), as in DOT [119], radial integral of P(r), as in

QBI [149], and the integral of P(r)T2 aS in DSI [160] are analytically computable. In this

experiment, the f P(r)T2dT iS chosen as the quantity to be compared and all the resulting

functions are represented by spherical harmonics expansions terminated after order 1 = 8.

First, to assess the noise resistance of these methods, for each noise level, we cal-

culated the similarity between the resulting probability profile and the corresponding

noiseless profile using the angular correlation coefficient formula [5, Eq. (71)]


TA"= =" (5-2)
[E m=- -U1 |wm2 1/2 1 1Im=-1l 2 1/2

where {wm}) and {<.. } are the spherical harmonic coefficients of the two functions to be

compared. The range of the angular correlation coefficient is from 0 to 1 where 1 implies

identical orientational profiles.

We also estimated the fiber orientations of each system by numerically finding the

maxima of the spherical functions with a QuI I-;-N. i.--ton numerical optimization algorithm

and computed the deviation angles between the estimated and the true fiber orientations.

As expected, the results of method (a), the pseudo-inverse method (P1), degrades quickly

as the noise level increases. It has also been observed that method (e), Min-L1 with

quadratic constraints (P4), produces much worse results than the other four methods

(b,c,d,f). In general, the results of method (d), Min-L1 with equality constraints (P3), arT

very close to the results of method (c), the Damped Least Squares method with damping










factor determined by GCV, due to the fact that method (d) starts with the solution of

method (c).

Figure 5-3 shows the mean and standard deviation of the angular correlations and

the deviation angles in the two-fiber experiment, respectively. In these two figures, we

do not include the results of the pseudo-inverse method (P1) and the results of Min-L1

with quadratic constraints (P4) since both methods are extremely sensitive to the noise

according to our observations. The angular correlations obtained using the four methods

shown in Figure 5-:3 are all very high (> 0.9). While the GCV and Min-L1 with equality

constraints are slightly worse with relatively larger standard deviations. It is also observed

that the angular correlations produced by the NNLS method are very close to 1 and have

the smallest standard deviation. The DLS method with the damping factor empirically

chosen to be So- also exhibits quite satisfactory stability. In terms of the accuracy of the

fiber orientation estimation, it is also clear from Figure 5-3 that the NNLS method gives

the most accurate results consistently. While there is no significant difference between the

other three methods. Note that in all the cases, Min-L1 methods with both equality and

quadratic constraints were initialized with the solution returned by the GCV method and

the NNLS method alv-a-l- started with a zero vector.

Finally, as a conclusion to our experiments on the simulated data, a quantitative

comparison was performed among the proposed method mixture of Wisharts (\l OW)

model and the two model-free methods, namely, the Q-hall ODF [149] and the DOT

[119]. All the resulting P(r) surfaces were represented by spherical harmonics coefficients

up to order 1 = 6 and the Q-hall ODF is computed using the formula in [5, Eq. (21)].

First, to gain a global assessment of these methods in terms of stability, we calculated

the similarity between each noisy P(r) and the corresponding noiseless P(r) using the

angular correlation coefficient formula given in [5, Eq. (71)]. The angular correlation

ranges from 0 to 1 where 1 implies identical probability profiles. Then, we estimated the

fiber orientations of each system by finding the maxima of the probability surfaces with










a CMI I-;-N E-ton numerical optimization algorithm and computed the deviation angles

between the estimated and the true fiber orientations. Figure 5-4 shows the mean and

standard deviation of the angular correlation coefficients, and error angles, respectively,

for the two-fiber simulation. Note that among the three methods examined, only MOW

results in small error angles and high correlation coefficients in presence of relatively

large noise. This trend also holds for the 1-fiber and the 3-fiber simulations. This can be

explained by noting that NNLS is able to locate the sparse spikes quite accurately even in

the presence of a lot of noise.

