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Analysis of Diffusion MRI Data in the Presence of …...Acknowledgements First of all, I'd like to...
Transcript of Analysis of Diffusion MRI Data in the Presence of …...Acknowledgements First of all, I'd like to...
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Analysis of Diffusion MRI Data in the Presence of Noise
and Complex Fibre Architectures
by
Ryan Fobel
A thesis submitted in conformity with the requirements
for the degree of Master of Science
Graduate Department of Medical Biophysics
University of Toronto
Copyright c© 2008 by Ryan Fobel
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Abstract
Analysis of Di�usion MRI Data in the Presence of Noise and Complex Fibre
Architectures
Ryan Fobel
Master of Science
Graduate Department of Medical Biophysics
University of Toronto
2008
This thesis examines the advantages to nonlinear least-squares (NLS) �tting of di�usion-
weighted MRI data over the commonly used linear least-squares (LLS) approach. A
modi�ed �tting algorithm is proposed which accounts for the positive bias experienced
in magnitude images at low SNR. For b-values in the clinical range (≈1000 s/mm2),
the increase in precision of FA and �bre orientation estimates is almost negligible,
except at very high anisotropy. The optimal b-value for estimating tensor param-
eters was slightly higher for NLS. The primary advantage to NLS was improved
performance at high b-values, for which complex �bre architectures were more easily
resolved. This was demonstrated using a model-selection classi�er based on higher-
order di�usion models. Using a b-value of 3000 s/mm2 and magnitude-corrected NLS
�tting, at least 10% of voxels in the brain exhibited di�usion pro�les which could not
be represented by the tensor model.
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Acknowledgements
First of all, I'd like to thank my supervisor and mentor, Greg Stanisz, for giving me
the freedom to �nd my own way, but always being available when I needed help,
and for teaching me to think like a scientist. To the other members of our research
group, Sharon, Wendy, Colleen, Kim, Lisa, Voytek and Emidio, for maintaining a
lighthearted mood in the o�ce and exposing me to lots of interesting science along
the way.
To the members of my supervisory committee, Simon Graham and Chuck Cun-
ningham, for all of their support and thoughtful feedback. To my fellow lab mates,
Gord, Patrick, General, Garry, Mathieu, Helen and Rachel; whether it was frisbee in
the park, hitting the pub after a hard days work, or long debates in the sixth �oor
lounge, you all helped to make these past couple of years at Sunnybrook both fun and
memorable. To Sadie Yancey, my wonderful lab/roommate, for her generous spirit
and for providing a welcome source of distractions. To the other DTI researchers
at Sunnybrook, Nancy and So�a, thanks for sharing your insight and always being
willing to lend an ear.
Thanks to my good friends and thrill seekers, Jon Lovell, Dave Crane, and Matt
Ellis. Tuckerman's and Burning man were legendary experiences, and really lit up my
imagination. To my younger brother, Christian, for our frequent nerdy conversations,
and for not rubbing it in too much when he �nished his thesis before me. To my best
friend and companion, Aislinn Clancy, for her cheesy jokes, love of life, and for never
failing to bring a smile to my face. Finally, a huge thanks to my parents, Richard
and Maribeth, for all of their love and support over the years. None of this would
have been possible without them.
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Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1 Introduction 1
1.1 Di�usion-weighted MRI . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Di�usion Tensor Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Image artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Optimal imaging parameters . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Thesis statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Comparison of least-squares �tting methods 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Linear least-squares . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Weighted linear least-squares . . . . . . . . . . . . . . . . . . 25
2.2.3 Nonlinear least-squares . . . . . . . . . . . . . . . . . . . . . . 26
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2.2.4 Correcting for magnitude bias . . . . . . . . . . . . . . . . . . 27
2.2.5 Comparing �tting algorithms . . . . . . . . . . . . . . . . . . 28
2.2.6 Optimal b-value . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.7 Number of non-di�usion-weighted images . . . . . . . . . . . . 31
2.2.8 Rotational bias . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 Fitting performance under clinical conditions . . . . . . . . . . 34
2.3.2 Optimal b-value . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.3 Number of non-di�usion-weighted images . . . . . . . . . . . . 36
2.3.4 Rotational bias . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 Fitting performance under clinical conditions . . . . . . . . . . 37
2.4.2 Optimal b-value . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4.3 Number of non-di�usion-weighted images . . . . . . . . . . . . 43
2.4.4 Rotational bias . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3 Testing the validity of the tensor model 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.1 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.2 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.3 Fitting higher-order models . . . . . . . . . . . . . . . . . . . 66
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
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3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 Conclusions and future work 87
4.1 Magnitude-correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Noise characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3 Improving SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4 Tractography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Bibliography 95
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List of Tables
2.1 b-values optimized for FA . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2 b-values optimized for ε1 . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Number of b0 images optimized for FA . . . . . . . . . . . . . . . . . 44
3.1 Elements of a rank-4 generalized DT . . . . . . . . . . . . . . . . . . 70
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List of Figures
1.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Stejskal-Tanner PGSE sequence . . . . . . . . . . . . . . . . . . . . . 3
1.3 Isotropic vs. anisotropic di�usion . . . . . . . . . . . . . . . . . . . . 8
1.4 Di�usion Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 displacement vs ADC pro�le . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Trace, FA and �bre orientation maps . . . . . . . . . . . . . . . . . . 13
1.7 Fibre tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8 Eddy currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.9 Eddy current distortions . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.10 Twice-refocussed spin echo sequence . . . . . . . . . . . . . . . . . . . 17
2.1 Log-transform of the di�usion signal . . . . . . . . . . . . . . . . . . 23
2.2 TEmin and TRmin vs. b-value . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Brain masking algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 SNR, trace, and FA histograms from experiment . . . . . . . . . . . . 38
2.5 Reduced chi-squared, FA, and directional statistics . . . . . . . . . . 39
2.6 Reduced chi-squared, FA, and directional statistic histograms . . . . . 40
2.7 Root mean squared error in Fractional Anisotropy vs. b . . . . . . . . 46
2.8 Near-optimal range of b-values . . . . . . . . . . . . . . . . . . . . . . 47
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2.9 Root mean squared error in ε1 vs. b-value . . . . . . . . . . . . . . . 48
2.10 Rotational bias in σFA versus number of unique directions . . . . . . 49
2.11 Rotational variance of σε1 versus number of unique directions . . . . . 50
2.12 Rotational variance of σFA vs. FA . . . . . . . . . . . . . . . . . . . . 51
2.13 Rotational variance of σε1 vs. FA . . . . . . . . . . . . . . . . . . . . 52
3.1 Crossing �bres schematic . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2 Crossing �bres vs. b-value . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Model selection by F-test . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5 Selected locations demonstrating non-Gaussian di�usion . . . . . . . 75
3.6 Voxel classi�cation maps vs. SNR . . . . . . . . . . . . . . . . . . . . 76
3.7 Voxel classi�cation maps vs. b-value . . . . . . . . . . . . . . . . . . 77
3.8 ADC pro�les vs. b-value . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.9 Crossing �bre-detection simulations vs. b-value (SNR=30 at b=1000 s/mm2) 80
3.10 Crossing �bre-detection simulations vs. b-value (SNR=70 at b=1000 s/mm2) 81
3.11 Crossing �bre-detection simulations at FA=0.7 . . . . . . . . . . . . 82
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List of Abbreviations
ADC Apparent Di�usion Coe�cient
DT Di�usion Tensor
DTI Di�usion Tensor Imaging
EPI Echo Planar Imaging
FA Fractional Anisotropy
GDT Generalized Di�usion Tensor
LLS Linear Least-Squares
MCNLS Magnitude-Corrected Nonlinear Least-Squares
MR Magnetic Resonance
MRI Magnetic Resonance Imaging
NEX Number of Excitations
NLS Nonlinear Least-Squares
pdf probability density function
RF radio frequency
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RMSE Root Mean Squared Error
SH Spherical Harmonics
SNR Signal-to-Noise Ratio
TE Echo Time
TR Repetition Time
WLLS Weighted Linear Least-Squares
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Chapter 1
Introduction
The brain is often classi�ed into two distinct tissue types: grey matter and white
matter. Grey matter is the part of the brain responsible for synthesizing and process-
ing information. White matter, on the other hand, constitutes the physical �wiring�
of the brain. It is largely composed of axonal �bre bundles which transmit signals
between various brain regions. The primary application for the work presented in
this thesis is the study of these white matter �bres. Both the physical layout and the
viability of white matter connection pathways are of great interest to researchers and
clinicians.
The ability to non-invasively probe the structural organization of white matter
is made possible by a speci�c type of Magnetic Resonance Imaging (MRI) called
di�usion-weighted MRI. This chapter introduces the historical development of this
technique, its underlying theory, and a broad overview of this rapidly developing �eld
of research.
1
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Chapter 1. Introduction 2
1.1 Di�usion-weighted MRI
In 1905, Einstein showed that the random motion of spherical particles suspended in
�uid, a phenomenon known as Brownian motion, was the result of thermal energy [1].
The following equation describes the displacement probability for a single molecule
along direction, x, at time, t:
P (x, t) =1√
4πDtexp
(−x2
4Dt
)(1.1)
where the self-di�usion coe�cient, D, describes the rate at which molecules tend to
spread out in a �uid medium. D = µpkBT , where µp is the mobility of the particles
(related to the particle size and viscosity), kB is the Boltzmann constant, and T is
the absolute temperature. In the case of free di�usion, the shape of the displacement
probability function, P (x, t), is Gaussian, and the average displacement is equal to√
2Dt, as demonstrated in Fig. 1.1.
Figure 1.1: (a) Brownian motion of a single water molecule modeled by a randomwalk simulation. (b) Displacement probability, P (x, t), along the x-axis for a singlemolecule starting at intial position x=0 (Eq. 1.1). <x> is the average displacementafter time t.
Almost �fty years after Einstein's discovery, Hahn [2] and Carr and Purcell [3]
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Chapter 1. Introduction 3
Figure 1.2: The Stejskal-Tanner Pulsed-Gradient Spin Echo (PGSE) sequence [4].The image is not to scale. δ is the pulse duration, ∆ is the di�usion time, and TE isthe time to echo (or echo time).
showed how the self-di�usion coe�cient, D, could be measured using Nuclear Mag-
netic Resonance (NMR). A modi�cation of this approach by Stejskal and Tanner used
pulsed-gradients to achieve a more accurate measurement [4]. This Pulsed-Gradient
Spin Echo (PGSE) sequence, shown in Fig. 1.2, is widely used in di�usion-weighted
MRI to measure the displacement of water molecules in tissue.
The PGSE sequence resembles a standard spin echo [2] with the addition of two
large di�usion-weighting gradients after the 90◦ and 180◦ Radio Frequency (RF)
pulses. The e�ect of this sequence can be explained in terms of a single hydrogen
atom, or spin. The �rst gradient causes the spin to accumulate phase in relation to its
position along the axis of measurement, x. After the gradient is turned o�, the spin
has accumulated phase φ = γGxδx1 (assuming negligible gradient ramp times), where
γ is the gyromagnetic ratio, Gx is the gradient amplitude, δ is the gradient duration,
and x1 is the initial position of the spin along the x-axis. A second identical di�usion
gradient is applied after time ∆, but because these gradients are separated by a 180◦
RF pulse, the phase accumulation now has the opposite sign, φ2 = −γGxδx2, where
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Chapter 1. Introduction 4
x2 is the position of the spin along the x-axis after the second gradient pulse. The
total phase accumulated by the spin is therefore:
φ1 + φ2 = γGxδ(x1 − x2) (1.2)
and is proportional to the spin displacement along the x-axis (x1 − x2). If the spin
remains stationary during the time between the two gradients (i.e. x1 = x2), the phase
terms cancel out and the echo amplitude is una�ected. If, however, the position of
the spin along the x-axis changes during di�usion time ∆, the phase terms do not
cancel. Comparing the signal to a reference signal without the di�usion gradients
allows quantitative measurement of the spin displacement.
In reality, it is not possible to measure the signal from a single spin. Instead, the
measured signal represents the net magnetization of all spins in the imaging voxel.
If the spins remain stationary throughout the experiment or if no di�usion gradients
are used, the signal is described by:
S0 =1
N
N∑j=1
µj exp(−iϕj) (1.3)
where µj and ϕj represent the magnetic moment and phase of spin j, respectively.
Vector rj describes the displacement of spin j along the direction of measurement,
and therefore the net di�usion-weighted signal is:
S =1
N
N∑j=1
µj exp(−iϕj) exp(iγGδrj) (1.4)
The signal can also be expressed as an integral in terms of the spin displacement
probability density function, P (r,∆):
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Chapter 1. Introduction 5
S = S0
∫P (r,∆) exp(iγGδr)dr (1.5)
where S0 is the signal in the absence of any di�usion gradients. A simple change of
variables, q = γGδ2π
, reveals the Fourier relationship between S and P (r,∆):
S = S0
∫P (r,∆) exp(i2πqr)dr (1.6)
This is referred to as the q-space formalism. In the case of free di�usion, where
P (r,∆) is Gaussian (Eq. 1.1), the inverse Fourier transform describes the net loss in
magnetization due to the di�using spins [5]:
S = S0 exp[−q2∆D
]= S0 exp
[−γ2G2δ2∆D
](1.7)
To account for di�usion that occurs during the �nite pulse width, ∆ is replaced with
the term ∆− δ3, or the �e�ective� di�usion time:
S = S0 exp
[−γ2G2δ2(∆− δ
3)D
](1.8)
The sequence parameters are commonly represented in the MRI literature by a single
di�usion parameter, b:
b = γ2G2δ2(∆− δ
3) (1.9)
which reduces Eq. 1.8 to the form:
S = S0 exp [−bD] (1.10)
By performing a logarithmic transform of Eq. 1.10 and rearranging, a linear rela-
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Chapter 1. Introduction 6
tionship between the log-normalized MRI signal, log(S/S0), and the self-di�usion
coe�cient, D, is established:
log(S/S0) = −bD (1.11)
The two unknowns in this equation are S0, the echo amplitude without di�usion
gradients, and the di�usion coe�cient, D. b is an independent experimental variable
controlled by pulse sequence parameters. If the signal is measured at two di�erent
b-values, calculation of the remaining parameters, S0 and D, is straightforward.
Discussion to this point has been limited to the case of free di�usion, in which
water molecules can freely move in any direction. In this case, the displacement
probability depends only on the temperature and mobility of the water molecules,
and the signal attenuation caused by di�using spins is monoexponential. However,
di�usion in biological tissue is complicated by the various barriers to water movement,
including cell membranes and organelles. While the self-di�usion coe�cient, D, may
be the same in both cases, the measured di�usion coe�cient in tissue is reduced. The
di�erence provides information about the cellular microstructure in the vicinity of
the water molecules and its ability to hinder and/or restrict the movement of water.
To address this distinction, the term Apparent Di�usion Coe�cient (ADC) [6] is
typically used. A further implication of di�usion in tissue is that the displacement
probability function is no longer necessarily Gaussian. The result is that the di�usion
signal in biological tissue is not monoexponential, though for b-values in the clinical
range of DTI (b≈1000 s/mm2), this is a reasonable approximation [7].
The Apparent Di�usion Coe�cient (ADC) describes the mobility of water molecules
in tissue, and this information can be used to develop insight into tissue microstruc-
ture. This is illustrated in Fig. 1.3, which shows a schematic of di�using water
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Chapter 1. Introduction 7
molecules in two di�erent environments. In the �rst case, there is no preferential
direction for the water molecules to move, meaning that the apparent di�usion coef-
�cient is approximately the same in every direction. This is referred to as isotropic
di�usion and it is typical of tissues that lack a dominant directional organization
(e.g. grey matter). Fig. 1.3b shows cylindrical tubes that represent a simpli�ed model
of white matter with axons running in parallel. The water molecules are relatively
free to di�use along the length of the axons, but their movement is highly restricted
perpendicular to the �bre orientation. This results in what is referred to as anisotropic
di�usion.
Di�usion in the majority of tissue types is only weakly anisotropic. Therefore,
measuring the Apparent Di�usion Coe�cient (ADC) along a single direction is usually
su�cient. In highly organized tissues such as white matter and muscle, it is necessary
to sample the ADC along several directions. Although some of the early research on
di�usion-weighted MRI recognized di�erences in the ADC along the x, y and z-axes
in white matter [8], these three measurements did not provide enough information to
completely describe the di�usion pro�le.
1.2 Di�usion Tensor Imaging
In 1994, Basser et al. introduced Di�usion Tensor Imaging (DTI) [9]. DTI enables full
three-dimensional characterization of the di�usion process in vivo, making it practical
to obtain detailed knowledge of white mater �bre orientation in the human brain. The
di�usion tensor is represented by a 3x3 symmetric matrix:
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Chapter 1. Introduction 8
Figure 1.3: Schematic of di�using water molecules in (a) isotropic (e.g. grey mat-ter) and (b) anisotropic (e.g. white matter) media. (c) Spin displacement probabilitydensity function for the case of free di�usion (i.e. no restriction). (d) Restricted dis-placement probability density function along directions parallel and (e) perpendicularto the �bres.
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Chapter 1. Introduction 9
D =
Dxx Dxy Dxz
Dyx Dyy Dyz
Dzx Dzy Dzz
(1.12)
The three diagonal terms represent the apparent di�usion along the x, y, and z axes
in the laboratory frame of reference. The o�-diagonal terms describe the degree
of correlation between the di�usion along the primary axes. Because the matrix is
symmetric (i.e.Dij=Dji), there are only six unique elements of the di�usion tensor, D.
Assuming that the molecular displacement probability density function is Gaussian,
it can be related to the di�usion tensor by the following equation:
P (r, t) =1√
(4πt)3|D|exp
(−rTD−1r
4t
)(1.13)
The measured signal along direction r is related to the di�usion tensor by the equation:
S = S0 exp(−brTDr) (1.14)
Eigenvalue decomposition of D determines the apparent di�usion coe�cients λ1,
λ2, and λ3, corresponding to eigenvectors ε1, ε2, and ε3. The primary eigenvector,
ε1, gives the estimated �bre orientation. If the eigenvalue and eigenvector terms are
written as 3x3 matrices, Λ and E,
Λ =
λ1 0 0
0 λ2 0
0 0 λ3
(1.15)
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Chapter 1. Introduction 10
E =
ε1x ε2x ε3x
ε1y ε2y ε3y
ε1z ε2z ε3z
(1.16)
The following equation demonstrates their relationship to the di�usion tensor, D:
D = EΛE−1 (1.17)
E is the rotation matrix relating the diagonalized tensor, Λ, to the laboratory frame.
Fig. 1.4 shows a graphical representation of the three-dimensional isoprobability dis-
placement pro�le for an anisotropic tensor in its standard from and aligned with the
laboratory axes.
Figure 1.4: The di�usion ellipsoid represents the isoprobability displacement surface.ε1 is the vector representing the estimated �bre orientation. The 3x3 eigenvectormatrix, E, rotates the di�usion tensor, D, such that its axes are aligned with thelaboratory frame, Λ.
Fig. 1.5a and b illustrate the relationship between the di�usion tensor and its
corresponding Apparent Di�usion Coe�cient (ADC) pro�le (i.e. a plot of the ADC
values versus orientation). The reason that the ADC pro�le is �peanut� shaped is
that it is a projection of the average displacement along the axis of measurement.
Fig. 1.5c shows that the in the case of multiple �bre bundles, the orientation of the
composite �bres is not obvious from the ADC pro�le.
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Chapter 1. Introduction 11
Figure 1.5: (a) Di�usion tensor ellipsoid representing the average water displacementpro�le and (b) the corresponding ADC pro�le. The "peanut" shape of the ADCpro�le stems from the fact that it is a projection of the average di�usion along agiven axis. (c) For a pair of crossing �bres, �bre orientation cannot be easily deducedfrom the ADC pro�le.
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Chapter 1. Introduction 12
Tensor eigenvalues can be used to calculate several rotationally invariant scalar
indices, including the trace and mean di�usivity, 〈λ〉:
trace(D) = λ1 + λ2 + λ3 (1.18)
〈λ〉 =λ1 + λ2 + λ3
3=
1
3trace(D) (1.19)
There are also several indices that describe anisotropy [10]. The most commonly
used is the Fractional Anisotropy (FA), which is calculated according to the following
equation:
FA =
√3[(λ1 − 〈λ〉)2 + (λ2 − 〈λ〉)2 + (λ3 − 〈λ〉)2]
2 (λ21 + λ2
2 + λ23)
(1.20)
These indices are often displayed using parametric maps. The orientation of the
principal eigenvector is visualized using a colour-coded map for which the x, y, and z
vector components are mapped to the red, green, and blue channels respectively, and
weighted by FA [11]. Examples of each of these maps are shown in Fig. 1.6.
It is also possible to process tensor information using �bre tracking algorithms.
These algorithms can be divided into two broad categories: streamline [12] and prob-
abilistic tractography [13]. Fig. 1.7 shows a schematic describing a simple streamline
tracking approach. These algorithms can be used to de�ne three-dimensional repre-
sentations of white matter �bre pathways. Probabilistic tractography algorithms are
an extension of this approach. They apply statistical tools and measures of uncer-
tainty to estimate the probability that two di�erent brain regions are connected.
The sensitivity of DTI to subtle changes in the cellular microstructure such as
axonal loss, demyelination, and/or in�ammation make it a valuable tool in the study
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Chapter 1. Introduction 13
Figure 1.6: (a) Trace(D) map, (b) FA map and (c) colour-coded �bre orientationmap for a single slice of a normal human brain. These are the most commonly usedmetrics for evaluating white matter integrity and pathology. Note the strong contrastbetween white and grey matter in the FA map, and the lack of contrast in the tracemap.
of white matter disease [14]. Because of its non-invasive nature, DTI is also well suited
for a host of research applications including the study of brain development [15] and
aging [16].
1.3 Image artifacts
The magnetic �eld in the scanner is constantly changing due to rapidly switching
gradients. These changing magnetic �elds induce electrical currents in conducting
materials within the scanner. The induced currents are referred to as eddy currents,
and result in magnetic �eld gradients whose direction is opposite to the change in
�eld as shown in Fig. 1.8b and c [17]. Eddy currents build up during the time varying
part of the gradient waveform and decay during stationary phases. The resulting
waveform looks as though it has gone through a low-pass �lter (Fig. 1.8d).
The gradients resulting from eddy currents are usually classi�ed according to
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Chapter 1. Introduction 14
Figure 1.7: Fibre tracking initiated at two di�erent seed voxels. The grey value ofeach voxel indicates its Fractional Anisotropy. Starting at a seed point (*), �bretracts follow the principal eigenvector of whatever voxel they are contained within.When they reach a voxel boundary, their direction is updated. Tracts are terminatedwhen either the change in direction is too great, or the FA drops below a certainthreshold [12].
their spatial dependence. B0 eddy current gradients are spatially constant over the
imaging volume. Linear eddy current gradients, gx(t), gy(t) and gz(t) vary linearly
with position in the x, y, and z direction, respectively. Each of these terms represents
a di�erent type of phase error, re�ected by their characteristic distortions shown in
Fig. 1.9. The net e�ect is a combination of all of these distortion types.
The di�usion-weighting gradients are very large relative to those used in most
other imaging sequences. When combined with the usual Echo Planar Imaging (EPI)
readout and its associated low bandwidth in the phase-encoding direction, this makes
DTI particularly susceptible to eddy current artifacts [17]. Because the tensor model
is �t on a per-voxel basis, any translation or deformation of the imaging volume can
cause the voxel-grouped measurements to originate from di�erent physical locations.
If the misalignment is too severe, it can render a data set useless.
To address this problem, Reese et al. developed a twice-refocused spin echo se-
quence (Fig. 1.10), which uses an additional refocusing pulse and carefully timed
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Chapter 1. Introduction 15
Figure 1.8: (a) Gradient waveform, Gx(t), (b) the �rst derivative of the gradientwaveform, dGx(t)/dt and (c) one of the induced eddy current terms, gx(t).
pulsed-gradients to null the dominant eddy-currents [18]. This substantially reduces
image distortion relative to the standard PGSE sequence.
Like all forms of MRI, DTI is subject to partial volume e�ects. This can a�ect
tensor estimation if voxels containing white matter also contain a mixture of grey
matter and/or cerebrospinal �uid. Crossing, bending, and diverging �bres also pose
a problem because the tensor model is insu�cient to describe the di�usion pro�les
resulting from these complex geometries [19]. The detection of voxels for which the
tensor model is invalid is the subject of Chapter 3.
1.4 Optimal imaging parameters
The quality of DTI results depends on many factors, including equipment, acquisi-
tion parameters, and post-processing methods. Much work has been done to optimize
pulse sequence parameters and gradient orientations to minimize the e�ects of sys-
tematic noise and improve the accuracy of anisotropy and �bre orientation estimates.
Jones et al. used �rst-order error propagation methods to derive an analytical expres-
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Chapter 1. Introduction 16
Figure 1.9: Distortion artifacts related to the di�erent eddy current gradient terms.(a) For an Echo Planar Imaging (EPI) readout, the B0 term causes an image shift inphase encode direction. The gz(t) term result in an image shift that is dependent onslice position. (b) The gy(t) term stretches or contracts the image along phase-encodedirection. (c) The gx(t) term produces image shear.
sion of the error in the tensor parameters [20]. This allows solving for the optimal
b-value with respect to the pulse sequence echo time (TE) and properties of the
sample (T2-relaxation and the apparent di�usion coe�cient). Alexander et al. ex-
amined the same problem using Monte Carlo simulations [21]. Both of these studies
suggest that a b-value in the neighbourhood of 1000 s/mm2 is optimal for healthy
white matter, though higher b-values may be useful for elucidating more complex
structures [21, 22, 23]. The ratio of the number of di�usion-weighted (NDW ) to non-
di�usion-weighted (Nb0) images is also important. An optimal value for the ratio
NDW :Nb0 depends on the parameter that is being measured. For example, to esti-
mate the principal direction of di�usion, only di�usion-weighted images should be
acquired [21] (i.e. NDW :Nb0 should be as high as possible). For estimating the frac-
tional anisotropy and/or trace, ratios in the range of 5:1 [21] to 8:1 [20] have been
proposed. It must be stressed that these and other �optimal� parameters apply only
to the speci�c conditions under which they were designed. Even then, they make a
number of assumptions which may not always be valid. They are intended to serve
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Chapter 1. Introduction 17
Figure 1.10: The Twice-refocussed spin echo sequence [18]. Gradient timings areadjusted to cancel out the dominant eddy current gradients.
only as a guideline.
There is an extensive body of research concerning the orientation of di�usion
gradients [24, 25, 20, 26, 27, 28]. Because �bre orientation is unknown a priori, it is
important that the gradient sampling scheme performs similarly across all possible
�bre orientations. This property is referred to as rotational invariance. While it is
impossible to design a truly rotationally invariant scheme [29], for a given number
of gradient orientations, rotational bias is minimized by spreading out the gradient
orientations evenly on a spherical shell. This is typically performed using either the
electrostatic repulsion algorithm [20, 27] or using the vertices of a set of geometric
volumes known as the platonic solids [25, 30].
While only six gradient directions are necessary to estimate all of the parame-
ters of the di�usion tensor, additional directions can reduce rotational bias. Monte
Carlo simulations by Papadakis et al. [27] show that increasing the number of gra-
dient orientations reduces rotational bias in the mean and standard deviation of FA.
Jones demonstrated that increasing the number of directions (with an equivalent
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Chapter 1. Introduction 18
total imaging time) reduces the rotational bias in �bre orientation, mean FA and
trace [26]. Both of these studies reported that for �tting a single tensor model, there
is a negligible bene�t to using more than thirty directions.
1.5 Thesis statement
One aspect of DTI that has been largely neglected by the research community is the
impact of data analysis on DTI results. Once di�usion-weighted images have been
collected from the MR scanner, computer algorithms are used to �t the tensor model
for each voxel in the data set. This thesis examines the implementation and di�erences
between various �tting algorithms, and proposes a new method to compensate for
signal bias at low SNR.
Chapter 2 addresses the following questions:
1. How important is the choice of �tting method to estimates of anisotropy and
�bre orientation?
2. Under what experimental conditions are the di�erences between �tting algo-
rithms most signi�cant?
3. Do optimal imaging parameters depend on the choice of �tting algorithm?
Chapter 3 explores a method for identifying voxels for which the di�usion tensor
model is invalid (i.e. those likely to contain complex �bre geometries). It extends an
existing model-selection framework [31] through the development of nonlinear �tting
algorithms for higher-order di�usion models. The resulting classi�er is appropriate
for high b-values (above 1000 s/mm2), for which complex �bre geometries are more
easily resolved [19, 22].
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Chapter 2
Comparison of least-squares �tting
methods
2.1 Introduction
Since the introduction of Di�usion Tensor Imaging in 1994 by Basser et al. [9],
much progress has been made in designing more robust and accurate DTI acqui-
sition strategies. However, clear consensus within the DTI community is lacking
with regards to the choice of �tting algorithm. Historically, most studies have used
simple and e�cient linear regression techniques [9, 25, 20, 26, 28]. A small mi-
nority have opted for more sophisticated and computationally expensive nonlinear
methods [32, 27]. From a theoretical standpoint, nonlinear techniques are more
sound. The added computational time has probably been the major barrier to
their adoption. Furthermore, the long history of using linear regression methods
in the �eld, combined with their wide availability in popular DTI software packages
(e.g. DTI Studio [33] and FSL [34]), has also meant that a transition to nonlin-
ear �tting has yet to gain wide acceptance. This leads to an important question:
19
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Chapter 2. Comparison of least-squares fitting methods 20
How much of an impact does the choice of �tting algorithm have on DTI results?