5.2 Real Data Experiments

The aforementioned reconstruction methods were tested on two real datasets provided

to us by researchers with the McE~night Brain Institute at the University of Florida. The

first dataset was acquired from a perfusion-fixed excised rat optic chiasm at 14.1 Telsa

using a Bruker Avance imaging system (Bruker NMR Instruments, Billerica, MA) with a

diffusion-weighted spin echo pulse sequence. Note that the rat optic chiasm is well suited

for the validation of the fiber reconstruction results because of its distinct myelinated

structure with both parallel and decussating (crossing) optic nerve fibers.

The imaging parameters used for the optic chiasm dataset are shown in Table 5-1. In

this optic chiasm dataset, there are 46 images acquired from 46 gradient directions at a

b-value of 1250s/mm2 and 6 additional images acquired at b a Os/mm2. Echo time and

repetition time were 23ms and 0.5s respectively; a and 6 values were set to 12.4ms and

1.2ms respectively; bandwidth was set to 35kHz; signal average was 10; a volume size of

128 x 128 x 5 and a resolution of 33.6 x 33.6 x 200pm3 WaS used. The optic chiasm images

were downsampled to 67.2 x 67.2 x 200pm3 TOSolution before the subsequent computation.

Two sets of experiments were performed on this optic chiasm dataset. The first

experiment was designed to compare the four different numerical methods discussed in

Section 4.4, namely, (a) damped least squares with GCV, (b) Min-L1 with quadratic

constraints, (c) damped least squares with a fixed regularization parameter, and (d)













-- I- IP2)Damped Least Squares (DLS)
IP2)DLSIGCV
-i (P3)M nLlwth equa ty
~ IPS)Nonnegatue Least Squares


O

S098~
















O


0.8


0.0 0



Figure 5-4.


std. dev. of noise std. dev. of noise

(a) (b)
Mean and standard deviation of (a) angular correlation coefficient and (b)
error angles for the two-fiber simulation. At each noise level, the four methods
compared here are (1) the DLS method as in (P2) (the damping factor is
empirically chosen to be So-); (2) the DLS method with the damping factor
determined using the GCV criterion; (3) the Min-L1 norm with equality
constraints as in (P3) initializedd with the solution obtained by DLS+GCV;
(4) the NNLS method as in (Ps). The di;11lai- II values are averaged over the
two fibers. Reproduced from [73] with permission. @[2007] IEEE.


-- DOT
QBI
*MOW


-.DOT
-QBI
*MOW
0.05
std. dev. of noise


0.05
std. dev. of noise


The plots on the left and on the right show the statistics of angular correlation
coefficients and error angles for the 2-fiber simulation, respectively. The
di1 pl i- 4I values for error angles are averaged over the two fiber orientations.
Reproduced from [77] with permission. @[2009] Springer.









Table 5-1. Imaging parameters used for the optic chiasm dataset.
Imaging parameters used for the optic chiasm dataset
magnetic field strength 14.1 Telsa
gradient directions 46 at high b-value, 6 at low b-value
high b-value 1250s/mm2
low b-value a Os/mm2
echo time (TE) 23ms
repetition time (TR) 0.5s
gradient pulse separation (A) 12.4ms
gradient pulse duration (5) 1.2ms
bandwidth 35kHZ
signal averaging 10
volume size 128 x 128 x 5
voxel size 33.6 x 33.6 x 200p~m3


non-negative least squares (NNLS). The resulting displacement probability profiles on a

region of interest are shown in Figure 5-5. For each method, the corresponding So image

is also shown in the upper left corner of each panel as a reference. For the sake of clarity,

we excluded every other voxel and overlaid the probability surfaces on the generalized

anisotropy (GA) maps [117]. GA is the variance of normalized diffusivity function. Higher

values of GA (brighter regions) indicate higher anisotropy.

As can be seen from Figure 5-5, the fiber-crossingf in the optic chiasm region is not

identifiable in either results obtained from the method using GCV (a) or the results

obtained from the method of Min-L1 with quadratic constraints (b), while both the

DLS method (c) and the NNLS method (d) are able to demonstrate the distinct fiber

orientations in the central region of the optic chiasm where ipsilateral m--- linated axons

from the two optic nerves cross and form the contralateral optic tracts. The failure of the

GCV method is due to the fact that the damping factors estimated by the GCV method

may not ak- -li- be optimal. The Min-L1 method with quadratic constraints only yields

satisfactory results in regions mostly populated with single fiber, which is consistent to

the previous observation on the 2-fiber simulation. It is also evident from Figure 5-5 that

compared to other three numerical methods, the NNLS scheme yields significantly sharper

displacement probability profiles.










Similar to the experiments on the simulated data, we also compare the proposed mix-

ture of Wisharts (\lOW) model with two model-free methods, namely, the QBI-ODF[149]

and the DOT [119]. The rest of this paragraph is reprinted from [77] with permission and

is copyrighted by Springer. Figure 5-6 shows the reconstruction results generated using

four different methods, namely, (a) QBI-ODF, (b) DOT, (c) MOW+Tikhonov regular-

ization, and (d) MOW+NNLS, on the same region of interest shown in Figure 5-5. The

fast combinatorial NNLS method [23] is used here as the NNLS solver. The computation

time for this region of interest containing 1024 voxels is less than 0.5 second for all four

methods on an Intel Core Duo 2.16 GHz CPU while the standard NNLS takes about 8

seconds. As seen in the Figure 5-6, the fiber-crossingfs in the optic chiasm region cannot

he identified by using the QBI-ODF method. Note that both the DOT method and the

MOW method with two different schemes are able to demonstrate the distinct fiber ori-

entations in the central region of the optic chiasm. However, it is evident from the figure

that compared to all other solutions, the MOW technique in conjunction with the NNLS

scheme yields significantly sharper displacement probability surfaces. This is particularly

horne out in the optic chiasm, in the center of each panel. The probability surfaces in the

QBI and DOT models are blurred, in part, because both vield a corrupted P(r) rather

than the actual displacement probability surfaces.The corrupting factor for the QBI is a

zeroth order Bessel function, for the DOT method it is a function that does not have an

analytic form. This corruption affects the accuracy of the reconstructed fiber orientations

as evidenced in the simulated data case where the ground truth was known. Note that

validating the precise angle of the fiber crossing in this real data set is non-trivial as it will

need a histology stack to be created and then fiber directions estimated from this stack to

be validated against those obtained from the DW-AIRI data.

To investigate the capability of diffusion weighted imaging in revealing the effects in

local tissue caused by diseases or neurologic disorders, further experiments were carried


















optic nervp- /)yj t;* & M// j
//////###1/// optic
&~s~b~9#J//QH~+$ ChiaSm

~"ff 4 optic tract
~--u~~idM/98~3ft

(a) Inisj J sis 3






optic nervp- / / 1' / # #207//

~u~p#p/)######ly~ ChiaSm


,~d~~U~9~ aft oWLptic tract


p ,,,,,,,/B ,, cmas

opti nr opictrc
(b) I st M I/I



opicneve/ / 4 / < *4/p


we .,47,ogyE. cmiasm

=ws+tit ptic tract


Probability maps computed using (a) damped least squares with GCV; (b)
Min-L1 with quadratic constraints (e = 1) initialized from (a);(c) damped least
squares with fixed regularization parameter (a~ = 0.6); ( d) non-negative least
squares from a rat optic chiasm data set overlaid on axially oriented GA [114]
maps. The decussations of myelinated axons from the two optic nerves at the
center of the optic chiasm are readily apparent. In all the plates, the
corresponding reference (So) image is shown in the upper left corner.
Reproduced from [73] with permission. @[2007] IEEE.


Figure 5-5.





















optic nerve ))J'di3///// L

&///s,~~uodgyj optic
JJ.~~~~C~~IeS ddjgJjIty cniasm



(a) QBI3J optic tract



opt Qicnene s //i fe ///


*** &// ess tge op4 c
*** ~44 &/O#$$# nam



R Iaic nev optic tract d
Figue 5-. Prbabiity surface coptdfr


optcne'e 7 944///

----**>>>Ms $$l cmasm







optic nerve ,)//b+/+/////




Nd MSOW optictrc



m~ ~ ~ a rat optic chas iae sig a


QBI-ODF, (b) DOT, (c) MOW+Tikhonov regularization, and (d)
MOW+NNLS. Note the decussation of row;linated axons from the two optic
nerves at the center of the optic chiasm. Reproduced from [77] with
permission. @ [2009] Springer.









out on two data sets collected from a pair of epileptic/normal rat brains. The rest of this

section is reprinted from [75] with permission and is copyrighted by Elsevier.