For b-values in the clinical range (b≈1000 s/mm2), Koay et al. recently demon-
strated that nonlinear techniques o�er a modest improvement in the accuracy of
Fractional Anisotropy (FA) and trace measurements for simulated data [35]. For
tractography applications, the principal direction of di�usion, ε1, is of critical impor-
tance, yet the e�ect of di�erent �tting algorithms on the estimation of this parameter
has not yet been studied.
Furthermore, bias introduced by the magnitude operation as signals approach
the noise �oor has been shown to a�ect DTI acquisitions with low SNR, high dif-
fusivity and/or high b-values [32]. This chapter describes the implementation of a
tensor �tting algorithm that compensates for this magnitude bias. This approach is
compared to linear, nonlinear and weighted least-squares �tting methods using both
simulations and in vivo data. The e�ect of �tting algorithms on the estimation of
Fractional Anisotropy and �bre orientation over a range of SNR, b-value, and gra-
dient orientation schemes is investigated to identify experimental conditions under
which di�erences between �tting algorithms are signi�cant.
Chapter 1 introduced several previous studies that examine optimization of the
DTI experiment. These focused on the b-value [21, 20], the ratio of di�usion-weighted
to non-di�usion-weighted images (NDW :Nb0) [21, 20], and the number of gradient
orientations necessary to achieve relative rotational invariance [26, 27]. All but one of
these studies [27] employ simulations and analytical expressions based on linear least-
squares regression. This chapter reevaluates the optimization of these parameters
using Monte Carlo simulations to determine whether or not the results are dependent
on the choice of �tting algorithm.
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Chapter 2. Comparison of least-squares fitting methods 21
2.2 Theory
The three �tting methods commonly used to �t di�usion tensor data are linear least-
squares (LLS), weighted linear least-squares (WLLS), and nonlinear least-squares
(NLS). These are sometimes referred to as linear, weighted linear, and nonlinear
regression methods. All of these methods are constructed in a similar fashion. For
each voxel in the imaging volume, there are six unique tensor elements as well as the
non-di�usion-weighted signal value, S0, that need to be estimated. These parameters
are represented by a 7x1 column vector:
x = [Dxx, Dyy, Dzz, Dxy, Dxz, Dyz, ln(S0)]T (2.1)
Each gradient sampling direction is represented by a three-dimensional unit vector,
ri, which has x, y, and z components: rix,riy, and riz. Each image also has an
associated b-value, bi. In most cases, this b-value is the same for all di�usion-weighted
images, but it is also possible to use multiple b-values in a single experiment. One
or more non-di�usion-weighted images are necessary to normalize the signal. These
non-di�usion-weighted images are commonly referred to as b0 images. The gradient
directions and b-values are used to construct an Nx7 experimental design matrix, B.
Each row of this matrix, Bi, corresponds to one of the N images.
Bi =[−bir2
ix, −bir2iy, −bir2
iz, −2birixriy, −2birixriz, −2biriyriz, 1]
(2.2)
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Chapter 2. Comparison of least-squares fitting methods 22
B =
B1
...
Bi
...
BN
(2.3)
The same design matrix can be used for any of the �tting techniques. The linear
least-squares, weighted linear least-squares and nonlinear least-squares algorithms
are described in the following sections.
2.2.1 Linear least-squares
Linear least-squares is both the simplest and most widely used �tting algorithm ap-
plied to DTI data. Measurements are �rst transformed by taking their natural log-
arithm. This converts the monoexponential di�usion signal (Eq. 1.14) into a simple
linear relationship as illustrated by Fig. 2.1. If the log-transformed measurements are
written as a column vector,
Y = [ln(S1), ln(S2), · · · , ln(SN)]T (2.4)
the linear relationship between the design matrix, B, and the model parameters, x,
is described in matrix form by the equation:
Y = Bx+ ln(e) (2.5)
The ln(e) term is a column vector representing the residuals between the tensor
model and the log-signal measurements. The linear least-squares algorithm seeks to
minimize the sum of the squares of these residual terms, thus solving for x. This
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Chapter 2. Comparison of least-squares fitting methods 23
is accomplished by calculating the Moore-Penrose pseudoinverse [36] of the design
matrix, B+, and multiplying it by the log-transformed measurements.
x = (BTB)−1BT = B+Y (2.6)
One performance advantage of the LLS approach is that the pseudoinverse of the
design matrix only needs to be calculated once. It is then multiplied by the log-
transformed signal from each voxel in the image.
In order for the linear least-squares solution to be optimal, there are three neces-
sary conditions. All of the residuals must be uncorrelated, have a mean of zero, and
have equal variances [37]. Although the errors should be independent in most cases,
satisfying the �rst condition, the second and third conditions are violated by a pair
of operations performed on DTI data; the magnitude and logarithmic transforms.
Figure 2.1: Simulated di�usion signal (a) before and (b) after the log-tranformation.The noisy data points give an indication of the noise behaviour across the full rangeof b-values (SNR=20 at b=0 s/mm2). The measured mean is positively biased due tothe magnitude operation. These graphs represent an Apparent Di�usion Coe�cientof 2x10−3 mm2/s, which is close to the maximum observed in healthy white matter.
Di�usion-weighted images have a real and imaginary component, though it is com-
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Chapter 2. Comparison of least-squares fitting methods 24
mon for the MR reconstruction software to apply the magnitude operation (i.e. |Si| =√Re(Si) + Im(Si)). With su�ciently high SNR, noise follows a roughly Gaussian
distribution, and it is common practice to treat it as such. In reality, the signal in mag-
nitude images follows a Rician distribution [38], and measurements approaching to
the noise �oor exhibit a positive bias. This is clearly illustrated in Fig. 2.1. Although
this problem exists in many other �elds, its applicability to MRI was �rst demon-
strated by Henkleman [39]. More recently, Jones and Basser examined the e�ect of
magnitude bias in the context of DTI [32]. Overestimation of the di�usion-weighted
signal causes an underestimation of the Apparent Di�usion Coe�cient (ADC), and
since this e�ect is most prominent for those measurements with a high ADC, it leads
to a �squashed peanut� e�ect [32].
The logarithmic transformation also introduces other problems to the linear least-
squares solution. This operation results in the variance being dependent on the mag-
nitude of the signal, and therefore on the b-value and Apparent Di�usion Coe�cient
(ADC). At low SNR, noise is ampli�ed as demonstrated in Fig. 2.1b. The expected
value for a log-transformed signal is also negatively biased and the variance is no
longer symmetric about the mean. These e�ects become increasingly important as
SNR decreases.
Considering the properties of both operations, the linear least-squares algorithm
can be expected to perform reasonably well at high SNR and in voxels where the
signal magnitude is relatively consistent across the entire set of measurements (nearly
isotropic di�usion). The following algorithms aim to address the limitations inherent
in the linear least-squares approach.
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Chapter 2. Comparison of least-squares fitting methods 25
2.2.2 Weighted linear least-squares
The weighted linear least-squares algorithm improves on the standard LLS method
by accounting for di�erences in variance about each of the signal measurements.
Following the application of the log-transform, the signal variance at each point is
proportional to the square of the signal magnitude divided by the square of the noise.
This can be used to construct the diagonals of the covariance matrix W. Because
individual measurements are independent, all of the o�-diagonal terms are equal to
zero.
W = diag(S2i /σ
2i ) (2.7)
The model parameters are then calculated according to the equation:
x = (BTWB)−1(BTW)Y (2.8)
Notice that Eq. 2.7 includes noise terms, σi. Although it is possible to estimate this
parameter, if the noise can be assumed to be consistent for all measurements, its
value will have no e�ect on the �t. Therefore, it can simply be set to one. Because
the diagonal entries in the covariance matrix depend on noisy signal measurements,
it is advantageous to recalculate W following the initial �t with the updated signal
estimates:
S = exp (Bx) (2.9)
Performing a second iteration of Eq. 2.8 with the updated covariance matrix produces
a better solution. Although the weighted least-squares method compensates for the
unequal variance caused by the logarithmic transform, it does not address the non-
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Chapter 2. Comparison of least-squares fitting methods 26
symmetric (negatively skewed) residuals. It is also susceptible to the magnitude bias.
Computational demands for the WLLS algorithm are slightly higher than those
of LLS because pseudoinversion of the weighted design matrix must be performed
twice for each voxel in the image, whereas for LLS, the pseudoinverse only needs to
be calculated once for the entire experiment.
2.2.3 Nonlinear least-squares
The nonlinear least-squares method di�ers from both WLLS and LLS by �tting the
model directly to the di�usion-weighted signal without any need for the logarithmic
transform. As a result, residuals have equal variance and an expected value equal to
zero in all situations where the magnitude bias is not signi�cant. In matrix form, the
signal equation is:
S = [S1, S2, · · · , SN ]T (2.10)
S = exp(Bx) + e (2.11)
A problem with nonlinear �tting is that there is no analytical solution. Since it is
not possible to solve the system of equations algebraically, an iterative minimization
of the sum of squared residuals is performed.
minN∑i=1
e2i = minN∑i=1
[Si − exp(Bix)]2 (2.12)
A consequence of nonlinearity is that there is no guarantee of �nding the global
minimum to the optimization problem. The search is usually initialized with the
either the LLS or WLLS solution, ensuring that the NLS solution is at least as
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Chapter 2. Comparison of least-squares fitting methods 27
good. The NLS algorithm also requires an e�cient means for searching the parameter
space. Several parameter search algorithms are available. The Levenberg-Marquardt
algorithm is widely used in the DTI community [27, 32, 40], though it has been
suggested that others may perform moderately better [41].
2.2.4 Correcting for magnitude bias
Jones and Basser proposed a correction scheme that introduces an additional noise-
estimation parameter to the nonlinear �tting algorithm [32]. This method was used
to independently �t ADCs along a single direction using multiple b-values. By adopt-
ing the same objective function, fMCNLS (Eq. 2.13), it is possible to �t the tensor
model. In addition, the noise parameter can be measured from a background region
of the image and incorporated as a �xed value. This reduces the number of model
parameters, improving the accuracy of the �t and computational e�ciency.
fMCNLS(x) =N∑i=1
[Si −
√exp2(Bix) + σ2
]2(2.13)
The MR signal consists of two independent channels, one real and one imaginary.
Assuming that they are both normally distributed with a standard deviation of σ, it
is possible to estimate σ from a background region of the magnitude image using the
following equation [39]:
σ =mean(Sbackground)√
π2
(2.14)
In regions of high SNR,√
exp2(Bix) + σ2 ≈ exp(Bix), and the magnitude-corrected
�t reduces to the standard nonlinear least-squares �t.
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Chapter 2. Comparison of least-squares fitting methods 28
2.2.5 Comparing �tting algorithms
The chi-squared statistic, χ2, is a measure of the overall �goodness� of �t [42]. A
lower chi-squared value indicates better agreement between the data and the model.
If the model is appropriate and the noise term, σ, is known, the expected value for
χ2 is equal to the degrees of freedom in the system, ν. The degrees of freedom is
the number of measurements minus the number of model parameters, (i.e. for �tting
the di�usion tensor model, ν=N -7). Normalizing χ2 by the degrees of freedom gives
the reduced chi-squared statistic, χ2r. This reduced form has the advantage that it
allows comparison between �ts with di�erent numbers of measurements and/or model
parameters. The expected value of χ2r is equal to one.
χ2 =N∑i=1
[Si − exp(Bx)]2
σ2i
=1
σ2
N∑i=1
e2i (2.15)
χ2r =
χ2
ν=
χ2
(N − 7)(2.16)
Besides knowing the expected value for chi-squared statistics, the distribution is also
well characterized [42]. This makes it possible to calculate the probability of an
obtained chi-squared value. For the purpose of comparing di�erent �tting algorithms,
assuming that the model is correct and using the same data, the best algorithm is
the one that produces the smallest chi-squared value on average.
Using computer simulations, the error in Fractional Anisotropy (FA) and �bre
orientation, ε1, can be determined because their true values are known. Metrics such
as the root mean square error (RMSE) or mean absolute error (MAE) can therefore
be used to compare the relative performance of di�erent �tting algorithms. In the
case of in vivo experiments, it is not possible to calculate these metrics because the
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Chapter 2. Comparison of least-squares fitting methods 29
truth data is unavailable. Instead, the variation in these parameters over repeated
experiments allows for estimation of experimental precision.
The variables σFA and σε1 represent the experimental precision in FA and �bre ori-
entation. σFA is de�ned as the standard deviation in FA measurements over repeated
experiments. σε1 is the average angular distance between the measured principal
eigenvector, ε1, and the dyadic tensor average, 〈ε1〉 [43]. This is a parametric ana-
logue to Jones' cone of uncertainty [44], where 2σε1 corresponds to a 95% directional
con�dence interval. σε1 can be interpreted as the root mean squared angular di�erence
between the measured direction of di�usion and the sample average. Mathematical
expressions for these parameters are:
σFA =
√√√√ 1
n− 1
n∑i=1
(FAi − 〈FA〉)2 (2.17)
σε1 =
√√√√ 1
n− 1
n∑i=1
(arccos |ε1i · 〈ε1〉|)2 (2.18)
When comparing the performance of di�erent �tting algorithms for the same data,
the best method is the one that minimizes χ2r, σFA, and/or σε1 .
2.2.6 Optimal b-value
The most common approach to collecting DTI data is to acquire one or more b0
images, followed by a set of di�usion-weighted images at a single b-value. This can
be thought of as acquiring points on a spherical shell. Increasing the b-value improves
contrast between signal measurements, but it also results in a loss of SNR, because
the di�usion signal decays in proportion to exp(-bD). Several groups have examined
this trade-o� in the context of tensor estimation [21, 20]. They reported that the
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Chapter 2. Comparison of least-squares fitting methods 30
optimal b-value depends heavily on the mean di�usivity, and to a lesser extent, the
anisotropy. For healthy white matter, a b-value in the neighbourhood of 1000 s/mm2
is recommended.
For the standard Stejskal-Tanner PGSE sequence (Fig. 1.2), b is a function of three
pulse sequence parameters: the pulsed gradient amplitude, G, gradient pulse width,
δ, and the di�usion time, ∆. Assuming negligible ramp times (i.e. high slew rates),
this relationship is described by Eq. 1.9. There are several important considerations
when increasing the value of b. First of all, the pulse width should be kept as small
as possible to minimize the amount of di�usion that occurs during application of the
gradients, enabling use of the short pulse-width approximation [5]. Secondly, b varies
linearly with the di�usion time, so increasing the di�usion time has less of an impact
relative to changing the gradient width or amplitude, which both have a second-order
relationship. Increasing the di�usion time also results in a longer echo time (TE),
which leads to reduced SNR due to T2-relaxation e�ects. Finally, if the di�usion time
is too long, intra/extracellular exchange and restricted di�usion e�ects can become
signi�cant [45]. For these reasons, it is preferable to increase b by adjusting the gra-
dient amplitude parameter, G. However, the maximum available amplitude depends
on scanner hardware, and this limitation is commonly reached. Therefore, achieving
very high b-values typically involves some form of compromise [7].