Table 5-2. Imaging parameters used for the rat brain dataset.
Imaging parameters used for the rat brain dataset
magnetic field strength 17.6 Telsa
gradient directions 46 at high b-value, 6 at low b-value
high b-value 1250s/mm2
low b-value 100s/mm2
echo time (TE) 28ms
repetition time (TR) 1.4s
gradient pulse separation (A) 17.5ms
gradient pulse duration (5) 1.5ms
bandwidth 750M~HZ
signal averaging 20 for high b-value and 5 for low b-value
volume size 200 x 100 x 32
voxel size 150 x 150 x 300p~m3


The imaging parameters used for the rat brain dataset are shown in Table 5-2. The

multiple-slice diffusion weighted image data were measured at 750 MHz using a 17.6 Tesla,

89 mm bore magnet with Bruker Avance console (Bruker NMR Instruments, Billerica,

MA). A spin-echo, pulsed-field-gradient sequence was used for data acquisition with a

repetition time of 1400 ms and an echo time of 28 ms. The diffusion weighted gradient

pulses were 1.5 ms long and separated by 17.5 ms. A total of 32 slices, with a thickness

of 0.3 mm, were measured with an orientation parallel to the long-axis of the brain (slices

progressed in the dorsal-ventral direction). These slices have a field-of-view 30 mm x 15

mm in a matrix of 200 x 100. The diffusion weighted images were interpolated to a matrix

of 400 x 200 for each slice. Each image was measured with 2 diffusion weigfhtingfs: 100

and 1250s/mm2. Diffusion-weighted images with 100s/mm2 Were measured in 6 gradient

directions determined by the vertices of an icosahedron in one of the hemispheres. The

images with a diffusion-weighting of 1250s/mm2 Were measured in 46 gradient-directions ,

which are determined by the tessellation of the icosahedron on the same hemisphere. The

100s/mm2 images were acquired with 20 signal averages and the 1250s/mm2 images were

acquired with 5 signal averages in a total measurement time of approximately 14 hours.










Figure 5-7 shows the displacement probabilities calculated front excised coronal

rat brain 1\RI data in a, (a) control and (b) an epileptic rat. The hippocanipus and

entorhinal cortex is expanded and depicts the orientations of the highly anisotropic and

coherent fibers. Note voxels with crossing orientations located in the dentate gyrus (dg)

and entorhinal cortex (ec). The region superior to CA1 represent the stratum lacunosuni-

moleculare and statunt radiatunt. Note that in the control hippocanipus, the molecular

1 .,-< c and stratum radiatunt fiber orientations paralleled the apical dendrites of granule

cells and pyramidal neurons respectively. In the epileptic hippocanipus, the CA1 subfield

pyramidal cell 1 .,-< c is notably lost relative to the control. The architecture of the dentate

gyrus is also notably altered with more evidence of crossing fibers. Future investigations

employing this method should improve our understanding of normal and pathologically

altered neum~ .Il 1Inyl~: in regions of complex fiber architecture such as the hippocanipus

and entorhinal cortex.












































22


s-- DG


S~tS$SSSA
dd ~iirid





Hilu


Figure 5-7.


Probability maps of coronally oriented GA images of a control and an epileptic
hippocampuus. Upper left corner shows the corresponding reference (SO) images
where the rectangle regions enclose the hippocampi. In the control
hippocampus, the molecular 1 ;r and stratum radiatum fiber orientations
paralleled the apical dendrites of granule cells and pyramidal neurons
respectively, whereas in the stratum lacunosum, molecular orientations
paralleled Schaffer collaterals from CA1 neurons. In the epileptic
hippocampus, the overall architecture is notably altered; the CA1 subfield is
lost, while an increase in crossing fibers can he seen in the hilus and dentate
gyrus (dg). Increased crossing fibers can also be seen in the entorhinal cortex
(ec). Fiber density within the statum lacunosum molecular and statum
radiale is also notably reduced, although fiber orientation remains unaltered.
Reproduced with permission from [75] @[2007] Elsevier.