In order to evaluate the performance of tensor estimation across a range of b-
values, it is important to compensate for changes in relative SNR caused by T2-
relaxation and variable echo times. While previous studies use analytical models
relating b to the minimum achievable TE [21, 20], this relationship is more convoluted
for the twice-refocused spin echo sequence [18] commonly used to reduce eddy current
artifacts. For this reason, minimum TE values were collected directly from the 3T GE
Signa console. For a given T2, SNR was normalized by exp(-TE/T2) in simulations
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Chapter 2. Comparison of least-squares fitting methods 31
to account for T2-relaxation e�ects.
Higher b-values are also associated with an increased scan time due to longer
TRs. For a �xed imaging time, this limits the total number of images that can be
acquired. Previous groups have ignored this e�ect [21, 20], but assuming that the
SNR is proportional to the square root of the total imaging time, SNR in simulations
can be normalized by√TR. The minimum TE, TR and the relative SNR factors are
shown in Fig. 2.2 for a T2 of 80 ms, which corresponds to healthy white matter [46].
Figure 2.2: SNR normalization curves for the 3T GE Signa scanner based on scanparameters described in section 2.3. (a) Minimum TE for a range of b-values. (b) Min-imum TR for a range of b-values. (c) SNR relative to b=1000 s/mm2 for a T2 of 80 ms,accounting for changes in minimum TE only, and for a combination of minimum TEand minimum TR.
2.2.7 Number of non-di�usion-weighted images
Non-di�usion-weighted images are extremely important because they provide the ref-
erence to which all di�usion-weighted images are normalized. As such, this mea-
surement has a large impact on all of the tensor parameters. Acquiring multiple b0
images can improve tensor precision, but for a �xed imaging time, this comes at the
cost of reducing the number of gradient directions. The number of b0 images can
be expressed as a ratio of the number of di�usion-weighted images to non-di�usion-
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Chapter 2. Comparison of least-squares fitting methods 32
weighted images, NDW :Nb0 , allowing this attribute to scale for experiments with a
di�erent number of total image acquisitions.
Jones et al. tested a range of NDW :Nb0 ratios in a water phantom experiment [20].
Their results indicated that a ratio of 8.33 minimized the standard deviation in the
trace of the di�usion tensor, and minimized the Fractional Anisotropy (for a water
phantom, FA should be equal to zero). However, this NDW :Nb0 ratio may not be
optimal for a clinical DTI exam, since the experiment measured isotropic di�usion
in �uid water and used a relatively small b-value (b=453 s/mm2). Alexander et al.
performed Monte Carlo simulations under conditions typical of clinical practice [21].
For determining the principal eigenvector, they showed that the optimal strategy is to
collect no b0 images at all. This is probably because the eigenvector does not depend
on the value of the tensor parameters, just their relative sizes (i.e. normalization is
not important). For a good trade-o� between directional information and size and
shape indices, they suggested a NDW :Nb0 ratio of about 5. Both of these experiments
used the linear least-squares �tting algorithm.
2.2.8 Rotational bias
Rotational bias refers to di�erences in the accuracy and precision of tensor-based
parameter estimation relative to �bre orientation. These di�erences are a form of
systematic error, i.e. they are not simply the product of random noise �uctuations,
but have an underlying, repeatable cause. Rotational bias applies to all tensor pa-
rameters, but its e�ects on Fractional Anisotropy and �bre orientation are the most
signi�cant for the majority of applications. There are two categories of rotational bias
to consider. First, there is rotational bias in the mean. This is a question of accu-
racy, and whether or not the level of accuracy depends on the �bre orientation. The
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Chapter 2. Comparison of least-squares fitting methods 33
second type has to do with the degree of uncertainty. In other words, is the precision
in Fractional Anisotropy or �bre orientation dependent on �bre orientation?
Papadakis et al. examined both types of bias with respect to anisotropy in-
dices [27]. They showed that rotational bias in the mean value of these indices was
relatively small compared to rotationally dependent di�erences in precision. Both
were reduced with the addition of more gradient directions, but there was negligible
bene�t beyond 18-21 directions. Interestingly, this early study employed nonlinear
�tting methods, but the question remains as to whether or not the results would
have been di�erent using LLS. It is also important to point out that the b-value used
in these simulations, b=1570 s/mm2, is higher than that used in standard clinical
practice. Finally, this study makes no mention of rotational bias in �bre orientation
estimates.
This second question was addressed by simulations performed by Jones [26]. These
were designed to examine the trade-o� between increasing the number of gradient di-
rections versus acquiring a smaller number of directions multiple times, for an equiv-
alent total imaging time. This study used LLS �tting and b=1000 s/mm2. Increasing
the number of unique directions dramatically reduced rotational bias in both mean
Fractional Anisotropy and uncertainty in the principal eigenvector, and the bias be-
came increasingly signi�cant at higher FA values. Jones concluded that to reduce
rotational bias e�ects to negligible levels, 20 directions were necessary for FA and 30
for the �bre orientation.
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Chapter 2. Comparison of least-squares fitting methods 34
2.3 Methods
2.3.1 Fitting performance under clinical conditions
Experimental DTI data were obtained from 3 healthy volunteers using a 3T GE
Signa system. Imaging parameters were as follows: 23 gradient orientations based
on the electrostatic-repulsion algorithm [20], b=1000 s/mm2, 2 b0 images, 2.6 mm
isotropic voxels, 48 slices, and a single excitation EPI readout. The maximum gradient
amplitude was 40 mT/m, and echo time and repetition time were 84.5 and 12 000 ms
respectively. The scan was repeated eight times to estimate experimental precision
in tensor parameters for a total scan time of 45 minutes. A twice-refocused spin
echo sequence was used to reduce eddy-current e�ects [18]. The noise parameter, σ,
was calculated from a 15x15 background region using all b0 images across all slices
(15x15x2x48=21 600 voxels). The region was carefully placed to avoid zero-padding
at the edges of the �eld of view, as well as regions potentially e�ected by Nyquist
ghosting [17]. All analysis was performed using Matlab (Mathworks, Natick, MA)
and the immoptibox �tting routines [47]. Complete source-code is available as the
dwi-toolbox package [48].
In order to facilitate comparison with simulations across a range of anisotropy
values, whole-brain masks were designed to avoid voxels likely to be a�ected by sus-
ceptibility artifacts and those containing CSF. First, b0 images were thresholded at
S0 > 13σ and a 3x3 median �lter was applied to remove isolated voxels [49]. This was
the minimum threshold for which no voxels were selected from outside of the brain.
An upper threshold of S0 > 30σ was used to select voxels containing CSF [31]. In
this case, the threshold was increased as far as possible such that the ventricles were
clearly removed. A 3x3 median �lter was also applied to the CSF mask. Finally,
edge detection was performed on the resulting binary mask, and this edge mask was
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Chapter 2. Comparison of least-squares fitting methods 35
convolved with a 3x3 box function. This de�ned a mask of boundary voxels, i.e.
brain/air, brain/bone or brain/ventricle borders. These voxels were removed in or-
der to reduce the impact of susceptibility artifacts and partial volume e�ects on the
whole-brain analysis. A graphical representation of this process is demonstrated in
Fig. 2.3. Note that although this masking algorithm is quite aggressive in removing
voxels from analysis, its purpose is to automatically de�ne a large set of voxels for
which artifacts are small. All of the �tting algorithms are being compared against
the same data. The goal of this experiment was to evaluate the di�erences between
�tting algorithms in a best-case scenario.
Figure 2.3: (a) Binary brain mask (SNR>13) �ltered with a 3x3 median �lter.(b) CSF mask (SNR>30) �ltered with a 3x3 median �lter. (c) Brain mask minusCSF. (d) Edge detection of c convolved with a 3x3 box function. (e) c minus d.
Di�usion tensors were �t for all voxels using the LLS, WLLS, NLS and MCNLS
routines. Reduced chi-squared (χ2r), Fractional Anisotropy (FA), and �bre orientation
(ε1) were calculated for each voxel. The means and standard deviations of these
parameters were estimated over the set of eight repeated measurements. Voxels were
binned by their mean Fractional Anisotropy values (as calculated by NLS) over the
eight repetitions, with bins centered at 0.1, 0.2, 0.3, ..., 0.9.
Monte Carlo simulations were performed across the same range of FA values
for comparison. The b-value, number of b0 images, and gradient orientations were
matched to the experiment. SNR of the b0 images was set to 20. For each FA value,
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Chapter 2. Comparison of least-squares fitting methods 36
100 instances of a reference tensor, D0, with trace 2.1 mm2/ms, were oriented uni-
formly in 3-D space. Di�usion-weighted signals were calculated and complex noise
was added in quadrature. The simulation was repeated 1 000 times producing 100 000
data sets (100 orientations x 1 000 repetitions) at each FA value.
2.3.2 Optimal b-value
Monte Carlo simulations were performed as in the previous section over a range of
b-values between b=300 to 3000 s/mm2 and FA values from 0 to 0.9. The full set
of simulations was repeated three times, once with no SNR normalization (i.e. the
same SNR for all b-values), once normalizing for minimum TE, and once normalizing
for the minimum TE and TR (based on Fig. 2.2). Optimal b-values were de�ned as
those which minimized the root mean squared error in Fractional Anisotropy (FA) or
�bre orientation. b-values for which the root mean squared error was within 5% of
its optimal value were also calculated, giving the range of near optimal performance.
2.3.3 Number of non-di�usion-weighted images
Simulations were performed for a total of N=25 images, with the number of b0 images
(Nb0) incremented from 1 to 10. The remaining images (NDW=N -Nb0) were used for
di�usion-weighting at a b-value of 1000 s/mm2. The optimal number of b0 images
was de�ned as that which minimized the root mean squared error in FA or ε1. As
in the previous experiment, the range of Nb0 values for which the root mean squared
error was within 5% of its optimal was also calculated.
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Chapter 2. Comparison of least-squares fitting methods 37
2.3.4 Rotational bias
Following Jones [26], a total of 60 di�usion-weighted images were broken into seven
possible scenarios: 6, 10, 12, 15, 20, 30, and 60 unique directions, with 10, 6, 5, 4, 3,
2, and 1 NEX respectively. In each case, 10 b0 images were simulated, resulting in an
NDW :Nb0 ratio of 6:1. Each experiment represents an equivalent total imaging time:
60 DW images plus 10 b0 image, for a total of 70 images. The remaining simulation
parameters were as follows: b=1000 s/mm2, trace=2.1 mm2/ms, and SNR of the b0
images was set to 20.
For each value of FA between 0 and 0.9, 10 000 repetitions were performed over
100 evenly spaced tensor orientations [50]. More repetitions were required relative
to previous experiments to obtain adequate precision in parameter values at each
tensor orientation. Simulated data were �t using the LLS, WLLS, NLS and MCNLS
algorithms. Images repeated along the same direction were included as separate points
in each �t.
2.4 Results
2.4.1 Fitting performance under clinical conditions
Fig. 2.4 shows the SNR, trace, and Fractional Anisotropy (FA) histograms, averaged
across eight repetitions for one of the subjects. The average trace and FA statistics
correspond to the NLS �t. The FA histogram had a peak at 0.2 and less than 5%
of the voxels had an FA greater than 0.7. Simulations used SNR and trace values
corresponding roughly to the peaks of the experimental distributions (SNR=20 and
trace=2.1 mm2/ms).
Fig. 2.5 shows the average reduced chi-squared, σFA, and σε1 over a range of FA
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Chapter 2. Comparison of least-squares fitting methods 38
Figure 2.4: Whole-brain mask (a) SNR (b) trace and (c) Fractional Anisotropy (FA)histograms, averaged across eight repetitions for a single subject.
values from simulations and for the combination of all 3 subjects. WLLS, NLS, and
MCNLS were virtually indistinguishable under these conditions. WLLS, NLS, and
MCNLS reduced chi-squared statistics were in agreement with the expected value of
one for all simulations. For the LLS �t, deviation from the expected value increased
with FA (Fig. 2.5a). This is consistent with other recent �ndings [35]. Similar, though
slightly higher, chi-squared values were observed in vivo. Experimentally measured
uncertainty in Fractional Anisotropy (FA) and �bre orientation agreed qualitatively
with simulations. For both parameters, LLS showed greater uncertainty versus other
�t types, and this di�erence became more pronounced with increased anisotropy
(Fig. 2.5).
Fig. 2.6 gives probability density functions for the reduced chi-squared, Fractional
Anisotropy, and �bre orientation from simulations with FA=0.9. This is the situation
under which di�erences between the �tting algorithms are most apparent. WLLS,
NLS and MCNLS are indistinguishable from one another under these conditions, so
only the linear least-squares and nonlinear least-squares results are shown. WLLS,
NLS, and MCNLS algorithms followed the theoretical reduced chi-squared distribu-
tion, while the LLS values were slightly higher. Uncertainty in measuring Fractional
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Chapter 2. Comparison of least-squares fitting methods 39
Figure 2.5: Mean reduced chi-squared, χ2r, from (a) simulations and (b) combined
data from 3 test subjects binned by Fractional Anisotropy (FA). Standard deviationin FA, σFA, from (c) simulations and (d) experiment. Standard angular deviationin the principal eigenvector, σε1 , from (e) simulations and (f) experiment. Imagingparameters: b=1000 s/mm2, 23 gradient orientations, and SNR≈20. Error bars show± one standard deviation con�dence intervals for the sample means.
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Chapter 2. Comparison of least-squares fitting methods 40
Figure 2.6: Simulation histograms for FA=0.9. (a) Reduced chi-squared, χ2r, (b)
Fractional Anisotropy (FA), and (c) angular di�erence of ε1 from the mean dyadictensor in degrees. σε1 is also labeled, representing the root mean squared angulardi�erence between the �bre orientation and the mean dyadic tensor.
Anisotropy was also increased using LLS, as evidenced by the broader distribution in
Fig. 2.6b. Fig. 2.6c shows a histogram of the angular di�erence between the estimated
direction, ε1, and the mean dyadic tensor, 〈ε1〉 [43]. The further this distribution shifts
to the right, the greater the degree of uncertainty in estimating the �bre orientation,
which corresponds to an increase in σε1 .
The average computational time for the four �tting algorithms was 0.01, 1.5,
4.9 and 5.2 ms/tensor for the LLS, WLLS, NLS and MCNLS �ts respectively using
Matlab 7.0 on a 2.13 GHz Intel Core 2 PC. The WLLS, NLS, and MCNLS �ts were
of the same order of magnitude, while the LLS �t was roughly 100 times faster. All
Matlab code was optimized where possible, though some di�erences may be related
to implementation.