.: ::. CA3
SSSrI Go
W~Iilus1


Oddestd//dd

d o as SSlllllltfijdS
IU~~1Y ~ Al1ddstaiddsra
**###ekkthdd 445St-
(l#ba~theb66 San--r~r __


(b) epileptic









CHAPTER 6
DISCUSSION AND CONCLUSIONS

6.1 Summary

Diffusion-weighted magnetic resonance imaging (DW-MRI) is a non-invasive imaging

technique that allows neural tissue architecture to be probed at a microscopic scale

in vivo. By producing quantitative measurements of on water molecular motion, DW-

MRI can be processed to map the fiber paths in the brain white matter. This valuable

information can be further exploited for neuronal connectivity inference and brain

developmental studies [87].

In this dissertation, a novel mathematical model for the diffusion weighted MR

signal attenuation is presented. The key postulation of the proposed model is that at

each voxel the diffusion of water molecules is characterized by a continuous mixture of

diffusion tensors. An interesting observation based on this continuous tensor distribution

model is that the MR signal attenuation can be expressed as the Laplace transform of the

associated tensor probability distribution function. It has also been show that when the

mixing distribution is parameterized by Wishart distributions, the resulting close form of

Laplace transform leads to a Rigaut-type fractal expression. This Rigaut-type function

exhibits the expected .-i-mptotic power-law behavior and has been phenomenologically

used in the past to explain the MR signal decay but never with a rigorous mathematical

justification until the development of the proposed model [74-76]. It is easy to show

that both the traditional diffusion tensor model and the multi-tensor model are limiting

cases of this continuous mixture of tensors model. Additionally, in the long wavelength or

low-q regime the proposed model leads to a quadratic decay which is consistent with the

traditional diffusion tensor model [14].

Another contribution of this work is on tackling the challenging problem of multi-fiber

reconstruction from the diffusion MR images. The mixture of Wisharts model, as a natu-

ral parametrization of the desired tensor distribution function, is used to describe complex










tissue structure involving multiple fiber populations. The multi-fiber reconstruction is

then formulated in a unified deconvolution framework [72, 73] which also includes some

other previously published approaches in this field. One central topic of this work is to

investigate several different deconvolution methods in the context of multi-fiber recon-

struction in diffusion AIRI. Experiments on both simulations and real data have shown

the favorable results from the proposed mixture of Wisharts (\lOW) model in conjunction

with the nonnegative least squares (NNLS) method, comparing with competing models

and methods from literature.

6.2 Open Problems

In this dissertation, we have developed a novel mathematical model which depicts the

diffusion AIR signal attenuation reasonably well, supported by both theoretical analysis

and experimental results. Regrettably but fortunately, there remain a number of open

problems .

6.2.1 Nonparametric Inverse Laplace Transform

The key postulation of the proposed mathematical model is that at each voxel the

diffusion of water molecules is characterized by a continuous mixture of diffusion tensors.

Based on this assumption, we also reveal the Laplace transform relationship between the

signal attenuation and the underlying tensor probability distribution. In our approach,

this tensor distribution is modeled by wishart distributions or mixture of Wisharts.

This choice is mainly justified by the existing closed form Laplace transform of Wishart

distributions which also yields the expected Rigaut-type fractal expression. Though it

is tempting and maybe more convincing to directly invert the Laplace transform in a

nonparametric way, one has to recognize the difficulties entailed in inverting Laplace

transform in the high dimension from the noisy data (see [54] for "cogent reasons for the

general sense of dread most mathematicians feel about inverting the Laplace ti Ion-1. .i ts ).