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Chapter 2. Comparison of least-squares fitting methods 41
2.4.2 Optimal b-value
Fig. 2.7 shows the root mean squared error in Fractional Anisotropy (FA) plotted
against the maximum b-value for three values of FA: 0.1, 0.5, and 0.9. Fig. 2.7a-c
represent the case where SNR is consistent across all b-values, while d-f shows the
case where the SNR is normalized for minimum TE and minimum TR. Normalizing
for just TE produced results that were intermediate to the two. For low anisotropy,
although the optimal b-value is the same across all �t types, NLS outperforms all
other algorithms for b-values greater than 1500 s/mm2. The bene�t of the MCNLS
�tting algorithm is only apparent at high b-values in the highly anisotropic case, and
it is relatively small (Fig. 2.7cf).
Fig. 2.8a-c shows the range of b-values for which the root mean squared error
in Fractional Anisotropy is within 5% of its minimum value under the three SNR
scenarios (SNR=20, normalized for minimum TE, and normalized for minimum TE
and minimum TR). MCNLS and WLLS (not shown) had results similar to NLS. Full
details are available in Table 2.1. The major di�erence between the �tting algorithms
relates to their behaviour relative to di�usion anisotropy. At low anisotropy, optimal
b-values are consistent across all �tting algorithms. However, for the LLS �t, the
optimal b-value decreases with increased anisotropy. This is consistent with previous
reports [21]. The other �tting algorithms show the opposite trend, with the optimal
b-value increasing with anisotropy. When trying to select a b-value that performs well
across the full range of anisotropies, the consequence is that the maximum b-value for
LLS is limited by performance at high anisotropies, while the other �tting algorithms
are limited by their performance at low anisotropies.
Fig. 2.8d-f, Fig. 2.9 and Table 2.2 show the analogous results for minimizing the
root mean squared error in ε1. In general, b-values are slightly lower than those opti-
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Chapter 2. Comparison of least-squares fitting methods 42
Table 2.1:
mized for Fractional Anisotropy (FA). In addition, SNR normalization has a stronger
impact on reducing the optimal b-value than it did in the case of FA. For estimating
ε1, the NLS �t has performance equivalent to or better than all other algorithms at all
b-values tested. The MCNLS �t shows no advantage for estimating �bre orientation
even at high b-values. LLS performs signi�cantly worse than all other algorithms in
cases of high b-value combined with high-anisotropy.
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Chapter 2. Comparison of least-squares fitting methods 43
Table 2.2:
2.4.3 Number of non-di�usion-weighted images
Table 2.3 shows the number of b0 images that minimize the root mean squared error
in Fractional Anisotropy across a range of anisotropies. Although WLLS, NLS, and
MCNLS showed a tendency towards a higher number of b0 images, this di�erence was
never greater than one. Lower anisotropies required a smaller relative number of b0
images, i.e. a higher NDW :Nb0 ratio.
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Chapter 2. Comparison of least-squares fitting methods 44
Table 2.3:
2.4.4 Rotational bias
Fig. 2.10 shows σFA (actual FA=0.9) plotted as a function of tensor orientation for the
LLS and NLS algorithms under 3 scenarios: 6 directions with 10 NEX, 10 directions
with 6 NEX, and 60 directions with 1 NEX. The results for NLS, WLLS, and MCNLS
were indistinguishable under these simulation conditions, so only the NLS and LLS
results are shown. In the case of 6 directions repeated 10 times, all �tting algorithms
had comparable performance, both in terms of the mean uncertainty in Fractional
Anisotropy and in the standard deviation of σFA with respect to tensor orientation
(Fig. 2.10g and h).
The standard deviation with respect to tensor orientation is a measure of how
much the uncertainty changes with respect to �bre orientation, indicating the rota-
tional bias in precision. Increasing the number of directions beyond 6 reduced the
mean value of σFA by about 20% in the case of NLS, WLLS, and MCNLS, whereas
no e�ect was observed for the LLS �t (Fig. 2.10g). The rotational bias in σFA was re-
duced with an increase in the number of directions for all �tting algorithms, although
the LLS asymptote was reached with 30 directions, while the asymptotic value for all
other algorithms was reached with just 15 directions (Fig. 2.10h).
The results for �bre orientation were similar. Moving beyond 6 unique directions
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Chapter 2. Comparison of least-squares fitting methods 45
reduced the mean value of σε1 by about 25% in the case of NLS, WLLS, and MCNLS,
whereas the mean value for LLS did not change (Fig. 2.11g). Rotational bias in σε1
was minimized with 30 directions for the LLS �t, and about 20 directions for all other
�t types. Fig. 2.12 and 2.13 show the mean value and rotational bias in σFA and σε1
plotted versus the number of unique directions for a range of di�usion anisotropies.
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Chapter 2. Comparison of least-squares fitting methods 46
Figure 2.7: Root mean squared error in Fractional Anisotropy plotted vs. b, for(a) FA=0.1, (b) FA=0.5, and (c) FA=0.9, with SNR of the b0 images held constantat 20 for all b-values. (d-f) The corresponding plots where SNR is normalized for TEand imaging time.
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Chapter 2. Comparison of least-squares fitting methods 47
Figure 2.8: Range of b-values for which the root mean squared error in FractionalAnisotropy is within 5% of its minimum value. (a) SNR=20 for all b-values, (b) SNRnormalized for minimum TE, and (c) SNR normalized for minimum TE and minimumTR. (d-f) The corresponding range of b-values for which the root mean squared errorin �bre orientation is within 5% of its minimum value.
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Chapter 2. Comparison of least-squares fitting methods 48
Figure 2.9: Root mean squared error in ε1 plotted vs. b, for (a) FA=0.5, (b) FA=0.7,and (c) FA=0.9, with SNR of the b0 images held constant at 20 for all b-values. (d-f) The corresponding plots where SNR is normalized for minimum TE and imagingtime. FA values below 0.5 are not shown because directional esimation is much lessreliable at low anisotropy.
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Chapter 2. Comparison of least-squares fitting methods 49
Figure 2.10: Uncertainty in Fractional Anisotropy as a function of �bre orientationusing the LLS �t for (a) 6 directions, (b) 10 directions and (c) 60 directions. (d-f) Analagous results for NLS �tting. (g) Mean uncertainty in FA (averaged over alltensor orientations) versus unique number of directions. (h) Standard deviation inσFA over all �bre orientations, representing the rotational bias in uncertainty. Notethat the vertical scale of g is 10x that of h.
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Chapter 2. Comparison of least-squares fitting methods 50
Figure 2.11: Uncertainty in ε1 as a function of �bre orientation using the LLS �t for(a) 6 directions, (b) 10 directions, and (c) 60 directions. (d-f) Analagous results forthe NLS �t. (g) Mean uncertainty in ε1 (averaged over all tensor orientations) versusunique number of directions. (h) Standard deviation in σε1 over all �bre orientations,representing rotational bias in uncertainty. Note that the vertical scale of g is 10xthat of h.
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Chapter 2. Comparison of least-squares fitting methods 51
Figure 2.12: Mean value of σFA averaged over all �bre orientations versus the numberof unique directions for an FA of (a) 0.1, (b) 0.5, and (c) 0.9. Standard deviation inσFA over all tensor orientations for an FA of (d) 0.1, (e) 0.5, and (f) 0.9 respectively.The vertical scale of a-c is 10x that of d-f.
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Chapter 2. Comparison of least-squares fitting methods 52
Figure 2.13: Mean value of σε1 averaged over all �bre orientations versus the number ofunique directions for an FA of (a) 0.5, (b) 0.7, and (c) 0.9. (d-f) Standard deviationin σε1 over all tensor orientations for an FA of 0.5, 0.7, and 0.9 respectively. Thevertical scale of a-c is 10x that of d-f.
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Chapter 2. Comparison of least-squares fitting methods 53
2.5 Discussion
This study demonstrated that at b=1000 s/mm2, DTI data analysis using the LLS al-
gorithm resulted in reduced directional reliability and precision in Fractional Anisotropy
(FA) especially for highly anisotropic voxels. At b=1000 s/mm2, there was very lit-
tle di�erence between any of the alternatives to LLS. Results in vivo qualitatively
agreed with the corresponding simulations. The average reduced chi-squared values,
χ2r, were slightly elevated in vivo, indicating the presence of voxels for which the dif-
fusion tensor model may have been insu�cient. Slightly higher uncertainty in both
FA and �ber orientation was also present in the in vivo data relative to simulations,
especially at high anisotropy. This was probably due to outliers, which had a dispro-
portionately large e�ect at high FA because there were relatively few voxels with FA
values in this range (Fig. 2.4c). Reduced precision in experimental results relative to
simulations could also be partially explained by variations in trace and SNR over the
brain volumes (in simulations, these variables were �xed).
Fig. 2.7, 2.8, and 2.9 clearly show that the optimal b-value depends on the �tting
algorithm being used. In general, the b-value that minimized the root mean squared
error in Fractional Anisotropy (FA) was slightly higher than the b-value optimized
for ε1. A b-value between 800 and 1000 s/mm2 o�ered a good trade-o� between
directional and anisotropy information for LLS (if SNR was normalized for minimum
TE and TR), while for WLLS, NLS, and MCNLS, the optimal range of b-values was
slightly higher; somewhere in the range of 900 to 1200 s/mm2.
Gradients with a higher maximum amplitude have potential to shorten the min-
imum TE necessary to achieve a given b-value (Fig. 1.2), which would reduce SNR
losses associated with higher b-values. This could shift the optimal b-value towards
the unnormalized SNR case. Ignoring SNR losses due to di�erences in TE and TR,
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Chapter 2. Comparison of least-squares fitting methods 54
the upper limit for LLS �tting was around 1100 s/mm2 for ε1, and 1400 s/mm2 for
FA. In the case of the WLLS, NLS and MCNLS �ts, the b-value value could poten-
tially be raised as high as 1800 s/mm2 while still maintaining near optimal accuracy
in parameters. In general, it seems that an added bene�t of the more advanced �tting
techniques is that they were better able to adapt to high b-values, a property that
will be exploited in Chapter 3.
The MCNLS algorithm essentially modulates signal measurements based on their
amplitude, i.e. the degree to which they are likely to experience a positive magnitude
bias. Low signals are especially prone to magnitude bias, so they are suppressed
more, whereas higher signals are modulated less. This technique estimated Fractional
Anisotropy more accurately than NLS for highly anisotropic voxels at high b-values,
but at low anisotropy, NLS showed better performance (Fig. 2.7a,d). A possible
explanation is that in the isotropic case, all measurements were suppressed almost
equally, which is analogous to a slight signal loss. The implication is that it may be
better to use the magnitude-compensation only for highly anisotropic voxels, possibly
using some sort of thresholding approach.
For a total of 25 images, 2-3 b0 images o�ered good performance across the full
range of anisotropies using LLS �tting, corresponding to a NDW :Nb0 ratio between
7 and 12. For WLLS, NLS, and MCNLS, 3-4 b0 images was optimal, which corre-
sponds to a NDW :Nb0 ratio between 5 and 8. While these guidelines may be useful for
minimizing the error in Fractional Anisotropy measurements, the acquisition of any
non-di�usion-weighted images at the expense of more sampling directions increases
the error in estimating �bre orientation. In this respect, these NDW :Nb0 ratios repre-
sent the minimum values that should be considered. Higher ratios improve the ability
to estimate the direction of ε1 at the expense of FA, which may be desirable in some
situations (e.g. tractography).
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Chapter 2. Comparison of least-squares fitting methods 55
WLLS, NLS, and MCNLS showed reduced rotational bias relative to LLS, and
reached their asymptotic values with fewer directions (15 for FA versus 30 with LLS,
20 for ε1 versus 30 with LLS). All of the alternatives to LLS also demonstrated lower
mean uncertainty in both �bre orientation and Fractional Anisotropy (FA) for any
number of directions greater than 6. For both FA and �bre orientation, the reduction
in mean uncertainty between LLS and the other algorithms was of a similar magnitude
to the reduction in the standard deviation versus orientation realized by acquiring an
increased number of unique directions (Fig. 2.12 and 2.13). This implies that using one
of the more advanced �tting algorithms had about the same degree of improvement
in results as using 20-30 directions versus 6; one reduced overall uncertainty, while
the other reduced rotational bias. Obviously, the greatest bene�t was achieved by
using one of the advanced �tting algorithms and acquiring more than 20 directions.
The combined results from this chapter illustrate several reasons for moving be-
yond the framework of linear least-squares �tting. Improved precision in tensor-based
parameters and reduced rotational bias both translate into increased con�dence in
DTI results. Although these performance gains are modest, the only expense is com-
putational time. With today's standard PC workstation, the time to �t DTI data
using nonlinear methods is roughly equivalent to the time it takes to acquire the data
from the MR scanner. Processor speed is no longer an excuse for suboptimal analysis
practices.
This chapter also demonstrated that a migration from LLS �tting requires a re-
thinking of data collection itself. Scan parameters that may have been optimal for
LLS algorithms would bene�t from minor adjustments. These results suggest that
slightly higher b-values and reduced NDW :Nb0 ratios are necessary to take full advan-
tage of the WLLS, NLS and MCNLS approaches.
It should be stressed that the optimal parameters reported in this thesis are only
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Chapter 2. Comparison of least-squares fitting methods 56
valid for data that can be fully described by the di�usion tensor model. Optimizing
di�usion-weighted MRI for the purpose of resolving crossing or bending �bres would
obviously require a di�erent approach. Even in the simplest case of parallel �bre bun-
dles, the phrase �optimal parameters� can be a bit misleading. Fitting performance
depends on a number of variables including, but not limited to: mean di�usivity, b-
value, tissue anisotropy, T2-relaxation, pulse sequence timing, etc. The fact that the
optimal b-value and NDW :Nb0 ratio vary considerably across the range of FA values
implies that it is impossible to optimize for voxels with low and high anisotropy si-
multaneously. A possible solution to this problem may involve the design of sampling
strategies that utilize multiple b-values. This is an area that deserves future study.
It should also be stressed that any choice of imaging parameters necessarily in-
volves trade-o�s, and choosing the best parameters involves a careful consideration
of the entire imaging system. The simulations presented in this chapter examined
some of these trade-o�s under a standard set of conditions. While the general trends
should translate to other situations, the speci�c �optimal� values may change. Dif-
ferences in scanner hardware will a�ect the SNR normalization factors used, and will
therefore have an e�ect on the recommended parameter values. It is also important
to consider biological di�erences in the subject population, which could change, for
example, mean di�usivity or T2 values. Source code for these simulations is freely
available [48], and the interested reader is encouraged to adjust it to suit their own
needs.
The optimization problem is further complicated by a host of potential image ar-
tifacts that include cardiac pulsatility, patient motion, susceptibility, eddy currents,
and partial volume e�ects. None of these e�ects were considered in the simulation
studies but each can have a major impact on experimental results. Once again, reduc-
ing these artifacts typically involves some form of trade-o�. For example, averaging
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Chapter 2. Comparison of least-squares fitting methods 57
multiple excitations can improve SNR, but this increases the potential for subject
motion and misregistration. A more thorough look at scan parameter optimization
is probably warranted, speci�cally to evaluate performance in vivo, where variability
beyond random noise exists. In any case, optimization of data collection should take
into account the intended �tting algorithm.
The presence of these additional sources of variation highlights a problem com-
mon to all least-squares methods: they are extremely sensitive to data outliers [51].
Other groups have implemented robust �tting techniques which are less susceptible
to outlier-related artifacts. Mangin et al. used the Geman-McLure M-estimator to
�t the di�usion tensor [52]. They showed improved performance in the presence of
corrupted data points, but the quality of �tting in the absence of outliers was re-
duced. Chang et al. used a similar technique to perform automated outlier-rejection,
after which they applied standard nonlinear �tting to the processed data [53]. The
application of robust statistics to the �eld of DTI shows promise, but there are several
concerns that still need to be addressed. Robust techniques must be able di�erenti-
ate between outliers caused by image artifacts and those that re�ect true deviations
from the tensor model, for example, those due to complex tissue structure and par-
tial volume e�ects. The latter case re�ects useful information that should not be
discarded.
Another area of interest that has not yet been explored is the impact of �tting
algorithms on bootstrap analysis [54]. Bootstrapping is a statistical technique that
involves resampling of experimental data to generate probability distributions for es-
timated model parameters. Model-based techniques such as the wild bootstrap [55]
and residual bootstrap [54] make an assumption that the residuals are either sym-
metric or similar across data points. These assumptions are not valid for LLS or even
WLLS �tting, though the degree to which this might in�uence bootstrap results is
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Chapter 2. Comparison of least-squares fitting methods 58
unknown. The results of this study suggest that it may be problematic for voxels
with high anisotropy and/or high b-values.
Finally, there are two important advantages to nonlinear �tting beyond those
explicitly tested in these experiments. First of all, NLS �tting produced chi-squared
statistics that were truer to the theoretical chi-squared distribution, as evidenced
by Fig. 2.5 and 2.6. This is important from the perspective of model validation,
as the chi-squared value is indicative of the agreement between the data and the
model. A �tting algorithm that arti�cially in�ates this statistic could obviously lead
to problems in this context. Secondly, while the performance of the LLS algorithm
drops o� dramatically at high b-values, nonlinear methods are less susceptible to
this e�ect. High b-values are of great interest because they accentuate non-Gaussian
di�usion processes for which the tensor model is insu�cient [19, 22]. This suggests
that an attempt to characterize non-Gaussian voxels may be facilitated by a move
to higher b-values. The combination of these two properties makes nonlinear �tting
particularly well suited for the study of model validation. This will be the focus of
Chapter 3.
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Chapter 3
Testing the validity of the tensor
model
�All models are wrong, some are useful.�
-George Box
3.1 Introduction
It is well known that the tensor model is limited to describing a single population
of �bres running in parallel. While this is generally su�cient for the majority of
voxels in white matter, more complex tissue architectures including crossing, bending,
and diverging �bres, are known to exist [19]. Applying the tensor model in these
situations can lead to misinterpretation of DTI results, as demonstrated in Fig. 3.1.
Heterogeneous �bre populations within a single voxel can result in erroneous �bre
directions and reduced Fractional Anisotropy (FA), e�ects which could be falsely
attributed to pathology or cause premature termination of tractography algorithms.
Therefore, it is important to realize the limitations of DTI and avoid making inferences
59
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Chapter 3. Testing the validity of the tensor model 60
Figure 3.1: (a,b) ADC pro�les for two distinct �bre populations separated by a 45-degree angle. (c) ADC pro�le from a voxel containing equivalent proportions of thesetwo �bre populations. (d) The di�usion tensor �t to the ADC pro�le in c. (e) FA mapwith overlayed ellipsoids representing the displacement pro�le for each voxel. Regionsi and ii contain homogeneous populations of �bres a and b, respectively. Region iiicontains a mixed population, resulting in an estimated �bre orientation that matchesneither of the included �bre populations, and an apparent reduction in FA.
based on the tensor model in situations where it is not justi�ed.
The inadequacy of the di�usion tensor model has motivated the development of
advanced reconstruction schemes including q-space [5], q-ball [56], and Persistent
Angular Structure (PAS) MRI [50]. These techniques require the acquisition of more
data, usually in the form of additional sampling directions. Although increased data
allows the �tting of more complex models, the question remains as to whether or not
these higher-order models are always necessary, or if the data quality is su�cient to
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Chapter 3. Testing the validity of the tensor model 61
warrant the �t. In voxels for which the underlying architecture is adequately described
by a tensor, the use of higher-order models can actually introduce additional error
due to over-�tting (i.e. �tting the noise). Therefore, even in those cases where it is
possible to �t higher-order models, one should always ask whether or not the increased
complexity is supported by the data.
Alexander et al. proposed a model-selection algorithm for classifying voxels into
categories of isotropic Gaussian, non-isotropic Gaussian (equivalent to the di�usion
tensor) and non-Gaussian di�usion [31]. This technique involves the �tting of a
hierarchical set of models based on the spherical harmonic series, and selecting the
most appropriate model using an F-test. It performs reasonably well for �bres crossing
at 90 degrees and in equal volume fractions, but su�ers as the separation angle is
reduced and/or the volume fractions become more unbalanced.
Because non-Gaussian di�usion is more apparent at high b-values [19, 22], it may
be possible to improve classi�er performance by imaging at b-values greater than
1000 s/mm2 (Fig. 3.2). However, the linear least-squares algorithm commonly used
to �t the spherical harmonic models performs poorly if the b-value is increased too
much, analagous to the results presented for the di�usion tensor in the previous
chapter. In addition, high b-values combined with high di�usivity can result in signal
measurements close to the noise �oor. In this situation, magnitude bias can result in a
�squashed peanut� artifact [32], which itself resembles non-Gaussian di�usion. Based
on the results of Chapter 2, a possible solution to this problem is an extension of the
magnitude-corrected nonlinear least-squares (MCNLS) �tting algorithm to higher-
order models. The ability to �t spherical harmonic models at high b-values should
result in an improved ability to detect voxels for which the di�usion tensor is invalid.
Note that throughout this chapter, di�usion pro�les for isotropic media and those
with parallel �bres are said to exhibit Gaussian di�usion. Technically speaking, all
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Chapter 3. Testing the validity of the tensor model 62
Figure 3.2: ADC pro�les of two �bres crossing at 90 degrees for 3 di�erent b-values:(a) b=1000 s/mm2, (b) b=2000 s/mm2, and (c) b=3000 s/mm2.
di�usion in biological tissue is non-Gaussian. Separate water compartments (e.g. in-
tra/extra cellular water), exchange, and restrictions to water mobility all violate the
Gaussian di�usion model [45]. However, deviation from Gaussian behaviour is rela-
tively minor for the aforementioned tissue types in the range of b-values available on
most clinical scanners (≤3000 s/mm2) [7]. Under these conditions, the Gaussian/non-
Gaussian distinction provides a useful way to di�erentiate between the voxels for
which the di�usion tensor is valid (e.g. isotropic di�usion and parallel �bre bundles),
and those with more complex geometries.
3.2 Theory
3.2.1 Hypothesis testing
Hypothesis testing is a commonly employed statistical technique to compare evidence
for a given hypothesis with the probability that the the results are a product of random
chance. In the context of this chapter, the test is whether or not the di�usion pro�le
in a given voxel is Gaussian in shape; that is, whether or not the tensor model is
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Chapter 3. Testing the validity of the tensor model 63
valid. The �rst step is to formulate the null hypothesis: that the di�usion pro�le
is Gaussian in shape. Next a metric that relates to this hypothesis is needed, for
example, the chi-squared statistic. The chi-squared value is calculated according to
Eq. 2.15, and it indicates how well the tensor model �ts the data.
The probability density function (pdf) of the metric should change if the null
hypothesis is false (e.g. in the case of non-Gaussian di�usion, the chi-squared value
should be higher). The pdf may be shifted, stretched, or have an entirely di�erent
shape. Ideally, it would be completely separated from that of the null hypothesis, in
which case it would be easy to separate the two scenarios. More likely, there will be
some degree of overlap and no matter where the threshold is drawn, some voxels will
be misclassi�ed. In this case, there are two types of error, false positives and false
negatives, as demonstrated in Fig. 3.3.
Figure 3.3: Hypothesis test schematic. The solid curve represents metric pdf if thenull hypothesis (Gaussian di�usion) is true . The dashed curve represents the pdf ofthe metric if the null hypothesis is false. The metric could be any measurement thatwould be expected to di�er between the two cases (e.g. chi-squared statistic). If themetric is greater than the decision threshold, the null hypothesis is rejected. Falsepositive and false negative errors are shown.
The theoretical distribution of the chi-squared statistic can be calculated based on
the number of model parameters and the number of data points. It is then possible to
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Chapter 3. Testing the validity of the tensor model 64
base the decision threshold on a speci�c p-value, for example, 0.01. This means that
if a voxel is imaged multiple times with the null hypothesis being true, its chi-squared
value would be less than the threshold 99% of the time. Any voxel with a chi-squared
value above this threshold would lead to a rejection of the null hypothesis.
While a hypothesis test based on chi-squared values can successfully identify some
voxels with complex architecture, performance is relatively poor except in the case
of a large separation angle. A more sophisticated approach involves reframing the
question of model validation as one of model selection. This strategy is examined in
the following section.
3.2.2 Model selection
The previously described work of Alexander et al. [31] utilized a hierarchical set of
models based on the spherical harmonic series. Assuming that di�usion is antipodally
symmetric, only the even-ordered terms are required. The 0th-order term alone rep-
resents isotropic di�usion. Truncating the series at the 2nd-order produces a model
that is equivalent to the di�usion tensor. The 4th-order model has �fteen parameters
and is capable of describing shapes that resemble crossing �bres [57]. Fig. 3.4 shows
a graphical representation of these models �t to four di�erent simulated test cases:
isotropic di�usion, prolate di�usion, oblate di�usion, and a pair of �bre bundles cross-
ing at 90 degrees. Once all of the models have been �t to each case, the simplest
model that can adequately describe the data is selected.
The method described by Alexander et al. relies on an F-test for model selection.
The F-test compares the chi-squared statistics from a pair of models and the relative
number of parameters in each [42]. It is calculated according to Eq. 3.1, where pa and
pb are the number of parameters for model a and b (pb > pa), and χ2a and χ
2b are the
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Chapter 3. Testing the validity of the tensor model 65
Figure 3.4: Noisy ADC pro�le and 0th, 2nd and 4th-order SH models �t to 4 testcases. The model selected by the F-test algorithm in each case is surrounded bya black box. Notice that each of the selected models has a similar shape to itscorresponding noisy ADC pro�le, and that increasing the model order further doesn'tsigni�canly change the shape.
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Chapter 3. Testing the validity of the tensor model 66
corresponding chi-squared values. The F-statistic can be thought of as a measure of
the relative improvement in the reduced chi-squared value a�orded by the additional
model parameters.
Fa,b =(N − pb − 1)(χ2
a − χ2b)
(pb − pa)χ2b
(3.1)
To utilize the F-statistic in a hypothesis testing framework, the null hypothesis is that
both models describe the data equally well; that is, there is no advantage to using
the more complex model. In this case, the F-statistic follows a known theoretical
distribution, and a threshold, Ta,b, is de�ned such that Fa,b is less than the threshold
99% of the time. Alternatively, thresholds can be based on performance criteria from
simulation studies. If the F-statistic is above the threshold, the null hypothesis is
rejected and the more complex model is deemed necessary.
The model selection algorithm works as follows. Each voxel is initially assigned
the 0th-order isotropic classi�cation. If F0,2 is greater than the T0,2 threshold, the
voxel is assigned to the 2nd-order class. Next, F2,4 is calculated and compared to
the T2,4 threshold. If it exceeds this threshold, its classi�cation is incremented to
4th-order. When the F-statistic fails to cross its respective threshold, the algorithm
is terminated.
3.2.3 Fitting higher-order models
The spherical harmonic (SH) series is a set of basis functions represented by Yl,m(θ, ϕ),
where θ and ϕ are the collatitude and longitude angles on the sphere, l = 0, 1, 2, ... is
the function order, and m = −l, ..., 0, ..., l are the indexed functions of order l. Any
complex-valued spherical function can be written as a linear combination of these SH
functions multiplied by a set of complex coe�cients, cl,m:
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Chapter 3. Testing the validity of the tensor model 67
f(θ, ϕ) =∞∑l=1
l∑m=−l
cl,mYl,m(θ, ϕ) (3.2)
Truncating this series at order l, leaves (l+1)(l+2)2
parameters.
Solving for the SH coe�cients is similar to �tting the di�usion tensor, described
in section 2.2. The SH coe�cients are written as a column vector, C.
C = [c0,0, c2,−2, c2,−1, c2,0, c2,1, ..., clmax,lmax ]T (3.3)
Each row of the experimental design matrix is setup using the b-values, bi, and SH
basis functions, Yl,m(θi, ϕi), that correspond to the applied di�usion gradients for each
of the N images:
Xi = [biY0,0(θi, ϕi), biY2,−2(θi, ϕi), ..., biYlmax,mmax(θi, ϕi), 1] (3.4)
The complete experimental design matrix is therefore:
X =
X1
...
Xi
...
XN
(3.5)
To solve for the SH coe�cients using linear least-squares regression, Y, is de�ned as
an Nx1 column vector of the log-transformed signal measurements,
Y = [ln(S1), ln(S2), · · · , ln(SN)]T (3.6)
resulting in the following equation:
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Chapter 3. Testing the validity of the tensor model 68
Y = XC (3.7)
The LLS solution is calculated by taking the pseudoinverse of the design matrix, X+,
and multiplying it by Y.
C = (X∗X)−1XT = X+Y (3.8)
X∗ is the conjugate transpose of matrix X.
It is also possible to calculate the weighted linear least-squares solution using the
same weighting matrix as in the di�usion tensor case (Eq. 2.7).
C = (X∗WX)−1(XTW)Y (3.9)
Nonlinear �tting is complicated by the fact that the SH coe�cients have an imag-
inary component. If signal measurements are written as an Nx1 column vector, S,
the di�usion equation takes the following form:
S = exp
0 0 1
......
...
−biRe(Xi) −biIm(Xi) 1
......
...
−bNRe(XN) −bNIm(XN) 1
Re(C)
Im(C)
ln(S0)
(3.10)
Dividing the SH coe�cients into their real and imaginary components doubles the
number of unknowns. The reason this problem is not encountered in the case of LLS
or WLLS is that these methods implicitly constrain S to be real-valued. While it is
possible to �t the system described in Eq. 3.10 using a nonlinear algorithm with N
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Chapter 3. Testing the validity of the tensor model 69
constraints, there is an alternative method that has several advantages.