Recent work by Leow et al. [90] is very similar to our model in that a tensor distri-

bution function (TDF) is associated to each voxel and the inverse problem there is also










to recover this tensor distribution function that best explains the observed diffusion AIR

signal. In [90], the domain of the TDF is a four-paranleter space which is reduced front the

original 6-dintensional tensor space by assuming that fiber tracts are cylindrical. Then a

gradient descent method is used to solve for an optimal TDF that nminintizes the difference

between the observed signal and the predicted signal in the least-square sense. The current

mixture of Wisharts (\lOW) model eniploi-. .1in this work actually reduces to a spherical

deconvolution model by assuming the cylinder syninetry and fixing the eigenvalues of the

tensor parameter in each Wishart components. These simplifying assumptions lead to a

linear system and hence can he solved efficiently. An alternative but much more compli-

cated deconvolution method would be a B li-o Io approach where all the parameters in

the MOW model including the tensor eigfenvalues and the weights can he imposed with

suitable prior distributions.

6.2.2 Adaptive Sparse Dictionary Learning

We have shown that several existing nmulti-fiber reconstruction models can he ex-

pressed in a unified deconvolution framework model and reformulated into the problem of

seeking a sparse and positive solution to a linear system [7:3]. In most existing approaches

that lie in this unified deconvolution framework, the system matrix, A, or the so-called

dictionary in the context of signal representation theory, is fixed depending on the choices

of the convolution kernel, the paranietrization of the volume fraction function, and the

discretization scheme. The exception is a recent work proposed in [1:3] where the shape of

the convolution kernel function is estimated simultaneously with the rest of the unknowns

of the model. However, the basis function of this adaptive kernel is still fixed as a spline

function. A significant extension to [1:3, 7:3] would be a supervised learning hased approach

that infers a learned dictionary front the training data instead of a pre-defined dictionary

as in [52, 100] and furthermore allows a sparse encoding of the diffusion signal using this

learned dictionary.









6.2.3 Sub-voxel Fiber Bundles Classification

It has been widely recognized that the diffusion tensor model does not perform

well in regions containing intra-voxel orientational heterogeneity. This limitation of the

diffusion tensor model has prompted the development of many multi-fiber reconstruction

methods, including those reviewed in Section 3.3, such as higher order tensor it., 1_;ng [7 14],

diffusion spectrum ill. 1_;hly [7 60], q-hall ill. 1_;ity [7 49], probabilistic PAS-AIRI[70], spherical

deconvolution[14:3], diffusion orientation transform[119], and the mixture of Wisharts

model [75] proposed in this work. However, to the best of our knowledge, none of these

approaches is able to differentiate between complex configurations of intra-voxel fiber

bundles, e~g fibers crossing, kissing, bending or twisting within a voxel. In other words,

these methods still suffer from topological ambiguity, which can not he easily resolved by

only estimating fiber orientations.

We believe that the ability of successfully labelling voxels into distinct sub-voxel

fiber bundle configurations would be crucial in the next generation of fiber tractography

algorithms. Recently, UCniie7lj~ v et al. [1:34] proposed a curve inference method that uses

differential geometric estimates in a local neighborhood to differentiate the structures

of fanning and curving fiber bundles. Barmpoutis et al. [11] presented a novel method

for estimating a field of .l-i-mmetric spherical functions, dubbed tractosemas, which can

he used to model .I-i-mmetries such as pl w-ing fibers. The mixture of Wisharts model

proposed in this work has been shown in [9] as an excellent generative model for depicting

the diffusion AIR signal attenuation. It would be an interesting future research topic to

investigate if the combination of this generative model and some discriminative models can

he potentially used to distinguish different sub-voxel fiber bundle categories and improve

the performance of the fiber pathway reconstruction.









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BIOGRAPHICAL SKETCH

Bing Jian was born in XinYu, JiangXi, CH.I. I After he completed his undergraduate

studies in computer science at the University of Science and Technology of CluI, I in July

1999, he chose to continue his studies at the same university, which earned him a master's

degree in electrical engineering in July 2002. He then joined the Ph.D. program of the

Department of Computer and Information Science and Engineering at the University of

Florida in August 2002. After his first year in the Ph.D. program, he started working

in the field of computer vision and medical image analysis. His research interests cover

a wide range of topics from computer vision and machine learning to medical imaging

and computer aided diagnosis. He hopes to make contributions to our society through

innovative research on where these disciplines meet.