There is a family of models based on higher-order tensors, or Generalized Di�u-
sion Tensors (GDTs) [58], that are theoretically equivalent to the spherical harmonic
models. Descoteaux et al. showed that it is trivial to convert between the two [59].
The advantage to working with GDTs is that the real and imaginary components
are completely separable. Therefore, if the measured signal is real-valued, all of the
GDT parameters are also real-valued, meaning that the GDT models can be �t using
nonlinear algorithms without the need to apply constraints. This makes �tting GDTs
conceptually easier and reduces computational demands relative to the spherical har-
monic models.
The �tting of a rank-l tensor will be described using the formalism of Ozarslan
et al. [60]. The apparent di�usion coe�cient along direction r for a rank-l tensor is
described by the equation:
D(l)(r) =3∑
i1=1
3∑i2=1
...3∑
il=1
Di1i2...ilri1ri2 ...ril (3.11)
where the indices ij represent the x, y, and z direction components respectively. To
maintain antipodal symmetry, l is forced to be an even number, as was the case with
the spherical harmonic models. Eq. 3.11 implies that there are 3l terms for a rank-l
tensor, however this number is greatly reduced by symmetry. This is analogous to
the case of the standard rank-2 tensor, where Dij is recognized as being equivalent
to Dji. For the general case, this property is expressed by the following equation:
Di1i2...in = D(i1i2...in) (3.12)
where (i1i2...in) represents all permutations of the tensor indices. The number of
unique elements is Nl = (l+1)(l+2)2
, which is the same number of parameters in a
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Chapter 3. Testing the validity of the tensor model 70
spherical harmonic model of order l. Each of the Nl unique elements are repeated µ
times, where:
µ =
l
nx
l − nx
ny
=l!
nx!ny!nz!(3.13)
nx, ny, and nz represent the number of x, y, and z indices in the tensor subscript.
The di�usion equation can now be written in terms of a rank-l GDT:
S = S0 exp
[−b
Nl∑k=1
µkDk
l∏p=1
rk(p)
](3.14)
where µk is the number of repeated instances of tensor elementDk, and k(p) represents
the p-th index in the subscript string for Dk. As an example, Table 3.1 shows the
unique elements for a rank-4 GDT and the number of times they are repeated in
Eq. 3.11.
Table 3.1: Elements of a rank-4 generalized DT
To solve for the GDT parameters, the unique tensor elements are written as a
column vector, x(l), where the bracketed superscript de�nes the tensor rank. Note
that x(2) is exactly the same as that used in Chapter 2 for �tting the standard di�usion
tensor.
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Chapter 3. Testing the validity of the tensor model 71
x(0) = [D, .., ln(S0)]T (3.15)
x(2) = [Dxx, Dyy, Dzz, Dxy, Dxz, Dyz, ln(S0)]T (3.16)
x(4) = [Dxxxx, Dyyyy, Dzzzz, Dxxxy, Dxyyy, ..., ln(S0)]T (3.17)
x(l) = [Di1,i2,i3,...,in , ..., ln(S0)]T (3.18)
The experimental design matrix, B(l), has N rows corresponding to the set of signal
measurements and (l + 1) columns. The �rst Nb0 rows represent the non-di�usion-
weighted images, and all elements in the rightmost column are equal to one. The
remaining elements are constructed from the b, µk, and rk(p) terms in Eq. 3.14. The
i-th row and the k-th column of the design matrix have the form:
−biµikl∏
p=1
rik(p) (3.19)
The design matrix for the rank-4 GDT has the following form:
B =
0 0 0 0 0 ... 1
......
......
......
−bir4x −bir4
iy −bir4iz −4bir
3ixriy −4bir
3ixriz ... 1
......
......
......
−bNr4Nx −bNr4
Ny −bNr4Nz −4bNr
3NxrNy −4bNr
3NxrNz ... 1
(3.20)
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Chapter 3. Testing the validity of the tensor model 72
From this point forward, �tting a GDT is exactly the same as �tting a standard
rank-2 tensor. The magnitude-corrected nonlinear least-squares (MCNLS) �t can be
applied to GDTs by minimizing Eq. 3.21, where σ is a noise-estimation parameter
measured from a background region of the image. Setting σ to zero results in the
standard nonlinear least-squares �t.
fMCNLS(x) =N∑i=1
[Si −
√exp2(Bix) + σ2
]2(3.21)
The estimated GDT parameters can be converted to the equivalent SH model [59],
or alternatively, the model selection algorithm can be applied to the GDT models
themselves.
3.3 Methods
3.3.1 Experiment
DTI data was obtained from four healthy volunteers using a 3T GE Signa system.
Voxels were 2.6 mm isotropic, and 42 slices were acquired to cover the entire brain.
10 b0 images and 55 gradient orientations were used. Half of the subjects were imaged
using a quadrature birdcage coil while the other half used an 8-channel head coil.
Three data sets were collected sequentially from each subject with b-values of 1000,
2000, and 3000 s/mm2 using the twice-refocused spin echo sequence [18]. TE and
TR were both minimized for each b-value resulting in TE/TR values of 85/12 000,
97/14 300 and 105/15 900 ms respectively. A brain mask was created by thresholding
the b=1000 s/mm2 images ≥ 10σ.
0th, 2nd and 4th-order GDT models were �t to the data using linear least-squares,
nonlinear least-squares and magnitude-corrected nonlinear least-squares �ts. A series
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Chapter 3. Testing the validity of the tensor model 73
of F-tests was performed to select the most appropriate model for each voxel. F-
test thresholds were set independently for each �t type, such that 99% of simulated
di�usion tensors with FA≤0.9 were correctly classi�ed by Monte Carlo simulations.
This is equivalent to �xing the false positive error rate to ≤1%.
3.3.2 Simulations
Monte Carlo simulations were performed for a two-tensor model with relative �bre
population ratios ranging from 1:9 to 5:5, and separation angles from 10 to 90 de-
grees. Each tensor had a trace of 2.1 mm2/ms, random orientation, and a Fractional
Anisotropy (FA) of 0.9 unless otherwise stated. Nb0 and the gradient orientations
were matched to the experimental setup. Each relative �bre population/separation
angle pair was simulated 10 000 times for six permutations of SNR (30 and 70 at
b=1000 s/mm2) and b-value (1000, 2000, and 3000 s/mm2). SNR was normalized for
minimum TE relative to the b=1000 s/mm2 case, as in section 2.3.2. GDT �tting
and model selection was performed as in the experiments.
3.4 Results
3.4.1 Experiment
Fig. 3.5 demonstrates three regions with clusters of voxels classi�ed as 4th-order at
b=3000 s/mm2 using the MCNLS �t and the 8-channel coil. Label 1 points to a
region in the pons where where the inferior-superior pyramidal tracts are crossed
by the transpontine tracts, which run in a lateral direction. Label 2 indicates the
crossing of the anterior-posterior optic radiation and the lateral �bres of the corpus
callosum. Label 3 points to �bre crossings in the corona radiata. The colour-coded
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Chapter 3. Testing the validity of the tensor model 74
direction maps in the �rst column help to anatomically identify the crossing �bre
tracts. The second column shows the F2,4-statistic maps, which compare the 2nd
and 4th-order models. The higher this F-statistic (i.e. the brighter the voxel), the
more likely that the voxel represented non-Gaussian di�usion. The labeled regions
demonstrate elevated F-statistics in the neighbouring voxels. These F-statistic maps
were thresholded to produce the classi�cation maps in column three. Images on the
far right show sagittal and coronal views of these locations.
All of the labeled regions were clearly present for both subjects using the 8-channel
head coil. Figure 3.6ac and e show the same slices from one of the subjects imaged
with the quadrature coil. 4th-order clusters at regions 1 and 3 are visible, though no-
ticeably smaller. The cluster corresponding to label 2 was absent from the quadrature
coil data.
Fig. 3.7 shows classi�cation maps for the same slices in Fig. 3.5, but for three
di�erent b-values. Clearly, the size of the 4th-order clusters increases with b. Across
the entire brain, the average proportion of voxels classi�ed as 4th-order was 1.9%,
3.9%, and 3.2% for b=1000, 2000, and 3000 s/mm2 using the quadrature coil. For the
8-channel coil, the mean 4th-order classi�cation rates were 4.4%, 10.8%, and 11.3%
at b=1000, 2000, and 3000 s/mm2.
The average SNR across the entire brain masks for the b0 images was 29, 25, and
24 at b=1000, 2000, and 3000 s/mm2 using the quadrature coil, and 71, 59, and 55
for the 8-channel coil. The reduction in SNR at the higher b-values can be attributed
to T2-relaxation e�ects resulting from the longer echo times. Fig. 3.8 demonstrates
noisy ADC pro�les for the voxels labeled in Fig. 3.5.
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Chapter 3. Testing the validity of the tensor model 75
Figure 3.5: Selected locations demonstrating clusters of voxels labelled as 4th-order.The top row is slice 7 (a-e). The middle row is slice 15 (f-j), and the bottom row is slice30 (k-o). All images are from the b=3000 s/mm2 data set using the MCNLS �t andthe 8-channel coil. The �rst column (a,f,k) shows colour-coded directional maps ofthe primary eigenvector. Column two (b,g,l) shows the F2,4-statistics. Column three(c,h,m) gives the classi�cation maps and the fourth column shows sagittal (d,i,n) andcoronal slices (e,j,o) of the labeled voxel locations.
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Chapter 3. Testing the validity of the tensor model 76
Figure 3.6: Voxel classi�cation maps for the same three slices as in Fig. 3.5at b=3000 s/mm2 using MCNLS �tting and (a,c,e) the quadrature birdcage coil(SNR≈24), (b,d,f) the 8-channel coil (SNR≈55).
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Chapter 3. Testing the validity of the tensor model 77
Figure 3.7: Voxel classi�cation maps for the same three slices as in Fig. 3.5 for(a,d,g) b=1000, (b,e,h) 2000, and (c,f,i) 3000 s/mm2 using the MCNLS �t and the8-channel head coil.
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Chapter 3. Testing the validity of the tensor model 78
Figure 3.8: Noisy ADC pro�les for voxels labelled in Fig. 3.5 at 3 di�erent b-valuesusing the 8-channel head coil. (a-c) are the voxel labeled as 1, (e-g) 2, and (h-j) 3.Note the increasing complexity of these shapes as the b-value is increased.
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Chapter 3. Testing the validity of the tensor model 79
3.4.2 Simulations
Fig. 3.9 shows the percentage of crossing �bres correctly identi�ed as 4th-order for
three di�erent b-values (SNR=30 at b=1000 s/mm2). This corresponds roughly to
the experimental SNR measured from the images acquired with the quadrature coil.
Fig. 3.10 shows the corresponding results for a reference SNR of 70, similar to the
8-channel coil experiments.
At both SNR levels, all �tting algorithms showed similar performance at b=1000 s/mm2.
The LLS-based classi�er improved at b=2000 s/mm2 but at b=3000 s/mm2 it was es-
sentially ine�ective. Both NLS and MCNLS improved dramatically when the b-value
was increased to 2000 s/mm2, though MCNLS showed a slight advantage over NLS
at low separation angles. When the b-value was increased to 3000 s/mm2, the advan-
tage of the higher b-value appeared to be partially canceled by the loss in SNR. NLS
performance was equivalent to, or slightly reduced b=2000 s/mm2 in all cases.
For MCNLS, the results were mixed. There was slightly better performance for
the highly unbalanced �bre populations at b=2000 s/mm2, while at b=3000 s/mm2,
the classi�cation algorithm seemed to do moderately better for smaller separation
angles. The performance gain between b=2000 and b=3000 s/mm2 was very small
relative to the di�erence between b=1000 and 2000 s/mm2.
The di�erence between the SNR=30 and SNR=70 results clearly demonstrates the
SNR-dependence of the algorithm. Higher SNR allowed the classi�er to reliably detect
smaller separation angles and unbalanced volume fractions. Fig. 3.11 shows detection
rates using the MCNLS algorithm across the three b-values when the Fractional
Anisotropy (FA) of the component tensors was reduced from 0.9 to 0.7. The lower
anisotropy dramatically reduced the ability to detect crossing �bres and illustrates
the need for high SNR in this application.
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Chapter 3. Testing the validity of the tensor model 80
Figure 3.9: Percentage of crossing �bres correctly classi�ed as 4th-order in simulationsfor SNR=30 at b=1000 s/mm2. SNR at b=2000 and 3000 s/mm2 was normalized forminimum TE as in section 2.3.2. F-test thresholds were set to limit the false positiverate to less than 1%. Results for (a) b=1000, (b) 2000, and (c) 3000 s/mm2 usingLLS �tting. (d-f) Analagous results using NLS and (g-i) MCNLS �tting. Results areaccurate to ±2%.
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Chapter 3. Testing the validity of the tensor model 81
Figure 3.10: Percentage of crossing �bres correctly classi�ed as 4th-order in simula-tions for SNR=70 at b=1000 s/mm2. This corresponds approximately to the exper-imental SNR using the 8-channel head coil. SNR at b=2000 and 3000 s/mm2 wasnormalized for minimum TE. Results for (a) b=1000, (b) 2000, and (c) 3000 s/mm2
using LLS �tting. (d-f) Analagous results using NLS and (g-i) MCNLS �tting. Re-sults are accurate to ±2%.
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Chapter 3. Testing the validity of the tensor model 82
Figure 3.11: Percentage of �bre crossings correctly classi�ed as 4th-order when theFractional Anisotropy (FA) of the component tensors was reduced to 0.7. Results at(a) b=1000 (SNR=30), (b) 2000 (SNR=25.7), and (c) 3000 s/mm2 (SNR=23.1) usingthe MCNLS �t. SNR was normalized for minimum TE. (d-f) Analagous results forSNR=70 (at b=1000 s/mm2). Results are accurate to ±2%.
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Chapter 3. Testing the validity of the tensor model 83
3.5 Discussion
The model selection framework introduced by Alexander et al. [31] provides an auto-
mated statistical approach to identify voxels that cannot be adequately represented
by the di�usion tensor. This has important applications both for avoiding misin-
terpretation of DTI results and providing evidence to justify more complex model-
ing/reconstruction procedures. The primary limitation of this method, one that it
shares with the majority of procedures for resolving complex �bre architectures, is
its reduced performance for �bres with small separation angles and/or unbalanced
relative �bre populations.
It has been previously reported that increasing the b-values can accentuate non-
Gaussian di�usion processes [19, 22]. A major reason that higher b-values have not
been widely adopted for this application is the limitation imposed by linear-least
squares �tting algorithms. The primary contribution of the work presented in this
chapter is the development of nonlinear and magnitude-corrected �tting algorithms for
the Spherical Harmonic and Generalized Di�usion Tensor models. These algorithms
enable robust modeling of higher-order di�usion processes at b-values much greater
than 1000 s/mm2.
The results of applying Magnitude-Corrected Nonlinear Least-Squares (MCNLS)
�tting to Alexander et al.'s model selection framework is a signi�cant improvement
in the ability to detect non-Gaussian di�usion, especially for voxels with reduced sep-
aration angles and/or unbalanced volume fractions. The F-test thresholds were very
conservative (false positive rate ≤1%), and as such, the model order is generally un-
derestimated in the results presented here. It is entirely possible to trade-o� increased
detection of crossing �bres at the expense of more false positives (more di�usion ten-
sors falsely classi�ed as 4th-order). More sophisticated thresholding methods may
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Chapter 3. Testing the validity of the tensor model 84
also be desirable to account for the multiple comparisons being performed [61].
The poor performance of the LLS-based classi�er was the results of its F-test
threshold increasing with the b-value to account for voxels that appear to show non-
Gaussian di�usion due to �tting error. This also explains the slight advantage of
magnitude-corrected nonlinear �tting versus standard nonlinear least-squares. The
primary advantage to MCNLS is that it can reduce false positives, or for the same
number of false positives, can improve the detection rate. In addition, Chapter 2 also
showed that at high b-values, MCNLS �tting reduces the error in estimating tensor
derived parameters relative to LLS, including FA and �bre orientation.
Increasing the b-value accentuates non-Gaussian di�usion, but increasing it too
much leads to SNR losses. Although it can be summarized from these results that the
b-value should be greater than 1000 s/mm2, a speci�c �optimal� b-value for detect-
ing non-Gaussian di�usion cannot be determined from this data. The SNR/b-value
relationship is primarily the result of exponential signal decay, exp(-bD), but there
is also a contribution from the longer echo time required to achieve higher b-values.
This e�ect is hardware dependent and improved gradients may alleviate these SNR
losses somewhat.
Classi�er performance was greatly in�uenced by SNR. This e�ect was demon-
strated in simulations and experimentally by the di�erences between the subjects im-
aged with the quadrature and 8-channel head coils. Although it is possible to increase
SNR through signal averaging, time constraints usually force a trade-o� between the
number of directions and the number of repeated acquisitions. Optimization of the b-
value, SNR, and number of directions is beyond the scope of this work, but it deserves
further study.
Using the 8-channel head coil at b=3000 s/mm2, 11.4% of voxels in the brain
were classi�ed as 4th-order on average. This is almost three times the number of
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Chapter 3. Testing the validity of the tensor model 85
4th-order voxels classi�ed at b=1000 s/mm2 (4.4%). The percentage of 4th-order
voxels at b=1000 s/mm2 was comparable to the 5% reported by Alexander et al. [31].
Experimental di�erences (SNR, number of gradient directions, etc.) and di�erences
in the model selection algorithm (�tting algorithm and F-test thresholds) do not allow
for a direct comparison of results, but the same clusters of 4th-order voxels were seen
in both studies.
Even with these improved �tting techniques and higher b-values, the simulation
results suggest that many voxels with complex �bre geometries may go undetected,
especially if the component �bres have low anisotropy, low separation angles, and/or
unbalanced �bre populations. Relatively speaking, this class of false negatives pose
a less signi�cant problem for DTI analysis because they have a much lower impact
on estimates of anisotropy and �bre orientation than crossings with high separation
angles and well balanced �bre populations.
Imaging at higher b-values has several implications that should be noted. Non-
monoexponential di�usion resulting from distinct populations of crossing �bres con-
tributes to the non-Gaussian pro�les detected by this algorithm. Even for those voxels
without crossing �bres, non-monoexponential di�usion can become signi�cant at high
b-values due to the multiple water compartments in biological tissue [7]. This means
that the shapes of ADC pro�les vary with the b-value. As a consequence, tensor
parameters can also change relative to b. For this reason, caution should be exercised
when comparing studies that use di�erent b-values.
The extended di�usion times necessary to achieve high b-values also increase the
relative e�ects of restricted di�usion and exchange [5]. While these e�ects may be sig-
ni�cant at very high b-values, they are unlikely to have a major impact on the current
experiment since the increase in di�usion time between b=1000 and b=3000 s/mm2
was less than 20 ms, corresponding to an average net water displacement of a few mi-
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Chapter 3. Testing the validity of the tensor model 86
crons. The e�ects of restricted di�usion and exchange may limit the upper value of b,
but once again, a higher maximum gradient amplitude could overcome this limitation.
Finally, this study looked speci�cally at non-Gaussian di�usion pro�les caused
by complex �bre architectures (i.e. partial volume e�ects). There are several other
sources that can potentially contribute to perceived non-Gaussian di�usion. These
include cardiac pulsatility, head motion, and eddy currents. While there are certainly
many established ways to reduce the overall impact of these contributions including
gating and image registration, others deserve consideration. First of all, robust �tting
techniques, which are less susceptible to outliers, have been applied to the �tting of
di�usion tensors [52, 53]. There is no reason that they could not also be extended
to higher-ordered models. Secondly, the antipodal symmetry of the ADC pro�le
may provide a means for separating true non-Gaussian di�usion e�ects from these
confounding artifacts.
The results of this study show that magnitude-corrected nonlinear �tting allows
for improved estimation of the SH and GDT models at high b-values. Combined
with the classi�cation scheme of Alexander et al. [31], this signi�cantly improves the
ability to detect regions of complex �bre architecture for which the di�usion tensor
is insu�cient. This information is critical for correctly interpreting DTI results, and
in providing justi�cation for the use of higher-order models on a voxel-by-voxel basis.
This �tting technique should also be useful to other areas of the di�usion community
that utilize the SH and/or GDT basis functions, e.g. spherical deconvolution and
q-ball imaging.
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Chapter 4
Conclusions and future work
Chapter 2 compared several algorithms used to �t the di�usion tensor model, while
Chapter 3 addressed the problem of model validation and the �tting of higher-order
models. This �nal chapter will describe the overarching conclusions that can be drawn
from this work and attempt to highlight speci�c areas that require further study.
4.1 Magnitude-correction
Both of the preceding chapters shared a common approach to the problem of Rician
noise, termed here as magnitude-corrected nonlinear least-squares �tting. Although
research on bias in magnitude MR images has a long history [62, 63, 64, 39, 65, 66],
relatively little attention has been paid to this issue in the context of DTI. Dietrich
et al. [67] examined the e�ect of magnitude bias in estimating ADC values, while
DTI artifacts caused by Rician noise were �rst reported by Jones and Basser [32].
The latter study examined bias in the measurement of Apparent Di�usion Coe�cient
pro�les at low SNR and proposed a simple correction method that could partially
compensate for this artifact. This method was adapted to �t the di�usion tensor in
87
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Chapter 4. Conclusions and future work 88
Chapter 2 (and higher-order models in Chapter 3) rather than single ADC values.
The noise parameter was based on characteristics of the image background and �xed
for model �tting. While simulation studies have shown that this approach is e�ective
in certain situations, it does have several important limitations. This section will
describe some of these limitations and suggest possible ways in which they may be
addressed.
The following objective function is minimized in MCNLS �tting:
N∑i=1
[Si −
√Si
2+ σ2
]2
(4.1)
where Si is the magnitude of the measured signal, Si is the signal estimation based on
the di�usion model, and σ is the average Gaussian noise, assumed to be equivalent in
the real and complex channels before application of the magnitude operation. This
method is similar in nature to an approximation originally proposed by Gudbjartsson
and Patz [64], which estimated signal amplitude, A, from the magnitude signal, M .
Their method, demonstrated in Eq. 4.2, can be rearranged to solve for M (Eq. 4.3).
The MCNLS correction scheme is based on a substitution of this result into the
standard least-squares di�usion signal equation (Eq. 2.12).
A =
√∣∣∣M2 − σ2
∣∣∣ (4.2)
M =√A2 + σ2 (4.3)
The Gudbjartsson and Patz approximation is known to overestimate the mean signal
at very low SNR (≤1), and therefore the MCNLS algorithm could reasonably be
expected to show reduced performance when one or more of the signal measurements
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Chapter 4. Conclusions and future work 89
are below this level.
Several other correction schemes have been proposed in the literature, and it may
be possible to integrate one or more into an improved �tting approach for di�usion
models. Miller and Joseph described a correction method for power images (i.e. mag-
nitude images that have been squared) [66]. This correction is described by the
following equation:
M2 = A2 + 2σ2 (4.4)
Note that M2 6= M2, and therefore applying this correction scheme to di�usion
models necessitates �tting the power signal directly. This can be accomplished using
the equation [66]:
S2 = S20 exp(−2bD) (4.5)
The advantage to this approach is that, in theory, extracting the true signal even
at very low SNR is possible. The problem is that the distribution of the corrected
power signal at low SNR is markedly non-Gaussian, making it unsuitable for least-
squares �tting [64]. Furthermore, this approach is not valid for multiple receiver coils
or multiexponential signals [66].
Dietrich et al. [67] described an exact means for correcting magnitude images
based on the theoretical Rician distribution:
M = σ
√π
2exp
(− A2
4σ2
)×[(
1 +A2
2σ2
)I0
(A2
4σ2
)+A2
2σ2I1
(A2
4σ2
)](4.6)
where I0 and I1 are the zeroth and �rst order Bessel functions. By measuring the
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Chapter 4. Conclusions and future work 90
average of the magnitude signal, M , as well as the noise parameter from the image
background, they showed that it is possible to obtain a numerical estimate of A. This
result could be applied to the �tting of di�usion models by implementing a method
based on the following:
minN∑i=1
[Si − σ
√π
2exp
(− Si
2
4σ2
)×
[(1 +
Si2
2σ2
)I0
(Si
2
4σ2
)+Si
2
2σ2I1
(Si
2
4σ2
)]]2
(4.7)
though this has not yet been attempted.
Koay and Basser recently presented a similar correction scheme which can simul-
taneously estimate the unbiased mean and variance from magnitude images [65]. This
approach requires estimates of both the mean and standard deviation of the magni-
tude signal, which necessitates multiple acquisitions. It is therefore not practical for
the purposes of this work.
One way to avoid the issue of magnitude bias altogether is to �t the complex
signal itself. Unfortunately, complex data and the necessary reconstruction algorithms
are not widely available on clinical scanners. The use of complex di�usion-weighted
images would not only solve the problems associated with magnitude bias, but would
also enable the measurement of non-symmetric di�usion pro�les [68]. If the demand
for complex DTI data were to increase, vendors may be convinced to provide this
capability.
4.2 Noise characterization
All of the prospects for magnitude correction that have been described to this point
rely on accurate characterization of signal noise. If it is assumed that the signal in the
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Chapter 4. Conclusions and future work 91
background region of the images follows a Rayleigh distribution, the noise parameter
can be calculated from either the mean or standard deviation of the signal [39]:
mean(Sbackground) =
√π
2σ (4.8)
std(Sbackground) =
√4− π
2σ (4.9)
For the 4-channel, 3T GE Signa system used in this study, the background signal
showed a distribution that was in relatively good agreement with the Rayleigh dis-
tribution. Another assumption was also made in this work: that the noise level was
equivalent for all voxels in the image set.
There are several scenarios under which these assumptions may be invalid. Images
reconstructed from multiple receiver coils show spatial variation in the noise level and
background signal characteristics [69]. A similar situation exists for parallel imaging
techniques [70, 71], meaning that more sophisticated noise estimation methods are
required, including those that can estimate noise on a per voxel basis [72]. It must be
stressed that these and any other changes that a�ect the noise properties of images
(e.g. reconstruction �lters) may also force modi�cations to the magnitude-correction
algorithm. This applies not only to image acquisition and reconstruction, but also
to post-processing steps. It is common practice to perform image registration and
eddy-current correction on di�usion-weighted images. Rohde et al. showed that the
interpolation step inherent in these processes alters signal variance properties [73].
This implies that estimating noise properties from post-processed images can lead to
erroneous results.
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Chapter 4. Conclusions and future work 92
4.3 Improving SNR
Robust detection of voxels with complex �bre architectures requires a high Signal-
to-Noise Ratio. There are several strategies which could potentially increase SNR.
Improved image hardware (i.e. higher �eld magnets, stronger gradients, better coils)
o�er one avenue towards achieving this goal. In addition, parallel imaging techniques
can reduce imaging time and therefore allow for a greater degree of temporal aver-
aging. Parallel imaging also has the added bene�t of mitigating several bandwidth-
related image artifacts [71]. As the quality of di�usion-weighted images improves, so
too will the performance of the proposed techniques.
4.4 Tractography
The model selection algorithm in Chapter 3 was used to construct classi�cation maps
identifying voxels for which the di�usion tensor was unsuitable. This was achieved
through an automated model selection based on F-statistics. These F-statistics could
also be integrated into an adaptive tractography algorithm that would modify its
behaviour relative to the evidence for the di�erent models. For example, strong
support for the isotropic model could be used as a stopping criterion. Evidence for
a 4th-order di�usion model could result in the application of a two-tensor model for
selected voxels within the brain.
Tractography o�ers a complimentary source of information which may help to
identify sites of �bre crossings. Because the �bre bundles in each voxel do not exist in
isolation (they connected to �bres in the surrounding voxels), �bre orientations from
voxels in the immediate neighbourhood may provide additional data which could be
integrated into the model selection algorithm. Other groups have attempted to re-
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Chapter 4. Conclusions and future work 93
solve complex �bre geometries based on this principle using Independent Component
Analysis [74].
4.5 Conclusions
The results of this work suggest that it is time to abandon linear least-squares as a
method for �tting di�usion models. In the case of the di�usion tensor, nonlinear algo-
rithms o�er reduced rotational bias and reduced uncertainty in Fractional Anisotropy
and �bre orientation. The only real drawback is added computational time, which at
this point, is equivalent to or less than the time of image acquisition.
Nonlinear �tting has been shown to be especially important for high di�usivity
and high b-value applications. Furthermore, a simple modi�cation to the nonlinear
�tting algorithm reduces bias caused by magnitude images with low SNR. This MC-
NLS �tting algorithm is primarily intended for high di�usivity and/or high b-value
applications, where low SNR is commonly encountered. In addition to demonstrat-
ing the advantages of nonlinear �tting, the results of this work show that optimal
imaging parameters (e.g. number of gradient directions and b-value) depend on the
chosen �tting algorithm, and therefore need to be adapted accordingly.
Finally, nonlinear �tting and magnitude-correction have been extended to higher-
order models including the spherical harmonic series and generalized di�usion tensors.
This enables improved �tting performance at high b-values, facilitating the identi�-
cation of voxels with complex �bre geometries. Looking ahead, beyond the di�usion
tensor, it seems likely that attempts at resolving these complex �bre trajectories will
increasingly involve high b-values, for which nonlinear �tting methods are essential.
The recommendations presented in this thesis, as a whole, will improve the overall
robustness of Di�usion Tensor Imaging, inspiring greater con�dence in results and
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Chapter 4. Conclusions and future work 94
therefore increasing the utility of DTI as a research and diagnostic tool.
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